algebraic logic.handbook

8/8/2019 Algebraic Logic.handbook

http://slidepdf.com/reader/full/algebraic-logichandbook 1/129

H. ANDREKA, I. NEMETI, I. SAIN

ALGEBRAIC LOGIC

We dedicate this work to J. Donald Monk who taught us algebraic logic and

more.

CONTENTS

Introduction

Part I. Algebras of relations

Getting acquainted with the subject

1. Algebras of binary relations

2. Algebras of relations in general

Connections with geometry

Connections with algebras of binary relationsConnections with logic

Algebras of infinitary relations

Proof theoretical connections

3. Algebras for logics without identity

Part II. Bridge between logic and algebra: Abstract Algebraic Logic

Introduction

4. General framework for studying logics

5. The process of algebraization

6. Equivalence theorems

7. Examples and applications

References

0The research of all three authors was supported by the Hungarian National Foundation for Basic

Research, grant No’s T16448, T23234.



2 ALGEBRAIC LOGIC

INTRODUCTION

Algebraic logic can be divided into two main parts. Part I studies algebras which

are relevant to logic(s), e.g. algebras which were obtained from logics (one way or

another). Since Part I studies algebras, its methods are, basically, algebraic. One

could say, that Part I belongs to “Algebra Country”. Continuing this metaphor,

Part II deals with studying and building the bridge between Algebra Country andLogic Country. Part II deals with the methodology of solving logic problems by

(i) translating them to algebra (the process of algebraization), (ii) solving the alge-

braic problem (this really belongs to Part I), and (iii) translating the result back to

logic. There is an emphasis here on step (iii), because without such a methodolog-

ical emphasis one could be tempted to play the “enjoyable games” (i) and (ii), and

then forget about the “boring duty” of (iii). Of course, this bridge can also be used

backwards, to solve algebraic problems with logical methods. We will give some

simple examples for this in the present work.

Accordingly, the present work consists of two parts, too. Parts I and II of

the paper deal with the corresponding parts of algebraic logic. More specifically,

Part I deals with the algebraic theory in general, and with algebras of sets of se-

quences, or algebras of relations, in particular. Part II deals with the methodologyof algebraization of logics and logical problems, equivalence theorems between

properties of logics and properties of (classes of) algebras, and in particular, dis-

cusses concrete results about logics obtained via this methodology of algebraiza-

tion. Since Part II deals with general connections between logics and algebras, a

general definition of what we understand by a logic or logical system is needed. Of

course, such a definition has to be broad enough to be widely applicable and nar-

row enough to support interesting theorems. The first section of Part II is devoted

to finding such a definition.

We need to make a disclaimer here. Algebraic logic, today, is an extremely

broad subject. We could not cover all of it. In Part II we managed to be broader

than in Part I. Even in Part II we could not come even close to discussing the

important research directions, but the definitions in Part II are general enough to

render the results applicable to all those logics which W. Blok and D. Pigozzi callalgebraizable (cf. e.g. [BP89], [FJ94]). Most of what we say in Part II can be gener-

alized even beyond this, e.g. to the equivalential logics of J. Czelakowski. Further

possibilities of generalizing Part II beyond algebraizable logics are in recent works

of Blok and Pigozzi, and others, cf. e.g. [BP86], [P91], [CzP], [ABNPS], [Cz97] .

In Part I we had to be more restrictive. We concentrated attention to those kinds

of algebras which are connected to the idea of “relations” (one way or another),

the idea of sets of pairs, or sets of triples, sets of sequences or something related

to these. An important omission is the theory of Boolean Algebras with Operators

( ¡ £

’s). ¡ £

’s are related to algebras of relations, and they provide an impor-

tant unifying theory of many of the algebras we discuss here. Another important



3

omission is Category Theoretic Logic. That branch of algebraic logic is not (at

all) unrelated to what we are discussing here, but for various reasons we could

not include an appropriate discussion here. In this connection more references

are given in the survey [N91]. Here we mention only Makkai [Mk87], [MkR],

[Z], [MkP]. We could not cover polyadic algebras, either. However, their (basic)

theory is analogous to that of cylindric algebras which we do discuss in detail.

There are a few exceptional points where the two theories wildly diverge, e.g. in[NS96] it was proved that the equational theory of representable polyadic algebras

is highly non-computable (while that of cylindric algebras is recursively enumer-

able). We refer the reader to the survey paper [N91] and to [HMTII] for modern

overviews of polyadic algebras. Cf. also [ST], [PS], [AGMNS]. Further important

omissions are: (i) the finitization problem (cf. [N91, beginning with Remark 2],

[S95], [Si93], [MNS97]); (ii) propositional modal logics of quantification, and

connections with the new research direction “Logic, Language and Information”

(cf. [V95], [MV], [MPM], [AvBN97], [vBtM], [vB97]); (iii) relativization as a

methodology for turning negative results to positive (cf. [N96], [M93], [Ma95],

[Mi95], [MV], [AvBN96]). Also there are strong connections between algebraic

logic and computer science, we do not discuss these here.

On the history: The invention of Boolean algebras belongs to the “prehistory”

of Part I. Algebras of sets of sequences (as in Part I) were studied by De Morgan,

Peirce, and Schroder in the last century; and the modern form of their theory was

created by Tarski and his school1. The history of Part II also goes back to Tarski

and his followers, but is, in general, more recent. For more on history we refer to

[AH], [ABNPS], [BP91a], [BP89], [HMTII], [Ma91], [Pr92], [TG].

1Relation and cylindric algebras were introduced by Tarski, polyadic algebras were introduced by

Halmos, algebras of sets of finite sequences were studied by Craig; for other kinds of algebras of sets

of sequences cf. e.g. [N91], [HMTII].



PART I

ALGEBRAS OF RELATIONS



6 ALGEBRAIC LOGIC

GETTING ACQUAINTED WITH THE SUBJECT OF PART I.

The algebraization of classical propositional logic, yielding Boolean algebras (in

short ¡

’s), was immensely successful. What happens then if we want to extend

the original algebraization yielding ¡

’s to other, more complex logics, among

others, say, to predicate logic (first–order logic)? 2

Boolean algebras can be viewed as algebras of unary relations. Indeed, the

elements of a¢ ¡

are subsets of a set ¥ , i.e. unary relations over ¥ , and the oper-

ations are the natural operations on unary relations, e.g. intersection, complemen-

tation. The problem of extending this approach to predicate logics boils down to

the problem of expanding the natural algebras of unary relations to natural alge-

bras of relations of higher ranks, i.e. of relations in general. The reason for this

is, roughly speaking, the fact that the basic building blocks of predicate logics are

predicates, and the meanings of predicates can be relations of arbitrary ranks. 3

Indeed, already in the middle of the last century, when De Morgan wanted to gen-

eralize algebras of propositional logic in the direction of what we would call today

predicate logic, he turned to algebras of binary relations. 4 That was probably the

beginning of the quest for algebras of relations in general. Returning to this quest,

the new algebras will, of course, have more operations than¢ ¡

’s, since between

relations in general there are more kinds of connections than between unary re-

lations (e.g. one relation might be the converse, sometimes called inverse, of the

other). So, our algebras in most cases will be Boolean algebras with some further

operations.

The framework for the quest for the natural algebras of relations is universal

algebra. The reason for this is that universal algebra is the field which investigates

classes of algebras in general, their interconnections, their fundamental properties

etc. Therefore universal algebra can provide us for our search with a “map and a

compass” to orient ourselves. There is a further good reason for using universal

algebra. Namely, universal algebra is not only a unifying framework, but it also

contains powerful theories. E.g. if we know in advance some general properties of

the kinds of algebras we are going to investigate, then universal algebra can rewardus with a powerful machinery for doing these investigations. Among the special

classes of algebras concerning which universal algebra has powerful theories are

the so called discriminator varieties and the arithmetical varieties. At the same

2The things we say here about predicate logic apply also to most logics having individual variables,

henceto all quantifierlogics. However, the present paperneed not be “predicatelogic centered” because

our considerations apply also to many propositional logics, e.g. to Lambek Calculus, propositional

dynamic logic, arrow logics, many-dimensional modal logics. C.f. e.g. [MV], [vB96], [vBtM], [Mi95],

[TG].3For more on this see Part II, sections 4, 7 of the present paper.4De Morgan illustrated the need for expandingthe algebras of unary relations(i.e. ¦ § ’s) to algebras

of relations in general (the topic of Part I of the present paper) by saying that the scholastics, after two

millennia of Aristotelian tradition, were still unable to prove that if a horse is an animal, then a horse’s

tail is an animal’s tail. (“©

is a tail of ©

” is a binary relation.)



ALGEBRAS OF RELATIONS 7

time, algebras originating from logic turn out to fall in one of these two categories,

in most cases. More concretely, more than half of these algebras are in discrimi-

nator varieties and almost all are in arithmetical ones. Certainly, all the algebras

studied in the present paper are in arithmetical varieties. Therefore, awareness of

these recent parts of universal algebra can be rewarding in algebraic logic. We will

not assume familiarity with these theories of universal algebra, we will cite the

relevant definitions and theorems when using them. 5

Moreover, as we already said, most of our algebras will be¢ ¡

’s with some ad-

ditional (extra-Boolean) operations. When these operations are distributive over

the Boolean join, as will be the case most often, such algebras are called Boolean

Algebras with Operators, in short ¡ £

’s. Many of our important classes of al-

gebras will be discriminator varieties of ¡ £

’s. The theory of ¢ ¡ £

’s is well-

developed. 6

Let us return to our task of moving from¢ ¡

’s of unary relations to expanded ¡

’s of relations in general. What are the elements of a ¡

? They are sets of

“points”. What will be the elements of the expanded new algebras? One thing

about them seems to be certain, they will be sets of sequences, because relations

in general are sets of sequences. These sequences may be just pairs if the relation

is binary, they may be triples if the relation is ternary, or they may be longer — oreven more general kinds of sequences. 7 So, one thing is clear at this point, namely

that the elements of our expanded ¡

’s of relations will be sets of sequences.

Indeed, this applies to all known algebraizations of predicate logics or quantifier

logics. 8

At this point it might be useful to point out that the most obvious approach (to

studying algebras of relations) based on the above observation (that the elements

of the algebra are sets of sequences) leads to difficulties right at the start. 9 So,

what is the most obvious approach? Consider some set¥

; let ¥

denote the

set of all finite sequences over¥

, and consider the ¡

# %(the powerset

5Some good introductions to universal algebra and discriminator varieties are [HMT, Chapter 0],

[BS], [C65], [Gr], [MMT], [W].6

Distributivity of the extra-Booleanoperations over join is usedin the theory to build a well-workingduality-theory for it (atom-structures or Kripke-frames, complex algebras). This duality theory is a

quite central part of algebraic logic. Because of the limited size of the present paper, we will not

deal with this here. Some references are Jonsson-Tarski [JT51], [HMT, section 2.5], Jonsson [J95],

Goldblatt [G90], [G91], Venema [V96],[V97], [AGiN95], [H97], [AGoN].7There is another consideration pointing in the direction of sequences. Namely, the semantics of

quantifier logics is defined via satisfaction of formulas in models, which in turn is defined via evalua-

tions of variables, and these evaluations are sequences. The meaning of a formula in a model is the set

of those sequences which satisfy the formula in that model. So we arrive again at sets of sequences.

For more on this see Part II, section 7 of the present paper.8As mentioned earlier, this also applies to the more complex propositional logics, like e.g. many-

dimensional modal logic.9With further work this approach can be turned into a fruitful approach to algebraizing logic, see

[N91, ' 7 (2–4) and the section containing Facts 2, 3 at the end of ' 4]; see also [HMTII, ' 5.6.(A

survey).3, p. 265], and the references therein. The approach originates with Craig, but already the

algebras in Quine [Q36] consist of sets of finite sequences.



8 ALGEBRAIC LOGIC

of ¥

conceived as a ¡

the standard way). Now if we are given any finitary

relation, say, ( ) ¥ 2 ¥ over ¥ , then ( 5

¥ % . So

¥ % contains all

relations over¥

independently of their ranks. Therefore it might be a candidate

for being the universe of an algebra of relations. Before thinking about what the

new, so called extra-Boolean operations on

¥ %should be, let us have another

look at its Boolean structure: If ( is a binary relation, we would like to obtain

its complement

¥ 2 ¥ % 9 ( as a result of applying a Boolean operation to ( .However, in our algebra

¥ %, the complement of

(is not

¥ 2 ¥ % 9 (but

something infinitely bigger.



CHAPTER 1

ALGEBRAS OF BINARY RELATIONS

The above difficulty with

¥ %motivates our concentrating first on the sim-

plest nontrivial case, namely that of the algebras of binary relations ( F ¡

’s). Ac-

tually, F ¡

’s will be strong enough to be called a truly first–order (as opposed

to propositional) algebraic logic, namely the logic captured by F ¡

’s is strong

enough to serve as a vehicle for set theory and hence for ordinary metamathemat-

ics. 1

Throughout this paper,

¥ %denotes the powerset of

¥, and

G

¥ %denotes the

Boolean algebra (in short ¡

) with universe

¥ %, for any set

¥. Thus

¥ %is

the set of all subsets of ¥

, and

G

¥ % H I

¥ % Q S Q U W

where S is the binary operation of taking union of two subsets of ¥ , and U is the

unary operation of taking complement (w.r.t. ¥ ) of a subset of ¥ . Then G

¥ % , as

well as any of its subalgebras, is a natural algebra of unary relationson¥

, because

a unary relation on¥

is just a subset of ¥

, hence an element of

¥ %.

A binary relation is a set of pairs. Thus the usual set-theoretic (or in other

words, Boolean) operations of union and complementation can be performed on

binary relations. First we consider two natural operations on binary relations that

use the fact that we have sets of pairs, namely relation-composition and relation

conversion. Let( Q `

be binary relations. Then their composition 2( a `

and the

converse( c e

of (

are defined as 3

( a ` H h I p Q r W t u w

p ( wand

w ` r %

(c e

H h I r Q p W t I p Q r W 5 ( .

1A very interesting class of algebras of relations which is halfway between ¦ § ’s and ¦ § ’s is the

class §

of cylindric algebras of dimension 2. They will be discussed at the beginning of section 2.2This is denoted by in part of the literature, e.g. in [HMTII]. The reason for this is that in

a large part of the literature,

is reserved for the case when

and

are functions and is written

backwards, i.e. what we denote by

is denoted by

.3Throughout this paper we will use the convention that if

is a binary relation, then

means

that

.

9



10 ALGEBRAIC LOGIC

By a concrete algebra of binary relations, aj

F ¡

, we understand an algebra

whose elements are binary relations having a greatest one among them, and whose

operations are the Boolean ones: union and complementation (w.r.t. this greatest

relation), relation-composition and relation conversion. Thus the universe of a

j

F ¡

is closed under these operations, e.g. the union and relation composition of

any two elements of the algebra are also in the universe of the algebra 4.

Formally, a j

F ¡

is of the form

k

H I m Q S Q U Q a Q

c e

W

where

mis a set of binary relations and

mhas a biggest element

,

S Q U Q a Qc e

are total operations onm

, which means that

h ( S ` Q –

( Q ( a ` Q ( c e ) mwhenever

( Q ` 5 m.

An algebra of binary relations, a F ¡

, is an algebra isomorphic to a concrete

algebra of binary relations. If k

is a F ¡

, then

denotes the greatest element of k

, which we shall sometimes call the unit of k

.

Throughout, we use abbreviations like¤ F ¡

also for denoting the corresponding

class itself, e.g. F ¡

also denotes the class of all F ¡

’s, and ¡

also denotes the

class of all ¡

’s.

The similarity type, or language, of our F ¡

’s should contain two binary func-

tion symbols for S and a , and two unary function symbols for U andc e

. In

this paper, for simplicity and suggestiveness, we use the symbolsS Q a Q U Q

c e

for

these. We hope, this will cause no confusion 5. Typical equations holding in F ¡

are

S z % a | H

a | % S

z a | % Q

S z %c e

H

c eS z

c e

. In the

literature Q are often used as function symbols for S , and likewise Q are

used as function symbols for a Qc e . Using these symbols, the above equations

look as

z % | H

| %

z | % Q

z %

H

z

, or

z % | H

| %

z | % Q

z %

H

z

.

So we will use the symbolS

also in abstract Boolean algebras. Moreover,

in abstract Boolean algebras we also will use

as derived operation:

z

def H

U

U

S U z %.

Q Q will denote the ordering

z

S z H z, smallest

element and biggest element, respectively. Thus

is an equation.

4To understand how (and why) the theory works, it would be enough to include only “ ” as “extra-

Boolean” operation. Inclusion of conversion is motivated by some of the applications. Cf. the discus-

sion of ¦

below Thm.1.9 (this section).5When seeing, say “

,” the reader will have to decide whether this denotes a term of

¦ §’s

built up from the variables

, or whether it denotes a set (the union of the sets

and

.)



1. ALGEBRAS OF BINARY RELATIONS 11

Having a fresh look at our F ¡

’s with an abstract algebraic eye, we notice that

they should be very familiar from the abstract algebraic literature. Namely, a¤ F ¡

k

consists of two well known algebraic structures, a Boolean algebraI m Q S Q U W

and an involuted semigroup I m Q a Qc e

W sharing the same universe m . The two struc-

tures are connected so that they form a normal Boolean algebra with operators, in

short a normal ¡ £

, which means that each extra-Boolean operation is distributive

over S (additivity) and takes the value whenever at least one of the arguments is

(normality). Alsoc e

is a Boolean isomorphism and

a

, where

is the

Boolean

, defines a complemented closure operation 6 onm

. The properties listed

in this paragraph define a nice variety¡ F ¡

containing F ¡

and is a reasonable

starting point for an axiomatic study of the algebras of binary relations. 7

DEFINITION 1.1 (¡ F ¡

, an abstract approximation of F ¡

)¡ F ¡

is defined to

be the class of all algebras of the similarity type of F ¡

’s which validate the

following equations.

(1) The Boolean axioms 8

S z H z S

,

S

z S | % H

S z % S | ,

U U

S z % S U

S U z % H

.

(2) The axioms of involuted semigroups, i.e.

a z % a | H

a

z a | % ,

a z %c e

H zc e

a

c e

,

c e c e H

.

(3) The axioms of normal ¡ £

, i.e. 9

6Closure operations are unary functions , where we have an ordering on . is

called a closure operation if it is order preserving, idempotent, and increasing, i.e. if for all ©

we have ª « ª

and © ® ª © ª

. Boolean orderings with closure

operations on them are one of the central concepts of abstract algebra, for example topological spaces

or subalgebras of an algebra areoftenrepresentedas such.

is calleda complemented closure operation

if ± ª « ± ª

, i.e. the complement of a closed element is closed. For more on these see e.g.

[HMT, p.38].7Most of these postulates already appear in De Morgan [D1864], and since then investigations of

§ § ’s have been carried on for almost 130 years.8Problem 1.1 in [HMT, p.245], originating with H. Robbins, asks whether this is an axiom system

for ¦ § . This problem has recently been solved affirmatively (by the theorem prover program EQP

developed at Argonne National Laboratory, USA). We will use this axiom system for¦ §

in Part II,

section 7.1.9We are omitting some axioms that follow from the already stated ones. E.g. here we omit

´

ª « ª

´

ª,

µ « µ.



12 ALGEBRAIC LOGIC

S z % a | H

a | % S

z a | % ,

S z %c e

H

c eS z

c e ,

a

H ,

c e H .

(4)c e

is a Boolean isomorphism and

a

is a complemented closure

operation, i.e.

U

c e % H

U

% c e ,

a

,

U

a

% H a U

a

%.

¶

If ·

is a set of equations, then¸ ¹ º

· %denotes the class of all algebras (of a

given similarity type) in which·

holds. A class»

of algebras is called a variety,

or an equational class, if » H ¸ ¹ º

· %for some set

·of equations. The following

theorem is due to A. Tarski.

THEOREM 1.2 F ¡

is an equational class, i.e. there is a set ·

of equations such

that ¤ F ¡

H ¸ ¹ º

· % .

To prove the above theorem, we will use the machinery of universal algebra.

First we prove that F ¡

is closed under taking subalgebras and direct products. If

»is a class of algebras, then

¼ »denotes the class of all subalgebras of elements of

»,

½ » Q ¾ »,

¿ »and

À Á »denote the classes of all algebras isomorphic to direct

products, isomorphic copies, homomorphic images, and ultraproducts of elements

of »

respectively. 10 Thus¤ F ¡

H ¾ j

¤ F ¡

.

LEMMA 1.3 F ¡

H ¼ ½ h I G

¥l 2 Â ¥ % Q a Qc e

W t ¥is a set

.

Proof. Let be a binary relation. We say that is an equivalence relation if

is symmetric and transitive, i.e. if

c eH

and a )

. The field of

is the

smallest set¥

such that ) ¥ 2 ¥

, i.e.¥ H h Ã t

u Ä % I Ã Q Ä W 5 or

I Ä Q Ã W 5

. The following three statements

Ç

%–

Ç Ç Ç

%will not be difficult to check:

Ç

%If

k

5 j

F ¡

, then

is an equivalence relation.

10Note that È is different from É Ê Ë and Ì Í in that È Î« Ê È , while É « Ê É etc. For our reasons

for definingÈ

this way see the remark after the definition of AlgÑin Part II, Def. 5.1. We will use

simple facts likeÊ È « È Ê

,Ê É « É Ê

,È È « È

, etc. without mentioning them.




Ç Ç

%If

is an equivalence relation, then

Ò

%

def H I G

% Q a Q

c e

W 5 j

F ¡ Ó

Ç Ç Ç

%Let

Ôbe a set and for all

Õ 5 Ôlet

×be an equivalence relation.

Assume that the fields of the ×

’s are pairwise disjoint. Then

Ò Ø Ù

× Ú Û

× Ü Ý

H Þ

× Ú Û

Ò

× %

Ó

Indeed, to see Ç

%, let

k

5 j

F ¡

and

def H

. Then 5 m

, hence a Q

c e

are

in m as well, hence a ) and c e

) , because is the biggest element of

m. But

c e ) is equivalent to

c e H , hence

is an equivalence relation.

To show Ç Ç

% , one has to check that for any ( Q ` ) also ( a ` ) and

(c e

) . These follow from

a ) ,

c e

) .

To show Ç Ç Ç

%, we define the function

á t

â

× Ú Û

× %

Þ

× Ú Û

× %by letting

for allã )

â

× Ú Û

×

,

á

ã %

def H I ã

×t Õ 5 Ô W

Ó

Then it is easy to check that thisá

is the required isomorphism.

We are ready to prove the lemma. First we show that F ¡

H ¼ ½

F ¡

. By

definition, j

F ¡

is closed under taking subalgebras, so F ¡

is also closed under

taking subalgebras (because F ¡

H ¾ j

F ¡

). Let Ô be a set, and letk

×5 j

¤ F ¡

with unit ×

for each Õ 5 Ô . We may assume that the ×’s have disjoint fields. Then

k

× )

Ò

× %, so

Þ

k

× )

Þ

Ò

× % Ý

H

Ò

â

× % 5 j

F ¡

by Ç

%–

Ç Ç Ç

%. This shows

thatÞ

k

×

is isomorphic to aj

F ¡

, i.e. F ¡

is closed under taking direct products.

Now letk

5 j

F ¡

with greatest element

. Then

is an equivalence relation,

let¥ ×

,Õ 5 Ô

be the blocks of this equivalence relation. Then¥ × 2 ¥ ×

are also equiv-

alence relations with pairwise disjoint fields, and

is the union of these. Hence

by

Ç Ç

%

–

Ç Ç Ç

%

we have that

k

)

Þ

I I G

¥ × 2 ¥ × % Q a Qc e

W t ¥ ×

is a block of W

.This completes the proof of Lemma 1.3.¶

To formulate our next lemma, we need the notions of a subdirect product and a

discriminator term.

Subdirect products of algebras, and subdirectly irreducible algebras are defined

in practically every textbook on universal algebra, cf. e.g. Gratzer’s book [Gr], or

[BS], [HMT], [MMT]. By a subdirect product we mean a subalgebra of a product

such that the projections of the product restricted to the subalgebra remain surjec-

tive mappings. An algebrak

is subdirectly irreducible if it is not (isomorphic to) a

subdirect product of algebras different fromk

. We note that the one-element alge-

bra is not subdirectly irreducible. By Birkhoff’s classical theorem, every algebra



14 ALGEBRAIC LOGIC

is a subdirect product of some subdirectly irreducible ones. Therefore, the sub-

directly irreducible algebras are often regarded as the basic building blocks of all

the other algebras. In particular, when studying an algebrak

, it is often enough to

study its subdirectly irreducible building blocks. For a class » of algebras, ¼ æ ç

» %

denotes the class of subdirectly irreducible members of »

. For» H

¢ ¡

,¼ æ ç

¡

%

consists of the 2-element Boolean algebra only (up to isomorphisms).

We say that a class»

of algebras has a discriminator term iff there is a term

Q z Q | Q Ã %in the language of

»such that in every member of

»we have

Q z Q | Q Ã % H è

| Q if

H z Q

Ã Qif

é

H z

Ó

The term

above is called a discriminator term. Sometimes instead of the four-ary

, the ternary discriminator term

ê

Q z Q | % H

Q z Q | Q

%is used. They are inter-

definable, since

Q z Q | Q Ã % H ê

ê

Q z Q | % Q ê

Q z Q Ã % Q Ã %. Therefore, it does not

matter which one is used. Moreover, in classes of algebras which have a Boolean

algebra reduct, like our F ¡

’s or¡ F ¡

’s, the discriminator term can be replaced

with the so called switching term

w

% H è

Q if é

H Q

Qif

H

Ó

By this we mean that in such a class of algebras, if

Q z Q | Q Ã % is a discriminator

term, thenw

% H

Q Q Q %is a switching term, and vica versa, if

w

%is a

switching term, then

Q z Q | Q Ã % H U w

í

z % | S w

í

z % Ã is a discriminator

term. Here, and later on,í

denotes symmetric difference, i.e. í

z

def H

U z % S

U

z % .

LEMMA 1.4¼ æ ç

F ¡

% H ¾ ¼ h I G

¥ 2 ¥ % Q a Qc e

W t ¥is a nonempty set

and

¼ æ ç

F ¡

%has a discriminator term.

Proof. Let »

def H ¾ ¼ h I G

¥7 2 ¥ % Q a Qc e

W t ¥ is a nonempty set . Letk

5

F ¡

.

Thenk

is isomorphic to a subalgebra of Þ

Ò

¥×

2 ¥×

% for some system I ¥×

t Õ 5 Ô W

of sets, by Lemma 1.3. If ¥ × H î

, thenÒ

¥ × 2 g ¥ × %is the one-element algebra

which can be left out from any product, so we may assume that each¥

×

above is

nonempty. But thenk

is a subdirect product of someï ×

,Õ 5 Ô

where eachï ×

is

a subalgebra of Ò

¥ × 2 ¥ × %. This shows that

¼ æ ç

F ¡

% ) ».

It is not difficult to check that

w

%

def H a

a

is a switching term onÒ

¥ 2 ¥ %for all

¥. Hence it is a switching term on

»also.

Thus,»

has a discriminator term.




Finally, if k

has a discriminator term, thenk

has no nontrivial congruences, i.e.k

is simple. This is a basic fact of discriminator theory. 11 Clearly, any simple

algebra is subdirectly irreducible, so» ) ¼ æ ç

¤ F ¡

%.

¶

We say that » is a pseudo-axiomatizable class if there are an expansion ð of

the language of »

, a setñ

of first–order formulas in this bigger languageð

and aunary relation symbol ¥ of ð such that

» H ò ó ô ¸ ¹ º ñ Q

where¸ ¹ º ñ

denotes the class of all models of ñ

, andò ó

ô denotes the operator

of taking reducts to the language of »

and restricting the universe to¥

at the same

time. In more detail: Letõ

be a model of the languageð

. Thenò ó õ

denotes

the reduct of õ

to the language of »

, andò ó

ôõ

denotes the restriction of the

modelò ó õ

to the interpretation¥ ö ) ø

of ¥

inõ

. I.e. whileõ

is a model

of the bigger language ð , ò ó ô õ is a model of the smaller language of » . If ù is

a class of models of the language of ð

, thenò ó

ôù

def H h ò ó

ôõ t õ 5 ù

.

It is known that pseudo-axiomatizable classes are closed under ultraproducts,

this is easy to show.

LEMMA 1.5¼ æ ç

F ¡

%is a pseudo-axiomatizable class.

Proof. The expansionð

of the language of F ¡

will be a many-sorted first-order

language with three sorts: ` Q ü and ( (for set, pairs, and relations), two unary

functionsý þ Q ý

e

fromü

to`

(first and second projections), a binary relationÿ

between ü and ( (for “is an element of”), and binary functions S Q a on ( , unary

functionsU Q

c e

on(

. The variables

Q z Q |are of sort

`, the variables

Ã Q Ä Q are

of sortü

, and the variablesp Q r Q w

are of sort(

. We also consider` Q ü Q (

as unary

relations. 12 See Figure 1.1.

The setñ

of axioms is as follows: In the following formulas we will writeÿ

in

infix mode, like Ã ÿ p . Also we will write comma in place of conjunction¢

. There

are free variables in the elements of ñ

, validity of an open formula is meant insuch a way that all the free variables are universally quantified at the beginning of

the formula.ñ

is defined to beh

p % Q

r % Q

Ó Ó Ó

Q

£

% , where

The “pair-axioms” are:

11The reason is the following. Assume that is a nonidentity congruence of ¥ . We will show that

then « ¦ ¨ ¦

. Let © ¦

, Î« ©

be such that ©

, and let ¦

be arbitrary. Then

« ª © ª « , so

.

12If one is not familiar with many-sorted models, then one can think of the above language as

having

as unary relation symbols, and e.g.

as a binary relation. Then to our axioms we

have to add statements like ª ª ª ª ª

´

ª «

´

ª # ª ª . Then the fact that the variable in the many-sorted language is of sort

while the variable

is of sort

means that one has to replace e.g. the formula& # ª «

with

& ª ª # ª 2 ª ª.



16 ALGEBRAIC LOGIC

5 6 8 6 A 6 B D

F G H

G

D

P Q R

S T U W X ` b d f g h S p X b T g h S r X g U g U ` h

t

6 u 6 w x 6 6 6 6

Figure 1.1.

(1a)

u Ã %

ýþ

Ã % H

Q ý

e

Ã % H z % .

(1b)ý þ

Ã % H ý þ

Ä % Q ý

e

Ã % H ý

e

Ä %

Ã H Ä.

Extensionality of sets of pairs:

(2) Ã

Ã ÿ p Ã ÿ r %

p H r

Ó

The definitions of the operations of j

¤ F ¡

:

(3a)Ã ÿ

p S r %

Ã ÿ p

orÃ ÿ r %

.

(3b)Ã ÿ

U p % Ã ÿ p.

(3c) Ã 5

p a r %

u Ä %

Ä ÿ p Q ÿ r Q ýþ

Ã % H ýþ

Ä % Q ý

e

Ä % H ýþ

% Q

ý

e

% H ý

e

Ã % %

Ó

(3d)Ã 5

p c e %

u Ä %

Ä ÿ p Q ý þ

Ã % H ý

e

Ä % Q ý

e

Ã % H ý þ

Ä % %

Ó

There are at least two elements in the relations sort:

(4)

u p r % p

é

H r

Ó

This finishes the definition of ñ

. We will show that

¼ æ ç

F ¡

% H ò ó ¸ ¹ º ñ

Ó

Indeed, letk

5 ¼ æ ç

F ¡

%, say

k

is isomorphic to a subalgebra of I G

¥ 2

¥ % Q a Qc e

W. We may assume that

k

) I G

¥ 2 8 ¥ % Q a Qc e

W. We define the three-

sorted modelõ

as follows.

` ö

def H ¥ Q ü ö

def H ¥ 2 ¥ Q ( ö

def H m Q

ý ö

þ

I Ã Q Ä W % H Ã Q ý ö

e

I Ã Q Ä W % H Ä for all Ã Q Ä 5 ¥ Q

I Ã Q Ä W ÿ ö p iff I Ã Q Ä W 5 p Q for all Ã Q Ä 5 ¥ Q p 5 m Q

p S ö r

def H p S r Qf U ö p

def H U p Q py a ö r

def H p a r Q p c e

ö

r

def H p c e

Ó




Then it is easy to check thatõ H ñ

andò ó õ H

k

. See Figure 1.2.

j

k l m l l

z { | } { ~

l m l

~ { } { z

Figure 1.2.

Conversely, let õ be such that õ H ñ . Let ¥

def H ` ö . Define the relation

between¥P 2 Â ¥

andü ö

as follows:

I Ã Q Ä W

iff ýþ

% H Ã Q ý

e

% H Ä

Ó

By (1a),(1b) inñ

then

is a bijection between¥ 2 ¥

andü ö

. Therefore we will

assume that

¥l 2 Â ¥ H ü

ö andÃ H ý

ö

þ

I Ã Q Ä W % Q Ä H ý

ö

e

I Ã Q Ä W %

Ó

We define now the function t ( ö

¥ 2 ¥ %as

p % H h I Ã Q Ä W t I Ã Q Ä W ÿ

ö

p

Ó

See Figure 1.3. Then is one-to-one by (2) in ñ . Axioms (3a)–(3d) in ñ say

that13

p S ö r % H

p % S

r % Q

Uö

p % H

¥ß 2 ¥ % 9

p % Q

p a ö r % H

p % a

r % Q

pc e

ö

% H

p % %c e

Ó

This shows that

is an isomorphism fromò ó õ

intoI G

¥ 2 ¥ % Q a Q c e W.

Finally, (4) inñ

implies that¥

is nonempty,ò ó õ

is nonempty.¶

We now are ready to apply the following theorem of universal algebra (cf. e.g.

[BS, Thm.9.4 (b,c)]).

THEOREM (universal algebra) If ¼ À Á »

has a discriminator term, then

13

def «



18 ALGEBRAIC LOGIC

ª «

¬ ¬

- ª

¯ ° ¯ ±

²

³ ´ µ ¶ ´ · µ µ ¸ ¹ ¶ º · » ¹ ¶ · ½ ¾ ¶ ½ · µ º ¿ À

Figure 1.3.

¼ ½ À Á » is an equational class.

Indeed, let» H ¼ æ ç

F ¡

%. Then

» H ¼ »by Lemma 1.4, and

» H À Á »by

Lemma 1.5, thus» H ¼ À Á »

. Also,»

has a discriminator term by Lemma 1.4.

Thus ¼ ½ À Á » is an equational class by the above theorem of universal algebra.

But ¼ ½ À Á » H ¼ ½ » H ¼ ½ ¼ æ ç

F ¡

% H

F ¡

by Lemma 1.3, and we are done

with proving that F ¡

is an equational class.

QED(Theorem 1.2)¶

In universal algebra, an equational class»

such that¼ æ ç »

has a discriminator

term is called a discriminator variety. So we proved that¤ F ¡

is a discriminator

variety.

The above proof of Theorem 1.2 uses techniques that can be applied in many

cases in algebraic logic. E.g. these same techniques work for cylindric and polyadic

algebras. See e.g. Thms 1.10, 2.3. I G

¥ß 2 ¥ % Q a Qc e

W is called the full F ¡

overthe set

¥. By Lemma 1.3 we could have defined

¤ F ¡

as

F ¡

H ¼ ½ h I G

¥ 2 ¥ % Q a Q

c e

W t ¥is a set

Qor as

F ¡

H ¼ ½ h I G

¥ 2 ¥ % Q a Q

c e


Ó

SetÁ Â Ã

F ¡ def H ¼ h I G

¥ 2 ¥ % Q a Q

c e


Ó

Then¤ F ¡

H ¼ ½

Á Â Ã

F ¡

. 14 This fact, and the class

Á Â Ã

F ¡

will be used in Part II

(section 7.4) when translating our algebraic results to logic.

14BecauseÈ É È « È É

, this is a basic theorem in universal algebra. See e.g. [HMT, 0.3.12].




In the following, we will define our classes of algebras of relations in this style.

So when defining new kinds of algebras of relations, we will first define the sim-

plest version (e.g. the one with top element¥ 2 ¥ 2

Ó Ó Ó

2 ¥ %, and then take all

subalgebras of all direct products of these.

Let» H

Á Â Ã

F ¡

. Then, as we have seen,¤ F ¡

H ¼ ½ »is a variety because

»has a discriminator term and

»is pseudo-axiomatizable. 15 In almost all our

cases, » , where » is the class of the corresponding set algebras, will be pseudo-axiomatizable because

»is defined to be a three-story structure like

¤ F ¡

, only

the operations on the third level will vary (and instead of ¥ 2 ¥

we may have

¥ 2 ¥ 2

Ó Ó Ó

2 ¥), and in most cases

»will have a discriminator term. 16

Theorem 1.2 indicates that F ¡

is indeed a promising start for developing a

nice algebraization of stronger logics (like e.g. quantifier logics), or in the non-

logical perspective, for developing an algebraic theory of relations. After Theo-

rem 1.2, the question comes up naturally whether we can strengthen the postulates

defining¡ F ¡

to obtain a finite set·

of equations describing the variety F ¡

, i.e.

such that F ¡

H ¸ ¹ º

· %would be the case. The answer is due to J. D. Monk:

THEOREM 1.6 F ¡

is not finitely axiomatizable, i.e. for no finite set ñ

of first-

order formulas is F ¡

H ¸ ¹ º

ñ % .

The idea of one possible proof is explained in Remark 2.9 in section 2 herein.

This idea is based on the proof of Thm 2.5 which is the reason why it is postponed

to that part of the paper. See Monk [M64], and also [HMTII, 5.1.57, 4.1.3], for the

original proof of Theorem 1.6 (in slightly different settings).

For a class»

of algebras, let Eq »denote the set of all equations valid in

».

THEOREM 1.7 Eq

F ¡

%

is recursively enumerable but not decidable.15We could have proved

Ä « È Ì Í Ämore directly, as follows. An ultraproduct of full

¦ §’s on

some sets Å

is isomorphic to the full¦ §

on the ultraproduct of the Å

’s, namely if Ç

is an ultrafilter

onÈ

, andÉ ª

denotes Ê ¨ ª Ì

, then

Í É Å ª Ç Ð

«

É Í Å Ç ª, and the isomorphism

Ó

is given by Ç Ô Ç © Ç Ö È Å © Å Å

Ç

. The reader is invited to

check thatÓ

is indeed an isomorphism. This method also is applicable in many places. We chose

the method of pseudo-axiomatizability for proving that Ä is closed under ultraproducts, because we

feel that this method reveals the real cause: our concrete algebras are usually pseudo-axiomatizable,

because “concrete” very much means this, i.e. “concrete” means that there is some extra structure not

coded in the operations, which means that this extra structure may disappear when taking isomorphic

copies.16Even if Ä would not have a discriminator term, then È É Ä would still be a quasi-variety, i.e.

definable by equational implications, because Ä will be pseudo-axiomatizable, hence Ä « Ì Í Ä , thus

È É Ä « È É Ì Í Äwill hold. It is a basic theorem of universal algebra that

Äis a quasi-variety iff

Ä « È É Ì Í Ä ×for some

Ä ×.



20 ALGEBRAIC LOGIC

Proof. An equation holds in¤ F ¡

iff it holds in¼ æ ç

F ¡

%by Lemma 1.3. Let

ñ

be the finite set of first-order formulas such that ¼ æ ç

F ¡

% H ò ó

¸ ¹ º

ñ % , from

the proof of Lemma 1.5. Thus an equation is valid in¼ æ ç

F ¡

%iff it is valid in

¸ ¹ º

ñ % (when all the variables of the original equation are considered to be of

sort(

). The consequences of any finite set of first-order formulas is recursively

enumerable by the completeness theorem of first-order logic. Thus Eq ¤ F ¡

% is

recursively enumerable (and an enumeration is given by the present proof).

The proof of undecidability of Eq F ¡

% goes via interpreting the quasi-equa-

tional theory of semigroups into Eq F ¡

%. The proof consists of two steps:

(Ç

) An equational implication (i.e. a quasi-equation) abouta

is valid in all semi-

groups iff it is valid in¤ F ¡

.

(Ç Ç

) To any equational implicationÙ

there is an equationÚ

in the language of ¤ F ¡

such that F ¡

H Ù iff F ¡

H Ú .

Proof of Ç

% : If Ù is true in all semigroups, then it is true in F ¡

because a is asso-

ciative in F ¡

. If Ù

fails in a semigroupI ` Q Û W

, then take the Cayley-representation

of this semigroup, this is an embedding of I ` Q Û W into I

` Ü 2 ` Ü % Q a W which is a

reduct of

Ò

` Ü 2 ` Ü % 5

¤ F ¡

. ThusÙ

fails in

F ¡

.

Proof of Ç Ç

%: The reason is that

¤ F ¡

is a discriminator variety, and in every

discriminator variety a quasi-equationÙ

is equivalent to an equationÚ

on the sub-

directly irreducibles 17. Now, by F ¡

H ¼ ½ ¼ æ ç

F ¡

%we have that

F ¡

H Ùiff

¼ æ ç

F ¡

% H Ùiff(by the above)

¼ æ ç

F ¡

% H Úiff

F ¡

H Ú.

Now (Ç

) and (Ç Ç

) above give an interpretation of the quasi-equations valid in all

semigroups into the equations valid in F ¡

. Since it is known that the former is

undecidable, we also have that the latter, EqÅ ¤ F ¡

%, is undecidable.

¶

The above method of proof for undecidability is also widely applicable in al-

gebraic logic. The above proof e.g. is in [CM]. For more refined uses of this

technique see e.g. [Ma80], [N85a] (finite dimensional part) [KNSS], [KNSS2],

[AGiN97, chapter II], [K97]. For more on (un)decidability in algebraic logic

we refer to the just quoted works together with Jipsen [Ji92], [Ma78a], [HMTII],

[N86], [N87], [N92], [MV], [Mi95], [N91].

We turn to determining the logic “captured by” F ¡

. We note that the con-

nection with logic will be much more lucid in the case of cylindric (and polyadic)

algebras of ß

-ary relations.

Letð à

áâ ã ä

denote first order logic without equality and using only three vari-

ables

Q z Q |, with countably many binary relation symbols

( þ Q (

e

Q

Ó Ó Ó

(so e.g.

17This is one of the basic facts of discriminator varieties. Assume that å is « æ 2

2 é «

æ é « æ . Then

ì ± í æ ª ï

ï ì ± é í æ é ª ± í æ ªcan be chosen for

ð,

whereì

denotes the switching term andí

denotes symmetric difference.




no ternary relation symbols), and the atomic formulas are( ×

Ã Q Ä %with distinct

variables Ã Q Ä (so atomic formulas of the form (×

Ã Q Ã % are not allowed).

THEOREM 1.8ð

à

áâ ã ä

can be interpreted into Eq F ¡

%. I.e. there is a recursive

function Ú mapping ð à

áâ ã ä

into the set of equations on the language of F ¡

such

that for every ñ 5 ðà

áâ ã ä

ñ is valid iff F ¡

H Ú

ñ %

Ó

Theorem 1.8 will be a consequence of the following, stronger Theorem 1.9. We

stated Theorem 1.8 because it states thatð

à

áâ ã ä

can be interpreted into EqÅ ¤ F ¡

%,

thus Eq F ¡

%is “at least as strong” as

ðà

áâ ã ä

. Set Theory can be interpreted inð

à

áâ ã ä

,

this is proved in Tarski-Givant [TG, ò 4.6, pp.127–134]. Thus the logic captured by F ¡

is strong enough to serve as a vehicle for set theory, and hence for ordinary

mathematics, as we mentioned at the beginning of this chapter. 18

We can characterize the expressive power of F ¡

in terms of ð

à

áâ ã ä

. This will be

stated and proved as Theorem 1.9 below. We need some preparations for stating

Theorem 1.9.

In the equational language of F ¡

let us use the variables Ä × Q Õ 5 ó , where

ó H h Q Q õ Q

Ó Ó Ó

is the set of natural numbers. For any model

õ H I ø Q ( ö

×

W × Ú

of ð à

áâ ã ä

letö

ö

denote the evaluation of the variablesÄ × Q Õ 5 ó

such that

ö

ö

Ä × % H (

ö

×

for all Õ 5 ó

Ó

Recall thatÒ

ø 2 ø % H I G

ø 2 ø % Q a Q c e W 5

Á Â Ã

¤ F ¡

.

If k

is an algebra,ö

is an evaluation of the variables,

is a term, andÚ

is an

equation, thenk

H Ú ö denotes that the equation

Úis true in the algebra

k

under

the evaluation ö of the variables, and

ã ø

denotes the element of m denoted by

the term

when the variables are evaluated according toö

.

Let Ã Q Ä be distinct elements of h

Q z Q | . Then ð ù ú

â

denotes the set of those

elements of ðà

áâ ã ä

which contain only Ã Q Ä as free variables. If ñ 5 ð û ü

â

and õ is a

model, then ñ ö denotes the following binary relation on ø :

ñ

ö

def H h I p Q r W 5 ø 2 ø t õ H ñ p Q r

Ó

The following Theorem 1.9 says 19 that, in a way, the expressive power of ¤ F ¡

isð û ü

â

. We included (i) for its simple content, and (ii) states a correspondence

between meanings of formulas inð û ü

â

and denotation of terms in elements of Á Â Ã

F ¡

. For more on the background ideas of this see Part II of the present paper.

18This also gives another proof for undecidability of Eq ¦ § ª

, because Set Theory is undecidable.19These statements and proofs are simplified versions of those in [TG]. Cf. also [HMTII,

'5.3, 4.3].



22 ALGEBRAIC LOGIC

THEOREM 1.9 (the expressive power of Eq F ¡

%)

(i) For anyñ 5 ð

à

áâ ã ä

there is an equationÚ

such that for all modelsõ

of ð

à

áâ ã ä

õ H ñiff

Ò

ø 2 ø % H Ú ö

ö

Ó

(ii) There are recursive functions ê t ð à

áâ ã ä

Ú ÿ ¢ and á t Ú ÿ ¢

ð à

áâ ã ä

such that for anyñ 5 ð û ü

â

and modelõ

ñ

ö

H ê

ñ %

¥ ¦ ¨

ã ø

Q and

for any term , set ¥ , and evaluation ö ,

¥ ¦ ¨

ô

ô

ã ø

H á

%

¥

ô

ã ø

¨

ú

! #

Ó

Proof. (i) follows from (ii), so it is enough to prove (ii).

The translation functioná t Ú ÿ ¢

ðà

áâ ã ä

is not hard to give. LetÃ Q Ä 5

h

Q z Q |

be distinct, and let

be the third variable, i.e.h Ã Q Ä Q H h

Q z Q |

. Wewill simultaneously define the functions á

ù ú

t Ú ÿ

¢

ð à

áâ ã ä

as follows:

á

ù ú

Ä×

%

def H (

×

Ã Ä % Q

á

ù ú

S %

def H á

% á

% Q á

U %

def H á

% Q

á

ù ú

a %

def H u

á

ù %

%

¢

á

% ú

% %,

á

ù ú

c e

%

def H á

ú ù

%.

For the other direction, we want to define, by simultaneous recursion, a term

ñ Q Ã Q Ä %for all distinct variables

Ã Q Ä 5 h

Q z Q | and

ñ 5 ð ù ú

â

such that for all

models õ we have Ç

% h I p Q r W 5 ø 2 ø t õ H ñ

Ã & p Q Ä & r % H

ñ Q Ã Q Ä %

¥ ¦ ¨

ã ø

Ó

So letñ 5 ð ù ú

â

.

Case 1. If ñ is an atomic formula, then ñ is (×

Ã Ä % or (×

Ä Ã % for some Õ 5 ó (by

ñ 5 ð ù ú

â

).

( ×

Ã Ä % Q Ã Q Ä %

def H Ä ×

,

( ×

Ä Ã % Q Ã Q Ä %

def H Ä ×

c e

.

Case 2. If ñ is a disjunction of two formulas, say ñ is ) 0 , then ) Q 0 5 ð ù ú

â

, and




) 0 Q Ã Q Ä %

def H

) Q Ã Q Ä % S

0 Q Ã Q Ä %.

Case 3. If ñ is a negation of another formula, then ñ is ) for some ) 5 ð ù ú

â

, and

we define

) Q Ã Q Ä %

def H U

) Q Ã Q Ä % .

Case 4. If ñ

begins withu Ã

, thenñ

isu Ã )

for some) 5 ð ù ú

â

, and then we define

u Ã ) Q Ã Q Ä %

def H a

) Q Ã Q Ä %.

Likewise we define

u Ä ) Q Ã Q Ä %

def H

) Q Ã Q Ä % a .

Case 5. Assume thatñ

begins withu

, i.e.ñ

isu )

. Then) 5 ð à

áâ ã ä

can be

arbitrary. It is easy to prove by induction that every element of ð à

áâ ã ä

is a Boolean

combination of formulas in ð û ü

â

Q ð

û 7

â

and ð ü 7

â

. Bring ) into disjunctive normalform )

e

Ó Ó Ó

) @ where each ) × is a conjunction of formulas with two free

variables. Nowu )

is equivalent to

u )

e

%

Ó Ó Ó

u )@

% Q

so by Case 2 we may assume that ) is of form ) ù ú

¢

) ù %

¢

) ú % where ) ù ú 5 ð ù ú

â

,

etc. Nowu )

is equivalent to

)

ù ú

¢

u

)

ù %

¢

)

ú %

%

Ó

We now define

u

) ù %

¢

) ú % % Q Ã Q Ä %

def H

) ù % Q Ã Q % a

) ú % Q Q Ä % .

It is not difficult to check that the so defined

ñ Q Ã Q Ä %satisfies our requirement

Ç

%.

¶

One can get very far in doing algebraic logic (for quantifier or predicate logics)

via F ¡

’s. 20 As we have seen, the natural logical counterpart of ¤ F ¡

’s is clas-

sical first-order logic restricted to three individual variables and without equality.

As shown in [TG, ò 5.3], this system is an adequate framework for building up

20If we want to investigate nonclassical quantifier logics, we can replace the Boolean reduct D of

¥ « D Ì

¦ §with the algebras (e.g. Heyting algebras) corresponding to the propositional

version of the nonclassical logic in question.



24 ALGEBRAIC LOGIC

set theory and hence metamathematics in it. One can illustrate most of the main

results, ideas and problems of algebraic logic by using only F ¡

’s.

We do not know how far F ¡

’s can be simplified without losing this feature. In

this connection, a natural candidate would be the class E F

of Boolean semigroups

of relations defined as

E F

H ¼ ½ h I G

¥ 2 ¥ % Q a W t ¥

is a set

Ó

So we require only one extra-Boolean operation “a

”.

The question is, how far E F

could replace F ¡

as the simplest, “introduc-

tory” example of Tarskian algebraic logic. We conjecture that the answer will

be “very far”. E F

is a discriminator variety with a recursively enumerable but

not decidable equational theory, and it is not finitely axiomatizable. Thus Theo-

rems 1.2–1.7 remain true if F ¡

is replaced with E F

in them. 21 We conjecture

that, following the lines of [TG,ò

5.3], set theory can be built up in E F

instead of F ¡

with basically the same positive properties (e.g. finitely many axioms) as the

present version [TG] has 22. It would be nice to know if this conjecture is true,

and, more generally, to see a variant of algebraic logic elaborated on the basis of E F

. We do not know what natural fragment of first-order logic with three vari-ables corresponds to E F

(if any). It certainly is difficult to simulate substitution

of individual variables using onlya

. The converse operation,c e

, is the algebraic

counterpart of substitution because, intuitively, (

Ä þ Q Ä

e

%c e

H (

Ä

e

Q Ä þ % . One can

simulate quantification bya

, and it is easily seen thata

is stronger than quantifica-

tion but withoutc e

it is not clear exactly how much stronger 23. Curiously enough,

these issues are better understood in the case of cylindric algebras to be discussed

in section 2.

If we want to algebraize first-order logic with equality, we have to add an extra

constant G º , representing equality, to the operations.F F ¡

denotes the class of

algebras embeddable into direct products of algebras of the form

I G

¥ 2 ¥ % Q a Q

c e

Q G º W

whereG º H G º I ¥ H h I Ã Q Ã W t Ã 5 ¥

is a constant of the expanded algebra. I.e.

F F ¡

H ¼ ½ Q I G

¥ 2 ¥ % Q a Q

c e

Q G º W t ¥is a set

T

Ó

F F ¡

abbreviates representable relation algebras.F F ¡

’s have been investigated

more thoroughly than F ¡

’s; actually, Theorems 1.2, 1.6 above were proved first

forF F ¡

’s.

21The proofs of Theorems 1.2, 1.7 given here go through for¦

with the obvious modifications.

Nonfinite axiomatizability of ¦

will follow from the later Thm.s 1.10, 1.11.22Perhaps here [N85], [N86] can be useful, because an analogous task was carried through there.

The last 12 lines of Jonsson [J82, p. 276], seem to be also useful here.23For applications in propositional dynamic logic,

¦ seems to be more relevant than

¦ §, be-

cause there converse (of programs or actions) is not an essential feature of the logic.




Letð

á

â ã ä

denote first-order logic with three individual variables

Q z Q |, with

equality, and with infinitely many binary relation symbols. (Thus the atomic for-

mulas are(

Ã Ä % Q Ã H Äfor any variables

Ã Q Ä 5 h

Q z Q | and the logical connec-

tives are Q Q u

Q u z Q u | .)

THEOREM 1.10 (basic properties of F F ¡

)

(i)F F ¡

is a nonfinitely axiomatizable discriminator variety with a recursively

enumerable but undecidable equational theory.

(ii) The logic captured byF F ¡

isð

á

â ã ä

, i.e. there are recursive functionsê t

ð

á

â ã ä

Ú ÿ¡ £ ¢ , and á t Ú ÿ ¢

ð

á

â ã ä

such that the “meanings” of ñ

and ê

ñ %as well as those of

and

á

%coincide, i.e. for any model

õ,

ñ 5 ð

á

â ã ä

with free variables

Q z , term and evaluation ö of variables,

ñ

ö

H ê

ñ %

¥ ¦ ¨

ã ø

and

¥ ¦ ¨

ô

ô

ã ø

H á

%

¥

ô

ã ø

¨

ú

! #

Ó

Proof. Obvious modifications of the proofs of Theorems 1.2, 1.7, 1.8 prove Theo-

rem 1.10, except for nonfinite axiomatizability of F F ¡

. For the proof of nonfinite

axiomatizability of F F ¡

see Remark 2.9.¶

The classes of algebrasF F ¡

Q

F ¡

Q

E F

have less operations in this order, they

form a chain of subreduct classes. Note that Eq

» %denotes the set of all equations

in the language of »

holding in»

. Thus

Eq E F

% VEq

¤ F ¡

% VEq

F F ¡

%

Ó

The next theorem says that these classes are finitely axiomatizable over the bigger

ones.

THEOREM 1.11 Let ·þ denote the following set of equations:

S z % c e H

c e S z c e Q

a z % c e H z c e a

c e Q

c e c e H

a U

c ea U z % z

a U

zc e

a U z %

U z %c e

a z %c e

Ó

Then Eq E F

% S · þaxiomatizes

F ¡

, and Eq E F

% S · þ S h G º a

H

axiom-

atizesF F ¡

.

The proof can be found in Andreka-Nemeti [AN93].¶

Theorem 1.11 talks about interconnections between the operations of F F ¡

. It

says, in a way, that the sole cause of nonfinite axiomatizability of F F ¡

is the



26 ALGEBRAIC LOGIC

operationa

, it is so strong that the other operations,c e

andG º

, are finitely axioma-

tizable with its help. This is in contrast with the case of cylindric algebras of ß -ary

relations, where the strength of the operations are “evenly distributed”.

The next figure, taken from [AN93] describes completely the interconnections

between the operationsa Q

c eQ G º

(in the presence of the Boolean operations). On

Figure 1.4, all classes represented by the nodes are varieties, except the ones insidea box (those are only quasi-varieties), and the classes inside a circle are not finitely

axiomatizable, except ¡

. 24

More on the equational theories of F F ¡

Q

¤ F ¡

and E F

:

Theorem 1.2 says that there is a set · of equations which defines¤ F ¡

. Let ·

be an arbitrary set of equations defining F ¡

. What do we know about·

? Theo-

rem 1.6 says that · is not finite, and Theorem 1.7 says that · can be chosen to be

recursively enumerable. By using the fact that F ¡

is a discriminator variety and

that Eq F ¡

%is recursively enumerable, and by using an argument of W. Craig,

one can show that·

can be chosen to be decidable 25 , i.e. there is a decidable

set·

defining F ¡

. On the other hand, we know that·

has to be complex in

the following sense: to any numberö

,·

must contain an equation that uses more

thanö

variables and all of the operation symbolsS Q U Q a

. There is an·

such that

c e

occurs only in finitely many members of ·

, by Theorem 1.11. The analogous

statements are true for E F

Q

F F ¡

. 26

Concrete decidable sets · definingF F ¡

are known in the literature, cf. e.g.

Monk [M69]. Lyndon [Ly] outlines another recipe for obtaining a different such

· . Hirsch–Hodkinson [HH] also contains such a set · . Some of these work for E F

Q

F ¡

. However, the structures of these·

’s are rather involved. 27 In this

connection, we note that the following is still one of the most important open

problems of algebraic logic:

24In Pratt [Pr90], the class ¦ X

of representable Boolean monoids is obtained from our¦

’s

by adding` a

as an extra distinguished constant. So the extra-Boolean operations of the ¦ X

’s are

` a , and thus ¦ ’s are the ` a -free subreducts of ¦ X ’s. All the results mentioned above for ¦ ’s

carry over to ¦ X ’s; e.g. ¦ X is a discriminator variety, hence the simple ¦ X ’s form a universallyaxiomatizable class, Theorems 1.2, 1.6 above apply to

¦ X.

§ ¦ § ¦ ¦ X ¦ §all occur

as nodes on Figure 1.4.25The idea is as follows. Let c be recursively enumerable, say c « ð d ª ð f ª

for a recursive

functionð

. For each numberg

, leth g ª

denote the conjunction of g

copies of ð g ª

. Since¦ §

is a

discriminator variety, there is an equation i h g ª ª which is equivalent to h g ª in È p r ¦ § . Moreover,

from i h g ª ª we cancompute back h g ª , see an earlier footnote. Then c

×

def « i h d ª ª i h f ª ª

is equivalent to c and c

×

is decidable. The decision procedure for c

×

is as follows: Take any equationv

. Decide whetherv

is i ª for some or no, and if yes, compute the . If we get an , check whether

is the conjunction of some, say

g, copies of an equation

Ó

. If yes, computeð g ª

and check whetherÓ

isð g ª

. If yes, v is inc

×

, otherwise not.26The need of infinitely many variables in any axiom system for

§was proved in Jonsson [J91],

the need of all the operation symbols ±

in addition is proved in Andreka [A94]. By Theorem 1.11

then the same hold for¦ § ¦

.27Cf. [HMTII, pp. 112–119], for an overview.




w x x x

w x x x x

Y

w x x

w x x x

j

w x x w x x x

Y k l

m n o m

w x

w x x

m { { {

o ~

o

o

~

o

~

o

~

{ ~

o

~

o

o

o o

~

~ ~ { ~

o

~

~

o

~

o

o

o o

~

o

~

Figure 1.4.

PROBLEM 1.12 Find simple, mathematically transparent, decidable sets·

of

equations axiomatizing E F

Q

F ¡

Q

F F ¡

. (A solution for this problem has to beconsiderably simpler than, or at least markedly different from the

·’s discussed

above.)

Equational axiom systems for algebras of relations like forF F ¡

, F ¡

Q

E F

are

interesting not only because of purely aesthetical reasons, but also because such an

axiom system gives an inference system for the corresponding logic. About this

logical connections see e.g. Theorems 5.4, 5.5, 6.3 in Part II.

Since the classesF F ¡

Q

F ¡

Q

E F

are not finitely axiomatizable, finitely axiom-

atizable approximations, or “computational cores” are used for them. For F ¡

we



28 ALGEBRAIC LOGIC

can take¡ F ¡

as such an approximation. ForF F ¡

, the varietyF ¡

of relation alge-

bras, defined by Tarski, is used in the literature as such an approximation. We get

the definition of F ¡

from the definition of ¡ F ¡

by replacing (4) with one stronger

equation (5), and by adding the equation G º a

H

.

DEFINITION 1.13 (

F ¡

, an abstract approximation of

F F ¡

)

F ¡

is defined to bethe class of all algebras of the similarity type of F F ¡

’s which satisfy the equations

(1)–(3) from the definition of ¡ F ¡

, together with (5), (6) below.

%

c e

a U

a z % U z

Ó

% G º a

H

Ó

¶

Equation (5) is equivalent, in the presence of the otherF ¡

-axioms 28 with the

following so called triangle-rule (5’)

Ü %

z a | % H iff

z

a |

c e

% H iff

|

z

c e

a

% H

Ó

Intuitively, (5’) says that the three ways of telling that no triangle

exists, are equivalent. 29 Thus, a relation algebra is a Boolean algebra together withan involuted monoid sharing the same universe, and the interconnection between

the two structures is that they form a normal¢ ¡ £

and the triangle rule (5’) holds.

Equation (6) says thatG º

is the neutral element of the semigroup operation “a

”.

We note that in algebraic logic this translates to the so called Leibniz law of equal-

ity in logic which says that equals cannot be distinguished 30.

28We note that µ « µ µÌ

« µ are usually omitted from the axiomatization of § , because

they follow from the rest of the axioms.29In a more algebraic language, (5’) says that the maps

Ô and

Ô

Ì

are conjugates

of each other, and likewise the maps Ô

and Ô Ì

are conjugates. We recall from

[JT51] that in a¦ §

the functions

v are conjugates of each other means that ï ª « µ

iff

ï

v

ª « µfor all

.

30For more on this see [BP89, p.10].




F ¡

is a very strong computational core forF F ¡

, almost all natural equations

aboutF F ¡

hold also inF ¡

. 31 AnF ¡

which is not inF F ¡

is called a nonrep-

resentableF ¡

. Equations holding inF F ¡

and not inF ¡

can be obtained from

each finite nonrepresentableF ¡

by using thatF ¡

is a discriminator variety of ¡ £

’s, as follows. Letk

5

F ¡

U

F F ¡

be finite. Thenk

cannot be embedded

in anyF F ¡

, and this can be expressed with a universal formula becausek

is fi-

nite. Now using the switching function, this universal formula can be coded asan equation

Ú. Then

Údoes not hold in

k

5

F ¡

, while it holds inF F ¡

. Many

finite nonrepresentableF ¡

’s are known in the literature. The smallest such has 16

elements. 32 A source for examples of finite nonrepresentableF ¡

’s is the so called

Lyndon algebras. A finite Lyndon algebra is a finiteF ¡

such thatG º

is an atom,

pc e

H p Q p a p H p S G º, and

p a r H U

p S r %hold for all other distinct atoms

p Q r.

Infinitely many of the finite Lyndon algebras are nonrepresentable (and infinitely

many are representable). Another way of getting finite nonrepresentableF ¡

’s is to

“distorte” a representable one. There are some known methods, like splitting and

dilating 33 with which we can obtain nonrepresentableF ¡

’s from representable

ones. NonrepresentableF ¡

’s are almost as important as representable ones. 34

Some special, interesting classes of F F ¡

’s turn out to be finitely axiomatizable,

below we list two such classes. These finite axiomatizations give (non-standard)finitary inference systems for

ð

á

â ã ä

, cf. Mikulas [Mi96], [Mi95]. The elegant,

purely algebraic proofs for the items in the next theorem are examples for signif-

icant applications of algebra to logic, via connections between algebra and logic

indicated in Part II of this paper.

THEOREM 1.14 Let ñ

and )

denote the following formulas, respectively.

u

z

c e a

G º

¢

z c e a z G º

¢

c e a z H %

u z

é

H

é

H z

¢

z

¢

zc e

a z G º %

Ó

ThenF F ¡

¸ ¹ º ñ H

F ¡

¸ ¹ º ñfor

ñ H h ñ and

ñ H h ) .

¶

For the proofs see Maddux [Ma78], Tarski-Givant [TG]. AnF ¡

in which ñ is trueis called a quasi-projective

F ¡

, or a

F ¡

, and anF ¡

in which ) is true is called a

functionally denseF ¡

. 35 We can look at Theorem 1.14 in two ways: on one hand

31In other words, only complicated equations can distinguish § and § .32This was found by McKenzie [McK].33For splitting in § see Andreka-Maddux-Nemeti [AMN], for dilation in § see Nemeti [N86],

Nemeti-Simon [NSi], Simon [Si97].34E.g. one proof of nonfinite axiomatizability of

§goes by finding infinitely many nonrepre-

sentable § ’s whose ultraproduct is representable. Investigating the structure of possible axiom sys-

tems for § often boils down to finding suitable nonrepresentable § ’s.35That any § is representable is a theorem of Tarski, an elegant algebraic proof was given by

Maddux [Ma78]. A different, illuminating proof is given in Simon [Si96]. For logical applications of

this area see [TG]. The proof that every functionally dense §

is representable is in Maddux [Ma78].

See also [AGMNS].



30 ALGEBRAIC LOGIC

it says that the class of quasi-projectiveF F ¡

’s is finitely axiomatizable (whileF F ¡

is not), and on the other hand it says that quasi-projectiveF ¡

’s are representable

(whileF ¡

’s in general are not). (And the same for functionally denseF ¡

’s.)



CHAPTER 2

ALGEBRAS OF RELATIONS IN GENERAL

By this point we might have developed some vague picture of how algebras

of binary relations are introduced, investigated etc. One might even sense that

they give rise to a smooth, elegant, exciting and powerful theory. However, our

original intention was to develop algebras of relations in general, which should

surely incorporate not only binary but also ternary, and in generalß

–ary relations.

Let us see how to generalize ourF F ¡

’s and F ¡

’s to relations of higher ranks.

Let us first fixß

to be a finite ordinal. As we said, we would like the new algebras

to be expansions of F F ¡

’s (and F ¡

’s). However, defining composition of ß –ary

relations forß õ

is complicated 1 . Therefore the following sounds like a moreattractive idea: We single out the simplest basic operations on ß –ary relations,

and hope that composition will be derivable as a term–function from these. Let

us see how we could generalize our generic or fullF F ¡

’s I G

¥ 2 ¥ % Q a Qc e

Q G º W

to relations of rank ß

. The obvious part is that these algebras will begin with

I G

¥ 2 ¥ 2

Ó Ó Ó

2 ¥ % Q G º Q

Ó Ó Ó

W , where

G º H h I Ã Q Ã Q

Ó Ó Ó

Q Ã W t Ã 5 ¥

is theß

–ary identity relation. Again,G º

is a constant, just as it was in theF F ¡

case. Let@

¥denote

¥ 2 ¥ 2

Ó Ó Ó

2 ¥, e.g.

â

¥ H ¥ 2 ¥ 2 ¥. The new operations

(besides the Boolean ones andG º

) we will need are the algebraic counterparts of

quantificationu Ä ×

, forÕ ª ß

. So, we want an operation that sends the relation

defined by(

Ä þ Q Ä

e

%to the one defined by

u Ä þ (

Ä þ Q Ä

e

%, and similarly for

u Ä

e

.

For( ) ¥ 2 ¥

let Dom

( %and Rng

( %denote the usual domain and range of

(.

Forß H õ

we define

j þ

( % H ¥ 2 Rng

( %and

j

e

( % HDom

( % 2 ¥

Ó

Now

I G

¥ 2 ¥ % Q jþ

Q j

e

Q G º W

is the full cylindric set algebra of binary relations over¥

, for short the full«

Á

ä

.

Before turning seriously toß

–ary relations, we need the following:

1Composition for g -ary relations is studied in Marx–Nemeti–Sain [MNS], Marx

[Ma95]. The definition is

é ª «

é #

é

Ì

¬

é

Ì

é

¬

¬

é

é

ª

.

31



32 ALGEBRAIC LOGIC

CONVENTION 2.1 Throughout we will pretend that Cartesian products and Car-

tesian powers are associative such that @

¥ 2 - ¥ H

@ ¯

- ¥ , and if e.g.

( )

â

¥then

ä

¥l 2 ( ) ° ¥ ± ( 2

ä

¥.

The full«

Á

@, i.e. the full cylindric set algebra of

ß–ary relations, is the natural

generalization of «

Á

ä

as follows. Let ( )

@

¥ . If Rng

( % H h I r

e

Ó Ó Ó

r@

c e

W t

I rþ

r

e

Ó Ó Ó

r@

c e

W 5 (

for somer

þ

, thenj

þ

( % H ¥ 2

Rng

( %

considered as aset of ß -tuples. Similarly, let Dom

( % H h I rþ

Ó Ó Ó

r@

c

ä

W t I rþ

Ó Ó Ó

r@

c

ä

r@

c e

W 5

(for some

r @

c e

, and let

j @

c e

( % HDom

( % 2 ¥. Generalizing this to

j ×with

Õ ª ß arbitrary, we obtain

j×

( % H

h I rþ

Q

Ó Ó Ó

Q r×

c e

Q p Q r× ¯

e

Q

Ó Ó Ó

Q r@

c e

W t I rþ

Q

Ó Ó Ó

Q r@

c e

W 5 ( and p 5 ¥

Ó

j ×is one of the most natural operations on relations. It simply forgets the

Õ-th

argument of the relation, or in other words, deletes theÕ-th column. However, since

deleting the Õ -th column would leave us with an

ß U % –ary relation, Dom

( % if

Õ H ß U , we replace the Õ -th column with a dummy column i.e. in the Õ H ß U

case we represent Dom

( % with the “pseudo ß –ary relation” Dom

( % 2 ¥ . The

“real rank” of an( )

@

¥is always easy to recover, namely it is

³

( % H h Õ ª

ß t j ×

( %

é

H ( . So j × is the natural operation of removing Õ from the (real) rank

of a relation.

For example, j father when applied to the “father, mother, child” relation gives

back the “mother, child” relation coded as “anybody, mother, child” (in which

the anybody argument carries no information i.e. is dummy). By a full «

Á

@we

understand an algebra

Ò ´ µ

@

¥ %

def H I G

@

¥ % Q j þ Q

Ó Ó Ó

Q j @

c e

Q G º W

for some set¥

. By a«

Á

@we understand a subalgebra of a full

«

Á

@with nonempty

2 base set¥

i.e.

«

Á

@

def H ¼ h

Ò ´ µ

@

¥ % t ¥is a nonempty set

Ó

By a representable cylindric algebra of ß

–ary relations, (anF

«

¡

@) we understand

a subalgebra of a direct product of full «

Á

@ ’s (up to isomorphism), formally:

F

«

¡

@H ¼ ½ h I G

@

¥ % Q jþ

Q

Ó Ó Ó

Q j@

c e

Q G º W t ¥ is a set

Ó

Note thatF

«

¡

@H ¼ ½ h

Ò ´ µ

@

¥ % t ¥is a set

H ¼ ½

full«

Á

@% H ¼ ½ «

Á

@

Ó

By the

same argument as in the case of ¤ F ¡

’s, everyF

«

¡

@is directly representable as an

algebra of ß

–ary relations (with the greatest relation a disjoint union of Cartesian

spaces).F

«

¡

@ is one of the “leading candidates” for being the natural algebra of

ß–ary relations.

2Excluding the empty base set here is not essential, it serves easier applicability in the second part

of this paper, in section 7.



2. ALGEBRAS OF RELATIONS IN GENERAL 33

The abstract algebraic picture is simple: anF

«

¡

@is a

¢ ¡

together withß

clo-

sure operations and an extra constant. Accordingly, an (abstract) cylindric algebra

of dimensionß

, a«

¡

@, is defined as a normal

¢ ¡ £

withß

self-conjugated and

commuting closure operations, and with a constant satisfying two equations. In

more detail:

DEFINITION 2.2 («

¡

@, an abstract approximation of

F

«

¡

@)

«

¡

@is definedto be

the class of all algebras of the similarity type of Ò ´ µ

@

¥ %which satisfy the follow-

ing equations for allÕ Q ¶ ª ß

.

(1) The axioms for normal ¡ £

, i.e.

the Boolean axioms,

j × H Q j ×

S z % H j ×

% S j ×

z %

Ó

(2) Axioms expressing that j × ’s are self-conjugated commuting closure operations,

i.e.

(i)

j×

H j×

j×

,

(ii)z j ×

H iff

j × z

H ,

(iii)j × j ·

H j · j ×

.

Because of the above axioms, the notation j

¨ ¸

H j × ¹

Ó Ó Ó

j × º

where » H

h Õ

e

Q

Ó Ó Ó

Q Õ

ø

makes sense. We will use that notation from now on. We will also use

the convention 3 that ß H h Q Q

Ó Ó Ó

Q ß U .

(3) The constant G º

has domain 1 and satisfies the “Leibniz-law”, i.e.

(i) j

¨

@ ¼ ¾ × ¿

G º H ,

(ii) j

¨

@ ¼ ¾ ×

ã

· ¿

G º j ×

H

whenever

j

¨

@ ¼ ¾ ×

ã

· ¿

G º and Õ

é

H ¶ .¶

An equivalent form of saying that j×

is self-conjugated4 is to say that the com-

plement of a closed element is closed. Thus, in the above definition, (2)(ii) can be

replaced with

j × U j ×

H U j ×

Ó

3For more on this see e.g. [HMT, Part I. pp. 31–32].4I.e. that it is the conjugate of itself, cf. the footnote to the triangle-rule (5)’ in section 1.



34 ALGEBRAIC LOGIC

Note that (2)(ii) is an analogon of the triangle rule (5)’ in the definition of F ¡

.

(3)(ii) expresses that the closure operator j×

is “discrete”, or is the identity, when

relativized to

G º× ·

def H j

¨

@ ¼ ¾ ×

ã

· ¿

G º Q

Õ

é

H ¶. Both (2)(ii) and (3)(ii) have equivalent equational forms, e.g. an equivalent

form of (3)(ii) is

G º × · j ×

G º × ·

% H G º × ·

whenÕ

é

H ¶

and an equivalent (together with the other axioms) form of (2)(ii) is

j ×

j × z % H j ×

j × z

Ó

Connection with geometry: The names in cylindric algebra theory come from

connection with geometry. Namely, an ß -ary relation is a set of ß -tuples, while

an ß -tuple is a point of the ß -dimensional space. E.g. I p Q r W is a point in the õ -

dimensional space with coordinatesp

andr

, whileI p Q r Q w W

is a point in the 3-

dimensional space with coordinatesp Q r

andw

. Thus a binary relation is a subset of

the 2-dimensional space, while anß

-ary relation is a subset of theß

-dimensional

space. Hence the name “cylindric algebra of dimension ß”.

If (

is a subset of theß

-dimensional space, thenj ×

( %is the cylinder above

(

parallel to theÕ-th axis, and

G ºis the main diagonal. Hence the name “cylindric

algebra”. The operationsj ×

andG º

are called “cylindrifications” and “diagonals”

in«

¡

-theory, andG º × ·

is usually denoted byº × ·

(for diagonal). Because of these

geometrical meanings, also the operations of F

«

¡

@are easy to draw. This is

illustrated on Figure 2.1, see also Figure 2.3.

How can we draw the operations of F F ¡

? Converse is easy to draw: the con-

verse of (

is the mirror image of (

w.r.t. the diagonal. However, relation compo-

sition of two relations ( Q ` is not so easy to draw. See Figure 2.2.

Thus cylindric algebras («

¡

’s) are simpler than relation algebras

F ¡

’s in twoways: «

¡

’s have only unary operations j×, while the central operation of

F ¡

is

the binary composition operationa

; and secondly, cylindrifications are easy to

draw, while composition is not so easy to draw. There are furher connections with

geometry, e.g. via projective planes. We do not discuss these herein, but cf. Monk

[M74], [Gi97], [NS97], [NS97], [AGiN97, Chapter II].

Connection between «

¡

’s andF ¡

’s: A natural question comes up: can these

“simple” «

¡

’s recapture the power of F ¡

’s? This was a requirement we expected

to meet, namely we expected that the theory of ß

-ary relations should be an ex-

tension of that of binary relations. The answer is thatF

«

¡

@with

ß Â Ãis strong

enough to recaptureF F ¡

, whileF

«

¡

ä

is not strong enough. In more detail:




Mirroring cannot be expressed by the diagonal and the cylindrifications in the

plane (i.e. in 2-dimensional space), but it can be expressed if we can move out to

Ã-dimensional space, see Figure 2.3.

Namely, by lettingÁ

×

·

%

def H j ×

G º × ·

%and

ü H ¥ 2 ¥l 2 h Ã

for some fixedÃ 5 ¥

, we have

( c e H ü

Á

ä

e

Á

e

þ

Á

þ

ä

j

ä

(

for( ) ü

. Here, we identified the binary( ) ¥ 2 ¥

with the ternary(P 2 4 h Ã

,

and similarly for (c e

. Composition also can be expressed:

( a ` H ü j

ä

Á

e

ä

j

ä

(

Á

þ

ä

j

ä

` %.

A more natural approach is based on identifying a binary relation ( ) ¥ 2 ¥

with the ternary relation

Dr

( %

def H ( 2 ¥

we call Dr

( %the dummy representation of

(as a ternary relation. Then Dr t

¥ 2 ¥ %

¥l 2 ¥ 2 ¥ %; and

Dr

(c e

% H

Á

ä

e

Á

e

þ

Á

þ

ä

Dr

( %,

Dr

( a ` % H j

ä

Á

e

ä

Dr

( %

Á

þ

ä

Dr

` % %.

So in a sense,F F ¡

’s form a kind of a reduct of F

«

¡

@

’s forß Â Ã

. In more

detail: Letk

5 «

¡

@, ß Â Ã . The relation-algebra reduct

Ò Æ k

of k

is defined as

Ò Æ k def H I ( p

k

Q S

Q U

Q a Q

c e

Q G º

þ

e

W Q

where

( p

k

H h p 5 m t j

·

p H pfor all

õ ¶ ª ß

and if p Q r 5 ( p

k

, then

p

c e

def H

Á

ä

e

Á

e

þ

Á

þ

ä

p Q

p a r

def H j

ä

Á

e

ä

p

Á

þ

ä

r %

Ó

Now, Dr t I G

¥ 2 ¥ % Q a Q c e Q G º W È

Ò Æ Ò ´ µ

@

¥ %is an isomorphism for

ß Â Ã.

We define

ò Ê «

¡

@

def H h

Ò Æk

t

k

5 «

¡

@

forÃ ß

Ó

For» ) «

¡

@,

ò Ê »is defined similarly.



36 ALGEBRAIC LOGIC

ThenF F ¡

H ¼ ò Ê

F

«

¡

@for all

ß õ. The classes

¼ ò Ê «

¡

@

ß Ã %form

a chain betweenF ¡

andF F ¡

, providing a “dimension-theory” for 5F ¡

. In more

detail,¼ ò Ê «

¡

â Í

F ¡

(Monk [M61]),¼ ò Ê «

¡ Î

H

F ¡

(Maddux, [Ma78a]),F ¡ Ï

¼ ò Ê «

¡

°

±

Ó Ó Ó

±

F F ¡

(Monk [M61]), and 6

F F ¡

H h ¼ ò Ê «

¡

@ t Ã ß ª ó H ¼ ò Ê «

¡

Ó

Investigating the connection betweenF F ¡ F ¡

%and

F

«

¡

«

¡

%is an interest-

ing subject. Some of the references are Monk [M61], Maddux [Ma78a], [Ma89],

[HMTII,ò

5.3], Nemeti–Simon [NSi], [Si96], [Si97], Hirsch–Hodkinson [HH97].

Recent developments in theF ¡

U«

¡

(– polyadic algebras) connection are reported

in [Si97], [NSi].

Thus the answer is thatF

«

¡

@’s,

ß õ, do recapture the power of

F F ¡

’s. (On

the other hand,F

«

¡

ä

’s do not7.)

Connection with logic: Cylindric algebras have a very close and rich connection

with logic. This connection is partly described in [HMTII,ò

4.3], and in Examples

6, 8, 9 in section 7 herein.

Summing up this connection very briefly:F

«

¡

-theory corresponds to model

theory of first-order like languages (or quantifier logics), while abstract «

¡

-theory

corresponds to their proof theory. Individual«

¡

’s correspond to theories in such

logics, homomorphisms between «

¡

’s correspond to interpretations between the-

ories, while isomorphism of «

¡

’s corresponds to definitional equivalence of mod-

els and/or theories. «

¡

-theoretic terms and equations correspond to first-order for-

mula schemas, an equationÚ

is valid inF

«

¡

if its corresponding formula schema is

valid, an equational derivation of Ú

corresponds to a proof of the formula (schema)

corresponding to Ú . More on this is written in section 7, Examples 6,8,9. Here,

first-order like languages encompass finite-variable fragments of first-order lan-

guage (FOL for short), usual FOL, FOL with infinitary relation symbols but with

finitary logical connectives, FOL considered as a propositional multi-modal logic,

FOL with several modified semantics etc.Most of the above is discussed in [HMTII, ò 4.3], especially when taken together

with [vB96]. Some other references illustrating the rich connection of «

¡

’s with

logic are the following. In Monk [M93] the connection with FOL is treated. In

[N87] and in [R] valid formula-schemas, in [N90] model theory of FOL with in-

finitary relation symbols, in [N96] FOL with generalized semantics, in [A], [SA]

algebras of sentences, in [Se85] and Biro-Shelah [BiSh] model theoretic notions

like saturated, universal, atomic models are investigated with the help of «

¡

’s, re-

spectively. [AvBN96], [vB96], [MV] connect«

¡

’s with modal logic, [S95], [SGy]

5As later, in Thm. 2.14(ii), we will see, this is analogous to the chainÈ Ð r

é §

é Ò Ó

Ô Õ µ ª

between §

é and §

é .6For the definition of § Ö see Def. 2.12.7For example, Eq §

is undecidable, while Eq § is decidable, see Thm. 1.10(i), Thm. 2.3(iii).




use«

¡

’s for searching for a FOL with nicer behaviour. Further references on this

line are e.g. [vB97], [V95a].

The connection of «

¡

’s with logic also sheds light on the above ways of ex-

pressing composition and converse of binary relations in«

¡

. Namely,F

«

¡

@is

the algebraic counterpart of first-order logic with ß variables (see Part II, example

6 in section 7), and in particularß

-variable first-order formulas and terms in the

language of F

«

¡

@are in strong correspondence with each other (see Corollary 5.5

in Part II). TheF

«

¡

-terms in the definition of anF ¡

-reduct are just the transcripts

of the 3-variable formulas defining composition and conversion of binary rela-

tions. (On the intuitive meaning of the terms

Á

×

·

see the remark after Example 7 in

section 7.)

At this point we can state the counterparts of Theorems 1.2–1.10. 8

THEOREM 2.3 (basic properties of F

«

¡

@) Let

ßbe finite.

(i)F

«

¡

@

is a discriminator variety, with a recursively enumerable equational

theory.

(ii)F

«

¡

@is not axiomatizable with a finite set of equations and its equational

theory is undecidable if ß õ .

(iii)F

«

¡

ä

is axiomatizable with a finite set of equations, and its equational the-

ory is decidable. The same is true for F

«

¡

e

. Any«

¡

ä

satisfying for all

Õ Q ¶ ª õ ,

Õ

é

H ¶

j ×

Û

Á

·

×

j ×

G º

H j þ

Û j

e

Q or

j þ

Û j

e

U

j ×

Á

·

×

j ×

U G º %

is representable 9.

(iv) The logic captured byF

«

¡

@is first-order logic with equality restricted to

ß

individual variables.

Proof. The proof of (i) goes exactly as in the previous section, cf. the proofs of Thm. s 1.2, 1.7.: The subdirectly irreducible members of F

«

¡

@are exactly the

isomorphic copies of the nontrivial«

Á

@’s (i.e. those with nonempty base set

¥),

and a switching term isj

¨

@

, i.e. in«

Á

@we have10

j

¨

@

Hè

if

é

H

if

H

Ó

8L. Henkin and A. Tarski proved that § é is a variety, J. D. Monk [M69] proved that § é is

not finitely axiomatizable, L. Henkin gave a finite equational axiom system for §

, and D. Scott

proved that Eq § ªis decidable.

9The above quasi-equation and equation then are equivalent to the so-called Henkin-equation (see

[HMTII, 3.2.65]), which was further simplified in Venema [V91,'

3.5.2]. On the intuitive meaning of

these see the paragraph preceding Thm.7.8 in this work.10Recall that

g « µ d f

g ± d

. Thusg

f « f ×

g ± d

.



38 ALGEBRAIC LOGIC

The proof of undecidability of F ¡

can be adapted here, too, by using the above

outlined connection betweenF ¡

and «

¡

. Namely, the quasi-equational theory of

semigroups can be interpreted inF

«

¡

@, e.g. by using the term

z H j

ä

Á

e

ä

j

¨

@ ¼

ä

Á

þ

ä

j

¨

@ ¼

ä

z %

Ó

The proof for nonfinite axiomatizability of

F

«

¡

@ Q ß õ

will be discussed inRemark 2.9. The proof of the first part of (iii) can be found in [HMTII, 3.2.65,

4.2.9]. It is not hard to check that in all«

¡

ä

’s, the quasi-equation and the equation

in (iii) are equivalent to each other. Also, the equation in (iii) is equivalent to

Ç

% j þ

Û j

e

j ×

Á

·

×

j ×

U G º %

which is then preserved under taking perfect extensions (because negation – occurs

only in front of a constant). Thus it is enough to show that any simple atomick

5

«

¡

ä

satisfying Ç

% is representable. Now, Ç

% implies that there are no defective

atoms ink

, in the sense of [HMTII, 3.2.59], and thenk

5

F

«

¡

ä

by [HMTII,

3.2.59]. For (iv) see example 6 in section 7 of Part II.¶

More on the fine-structure of the equational theory of F

«

¡

@will be said later,

after Problem 2.11.

How far did we get in obtaining algebras of relations in general (binary, ternary,Ó Ó Ó

, ß –ary,Ó Ó Ó

)?F

«

¡

@is a smooth and satisfactory algebraic theory of ß –ary

relations. So, can our theory handle all finitary relations? The answer is both yes

and no. Namely, since ß is an arbitrary finite number, in a sense, we can handle

all finitary relations. But, we cannot have them all in the same algebra or in the

same variety. For any finite family of relations, we can pick ß such that they are

all inF

«

¡

@. But this does not extend to infinite families of relations. To alleviate

this, we could try working in the system I

F

«

¡

@ t ß 5 ó W of varieties instead of

using just one of these. To use them all together, we need a strong coordination

between them. This coordination is easily derivable from the embedding function

Dr sending(

to( 2 ¥

for( )

@

¥

defined above. Let

k

)

Ò ´ µ

@

¥ % H

I G

@

¥ % Û Û Û Wbe a

«

Á

@and let

ïbe the

«

Á

@ ¯

e

generated by the Dr image of k

,

i.e. ï )

Ò ´ µ

@ ¯

e

¥ % H I G

@ ¯

e¥ % Û Û Û W is generated by h Dr

( % t ( 5 m . The

biggestk

yielding the sameï

is called theß

–ary neat-reduct of ï

, formallyk

H Ù ç @

ï %. Then

Ú

ÿ@

ï % H h r 5 Û t j@

r % H r

Ó

Intuitively,Ù ç @

ï %is the algebra of

ß–ary relations “living in” the algebra

ï

of ß –ary relations. It is not hard to see that Ù ç @ t

F

«

¡

@ ¯

e

U

F

«

¡

@

is a functor, in the category theoretical sense, for everyß

. Now, we can use the

collection of varietiesF

«

¡

@for all

ß 5 ó, synchronized via the functors

I Ù ç @ t

ß 5 ó W, as a single mathematical entity containing all finitary relations.




Another possibility is to insist that we want all finitary relations over¥

repre-

sented as elements of a single algebra. In other words, this goal means that instead

of a system of varieties we want to consider a single variety that in some sense

incorporates all the original varieties taken together. Indeed, eachF

«

¡

@can be

viewed as incorporating all theF

«

¡

ø

’s forö ß

, since the latter can be recov-

ered fromF

«

¡

@ by using the functors Ù ç @

c e

, Ù ç @

c

ä

etc. So as ß increases,F

«

¡

@ gets closer and closer to the variety we want. Indeed, we take the limit of this sequence. There are two ways of doing this, the naıve way we will follow

here and the category theoretical way we only briefly mention. It is shown in the

textbook Adamek–Herrlich–Strecker [AHS] that the system or “diagram”

F

«

¡

e

Ü Ý

¹

Þ

U

F

«

¡

ä

Ü Ý à

Þ

U Û Û Û

F

«

¡

@

Ü Ý á

Þ

U

F

«

¡

@ ¯

e

Û Û Û

is “convergent” in the category theoretic sense, i.e. that it has a limitâ

. Indeed, it

is this classâ

of algebras that we will construct below in a naıve way that does not

use category theoretic tools or concepts.

We first extend our Convention 2.1, stated at the beginning of the present section

concerning associativity of Cartesian products and powers. In the sequel, ó is the

smallest infinite ordinal, as well as the set of all finite numbers, and ¥

is the

set of ó

-sequences over¥

. Furthermore,@

¥ 2 ¥ H ¥, and if

( )

@

¥then

(1 2 4 ¥ ) ¥, for

ß ª ó. We will also have to distinguish the constant

G ºof

F

«

¡

â

from that of F

«

¡Î

. Therefore we let

G º

@ def H h I p Q

Ó Ó Ó

Q p W t p 5 ¥

denote theß

–ary identity relation on¥

.

How do we obtain an algebra containing all finitary relations over ¥ ? If ( is

binary, but we want to treat it together with a 5–ary relation, then we represent(

by ( 2 ¥D 2 X ¥ 2 ¥ H ( 2

â

¥ in a «

Á

°

. Taking this procedure to the limit, if we want

to treat(

together with relations of arbitrary high ranks, then we can represent(

with (C 2 ¥ . This way we can embed all finitary relations into relations of rank

ó, and relations of different ranks become “comparable” and “compatible”.11 We

still haven’t obtained the definition of «

Á

’s from that of «

Á

@ ’s because we do not

know what to do with the constantG º

. More specifically, we want to be able to use

the neat reduct functorÙ ç @

, as the inverse of (

( 2 ¥for

( )

@

¥, in order

to recover the original «

Á

@’s from the new «

Á

. This means that for G º

@

)

@

¥

we wantG º

@

2 ¥to be a derived constant (distinguished element) in our algebra.

11In particular we avoid the problem we ran into at the end of the introduction to this part in con-

nection with the Boolean algebra Ê ä

Ö

ª , because e.g. the complement of ¨

Ö

is ¨

Ö

where denotesé

if is an g -ary relation. Instead of trying to tame the Boolean-like algebra

å

é

for some

g

å æ ä

Ö

ª, we simply represent

å

é

by

¨

Ö

which is an

element of Ê

Ö

ª ç

çé

é

ä

Ö.



40 ALGEBRAIC LOGIC

AddingG º

H h I p Q

Ó Ó Ó

Q p Q

Ó Ó Ó

W t p 5 ¥ as an extra constant does not ensure this

any more. One of the most natural solutions is letting

G º

× ·

H G º × · H h Ù 5

¥ t Ù × H Ù ·

and defining a full«

Á

as

Ò ´ µ

¥ %

def H I G

¥ % QT j × Q G º × · W ×

ã

·

Q

where the G º × · ’s are constants. The price we had to pay for replacing the finite

boundß

on the ranks of relations we can treat with the infinite boundó

is that we

had to break up our single constantG º

to infinitely many constantsG º × ·

Õ Q ¶ 5 ó %.

F

«

¡

is defined to consist of all subalgebras of direct products of full«

Á

’s

(up to isomorphisms),

F

«

¡

H ¼ ½ h I G

¥ % Q j × Q G º × · W ×

ã

·

t ¥is a set

Ó

Again, as it was the case with F ¡

’s andF

«

¡

@ ’s,F

«

¡

’s are directly repre-

sentable as algebras whose elements areó

–ary relations.

The elements of

¥ % are all the ó -ary relations over ¥ , and not only the

“representations”( 2

¥

of finitary relations(

over¥

. Can we actually recover

the algebras of finitary relations from the huge full «

Á

’s? Let

F è

¥ % H h ( 2

¥ t ( )

@

¥for some

ß 5 ó

Ó

ThenF è

¥ % )

¥ % ; moreover it is a subalgebra of the full «

Á

Ò ´ µ

¥ % with

universe

¥ %. We will denote this subalgebra by

Ò é

¥ %

Ó

Now, we set

«

Á

è

def H ¼ h

Ò é

¥ % t ¥is a nonempty set

Ó

In a sense,«

Á

è

is the narrowest reasonable class of algebras of finitary relations.12

If ( 5

F è

¥ % , then ³

( % H h Õ 5 ó t j×

(

é

H ( is finite, in short ( is

finite-dimensional. We note that the converse is not true, there are ( ) ¥ with

³

( % H î , yet ( &5

F è

¥ % . Indeed, fix Ã 5 ¥ and set

( H h ¢ 5

¥ t h Õ ª ó t ¢ ×

é

H Ã is finite

Ó

12The letters ì (and í î ) refer to “finitary relations”. ï ìÖ

is the a bove mentioned category theoret-

ical limit ñ . More precisely, for this equality to be literally true, when forming the category theoretic

limitñ

, instead of the varieties §

é we have to start out from their subdirectly irreducible mem-

bers, which are nothing but ï é ’s. So ï ìÖ

is the limit of the sequence ï

ï é

. The class

ï ìÖ

and its relationship with §Ö

was systematically investigated in Andreka [A73], [AGN73],

[AGN77], [HMTAN], [HMTII]. In the first three works the class was denoted byñ ò

orñ ó

, while in the

last two by ï ô õ ÷

Ö

ï ñ ì Ö, the latter being the standard notation today.




Then( &5

F è

¥ %if

¥ Â õ, while

j × ( H (for all

Õ ª ó. A relation

( ) ¥is

called regular if

¢ 5 (iff

| 5 ( %whenever

¢ Q | 5

¥ Q ¢ I ³

( % H | I ³

( %

Ó

Then the elements of F è

¥ %are exactly the finite-dimensional regular relations on

¥.

Now we turn to the connections between the classes«

Á

è

,«

Á

andF

«

¡

. In-

tuitively, the elements of «

Á

è

are algebras of finitary relations, while the elements

of «

Á

(as well as those of F

«

¡

) are algebras of ó

–ary relations13.

THEOREM 2.4 (basic properties of F

«

¡

)

(i)F

«

¡

is a variety with recursively enumerable and undecidable equational

theory.

(ii) Eq F

«

¡

% H

â

h Eq F

«

¡

@ % t ß 5 ó

Ó

I.e. in the language of F

«

¡

@, the

same equations are true inF

«

¡

@and in

F

«

¡

.

(iii)F

«

¡

H ¼ ½ «

Á

H ¼ ½ À Á «

Á

è

é

H ¼ ½ «

Á

è

. I.e.F

«

¡

is both the vari-

ety and quasi-variety generated by «

Á

è

; the same equations and quasi-equations are true in

«

Á

è

and in«

Á

, but there is an infinitary quasi-

equation distinguishing«

Á

è

and «

Á

.

Proof. (ii) follows from [HMTII, 3.1.126]. Recursive enumerability and undecid-

ability of Eq F

«

¡

%follows from (ii) and Thm. 2.3 (for recursive enumerability

one also has to use the proof of Thm. 2.3, namely that the recursive enumerations

of Eq F

«

¡

@ %given there are “uniform” in

ß). That

F

«

¡

is a variety follows

e.g. from [HMTII, 3.1.103], (where it is proved directly thatF

«

¡

is closed under

taking homomorphic images).F

«

¡

H ¼ ½ À Á «

Á

è

follows from [HMTII, 3.2.8,

3.2.10, 2.6.52]. To showF

«

¡

é

H ¼ ½ «

Á

è

consider14 the following infinitary

quasi-equation Ù : ù

h j ×

H

t Õ ª ó

¢

G º þ

e

H

Ó

ThenÙ

is valid in¼ ½ «

Á

è

while it is not valid inF

«

¡

.¶

We note that¼ À Á «

Á

è

é

H

F

«

¡

. Indeed, consider the following universal

formula0

G º

¨

â

é

H

U G ºþ

e

j

ä

G º

¨

â

Qwhere

G º

¨

â

def H U G º þ

e

U G º þ

ä

U G º

e

ä

Ó

13Thm. 2.4(i) is due to L. Henkin and A. Tarski. For the rest of the credits in connection with

Thm 2.4 we refer the reader to [HMT, I, II] and [HMTAN].14Another proof, exporting logical properties to algebras, can be found at the end of Example 9 in

section 7.



42 ALGEBRAIC LOGIC

(The intuitive content of 0

is that if there is a “subbase” of cardinalityÂ

3, then

there are no subbases of size 2.)F

«

¡

is not a discriminator variety, e.g. because there are subdirectly irre-

ducible but not simpleF

«

¡

’s. But it is still an arithmetical variety of ¡ £

’s,

from which many properties of F

«

¡

follow by using theorems of universal al-

gebra. It is not true that¼ æ ç

F

«

¡

% H ¾ «

Á

, in fact no intrinsic characterization

of ¼ æ ç

F

«

¡

% is known.15 The varietyF

«

¡

is very well investigated, perhaps

the most detailed study is in [HMTAN], [HMTII]. For more recent results see e.g.

Goldblatt [G95], Monk [M93], Shelah [Sh], Sereny [Se85], [Se97], Hodkinson

[H97].

The theorems which say that¤ F ¡

,F F ¡

, andF

«

¡

@are not finitely axiomati-

zable, carry over toF

«

¡

too. However, to avoid triviality, instead of non-finite

axiomatizability we have to state something stronger, becauseF

«

¡

has infinitely

many operations and finitely many axioms can speak about only finitely many op-

erations. Taking this into account, when trying to axiomatizeF

«

¡

, one could

still hope for a finite “schema” (in some sense) of equations treating the infin-

ity of theF

«

¡

-operations uniformly. A possible example for a finite schema is

j × j ·

H j · j ×

Õ Q ¶ 5 ó %. The following theorem16 implies that it will be hard to

find such a schema, and that certain kinds of schemata are ruled out to begin with.

THEOREM 2.5 (nonfinite axiomatizability of F

«

¡

) The varietyF

«

¡

is not ax-

iomatizable by any set ñ

of universally quantified formulas such that ñ

involves

only finitely many variables.

Proof. Plan: For all ª ó

we will construct an algebrak

-

such that

a)k

-

&5

F

«

¡

b) every

–generated subalgebra of k

-

is inF

«

¡

.

This will prove the theorem because of the following. Assume thatñ

is a set

of quantifier-free formulas such that ñ involves at most variables ( Ä p ÿ

ñ %

ª ó) and

F

«

¡

H ñ. Then

ñis valid in an algebra

ïiff

ñis valid in every

-

generated subalgebra of ï , because Ä p ÿ

ñ % and ñ contains no quantifiers.

Thusk

-

H ñby b) and by

F

«

¡

H ñ. Then

k

-

&5

F

«

¡

shows thatñ

does

not axiomatizeF

«

¡

.

Construction of k

-

: Letû Â õ

-

be finite, and letI ¥ × t Õ ª ó W

be a system of

pairwise disjoint sets each of cardinalityû

. Let

15It is known that È p r §Ö

ª is a proper subclass of Ê ü ïÖ

. More on this see [HMTII, 3.1.83-

3.1.88], [ANT].16Monk [M69] proves that §

Öcannot be axiomatized with a finite number of schemas of equa-

tions like those in the definition of §

Ö

. See [HMTII, 4.1.7]. Thm. 2.5, due to Andreka, is a general-

ization of that result and can be found in [A97] or in [M93].




¥ H

â

h ¥ × t Õ 5 ó Qlet

Ù 5

Þ

× Ú

¥ ×

def H h ¢ 5 ¥ t

Õ 5 ó % ¢ × 5 ¥ × be arbitrary,

( H h | 5

Þ

× Ú

¥ × t h Õ 5 ó t | ×

é

H Ù × ª ó , and let

k

Ü

be the subalgebra of I G

¥ % Q j × Q G º × · W ×

ã

· Ú

generated by the element(

.

Then ( is an atom of k

Ü because of the following. For any two sequences ¢ Q | 5 (

there is a permutation t ¥ È

¥of

¥taking

¢to

|and fixing

(, i.e.

¢ a H |

and( H h ý a t ý 5 (

(the obvious choice for

, interchanging¢ ×

and| ×

for all

Õ 5 óand leaving everything else fixed, works). If

is a permutation of

¥fixing

(, then

fixes all the elements generated by

(because the operations of

F

«

¡

are permutation invariant. Thus if î

é

H p 5 mÜ and

¢ 5 p (then

( ) p, showing

that(

is an atom of k

Ü.

We now “split(

intoû

new atoms(

·

each imitating(

” obtaining a new,

bigger algebrak

from our oldk

Ü. I.e. we choose a larger algebra

k

such thatk

satisfies the conditions below and is otherwise arbitrary.

k

Ü )

k

, the Boolean reduct of k

is a Boolean algebra,

( · are atoms of k

and j × ( · H j × ( for ¶ û Q Õ ª ó ,

each element of k

is a Boolean join of an element of k

Üand of some

( ·’s

j×

distributes over joins, for any Õ ª ó , i.e.k

H j×

S z % H j×

S j×

z .

Note that ink

“ S ” is only an abstract algebraic operation and not necessarily set

theoretical union. It is easy to see that such an extensionk

of k

Üexists. See Figure

2.4.

By the above, we have constructed our algebrak

-

which in the following we will

denote just byk

.¶

CONVENTION 2.6 In this proof we use the symbols , S denoting the concrete

operations of our set algebras («

Á

’s) also as the corresponding abstract algebraic

operation symbols (denoting themselves in «

Á

’s). So, if

, z are variable symbols,

then

zis a term. We hope context will help in deciding whether

zis meant

to be a term or a concrete set.

U z is the Boolean term

U z % denoting the

set

9 zin

«

Á

’s. It is especially important to note that since, for the algebrak

constructed above,k

&5

F

«

¡

was not excluded, the operations denoted by S ,

,

j ×etc. in

k

are not assumed to be the real, set theoretic ones. They are just

abstract operations despite of the notation “S

” etc. The Boolean ordering onk

will be denoted by

.



44 ALGEBRAIC LOGIC

CLAIM 2.7k

&5

F

«

¡

.

Proof. ForÕ Q ¶ ª ó

,Õ

é

H ¶, let

Á

×

·

% H j ×

G º × ·

%and

Á

×

×

% H

. Let the term

be defined by

%

def H þ

× ¢

Á

þ

×

j

e

Ó Ó Ó

j¢

þ

×

· ¢

U G º× ·

Ó

Let ¥

¨ ¦

H h | 5 ¥ t h Õ 5 ó t | ×

é

H Ù × is finite

. (

¥

¨ ¦

is usually called the

weak Cartesian space determined by ¥ and Ù.) Then, for our concrete choice of

( ,k

Ü H

( % H because of the following:

j

e

Ó Ó Ó

j ¢ ( H

¥ þ 2

¢

¥ 2 ¥ ¢ ¯

e

2

Ó Ó Ó

%

¥

¨ ¦

Q

Á

þ

×

j

e

Ó Ó Ó

j ¢ ( H

©

×

¥ 2 ¥ þ 2

¨

¢

c

×

¥ 2 ¥ ¢ ¯

e

2

Ó Ó Ó

¥

¨ ¦

Q

þ

× ¢

Á

þ

×

j

e

Ó Ó Ó

j ¢ ( H

©

¨

¢ ¯

e

¥ þ 2 ¥ ¢ ¯

e

2

Ó Ó Ó

¥

¨ ¦

Ó

Then by ¥þ

û we have that there is no repetition–free sequence in¨

¢ ¯

e

¥þ

.

Thusk

Ü H

( % H

Ó

Then

k

H

( % H

by

k

Ü )

k

and( 5 m Ü

. Assume that

k

5

F

«

¡

. Thenthere is a homomorphism

t

k

ï 5 «

Á

such that

( %

é

H , for some

ï. By

( %

é

H , there is ¢ 5

( % . By ( jþ

(·

we have

( % ) jþ

(·

% , so there is

Ã ·such that

I Ã · Q ¢

e

Q ¢

ä

Q

Ó Ó Ó

¢ ×

Ó Ó Ó

W 5

( · %, for all

¶ û. These

Ã ·’s are different

from each other since the ( · ’s are disjoint from each other, and so the

( · % ’s are

disjoint from each other. Consider the sequence

| H I Ã þ Q Ã

e

Q

Ó Ó Ó

Q Ã ¢ Q ¢ ¢ ¯

e

Q

Ó Ó Ó

W

Ó

Then| 5

( % %is easily seen as follows. Obviously

| 5 U G º× ·

, if Õ ª ¶ û

.

Further

I Ã×

Q |

e

Q

Ó Ó Ó

|·

Ó Ó Ó

W 5 G ºþ ×

j

e

Ó Ó Ó

j¢

( %, hence

| 5

Á

þ

×

j

e

Ó Ó Ó

j¢

( %if

Õ û.

Thus | 5

( % % , a contradiction.¶

CLAIM 2.8 The

–generated subalgebras of k

are inF

«

¡

.

Proof. Let ) m Q . For all Õ Q ¶ û define

(×

(·

iff

5 % (×

(·

Ó

Then

is an equivalence relation onh ( · t ¶ û

which has õ

-

blocks by

. Let

ýdenote the number of blocks of

, i.e.

ý H h ( · & t ¶ û

õ-

û. Define

Û H h p 5 m t

Õ Q ¶ û %

(×

(·

and (·

p

(×

p %

Ó

We now show that Û is closed under the operations of k

.: Let ö ª $ ª ó .




1)Û

clearly is closed under the Boolean operations.

2)G º

ø &

5 Ûsince

¶ û % ( ·

é

G º

ø &

.

3) Clearly,m Ü ) Û

(since(

is an atom of k

Ü), and

j

ø

p 5 m Üfor all

p 5 m.

Thus j

ø

r 5 Û (for all r 5 m ).

Letï )

k

be the subalgebra of k

with universeÛ

. By ) Û

, it is enough to

show thatï 5

F

«

¡

.

We will define an embedding t ï È I G

¥ % Q j × Q G º × · W ×

ã

·

. Leth z · t ¶ ª

ý H h (

( · & % t ¶ û . Then

h z · t ¶ ª ý is a partition of

(in

ï, i.e. they

are pairwise disjoint and sum up to ( , j×

z·

H j×

( for all ¶ ª ý and Õ ª ó and

every element of ï is a join of some element of m Ü and of some z·

’s. So, ï looks

like the algebra on Figure 2.5.

First we define the images of thez

·

’s. Let H h Q Q

Ó Ó Ó

Q û U and let

Q Q % be a commutative group. For each Õ ª ó let á×

t ¥×

È

be a bijection

such thatá

×

Ù×

% H . For

¶ ª ûdefine

(

Ü Ü

·

H 0 | 5 ( t 2 I á ×

| × % t Õ ª ó W H ¶ 4 Q

where(

denotes the group theoretic sum in

Q Q %. Then it is not difficult to

check that the (Ü Ü

·

’s are disjoint from each other and

j × (

Ü Ü

·

H j × (for all

Õ ª ó

for the concrete set theoreticj ×

’s. Define for all¶ ª ý U

( Ü

·

H ( Ü Ü

·

( Ü

5

c e

H

â

Q( Ü Ü

·

t ý U ¶ ª ûT

Ó

We are ready to define the embedding of Û .: We define for all r 5 Û

r % H

r U ( % S

Ù

h (

Ü

·

t ¶ ª ý Q z·

r Q

where r U ( is computed ink

, and since

r U ( % 5 m Ü )

# % , the rest of the

operations are the concrete set theoretic ones. Now it is not difficult to check that

is an embedding t ï È I G

¥ % Q j × Q G º × · W ×

ã

·

as follows.

Clearly

preservesS Q U

.

r % H implies

r H , hence

is one–one.

G º

ø &

% H G º

ø &

. Now we check j

ø

r % H

j

ø

r % .

j

ø

r % H j

ø

r U ( % S

Ù

h (

Ü

·

t z · r



46 ALGEBRAIC LOGIC

H j

ø

r U ( % S

Ù

h j

ø

(

Ü

·

t z·

r

H j

ø

r U ( % S

Ù

h j

ø

z·

t z·

r

H j

ø

r U ( % S

Ù

h z · t z · r

H j

ø

r Q

where the operations in the first two lines are set theoretic while those in the last

three lines are understood in the abstract algebrak

.

j

ø

r % H

j

ø

r U ( % S

â

h ( Ü

·

t

z·

j

ø

r H j

ø

r , since

u ¶ % z·

j

ø

r iff ( j

ø

r , and (

é

j

ø

r iff j

ø

r H

j

ø

r U (.

QED(Theorem 2.5)¶

Remark 2.9 below describes the modifications needed for obtaining proofs for

the analogous (with Thm. 2.5) non-finitizability theorems forF

«

¡

@

ß õ %and

F F ¡

.

REMARK 2.9 Here we outline the modifications of the above proof of Theo-

rem 2.5 yielding proofs for non-finite axiomatizability of F

«

¡

@ andF F ¡

.Let

8be

ßor

ó. An algebra

k

similar toF

«

¡ 9

’s is said to be representable if k

5

F

«

¡ 9

. Thus representability means thatk

is isomorphic to an algebrak

¯

whose elements are 8 –ary relations and whose greatest element is a disjoint union

of Cartesian spaces.k

¯

is called a representation of k

and sometimes the iso-

morphism t

k

È

k

¯

too is called the representation of k

. By a homomorphic

representation we understand a homomorphism mappingk

into some«

Á

9

. This

concept receives its importance from the simple but useful fact that representability

of k

is equivalent with the existence of a set@

of homomorphic representations

of k

such that

nonzero

5 m %

u 5 @ %

%

é

H .

The intuitive idea of the above proof of Theorem 2.5 was the following. We

found two different ways of “counting” the elements of the domainh ¢ þ t ¢ 5

(

of the relation(

. This counting was done by looking only at the abstract,i.e. isomorphism invariant properties of k

. The two ways of counting were: (1)

Looking at the number of the disjoints elements( ·

below(

. This allowed us to

conclude that the domain of ( must be big. (2) Using the G º× ·

’s exactly as one uses

equality in first–order logic to express that a certain finite set is smaller than some

û , we concluded that the domain of ( must be small. (This was done by the term

( %in the proof of Claim 2.7.)

We started out from ank

Ü 5

F

«

¡

in which the counting (2) said that “ Dom

( % ”

is small. Then by splitting, we enlargedk

Üto

k

, such that in this bigger algebrak

the counting (1) said that “ Dom

( % ” is big. Thus ink

the two countings (1) and

(2) contradict each other, ensuringk

&5

F

«

¡

.

This is how we constructed one nonrepresentable algebra (k

-

). We were able

to construct an infinite sequence of such algebras in such a way that as

increases,




the contradiction between (1) and (2) becomes weaker and weaker. Actually, as

approaches infinity, the contradiction between (1) and (2) vanishes. So in the

ultraproduct of thek

-

’s, (1) and (2) do not contradict each other any more, and

this ultraproduct is inF

«

¡

. In our construction the conflict between (1) and (2)

became weaker and weaker in the sense that more and more elements had to be

inspected for discovering this contradiction.17 This finishes the intuitive idea of

the proof of Theorem 2.5.

Next we would like to repeat this proof forF

«

¡

@in place of

F

«

¡

, withõ ª

ß ª ó. If we simply replace

óeverywhere with

ß, the proof does not go through

because the counting in (2) needs an arbitrarily large number of G º × ·

’s and we have

onlyß 2 ß

many.18 So we need a new method for doing (2). This amounts to

looking for an abstract algebrak

together with its element(

and concluding that

in any (homomorphic) representation t

k

ï 5 «

Á

@of

k

, the domain¥ þ

of

( %must be of smaller size than a certain

û. (The difficulty is that we have to

be able to repeat this for arbitrarily large û 5 ó .) We also need to keep in mind

that we will want to have a contradiction with (1), which means that we will want

to split ( . In order to be able to do this, we only need that ( remain an atom.

There are many natural ways for ensuring (by abstract properties) smallness of a

set. Perhaps the simplest way is the following. If we could “see” by looking atk

“abstractly” that¥ þ 2 ¥ þ

is a union of fewer thanû

functionsá ×

Õ ª û U %

each of which is coded by an element of k

, then the domain ¥ þ of ( must be

of smaller cardinality thanû

in any representation of k

. E.g. we can take these

functions (the á × ’s) to be powers of a single suitable permutation á of ¥ þ ; say let

¥ þ H û , and let á be the usual successor modulo û . Let D

def H á 2

@

c

ä

¥ . Then

D )

@

¥. We include into our algebra

k

Ü, besides

(, also

Das a new generator

element. It can be checked that ( remains an atom (because no subset of ¥ þ

became “definable”). Now, similarly to the way we used the equation

( % H in

the proof of Claim 2.7, by studying the abstract properties of D

and(

in the newk

Üwe can conclude that in any homomorphic representation

of

k

Ü, the Cartesian

square of the domain of

( %is contained in the union of fewer than

ûpowers

of a function coded by

D %. But then this domain must be of cardinality

û.

(Exactly what we proved in Claim 2.7 of the old proof. So we can prove our newClaim 2.7.) After this modification, the whole proof goes through by replacing

all occurrences of ó

withß

.19 This completes the outline of the proof thatF

«

¡

@

cannot be axiomatized with quantifier free formulas using finitely many variables,

if ß õ

.

Let us turn to theF F ¡

case, i.e. to Theorems 1.6, 1.10. The idea is basically the

17This allowed us to avoid ultraproducts in the final argument. We find it more natural to explain the

intuitive idea in terms of ultraproducts, which incidentally happens to be the way the original proof of

nonfinitizability went.18We can construct the algebras

¥

Ñ as in the proof of Thm. 2.5 forE F H P

v

g ª. But for the

contradiction to vanish, we need¥

Ñ for arbitrarily largeE

.19We need

g Õ ×to be able to see abstractly that the

Å’s are functions. This proof is worked out in

detail in [A97, Thm.1].



48 ALGEBRAIC LOGIC

same as in the above outlinedF

«

¡

@case. Exactly as in the

F

«

¡

@case, here too we

use two counting principles (1), (2) and construct algebrask

-

in which (1) and (2)

contradict each other. Again we want a controllable contradiction such that as

approaches infinity the contradiction vanishes. Here we have to take a less obvious

principle for counting in (2), because inF F ¡

, functions interfere with splitting20

elements ( H ¥ þn 2 ¥

e

. E.g. we can use colorings of the full graph ¥ þ 2 ¥ þ with

finitely many colors without monochromatic triangles, and then apply Ramsey’stheorem. This means that we arrange

¥ þy 2 ¥ þto be a disjoint union of symmetric

relations þ Q

Ó Ó Ó

Q Rsuch that

R H G º I ¥ þ(symmetric means

× H

c e

×

) and

× a × % × H . To ensure splittability of

(we also arrange that

× a · ±

ø

whenever h Õ Q ¶ Q ö

,Õ Q ¶ Q ö ª ÿ

. We let ourk

Übe generated in this case

byh þ Q

Ó Ó Ó

Q R Q ( . All these properties of the

×’s were abstract, “equational”

ones.21 This ensures that in every representation of k

Ü , the domain ¥þ

of ( must

be finite (by Ramsey’s theorem). We split(

intoó

many( ×

’s obtainingk

fromk

Ü

as we did in theF

«

¡

,F

«

¡

@cases before. The rest of the proof goes through as

before with replacingó

(orß

) everywhere by 2, except for the following change.

In theF F ¡

case we have to look at the ultraproduct of thek

-

’s and observe that

it is representable (since the contradiction between (1) and (2) disappeared as both

counting gives us continuum many elements). Therefore this proof gives only non-

finite axiomatizability of F F ¡

(i.e. Monk’s theorem) without proving (Jonsson’s

result saying) that infinitely many variables are needed. For the latter, one has to

fine-tune the construction some more.22 ¶

In section 1, Theorem 1.6 leads to Problem 1.12 in a natural way. Exactly the

same way our present Theorem 2.5 leads to the following important open problem.

PROBLEM 2.10 Find simple, mathematically transparent, decidable sets · of

equations axiomatizingF

«

¡

. TheF

«

¡

@, õ ª ß ª ó version of this problem is

open and interesting, too.

TheF

«

¡

@ version is strongly related to Problem 1.12 in section 1. On the

other hand, the present,F

«

¡

version has a logical counterpart, cf. e.g. [HMTII,

Prob.4.16, p.180]. This is one of the central problems of Algebraic Logic, cf. [HMTII,

Prob.4.1], Henkin–Monk [HM74, Prob.5], etc. For strongly related results (or for

partial solutions) see [HMTII, pp.112–119], [V91], [V95], [Si91], [Si93], [HH].

20Splitting in § is defined, and the conditions for splittability are described, in [AMdN].21In fact, it is an open problem in § -theory (see e.g. [AMN, section on open problems], whether

there are such concrete relations W

W

ô

on some set or not. What we should do here is that

we state these properties abstractly on some abstract relationsW

W

ô

. The only difference from

the previous proof will then be that we do not know whether¥

×

§. But this does not matter, what

we need is that¥

Ñ

§and

Í ¥

Ñ

Ç §.

22This is done in [AN90], where we use projective geometries for the purposes of counting in (2).




PROBLEM 2.11 Is there a finite schema axiomatizable quasi-variety»

such that

Eq

» % H Eq F

«

¡

% , i.e. the variety generated by » isF

«

¡

? The same for F

«

¡

@ for

ß ª ó. I.e. is there a finitely axiomatizable quasi-variety

» )

F

«

¡

@

such that F

«

¡

@H ¿ » ?

This problem is related to the existence of weakly sound Hilbert-style inferencesystems for first-order logic, see Part II, Thm.6.5 and Open Problem 7.2.

On the structure of the equational axiomatizations of F

«

¡

@ ,F

«

¡

:

Let·

be an arbitrary set of equations axiomatizingF

«

¡

@. As in the

F F ¡

-case,

· must be infinite, but it can be chosen to be decidable. Unlike theF F ¡

-case, here

every operation symbol has to occur infinitely many times in·

(in theF F ¡

-case,

only the Booleans and a had to occur infinitely many times). A similar statement

is true forF

«

¡

in place of F

«

¡

@

. For more on this see Figure 3.1 and [A97],

[A94]. Concrete decidable sets · are known, see e.g. [HMTII, pp.112-119], cf.

also [V91], [V95], [Si91], [Si93], [HH97]. However, it would be important to find

choices of · with more perspicuous structures, see Problem 2.10.

Let us turn to the relationship betweenF

«

¡

and its abstract approximation

«

¡

. These investigations yield information on proof theoretical properties of

first-order logic and of some related logics. See Examples 6, 8, 9 in section 7,

especially theorems 7.4 - 7.7.

DEFINITION 2.12 ( «

¡

, an abstract approximation of F

«

¡

) A «

¡

is a nor-

mal ¡ £

of the same similarity type asF

«

¡

in which thej ×

’s are self-conjugated

commuting closure operations, and in which the constants G º× ·

satisfy the follow-

ing equations:

(3)’ For allÕ Q ¶ Q ö ª ó

(i)j × G º × · H Q j

ø

G º × · H G º × ·if

ö

é

H Õ Q ¶,

G º × × H Q G º × · H G º · × Qand

G º × · G º ·

ø

G º ×

ø

.

(ii) G º× ·

j×

H

whenever

G º× ·

and Õ

é

H ¶ .

¶

To treatF

«

¡

@ Q «

¡

@and

F

«

¡

Q «

¡

in a unified manner, we replaceó

in the

definitions of F

«

¡

and«

¡

with an arbitrary but fixed ordinal8

, obtainingF

«

¡ 9

Q «

¡ 9

(here8 H ß

and8 H ó

are of course permitted).23

23This generalization will also be useful in algebraizing various quantifier logics different from clas-

sical first–order logic.



50 ALGEBRAIC LOGIC

If 8 H ß ª ó

, then the newly definedF

«

¡

@and

«

¡

@are only definitionally

equivalent with the previously defined ones, because in Def. 2.2 we had only one

constantG º

in place of the presentß 2 ß

–many constantsG º × ·

. This definitional

equivalence is given by

G º H

þ

×

ã

·

@

G º × ·and

G º × · H j

¨

@ ¼ ¾ ×

ã

· ¿

G º

Ó

Since definitional equivalence is a very close connection between classes of alge-

bras, we did not give new names forF

«

¡

@ and «

¡

@ .

DEFINITION 2.13 (Locally finite, dimension-complemented «

¡

’s, and neat-re-

ducts) Let 8 Q Y

be any ordinals.

(i) Let k

5 «

¡9

,

5 m. Then

³

%

def H h Õ 5 8 t j ×

é

H

.

â

è 9 def H h

k

5 «

¡ 9

t

5 m %

³

%is finite

%

Ó

`

j

9 def H h

k

5 «

¡ 9

t

5 m %

8 9 ³

%

is infinite%

Ó

(ii) Assume8 Y

and k

5 «

¡ a

. Then

Ú

ÿ

9 k def H h

5 m t ³

% ) 8 ,

c d

9 k def H I

Ú

ÿ

9 k

Q j

×

Q G º

× ·

W ×

ã

·

9

.

In the above,j

×

Q G º

× ·

denote the corresponding operations of k

. It can be checked

that c d

9 k

5 «

¡ 9

.

Ù ç

9

«

¡a def

H h c d

9k

t

k

5 «

¡a

.

¶

The elements of â

è9

and`

j

9

are called locally finite and dimension–comple-

mented «

¡ 9

’s respectively. c ÿ

9 k

is called the 8 -neat reduct of k

. We note that

`

j

9

H h

k

5«

¡ 9

t

5 m %

8 9³

%

é

H î %

Ó

THEOREM 2.14 (Relationships 24 betweenâ

è i

Q

`

j

9

Q Ù ç

9

«

¡ a

andF

«

¡ 9

.)

24The classes È Ð r r § s were introduced by Henkin, and Monk [M61] proved that they are vari-

eties. Thm. 2.14(i) is due to J. D. Monk. The equalities in (ii) are due to L. Henkin [H55], while the

inequality in (ii) was proved by J. D. Monk [M69].




(i)F

«

¡

H ¼ À Á â

è

é

H ¼ ½ â

è

and `

j

)

F

«

¡

. I.e. there is no universal

formula distinguishingF

«

¡

fromâ

è

, and every`

j

is representable. The

same hold for all 8 Â ó in place of ó .

(ii)F

«

¡ 9

H ¼ Ù ç

9

«

¡ 9

¯

H t h ¼ Ù ç

9

«

¡ 9

¯

-

t 5 ó

é

H ¼ Ù ç

i

«

¡ 9

¯

-

for all8

and for all finite

.¼ Ù ç

9

«

¡ 9

¯

-

is a variety for all8

and

.

Proof. The positive statements follow from [HMTII, 3.2.10, 2.6.32(ii), 2.6.50,

2.6.52 3.2.11]. The negative statements are also proved in [HMTII] taken together

with [HMTAN].¶

The above theorem gives information on the proof theory of first order logic

(FOL) and on its ß -variable fragment v@

. Intuitively, it says (in several different

forms) that the important feature of FOL is not that each formula involves only

finitely many variables, but that given any formula, there are infinitely many vari-

ables it does not involve 25.

Based on the above theorem, an inference system is given both for FOL andv @

which uses the finite-schema axiomatization of «

¡ 9

together with a supply of

variables which do not occur in our original formulas.26 I.e. these variables can

occur in a proof, but not in the final formula we want to prove.

25The earlier mentioned theorem saying that quasi-projective § ’s are representable, also speaks

about this phenomenon: the projections are used for coding together the already involved variables, so

that we get one more “unused” variable. This idea comes through clearly in [Si96]. The same idea is

used for proving finite schema axiomatizability (i.e. completeness of the corresponding logic) in [S95],

[SGy]. The same idea is used for obtaining an unorthodox completeness theorem in [Si91].26Cf. e.g. [HMTII, p.157], [N96], [AGN77, Thm. 3.15] and [Si91]. Cf. also section 7 herein.



w 1

x

0

y

y

y

y

y

y

y

y

y

y

y

y

j

e

%

j þ

%

G º

ä

w 0

2

1

G º

â

w

j

e

%

Figure 2.1.



j l n l n j j j n

l n n n l l } l ~ j }

Figure 2.2.



Figure 2.4.

ª «

¬

- ® ° ² ³ ´ µ

Figure 2.5.



CHAPTER 3

ALGEBRAS FOR LOGICS WITHOUT IDENTITY

We start fromF

«

¡9

, and would like to consider its “natural” G º -free reducts. If

we simply omit G º (or G º× ·

), then we lose not only identity (or equality), but also

our ability to “algebraize” substitution of individual variables like ñ

ñ Ä×

& Ä·

in the logic to be algebraized. Therefore, before dropping theG º × ·

’s, we first add

our term functions

Á

×

·

% H j×

G º× ·

%for

Õ

é

H ¶,

Á

×

×

% H

. Now,

F E

«

9 def H ¼ ½ h I G

9

¥ % Q j×

Q

Á

×

·

W×

ã

·

9

t ¥ is a set

Ó

F E

«

9

’s are called representable substitution-cylindricalgebras, cf. [N91], [AGMNS].

They are the simplest kind in the family of polyadic-style algebras. The theory of F E

«

9

is analogous with that of F

«

¡ 9

, in particular, if 8 õ

, thenF E

«

9

is not

finitely axiomatizable, cf. [ST].

Letò ó ¶ ·

be the operator which to any cylindric-type algebrak

H I m Q S Q U QT j × Q

G º× ·

W×

ã

·

9

associates theF E

«

9

-type algebra

ò ó ¶ ·

k

%

def H I m Q S Q U Q j

×Q

Á

×

·

W×

ã

·

9

where

Á

×

·

is the derived operationof k

defined above. Now,F E

«

9

H ¼ ò ó¶ ·

F

«

¡ 9

.

The finitely axiomatizable approximationE

«

9

of F

«

¡ 9

is defined analogously,

E

«

9 def H ¼ ò ó ¶ · «

¡ 9 Ó

E

«

9

’s are called substitution-cylindric algebras.

THEOREM 3.1E

«

9

is a finite schema axiomatizable variety containingF E

«

9

.¶

For the simple set of axioms, and for information on the proof we refer the

reader to [AGMNS]. Cf. also [N91,ò

8].

The theory of the pairE

«

9

Q

F E

«

9

is almost completely analogous with that

of «

¡ 9

Q

F

«

¡ 9

. The connection betweenE

«

9

-theory and first order logic with-

out equality is analogous with the connection between«

¡ 9

-theory and logic with

equality. In particular, the logic counterpart of the algebraic operation

Á

×

·

is the

57



58 ALGEBRAIC LOGIC

“substitution-modality” Ä × & Ä ·

, cf. Part II, section 7. The logicsv

¶

@

andv

¶

á

@

are

introduced in section 7, example 7. Let v

7@

and v

7

á

@

denote the fragments of these

logics obtained by dropping the connective Ä × Q Ä ·

. Then the«

¡

@ Q

F

«

¡

@pair is

strongly connected to v

7

á

@

while theE

«@

Q

F E

«@

pair is completely analogously

connected tov

7@

. More on the connection betweenE

«

9

and logic can be found in

[SN], [MV], [V95a], [vB96], [N91].

The classes«

¡ 9

Q

F

«

¡ 9

andE

«

9

Q

F E

«

9

introduced so far constitute the hearts

of the following two “worlds”: the algebraic counterpart of logics with equality

(the “cylindric world”), and the algebraic counterpart of logics without equality

(the “polyadic world”). In both worlds one can introduce natural extra operations

like e.g. cardinality quantifiers, generalized cylindrifications, but what determines

the most basic theorems (remaining true for the expanded algebras) remains the

«

¡9

-structure or theE

«

9

-structure. Therefore it seems reasonable to pay some-

what more attention (say, as a default) to «

¡9

andE

«

9

than to their versions

enriched with extra operators. 1

E

«

9

’s with extra operators (like e.g. Ä × Q Ä · ) are discussed in the literature under

the names quasi-polyadic algebras (

Þ

¡

9

’s) and polyadic algebras respectively.2

The most important extra operator ¹

× ·

is a substitution operator like

Á

×

·

( ¹

× ·

corre-

sponds to the logical connective Ä × Q Ä ·

in section 7, Example 7). Letã )

9

¥

.Then

¹

þ

e

ã %

def H h I Ù

e

Q Ù þ Q Ù

ä

Q

Ó Ó Ó

W t Ù 5 ã

Ó

I.e., ¹

þ

e

interchanges Ùþ

and Ù

e

in a sequence Ù . For Õ Q ¶ ª 8 , ¹

× ·

is defined

completely analogously. Now,F

Þ

¡

9

’s are defined to beF E

«

9

’s enriched with

the¹

× ·

’s (Õ Q ¶ ª 8

):

F

Þ

¡

9 def H ¼ ½ h I G

9

¥ % Q j × Q

Á

×

·

Q ¹

× ·

W ×

ã

·

9

t ¥is a set

Ó

The abstract class

Þ

¡

9

approximatingF

Þ

¡

9

is defined by finitely many axiom-

schemes analogously to the definition of «

¡ 9

orE

«

9

, cf. [N91], [ST], [AGMNS].

Polyadic algebras (

F

Þ

¡ 9

andÞ

¡ 9

) are obtained from

F E

«

9

and

E

«

9

by addinginfinitary substitutions and infinitary cylindrifications denoted as

Á ¼

,j

¨ ¸ , for t

8

8 and » ) 8 . For the theory of these algebras we refer to [Ha62], [HMTII],

[N91]. Cf. also [NS96].

We note that the theory of

Þ

¡

’s seems to be very strongly analogous with that

of E

«

9

’s. (However, in certain studies, e.g. when investigating the connection

betweenF ¡

’s and«

¡

’s,«

¡

’s enriched with the¹

× ·

’s play a very illuminating role

1Of course, no such rule is valid in general. Actually, pushing the above considerations further,

r ’s seem to be at the heart of § r -theory, too, therefore they could be considered as the core of

(or basis for) the algebraizations of quantifier logics in general. This unified perspective for algebraic

logic has not been elaborated yet.2

Í §’s,

Í §’s, and their versions with equality like

Í ¾ §’s, originate with P. Halmos (cf. [Ha62]).

’s originate with C. Pinter, cf. e.g. [AGMNS] or [N91].



3. ALGEBRAS FOR LOGICS WITHOUT IDENTITY 59

(cf. [NSi]). The latter algebras are called

Þ

¡

’s with equality, or

Þ À

¡

’s. Cf.

e.g. [HMTII] for their theory.)

For lack of space, we do not discuss further the theory of E

«

9

’s with extra

operators (like

Þ

¡

’s,Þ

¡

’s, etc).

The next figure, taken from [A97] describes the interconnections between the

operationsj × Q G º × · Q

Á

×

·

and¹

× ·

(in the presence of the Boolean operations). (On

the figure,G º × · Q

Á

×

·

are denoted asº × · Q

Á

× ·, respectively.) On Figure 3.1, nodes rep-

resent classes of algebras of relations where the units are Cartesian spaces@

¥

(Ã ß ª ó

), and the operations are those along the path leading to the node. A

broken edge between two nodes means that the second class is finitely axiomati-

zable over the first one, a bold edge means non-finite axiomatizability over, and a

normal line means that it is unknown (to the authors) whether finite or non-finite

axiomatizability holds.



Á Â Ã

Ä Å Æ

Ç

Å

È

Å Æ

É

Å Æ

Ç

Å

È

Å Æ

É

Å Æ

Ä Å Æ

È

Å Æ

É

Å Æ

Ä Å Æ

Ç

Å

É

Å Æ

ÄÅ Æ

Ç

Å

È

Å Æ

É

Å Æ

Ç

Å

É

Å Æ

Ç

Å

È

Å Æ

Ä Å Æ

É

Å Æ

Ä Å Æ

È

Å Æ

Ä Å Æ

Ç

Å

Ç

Å

Ä Å Æ

Figure 3.1.



PART II

BRIDGE BETWEEN LOGIC AND ALGEBRA:

ABSTRACT ALGEBRAIC LOGIC



62 ALGEBRAIC LOGIC

INTRODUCTION TO PART II

Let us start by putting the subject matter of the present part (i.e. Part II of the

present paper) into perspective.

The idea of solving problems in logic by first translating them to algebra, thenusing the well developed methodology of algebra for solving them, and then trans-

lating the solution back to logic, goes back to Leibnitz and Pascal. Papers on the

history of Logic (e.g. Anellis–Houser [AH], Maddux [Ma91]) point out that this

method was fruitfully applied in the Ê

th century not only to propositional logics

but also to quantifier logics (De Morgan, Peirce, etc. applied it to quantifier logics

too). The number of applications grew ever since. (Though some of these re-

mained unnoticed, e.g. the celebrated Kripke–Lemmon completeness theorem for

modal logic w.r.t. Kripke models was first proved by Jonsson and Tarski in 1948

using algebraic logic.)

For brevity, we will refer to the above method or procedure as “applying Alge-

braic Logic (AL) to Logic”. This expression might be somewhat misleading since

AL itself happens to be a part of logic, and we do not intend to deny this. We will

use the expression all the same, and hope, the reader will not misunderstand ourintention.

In items (i) and (ii) below we describe two of the main motivations for applying

AL to Logic.

(i) This is the more obvious one: When working with a relatively new kind of

problem, it often proved to be useful to “transform” the problem into a well under-

stood and streamlined area of mathematics, solve the problem there and translate

the result back. Examples include the method of Laplace Transform in solving

differential equations (a central tool in Electrical Engineering). 3

In the present part we define the algebraic counterpart Alg

ð %of a logic

ðto-

gether with the algebraic counterpart Alg-

ð % of the semantical-model theoretical

ingredients of ð

. Then we prove equivalence theorems, which to essential logical

properties of ð associate natural and well investigated properties of Alg

ð % suchthat if we want to decide whether

ðhas a certain property, we will know what to

ask from our algebraician colleague about Alg

ð % . The same devices are suitable

for finding out what one has to change inð

if we want to have a variant of ð

having

a desirable property (whichð

lacks). To illustrate these applications we include

several examples (which deal with various concrete logics) in section 7. For all

this, first we have to define what we understand by a logic ð in general (because

3At this point we should dispell a misunderstanding: In certain circles of logicians there seems to be

a belief that AL applies only to syntactical problems of logic and that semantical and model-theoretic

problems are not treated by AL or at least not in their original model theoretic form. Nothing can be

as far from the truth as this belief, as e.g. looking into the present part (i.e. Part II) should reveal. A

variant of this belief is that the main bulk of AL is about offerring a cheap pseudo semantics to Logics

as a substitute for intuitive, model theoretic semantics. Again, this is very far from being true.



BRIDGE BETWEEN LOGIC AND ALGEBRA: ABSTRACT ALGEBRAIC LOGIC 63

otherwise it is impossible to define e.g. the function Alg associating a class Alg

ð %

of algebras to each logic ð ).

(ii) With the rapidly growing variety of applications of logic (in diverse areas

like computer science, linguistics, AI, law, physics, etc.) there is a growing number

of new logics to be investigated. In this situation AL offers us a tool for economy

and a tool for unification in various ways. One of these is that Alg

ð % is always

a class of algebras, therefore we can apply the same machinery, namely universal

algebra, to study all the new logics. In other words, we bring all the various logics

to a kind of “normal form” where they can be studied, compared, and even com-

bined by uniform methods. Moreover, for most choices of ð

, Alg

ð %tends to ap-

pear in the same “area” of universal algebra, hence specialized powerful methods

lend themselves to studyingð

. There is a fairly well understood “map” available

for the landscape of universal algebra. By using our algebraization process and

equivalence theorems, we can project this “map” back to the (far less understood)

landscape of possible logics.

In section 7, we will illustrate the above outlined “application of AL to logic”

by using the AL-results of Part I, as follows. In Part I, we studied various dis-

tinguished classes of algebras, like e.g.F

«

¡

@. Here, after studying the bridge

(ð

Alg

ð %

etc.) between the world of logics and that of algebras, we look up those distinguished logics to which the distinguished algebras of Part I belong.

E.g. we will find a certain logicv @

for whichF

«

¡

@ HAlg

v @ %. Then we will use

results in Part I aboutF

«

¡

@to establish properties of

v @. For this, we will use the

“equivalence theorems” established in section 6 (of Part II). BesidesF

«

¡

@and

v @, a similar procedure will be applied to other distinguished classes of algebras

(from Part I) and other distinguished logics.

The approach reported here is part of a broader, joint approach with W. Blok

and D. Pigozzi outlined in [ABNPS]. The present part contains only a somewhat

specialized version of that general approach, in order to suit the special needs

of the present work. Besides [ABNPS], we refer to [BP89], [BP91], [Cz97],

[P91], [FJ94], [FJ97], [PP], [CzP], as well as [ANSK], [HMTII, sections 5.6, 4.3],

[AKNS], [NA], [M96], [H96], [Mi95] for the more general approach. The seman-tic aspect of this approach goes back to e.g. [AN75], [AGN77], [AS].



CHAPTER 4

GENERAL FRAMEWORK FOR STUDYING LOGICS

DEFINITION 4.1 (logic) By a logicð

we mean an ordered quadruple

ð

def H I D Q Ë Q ø Q H W Q

where (i)–(iv) below hold.

(i) D (called the set of formulas of ð ) is a set of finite1 sequences (called words)

over some set ã

(called the alphabet of ð

).

(ii) Ë (called the provability relation of ð ) is a relation between sets of formulas

and formulas, that is, Ë )

D % 2 D . Following tradition2 , instead of

“ I ñ Q ñ W 5 Ë ” we write “ ñ Ë ñ ”.

(iii) ø is a class3 (called the class of models of ð ).

(iv) H

(called the validity relation) is a relation betweenø

and D

that is, H )

ø 2 D. Instead of “

I õ Q ñ W 5 H” we write “

õ H ñ”.

If ð is a logic, then by D Ì , Ë Ì , ø Ì , H Ì we denote its corresponding parts.

¶

Intuitively, D is the collection of “texts” or “sentences” or “formulas” that canbe “said” in the language

ð. For

» ) Dand

ñ 5 D, the intuitive meaning of

» Ë ñ

is thatñ

is provable (or derivable) from»

with the syntactic inference system (or

1With this we exclude infinitary languages like Ï Ð Ñ Ò Ï

é Ó

Ñ

Ö

. This exclusion is not necessary, all

the methods go through with some modifications. Actually, occasionally we will look into properties

of the finite variable fragment Ï

éÓ

Ñ

Ö

of infinitary logic, because it naturally admits applications of our

methods and plays an essential role in finite model-theory and in theoretical computer science.2This tradition is used for all binary relations: if

is a binary relation, then instead of

we sometimes write .3Although it is not automatically permitted in the “most official” version of set theory (ZF), we may

assume that for any four classes Ô

Ô Õ the tuple Ô

Ô Õ exists and is again a class. This

does not lead to set theoretical paradoxes. What one should avoid is assuming that the collection of all

classes would (exist and) form a class again (and variants of this). For more on this cf. [HMT, p.34,

first 10 lines], [HMT, p.25], [AHS,'

2, pp.5-8], or almost any work on abstract model theory.

65



66 ALGEBRAIC LOGIC

deductive mechanism) of ð

. In all important cases,Ë

is subject to certain (well-

known) conditions like » Ë ñ and » S h ñ Ë ) imply » Ë ) for any » ) D

andñ Q ) 5 D

. The classø

of models is understood in the spirit of model theory:

The models õ 5 ø of ð are thought of as “possible environments” or “possible

interpretations” or “possible worlds”, cf. Allen [Al]. Here a possible world is not

the same as the technical devices called possible worlds in a Kripke model. The

validity relation tells us which texts are “true” in which possible environments (orworlds or models) under what conditions. Usually

Dand

Ëare defined by what

are called grammars in mathematical linguistics.I D Q Ë W

together with the grammar

defining them is called the syntactical part of ð

, whileI ø Ì Q H W

is the semantical

part or model theoretical part of ð

4.

As a binary relation betweenø

andD

, H

induces a Galois-connection between

øand

D, and in particular, it defines two closure operators, one on

øand one on

D. Next, we collect some of the relevant definitions.

DEFINITION 4.2 (theory, models, consequences)

(i) Let Ú ) ø

and ñ ) D

. Then

Ú H Ì ñiff

õ 5 Ú %

ñ 5 D % õ H Ì ñ

Ó

We will write Ú H ñ in place of Ú H h ñ and similarly when Ú H h õ .Û Ü

Ì

Ú %

def H h ñ 5 D t Ú H ñ

, the theory of Ú

, and

¸ ¹ º Ì

ñ %

def H h õ 5 ø t õ H ñ

, the class of models of ñ

.

(ii) Semantical consequence, valid formulas: Let ñ S h ñ ) D . Then

ñ ÝH Ì ñiff

¸ ¹ º

ñ % H Ì ñ Q

ñ is a semantical consequence of ñ .

H Ì ñiff

ø H Ì ñ. In this case we say that

ñis a valid formula of

ð.

4Cf. Sections 14, 15 of Gabbay [Gab3] for more intuitive motivation on how and why these parts

are highlighted in a logic.

At this point a natural objection suggests itself: Why is Ô Þ an arbitrary class? Why did we not

assume (like in Barwise-Feferman [BF]) that Ô Þ is a class of first order structures or of algebraic

systems? The answer is (i)-(iii) below. (i) In institutions theory they do the same what we do and

for the same reasons. Cf. the subsection “Connections with the literature” at the very end of the

present section. (ii) We are developing a general theory, and we do not know in advance what kinds of

structures will be the models of our Ï . E.g. they may be classical first order models, they may or may

not have a topological structure too, they may be propositional Kripke-models, they may have infinitary

relations on them (cf. [HMT,'

4.3]), they may be models of intensional logic in the sense of Montague

etc. Therefore, at the very beginning, we do not want to commit ourselves on exactly what kinds of

mathematical objects will the elements of Ô Þ be. (iii) All the same, during the development of our

theory, we will impose some structure onÔ Þ

(but only gradually). This structure-imposing process is

carried even further in [ABNPS], cf. e.g. “concrete semantical systems” therein.



4. GENERAL FRAMEWORK FOR STUDYING LOGICS 67

(iii) Axiomatizable classes of models:

¸ ¹ ºÌ

Û Ü

Ì

Ú % is called the axiomatizable hull of Ú . Ú is axiomatizable iff

Ú H ¸ ¹ º Ì

ñ % for some

ñ ) D. In this case we also say that

ñdescribes

or defines Ú .

(iv) Provability, or derivability:

ËÌ

ñ iff î ËÌ

ñ , in this case we say that ñ is provable or derivable in ð . If

ñ Ë Ì ñ, then we say that

ñis provable from

ñ(in

ð).

If there is no danger of confusion, we will omit the subscript ð from ÝHÌ

,Û Ü

Ì,

¸ ¹ ºÌ

etc.¶

REMARKÛ Ü

¸ ¹ ºand

¸ ¹ º

Û Ü

are the two closure operators induced by H

. The

semantical consequence relation ÝH

is a binary relation between

D %and

D,

just like Ë is. To treat Ë and ÝH uniformly, in some places a logical system is

defined to be I D Q

W where

)

D % 2 D . E.g. Blok-Pigozzi [BP89] uses

this definition. In this notion,

can mean either the derivability relation or the

semantical consequence relationá Ý

â

. Connections between our conception of alogic and the literature will be discussed at the end of this section. ã

The definition of a logic in Def. 4.1 is very broad. Actually, it is too broad for

proving interesting theorems about logics. Now we will define a subclass of logics

which we will call algebraizable semantical logics. The notion of an algebraizable

logic is broad enough to cover a very large part of the logics investigated in the lit-

erature 5. On the other hand, the class of algebraizable logics is narrow enough for

proving interesting theorems about such logics, that is, we will be able to establish

typical logical facts that hold for most logics studied in the literature.

Below, in Definitions 4.3–4.10, we collect some common features of logics.

We will discuss the usual extra assumptions one usually makes about a logicä

in the following order. First we discuss (assumptions on) the distinguished parts

of ä

beginning withD Ì

and ending withá

â

Ì. Then in Def. 4.10, we will discuss

(assumptions on) how these parts are put together. Often, what we call “extra

assumptions” here will also imply “extra structure”.

The set D of formulas is usually defined by fixing a set æ ç of logical connec-

tives and a set è of atomic formulas:

5Moreover, in Remark 4.11 we will indicate how to extend the methods of the present work from

algebraizable logics to a broader class called protoalgebraic semanticallogics. The latter is really broad

enough to cover most logics in the literature.



68 ALGEBRAIC LOGIC

DEFINITION 4.3 (ä

has connectives) (i) Assume that two sets,è

and æ ç

are

given, such that every element of æ ç has a finite rank. Then é ê è ë æ ç í denotes the

smallest set î

satisfying (1),(2) below:

(1) è ï î , and

(2) for everyð ñ ë ò ò ò ë ð ó ô î

and õ ô æ ç

of rank ö

,õ ê ð ñ ë ò ò ò ë ð ó í ô î

.

Note that é ê è ë æ ç í

is the universe of the word-algebra of typeæ ç

generated by

è.

(ii) We say that é ÷ is given by ø è ë æ ç ù if é ÷

â

é ê è ë æ ç í . In this case we

say that è is the set of atomic formulas or atomic propositions of ä , and æ ç is

the set of logical connectives of ä . è is also called the vocabulary of ä

6. The

word-algebra generated by è and using the logical connectives of æ ç as algebraic

operations is denoted by ú , and is called the formula algebra of ä . Note that ú

â

ø é ë õ û ù ü ý þ ÿ where õ û ê ðñ

ë ò ò ò ë ðó

í

def â

õ ê ðñ

ë ò ò ò ë ðó

í ô é for all ðñ

ë ò ò ò ë ðó

ô é

and ö

-ary connectiveõ ô æ ç

.

(iii) We say that ä

has connectives if é ÷

is given byø è ë æ ç ù

for someè ë æ ç

.

In this case usually we assume that ø è ë æ ç ù is given together with ä . ã

Next we turn to inference systems

÷ . Inference systems (usually denoted as

) are syntactical devices serving to recapture (or at least to approximate) the se-

mantical consequence relation of the logic ä . The idea is the following. Suppose¡

á

¢

â

ð. This means that, in the logic

ä, the assumptions collected in

¡

semanti-

cally imply the conclusion ð . (In any possible world or model £ of ä whenever¡

is valid in£

, then alsoð

is valid in£

.) Then we would like to be able to repro-

duce this relationship between¡

andð

by purely syntactical, “finitistic” means.

That is, by applying some formal rules of inference (and some axioms of the logic

ä ) we would like to be able to derive ð from¡

by using “paper and pencil” only.

In particular, such a derivation will always be a finite string of symbols. If we can

do this, that will be denoted by¡

ð .

Inference systems are usually given by axioms and inference rules. These ax-

ioms and rules use formula-schemes in place of concrete formulas. A formula-

scheme is just like a formula, the only difference is that it is built up from formula-

variables (i.e. metavariables ranging over formulas) in place of atomic formulas.

DEFINITION 4.4 (formula-scheme, Hilbert-style inference system) (i) Assume that

é ÷ is given by ø è ë æ ç ù . We will call ¤ ¥ , § ¨ formula-variables , and é will

6Sometimes, informally,

is also called the set of “propositional variables”. Here one should

emphasize that the connotations of this name are misleading.




denote the set of all formula-variables, i.e.é

â

¤ ¥ § ¨ ". The elements of

é ê é ë æ ç í are called formula-schemes , and é $ will denote the set of formula-

schemes of ä

. I.e.é $

def â

é ê é ë æ ç í. An instance of a formula-scheme is obtained

by substituting formulas for the formula variables in it. A formula-scheme is called

valid if all of its instances are valid.

(ii) An inference-rule forä

is a pair % ø &

ñë ò ò ò ë &

óù ë & 0 2

, where every& ¥ ê § 5

ö íis a formula scheme of

ä. This inference rule will be denoted by

&ñ

ë ò ò ò ë &ó

&0

ò

An instance of an inference rule is obtained by substituting formulas for the

formula variables in the formula schemes occurring in the rule. An inference

rule ø ø &ñ

ë ò ò ò ë &ó

ù ë & 0 ù is called valid if

ðñ

ë ò ò ò ë ðó

" á

¢

â

ð 0 for all instances

ø ø ðñ

ë ò ò ò ë ðó

ù ë ð 0 ù of it. Valid inference rules are also called admissible rules or

strongly sound rules in the literature.

(iii) A Hilbert-style inference system (or calculus) for ä is a pair ø 9 A ë B D ù

where 9 A is a finite set of formula-schemes and B D is a finite set of inference rules

for ä .

(iv) A Hilbert-style inference system G

â

ø 9 A ë B D ù defines a provability or

derivability relation as follows. Assume¡ P

ð " ï é . We say that ð is -

derivable (or -provable) from

¡

iff there is a finite sequence ø ðñ

ë ò ò ò ë ð ÿ ù of

formulas (an -proof of ð from¡

) such that ð ÿ is ð and for every R 5 § 5 ç

U

ð¥

ô

¡

or

U

ð¥ is an instance of an axiom scheme (an axiom for short) of

Gor

U

there are Vñ

ë ò ò ò ë Vó

¨ § , and there is an inference rule of G such that X Y ` a c cd c a X Y f

X g

is an instance of this rule.

We write

¡

ð

if ð

is

-provable from

¡

. Now

â

ø

¡

ë ð ù

¡

ð "

. We saythat

is given by

ø 9 A ë B D ù. Throughout, we identify

Gwith

, e.g. we say that

ð

is an axiom of

.ã

Next, we turn to the semantics q ÷ , á

â

÷ of ä . Usually, validity of formulas

in models, i.e.á

â

÷is defined indirectly by first defining something more basic,

namely the meanings or denotations of formulas (and of other kinds of syntactic

entities belonging to the language) in models. The idea is that the meaning of

some syntactic entity (like a noun-phrase, or a sentence) need not always be a

truth value. Therefore, first we define a so-called meaning function which to each

syntactic entityð

and each model£

associates some semantic entityr ç s ê ð ë £ í

called the meaning of ð

in£

. After knowing what the syntactic entities mean in



70 ALGEBRAIC LOGIC

the models, one may be able to derive information about which sentences are true

or valid in which models.

DEFINITION 4.5 (meaning function,ä

is compositional) (i) Let r ç s be any func-

tion mapping é u q into a class; and let us call r ç s ê ð ë £ í the meaning of ð in

£

. For a fixed £ ô q

, the functionr ç s y

mappingé

to the set of meanings7

is defined by letting for allð ô é

r ç sy

ê ð í

def â

r ç s ê ð ë £ í ò

We say that r ç s is a meaning-function for ä , if validity of a formula depends

only on its meaning, i.e. if ê í

below holds:

ê í r ç sy

ê ð í

â

r ç sy

ê í

â

£ á

â

ð iff £ á

â

ò

(ii) Assume that ä

has connectives. We say that the meaning-functionr ç s

is

compositional if the meanings of formulas are built up from the meanings of their

subformulas, i.e. if the condition below is satisfied for allð

¥ë

¥ô é

,R § ö

and ö

-ary connectiveõ ô æ ç

:

ó

¥ ñ

r ç s y ê ð ¥ í

â

r ç s y ê ¥ í

â

r ç s y ê õ ê ð ñ ë ò ò ò ë ð ó í í

â

r ç s y ê õ ê ñ ë ò ò ò ë ó í í ò

This condition says exactly that r ç s y

is a homomorphism on the formula

algebra. We say that ä is compositional if it has connectives and a compositional

meaning-function (w.r.t. the connectives of ä ). This property is traditionally called

Frege’s principle of compositionality.

(iii) From now on, by a logicä

we understand a logic with a meaning-function,

i.e.ä

â

ø é ë q ë r ç s ë á

â

ù

, wherer ç s

is a meaning-function for the rest of ä

.ã

On the ingredients of a logic: Letä

â

ø ò ò ò r ç s ë á

â

ùbe a logic. Then, using the

terminology of Frege, Carnap, Montague as in [vBtM], r ç s represents the inten-

sional (or denotational) aspects of semantics, whileá

â represents the extensional

7We use the word “meaning” in the sense Frege used “intension” or “sense”. It is important to

emphasize that “meaning” is much more general than “extension” or truthvalue (though for some logics

the two may coincide). Further denotation (e.g. in Partee [Pa]) can be identified with what we call

“meaning”. As an example, let be Richard Montague’s intensional logic. Then is

exactly what Montague calls the intension of in . See also [Pa, pp. 1–18] on meaning(s). It seems

that we are using the word “meaning” in the same sense as Janssen [Ja, pp. 419–470] does. What we

call the meaning function j

is called a meaning assignment in [Ja, p. 423].




(or truth-value oriented) aspect of semantics, inä

. Therefore, those approaches to

general logics in which r ç s is repressed, seem to be extensionally oriented (cf.

e.g. institutions theory e.g. in Gabbay [Ga94, p. 359]), while the ones emphasiz-

ing r ç s (e.g. Andreka-Sain [AS], [HMTII], Epstein [E], Sain [S79]) seem to be

intensions-oriented (or sense or denotations oriented).

In many logics we have a derived connective l and a formula denoted as m n D o

which establish a strong connection betweenr ç s

andá

â , namely

(i)r ç s y ê ð í

â

r ç s y ê íiff

£ á

â

ð l , and

(ii)£ á

â

ðiff

r ç sy

ê ð í

â

r ç sy

ê m n D o í.

In these logics there is a strong connection between ê £ í ï é

and the kernel

of the meaning functionö o n ê r ç s y í ï é u é

, namely the kernel of r ç s y

and

ê £ íare recoverable from each other.8 We will say that

ähas the filter-property

iff there are derived connectives that generalize the above situation.

DEFINITION 4.6 (ä

has the filter-property) (i) Aö

-ary derived connective is a formula-scheme

ô é $using the formulavariables

¤0

ë ò ò ò ë ¤ ó ñonly. If

ð 0 ë ò ò ò ë ð ó ñ are formulas, then ê ð 0 ë ò ò ò ë ð ó ñ í denotes the instance of when

we replace¤

0ë ò ò ò ë ¤ ó ñ

byð

0ë ò ò ò

,ð ó ñ

respectively.9

(ii) We say that ähas the filter-property iff there are derived connectives 0 ë ò ò ò ë

ñand

{0

ë ò ò ò ë { ñ(unary) and

0

ë ò ò ò,

ÿ

ñ(binary) (

r ë ç ô ) of

ä

with the following properties: For all ð ë ô é and for all £ ô q

(1)r ç s y ê ð í

â

r ç s y ê í ~

ê § ¨ ç í

£ á

â

ð ¥

.

(2)£ á

â

ð ~

ê V ¨ r í

r ç s y ê ê ð í í

â

r ç s y ê { ê ð í í .

ã

In the case of classical logic, we can choose the above derived connectives such

that is “ l ”, ê ð í is ð , and { ê ð í is “True”.

Finally, most logics have some substitution-invariance properties. In this re-

spect we usually expect a logic to be substitutional in the sense of Def. 4.7 below.

If it is not, then we rather treat it as a “theory” of a substitutional logic. For such

examples see section 7 (Examples) or [AKNS].

8If

is a function, then

.9The algebraic counterpart of “derived connective” is “term function”. If

is binary, then we will

write

in place of

.



72 ALGEBRAIC LOGIC

DEFINITION 4.7 (ä

has the substitution property) (i) By a substitution$

we un-

derstand a function $ è é (we will “substitute” ô è with $ ê í ô é ).

If ð ô é

, thenð ê $ ê í í

denotes the formula we obtain fromð

after simulta-

neously substituting $ ê í for every occurrence of , for all ô è in ð . In other

words,ð ê $ ê í í

def â

$ ê ð í , where

$is the (unique) extension of

$ è éto a

homomorphism

$ ú ú. 10

(ii) ä has the (syntactic) substitution property (or ä is substitutional) iff for any

formula ð ô é and substitution $ è é

á

â

ðimplies

á

â

ð ê $ ê í í ò

This means that a formula of ä

is valid iff the corresponding formula scheme of

ä is valid (where we get the corresponding formula scheme by substituting atomic

formulas ¥ ô è

with formula variables¤ ¥ ô é

).

(iii)ä

has the semantical substitution property iff for any model£ ô q

and

substitution $ è é there is another model ô q such that

r ç s ª ê í

â

r ç s y ê $ ê í í for all ô è ò

Intuitively, the model

is the substituted version of £

along$

.ã

The semantical substitution property says that the atomic formulas can have the

meanings of any other formulas. (This statement will be made precise in Propo-

sition 5.2.) Examples where we have and do not have this property are given in

section 7. We state the next proposition without its simple proof.

PROPOSITION 4.8 If a logicä

has the filter-property and the semantical substi-

tution property, then it has the syntactic substitution property, too. ã

Thus, a “fully-fledged” logic ä

â

ø é ë ë q ë r ç s ë á

â

ù sometimes is given as

ä

â

ø ø è ë æ ç ù ë ø 9 A ë B D ù ë q ë r ç s ë á

â

ù . Often, not all parts of a logic are given.

Sometimes we have onlyø é ë ù

and we are searching for a “semantics”ø q ë r ç s ë á

â

ùfor it such that e.g.


â

ùis complete.11 Or, even more often, we

haveø é ë q ë r ç s ë á

â

ùand we are searching for a provability relation

such that


â

ùwould be complete. Sometimes

ø é ë ùis called a “deductive

logic” (or syntactic one), whileø é ë q ë r ç s ë á

â

ùis called a “semantic logic” (cf.

e.g. [ABNPS]). (Though, one should keep in mind thatø é ë á

¢

â

ùis a “deductive

logic” in this sense.) From now on we will often omit some parts of a logic. Most

10Such a unique extension exists because¬

is the word-algebra generated by

, i.e., in algebraic

terms, it is freely generated by

.11Completeness of a logic will be defined in Def. 6.1 in section 6.




often we will deal withø é ë q ë r ç s ë á

â

ùand we will say that

ä

â


â

ù

is a logic or more carefully, a semantical logic. Most of the notions are meaningful

for it, e.g. thatä

is compositional etc.

DEFINITION 4.9 (Algebraizable semantical logic, structural logic) Let ä

â


â

ù

be a logic in the above sense.U We say that

äis structural if

äis compositional and has the semantical

substitution property.

U We say that ä

is an algebraizable 12 semantical logic if ä

is structural and

has the filter-property.

ã

In most cases, the setè

of atomic formulas is a parameter in the definition of the

logicä

. Namely,è

is a fixed but arbitrary set. So in a sense,ä

is a function of è

,

and we could writeä -

(instead of ä

) to make this explicit. Most often the choice

of è has only limited influence on the behaviour of ä . However, we will have to

remember thatè

is a freely chosen parameter because in certain investigations,

the choice of è does influence the behaviour of ä - .

DEFINITION 4.10 (General logic, algebraizable general logic)

(i) A general logic is a function (or indexed family)

® def â

ø ä

-

èis a set

ù ë

where for each set è

,ä -

â

ø é - ë q - ë r ç s - ë á

â

- ùis a logic in the above

sense.

(ii) We say that ®

has connectives iff there is a set æ ç of connectives such that

for every set è

,æ ç

is the set of connectives of ä -

in the sense of Def. 4.3 and è

is the set of atomic formulas 13 of ä - , i.e. é -

â

é ê è ë æ ç í for all è . Sometimes

èis called the vocabulary of

ä -.

(iii)®

is compositional if it has connectives and ä

- is compositional for allè

.

(iv)®

has the filter-property iff there are derived connectives 0 ë ò ò ò ë ñ ,

{0

ë ò ò ò ë { ñ ,

0

ë ò ò ò ë ÿ

ñ(common for all possible choices of

è) such that

ä -has the filter-property with these, for all

è.

12The definition of algebraizability originates with Blok and Pigozzi, cf. e.g. [BP89].13We are making simplifications now. It is not necessary to have ° in ± for all sets . What

is needed for our investigations is that for all cardinality²

there be a

such that the set of atomic

formulas of

° has cardinality at least²

.



74 ALGEBRAIC LOGIC

(v)®

has the substitution property iff for allè ë ³

,$ è é ´

, and ð ô é -

,

á

â

-

ðimplies

á

â

´

ð ê $ ê í í ò

(vi)®

has the semantical substitution property iff for all setsè ë ³

,$ è é ´

and £ ô q ´ there is ô q - such that r ç s

´

y µ

$

â

r ç s -

ª

.

(vii)®

is an algebraizable general logic iff ®

is compositional, has the filter-

property, and has both substitution properties,®

is structural if it is compositional

and has the semantical substitution property.

(viii) The notions of a formula-scheme, valid formula-scheme, valid rule, and

Hilbert-style inference system for a general logic are the obvious generalizations

of their versions given for (non-general) logics ä , cf. e.g. Def. 4.4.

(ix) By a fully fledged general logic we understand a function

®

â

ø ä

-

è is a set ù

such that for each set è

,ä -

â

ø é - ë - ë q - ë r ç s - ë á

â

- ù

is a fully fledged logic i.e. ø é ë ù is a deductive logic, ø é ë q ë r ç s ë á

â

ù is a semantical logic, and

-

ï á

¢

â - . Items (ii)–(vii) above extend to the fully fledged case the natural way.

ã

REMARK 4.11 We note that ®

is an algebraizable general logic iff ä - is an alge-

braizable semantical logic for allè

, the connectives and the derived connectives

for the filter-property are the same for all è , and the condition below holds for all

è ï ³:

¶

r ç s

-

y

£ ô q

- ·

â¸

ê r ç s

´

y

í ¹ é

-

£ ô q

´ º

ò

Intuitively, this condition says that ä -

is the natural restriction of ä ´

.ã

REMARK 4.12 The theories of semantical logics ä »

â


â

ù and de-

ductive logicsä ½

â

ø é ë ë ùare best when developed in a parallel fashion, cf. e.g.

[ABNPS]. Throughout this remark we assume that ä » is structural (cf. Def.4.9).

We called ä

» algebraizable iff it has the filter property. An analogous defini-

tion for algebraizability of ä ½ was given in the papers of Blok and Pigozzi, cf.

[BP89]. There are weaker properties of ä

½ studied in the Blok and Pigozzi papers

which properties already enable one to apply (at least part of) the methodology of

algebraic logic to the logics in question.




Logics with these properties are called “protoalgebraic”, “equivalential”, and

“weakly algebraizable”.14 (There are other such properties in the literature, but

the weakest one facilitating application of our methodology seems to be being

protoalgebraic.) As we implied, the properties of being protoalgebraic and equiv-

alential naturally extend to semantical logics.

We call a semantical logic ä» protoalgebraic iff there is a set ê ð ë í

â

¥

ê ð ë í § ô G " of derived connectives such that

ê í á

â

ê ð ë ð íand

£ á

â

ê ð ë í

£ á

â

ð ¿ £ á

â

ë

for all £ ô q , and ð ë ô é .

ä »is called equivalential iff there is

as in

ê íabove, but such that Condition

(1) in Def.4.6 holds for this and ä » .

ä» is called weakly algebraizable iff it is protoalgebraic and there are sets

ê A í

â

¥ ê A í § ô G " and { ê A í

â

{ ¥ ê A í § ô G " as in Def.4.6(2).

Clearly, if ä

» is both equivalential and weakly algebraizable, thenä

» is in-

finitely algebraizable, where “infinitely” means that ë ë { may be infinite sets of

derived connectives. (If they are finite, thenä

» is algebraizable.) To keep the rest

of this discussion short, we concentrate on protoalgebraic and equivalential (but

all what we say can be extended to weakly algebraizable, too).We note that if ä » is protoalgebraic (or equivalential), then so is its deductive

counterpart ø é ë á

¢

â

ù in the sense of e.g. [CzP]. Moreover, ä ½ is protoalge-

braic/equivalential in the sense of [CzP] iff there is a semantical logic ä » such

that ä ½

â

ø é ë á

¢

â

÷ Ã ù and ä » is protoalgebraic/equivalential, respectively. For the

method of proving such equivalence theorems we refer to Font-Jansana [FJ94]

and [ABNPS]. We also note that

protoalgebraicÄ

equivalentialÄ

algebraizable

for semantical logics (the same applies to deductive ones, too). The machinery

(algebraization process, equivalence theorems etc) developed in the present work

does extend to protoalgebraic and equivalential semantical logics (from algebraiz-

able ones). Cf. e.g. Hoogland [H96], Hoogland-Madar asz [HoM] for part of thisextension. For brevity, in this work we present the above mentioned machinery

for the case of algebraizable logics only, at the same time inviting the interested

reader to extend this machinery to the protoalgebraic and/or equivalential cases,

too.ã

Connections with the literature: What we call a fully fledged logicä

â

ø é ë

ë ò ò ò ë á

â

ùwas called15 a “formalism” in Tarski-Givant [TG, p.16 (section 1.6)],

an “axiomatic system with semanticsø 9 ë ë Å Æ Ç ë á

â

ù” in Aczel [Ac, p.265] (here

14Cf. Czelakowski-Pigozzi [CzP].15A difference is that the intensional part

is suppressed in most of the quoted works but it is

not suppressed in e.g. Epstein [E].



76 ALGEBRAIC LOGIC

9coincides with our

é), and a “logic

øSign

ë $ o ç ë q È É ë ë á

â

ù” in Marti-Oliet–

Meseguer [MOM]. More precisely, the latter corresponds to our fully fledged

general logics (cf. the subitem below).

What we call a semantic logic was called a “semantical system” in [ABNPS],

a “semantical system ø 9 ë Å Æ Ç ë á

â

ù ” in Aczel [Ac, p.265]; and our general logic®

â

ø ä - ò ò ò ùcorresponds to an institution

øSign

ë $ o ç ë Å Æ Ç ë á

â

ùin Marti-Oliet–

Meseguer [MOM, p.358] (the latter will be elaborated below).

Connection with institutions: Institutions theory (e.g. [MOM]) emphasizes the

category theoretic aspects of a general logic®

â

ø ä-

ò ò ò ù(which are down-

played here), and suppresses the intensional aspects represented by rç

s . To

see that such a general logic is a category whose objects are the logicsä -

, let

Sign â

è èoccurs as a set of atomic formulas in

®

"be fixed16. (We know

that according to our conventions, Sign â Ê all sets”, but let us abstract away from

this, and just assume that Sign is a fixed proper class.) Let è ë è ñ ô Sign. Next

we define what a logic morphismÌ ä - ä -

`

is. A logic morphism is a pair

Ì

â

ø Î ë Ï ùsuch that

Î ú - ú -

`

is a homomorphism of the formula-algebras

andÏ q

-

`

q- “makes everything commute”, e.g.

£ á

â

Î ê ð í ~

Ï ê £ í á

â

ð.17 Now, if

ø Î ë Ï ùis such a logic morphism, then

ê Î ¹ è íis called

a signature morphism. Let Sign be the category of these signature-morphisms (as

arrows, and Sign itself is the class of objects).

Let Log be the category of logicsä -

and logic morphismsÌ

â

ø Î ë Ï ùoccurring

in®

. Then there is a functorÓ Sign Log sending

èto

ä -. This

Óis

almost the institution we are looking for. There is one ingredient missing, though.

Namely,q -

is not only an arbitrary class, but is a category in all the applications

we know of. This category character of q - is de-emphasized in the present paper,

but it does show up in the theory later, cf. e.g. the category of concrete semantical

systems in [ABNPS]. So, let us assume that each q - is a category. Then our

functorÓ

above induces two functorsÎ r Õ Sign Fmla and

Å Æ Ç Sign

CatÖ ×

, where Fmla is the category of formula-algebras (of the formú -

), and for

eachè

,

Ó ê è í

â

ø Î r Õ ê è í ë Å Æ Ç ê è í ë á

â

-

ù ë

while

Ó ê Ø í

â

ø Î r Õ ê Ø í ë Å Æ Ç ê Ø í ù

for the morphismsØ

of Sign. Now the tuple

G ê

®

í

â

øSign

ë Î r Õ ë Å Æ Ç ë á

â

ù

is called an institution, where

á

â â

ø á

â

-

è ô Sign ù ò

16What we call here a set

of atomic formulas is called a signature in institutions theory.17If meaning functions are also present, then

Ùshould induce a function

Ù Úon the meanings such

that Û Ü

j Ý

Ù Ú j , cf. [ABNPS].




At this point one can see that to a general logic®

there is an equivalent insti-

tution G ê

®

í , and conversely to an institution ß there is a general logic à ê ß í such

that the two can be recovered from each other. (Here we assumed that inà ê ß í

the models still form a category.) Therefore, institutions and general logics can be

studied interchangeably, depending on the kind of mathematical tools (universal

algebra or categories) one wants to use.18

In institutions theory, our “ Î r Õ ” is denoted by “ $ o ç ”, exactly because therer ç s

is suppressed and therefore meanings are replaced by truthvalues. So, when

the theory is applied to e.g. first-order logic, then attention has to be restricted to

sentences (=closed formulas) because meanings of open formulas are more com-

plex objects than just truthvalues.

We do not treat here the different notions of equivalence of logics, morphisms

acting between logics, concrete semantical logics, cf. e.g. [ABNPS]. Also, we do

not treat here interpretability between logics, and combining logics, [Gab], [Gab2],

[JKE], [ABNPS]. These are important and very interesting subjects.

For the rest of this work, one of the most important definitions of section 4 is

that of an algebraizable general logic. It is summarized in Remark4.11.

18To make this statement hundred percent true, one should include the intensional aspect into

institutions and makeá

° into a category in general logics. We do not see why one would not do these

amendments.



CHAPTER 5

THE PROCESS OF ALGEBRAIZATION

The algebraic counterpart of classical sentential logic ä â is the variety ã ä of

Boolean algebras. Why is this so important? The answer lies in the general expe-

rience that sometimes it is easier to solve a problem concerningä

â by translating

it toã ä

, solving the algebraic problem, and then translating the result back toä

â

(than solving it directly inä

â ).

In this section we extend applicability of ã ä

toä

â to applicability of algebra in

general to logics in general. We will introduce a standard translation method from

logic to algebra, which to each logicä

associates a class Alg ê ä íof algebras. (Of

course, Algê ä

âí

will beã ä

.) Further, this translation method will tell us how tofind the algebraic question corresponding to a logical question. If the logical ques-

tion is about ä , then its algebraic equivalent will be about Alg ê ä í . For example,

if we want to decide whetherä

has the property called Craig’s interpolation prop-

erty, then it is sufficient to decide whether Alg ê ä í has the so called amalgamation

property (for which there are powerful methods in the literature of algebra). If

the logical question concerns connections between several logics, say between ä ñ

andä å

, then the algebraic question will be about connections between Alg ê ä ñ í

and Alg ê ä å í . (The latter are quite often simpler, hence easier to investigate.) This

“bridge” also enables us to solve algebraic problems by logical methods (for an

example see section 7).

DEFINITION 5.1 (meaning algebra, Alg

, Alg) Let ä

â


â

ù

be acompositional logic withé æ

â ç .

(i) First we turn every model into an algebra. Compositionality of r ç s y

means

that we can define an algebra of typeæ ç

on the set

r ç s y ê ð í ð ô é "of

meanings. This algebra isr ç s y ê ú í

, it will be called the meaning algebra of £

and it will be denoted by£ ê ë ê £ í

. In more detail, to any logical connectiveõ

of

arityö

we can define aö

-ary functionõ

y

on the meanings in£

by setting for all

formulas ðñ

ë ò ò ò ë ðó

õ

y

ê r ç sy

ê ðñ

í ë ò ò ò ë r ç sy

ê ðó

í í

def â

r ç sy

ê õ ê ðñ

ë ò ò ò ë ðó

í í ò

(We could say that õ

y

is the meaning of the logical connective õ .) Then £ ê ë ê £ í

def â

ø

r ç s y ê ð í ð ô é " ë õ

y

ùü ý þ ÿ .

79



80 ALGEBRAIC LOGIC

(ii) Alg

ê ä ídenotes the class of all meaning-algebras of

ä, i.e.

Alg

ê ä í

def â

r ç sy

ê ú í £ ô q ÷ "

â

£ ê ë ê £ í £ ô q ÷ " ò

(iii) Let í ï q ÷

. Then for everyð ë ô é

ð î ï

def ~

ê £ ô í í r ç s y ê ð í

â

r ç s y ê í ò

Thenî ï

is an equivalence relation, which is a congruence onú

by composition-

ality of ä

.ú î ï

denotes the factor-algebra of ú

, factorized byî ï

. It is called

the Lindenbaum-Tarski algebra of í

. Now,

Alg ê ä í

def â

ß

ú î ï í ï q÷

" ò

Thus, Algê ä í

is the class of isomorphic copies of the Lindenbaum-Tarski algebras

of ä

.

(iv) Let ®

â

ø ä - è is a set ù be a general logic. Then

Alg

ê

®

í

def â ò Alg

ê ä

-

í èis a set

ë é

-

æ

â ç

" ë

and

Alg ê

®

í

def â

ò Alg ê ä

-

í è is a set ë é

-

æ

â ç

" ò

ã

REMARK In thedefinition of Alg

ê ä í above, it is important that Alg

ê ä í is not

an abstract class in the sense that it is not closed under isomorphisms. The reason

for defining Alg

ê ä íin such a way is that since Alg

ê ä íis the class of algebraic

counterparts of the models of ä

, we need these algebras as concrete algebras

and replacing them with their isomorphic copies would lead to loss of information

(about semantic-model theoretic matters). See e.g. the algebraic characterization

of the weak Beth definability property , Theorem 6.12 in the next section.ã

For a logic ä , let q ç s ÷

def â

r ç sy

£ ô q ÷ " . That is, q ç s ÷ is the class

of “meaning-homomorphisms” of the logicä

(or equivalently, the unary meaning-

functions induced by the models of ä ). If ô is an algebra and õ is a class of

algebras, thenî È r ê ô ë õ í

denotes the class of all homomorphismsÌ ô ö

where ö ô õ .



5. THE PROCESS OF ALGEBRAIZATION 81

PROPOSITION 5.2 (characterization of structural logics) Let ä

and ®

be a com-

positional logic and a compositional general logic, respectively. Then (i)–(ii) be-

low hold.

(i) ä has the semantic substitution property iff

q ç s ÷

â

î È r ê ú ë Alg

ê ä í í, iff

q ç s ÷

â

î È r ê ú ë õ í for some class õ of algebras.

(ii)®

has the semantic substitution property iff

q ç s -

â

î È r ê ú - ëAlg

ê

®

í í, for all

è, iff

q ç s -

â

î È r ê ú - ë õ í , for all

è, for some

õ.

Proof. To prove the first equivalence in (i), assume that ä has the semantic sub-

stitution property, andÌ ú r ç s

yê ú í

for some£ ô q ÷

. We want to

show that Ì

â

r ç sª

for some ô q ÷ . For each ô è let ð

×

ô é

be such thatÌ ê í

â

r ç sy

ê ð

×

í, and let

$ è ébe defined such that

$ ê í

â

ð

×

for all ô è . By the semantical substitution property of ä , there

is ô q ÷

such that for allð

,r ç s

ªê ð í

â

r ç sy

ê

$ ð í. Then for all

ô è,

r ç s ª ê í

â

r ç s y ê $ ê í í

â

Ì ê í, i.e.

r ç s ªand

Ìagree on

è. Since

úis

generated by è , this implies that Ì

â

r ç s ª . The other direction of the firstpart of (i) is trivial: Let

£ ô q÷ and

$ è é. We have to show that

r ç s y

µ

$ ô q ç s, which is true by

r ç s y

µ

$ ú r ç s y ê ú í ô Alg

ê ä í.

To prove the equivalence of the second and third statements in (i), assume that

q ç s÷

â

î È r ê ú ë õ í. We want to show that

q ç s÷

â

î È r ê ú ë Alg

ê ä í í.

Notice first thatq ç s

÷

â

î È r ê ú ë õ íimplies that Alg

ê ä í ï û õ. So let

Ì ú ô,

ô ô Alg

ê ä í. Then

Ì ú öfor some

ö ô õ, by Alg

ê ä í ï û õ.

Thus Ì ô q ç s ÷ by q ç s ÷

â

î È r ê ú ë õ í .

The proof of (ii) is completely analogous, we omit it. ã

THEOREM 5.3 (connection between Alg

and Alg)

(i) Let ä be a compositional logic. Then

û ý Alg ê ä í

â

û ý Alg

ê ä í ò

(ii) Let ®

be a structural general logic. Then

Alg ê

®

í

â

û ý Alg

ê

®

í ò

Proof. Proof of (i): First we show Alg

ê ä í ï ß Alg ê ä í . Let ô ô Alg

ê ä í ,

sayô

â

r ç s y ê ú í. Set

í

â

£ ". Then

î ï

â

ö o n ê r ç s y í, so

ú î ïis

isomorphic toô

. To show the other direction, letô ô

Algê ä í

, sayô

â

ú î ïfor

someí ï q

÷ . Then there is a subsetí

¡

ï ísuch that

ú î ï

â

ú î ï ¢(this



82 ALGEBRAIC LOGIC

holds becauseé

is a set). Thus we may assume thatí ï q ÷

is a set. We define

for all ð ô é

Ì ê ð îï

í

def â

ø r ç sy

ê ð í £ ô í ù ò

This is a sound definition by the definition of îï

. It is not difficult to check that

Ìis one-to-one and a homomorphism, so

Ì ô ¤

¦

ø r ç sy

ê ú í £ ô í ù,

showing thatô ô û ý

Alg

ê ä í

. Sinceû ý

is a closure operator, we are done withproving (i).

Proof of (ii): First we note that, by (i), Alg ê ä - í ï û ý Alg

ê ä - í for any set è ,

thus Alg ê

®

í ï û ý Alg

ê

®

í holds.

We are going to proveû ý Alg

ê

®

í ï Alg ê

®

í. Let

ô ô û ý Alg

ê ä í, say

ô ï

¦

¥ ý § ô ¥ for a set G and algebras ô ¥ ô Alg

ê

®

í . Let Ì ú © ô be any onto

homomorphism (e.g. we can take forÌ

the homomorphic extension of the identity

mappingÌ

¡

9 9). For each

§ ô Glet ¥ denote the projection function onto

ô ¥ , and let Ì ¥

def â

¥

µ

Ì . Then Ì ¥ ú © ô ¥ ô Alg

ê

®

í . By Proposition 5.2 (ii)

thenÌ ¥

â

r ç sy

g for some£ ¥ ô q ©

. Letí

â

£ ¥ § ô G ". Then it is easy to

check that Ì ê ð í

â

Ì ê í iff ð îï

for all ð ë ô é © . Thus ô is isomorphic to

ú

©

î ï ô Alg ê

®

í, and we are done.

ã

We note that we also proved that for structural logicsä

,

Algê ä í

â

û ýAlg

ê ä í

ô á 9 á á é÷

á " ò

Now we turn to proving that the equations valid in Algê ä í

correspond to the

valid formula-schemes of ä

, and the quasi-equations valid in Alg ê ä ícorrespond to

the valid rules of ä

. Here we will use the filter-property. If ä

is algebraizable, then

the equational and quasi-equational theories of Alg ê ä írecapture the validities and

the semantical consequence relationá

¢

â of ä

, respectively. Thus, when a logicä

is given, it is interesting to investigate the equational and quasi-equational theories

of Alg ê ä í. Note that by Theorem 5.3 above, Alg ê ä í

and Alg

ê ä íhave the same

equational and quasi-equational theories.

First we note that formulas and formula-schemes are terms in the language of

Alg ê ä í. Hence if

ð ë ô é $, then

ð

â

is an equation in the language of

Algê ä í

where we consider the formula-variables¤

¥ as algebraic variables (ranging

over the elements of the algebras). Similarly, if ð ë ô é

, thenð

â

is also

an equation in the language of Algê ä í

, where we consider the elements of è

as

algebraic variables.

THEOREM 5.4 (valid rules of ä

and quasi-equations of Alg ê ä í) Let

äbe a com-

positional logic with filter-property. Let ë { ë ë r ë ç

be as in the definition of

filter-property. Then (i)–(ii) below hold.



5. THE PROCESS OF ALGEBRAIZATION 83

(i) A ruleø ø ð 0 ë ò ò ò ë ð ó ñ ù ëm ð ó ù

of ä

is valid (or admissible) iff

Alg ê ä í á

â

!"

f

g

"$

¥

ê ð % í

â

{¥

ê ð % í ê ð ó í

â

{ ê ð ó ífor each

V ¨ ç.

(ii) A quasi-equationð 0

â

0 & ò ò ò & ðó ñ

â

ó ñ

ðó

â

ó

(with variables from

é ) is valid in Alg ê ä í

iff

the rules

ð 0 0 0 ë ò ò ò ë ð 0 ÿ ñ 0 ë ò ò ò ë ð ó ñ| 0 ó ñ ë ò ò ò ë ð ó ñ ÿ ñ ó ñ

ðó

ó

are valid inä

for allV ¨ ç

.

Proof. Assume that ) is a valid rule of ä

of the formø ø ð 0 ë ò ò ò

,

ð ó ñ ù ë ð ó ù. Let 0

o 0 denote the quasi-equation associated to it in (i). We want

to show that 0o 0

is valid in Alg ê ä í. By Theorem 5.3 it is enough to prove that it

is valid in Alg

ê ä í . Let ô ô Alg

ê ä í , and let Ì é 9 be an evaluation

of the variables in 0o 0

such that the hypothesis part of 0o 0

is true inô

under the

evaluation Ì , i.e. assume that

ê 3 í ô á

â

!"

f

g

"$

¥ ê ð%

í

â

{ ¥ ê ð%

í

Ì ò

We want to show

ô á

â

ê ð ó í

â

{ ê ð ó í

Ì ò

Byô ô Alg

ê ä í, there is

£ ô q ÷such that

ô

â

r ç sy

ê ú í. For any

¤ ¥ ô é

take ¥ ô é such that Ì ê ¤ ¥ í

â

r ç sy

ê ¥ í and let ø ø ð

¡

0

ë ò ò ò ë ð

¡

ó ñ

ù ë ð

¡

ó

ù be the

instance of our rule ) by replacing each¤ ¥

with ¥

. Then for each§ ¨ r

and 5 ö

we have that

Ì ê ¥ ê ð % í í

â

r ç s y ê ¥ ê ð

¡

%

í í and the same for { , i.e.

Ì ê { ¥ ê ð % í í

â

r ç sy

ê { ¥ ê ð

¡

%

í í . (Here

Ì denotes the homomorphic extension of Ì to é $ .) Then

by the filter-property of ä

, and by our assumption ( 3 ), we have£ á

â

÷ ð

¡

%

for

all 5¨ ö . Since ø ø ð 0 ë ò ò ò ë ð

ó ñù ë ð

óù is a valid rule, and ø ø ð

¡

0

ë ò ò ò ë ð

¡

ó ñ

ù ë ð

¡

ó

ù

is an instance of it, this implies£ á

â

÷ ð

¡

ó ñ

. Then by the filter-property again,

r ç s y ê ê ð

¡

ó

í í

â

r ç s y ê { ê ð

¡

ó

í í, i.e.

Ì ê ð ó í í

â

Ì ê { ê ð ó í íand we are done.

Conversely, assume that the quasi-equation is valid in Alg ê ä í , and we want to

show that the rule is valid. Letø ø ð

¡

0

ë ò ò ò ë ð

¡

ó ñ

ù ë ð

¡

ó

ùbe an instance of the rule

that we got by substituting ¥ to the formulavariables ¤ ¥ , for all § ¨ . Assume

£ ô q÷ and

£ á

â

÷

ð

¡

0

ë ò ò ò ë ð

¡

ó ñ

". We want to show

£ á

â

÷ð

¡

ó

. By the

filter-property we haver ç s y ê

¥ê ð

¡

%

í í

â

r ç s y ê {¥

ê ð

¡

%

í í. Let

Ì é $ é

be a homomorphism such thatÌ ê ¤

¥í

â

¥ for all

§ ¨ . Then

Ì ê ¥

ê ð % í í

â



84 ALGEBRAIC LOGIC

r ç s y ê ¥ ê ð

¡

%

í íand the same for

{ ¥, thus ( 3 ) above holds with

ô

â

r ç s y ê ú í.

Thus ô á

â

ê ðó

í

â

{

ê ðó

í

Ì , since the quasi-equation is valid in ô ô Alg ê ä í ,

i.e.r ç s y ê ê ð

¡

ó

í í

â

r ç s y ê { ê ð

¡

ó

í í, for all

V ¨ r. By the filter-property then

£ á

â

÷ ð

¡

ó

as was to be shown.

We omit the proof of (ii). It is analogous to the above proof of (i).ã

COROLLARY 5.5 (Valid formula-schemes, validities, and Eq ê Alg ê ä í í )

(i) Let ä be a compositional logic with filter-property. Let ë { ë ë r ë ç be as in

the definition of the filter-property. Then for every formula-schemeð

of ä

ð is a valid formula-scheme of ä iff

Alg ê ä í á

â

ê ð í

â

{

ê ð í for all V ¨ r .

(ii) Assume further that ä is algebraizable. Then for any formulas ð ë ð 0 ë ò ò ò ë ð ó

of ä ,

á

â

÷ð

iff

Algê ä í á

â

ê ð í

â

{ ê ð í

for each V ¨ r .

ð0

ë ò ò ò ë ð ó ñ " á

¢

â

÷ð ó

iff

Alg ê ä í á

â

!"

f

g

"$

¥ ê ð % í

â

{ ¥ ê ð % í

ê ðó

í

â

{

ê ðó

í for each V ¨ r .

(iii) The set of valid formula-shemes of ä

is decidable (recursively enumerable)

iff Eq ê Alg ê ä í í is decidable (recursively enumerable). The set of valid (admissible)

rules of ä

is decidable (recursively enumerable) iff the quasi-equational theory of

Alg ê ä í is decidable (recursively enumerable).

(iv) Statements (i) and (ii) above hold for general logics®

in place of ä

.ã

We say that the validity problem of the logic ä is decidable iff the set of valid

formulas of ä

is decidable. If ®

â

ø ä - èis a set

ùis a general logic, then the

validity problem of ®

is decidable iff it is decidable for ä - , for all è .

COROLLARY 5.6 Let ä

be an algebraizable logic withá è á 6

or an algebraiz-

able general logic. Then the validity problem of ä

is decidable iff Eq ê Alg ê ä í íis

decidable. ã



86 ALGEBRAIC LOGIC

U

is strongly complete and sound for

äiff the derivability and semantic

consequence relations coincide, i.e. iff

ê

¡

ï é í ê ð ô é í A

¡

ðiff

¡

á

¢

â

ð C ò

Let Q

â

ø 9 A ë B D ù

be a Hilbert-style inference system. We say that Q

is (weakly, finitely, strongly) complete for ä if the derivability relation given by ø 9 A ë B D ù is

such for ä

. We say that Q is (weakly, finitely, strongly) complete for a general logic®

â

ø ä - è is a set ù if Q is such for all ä - . We use an analogous terminology

for the soundness properties.ã

The next theorem is a characterization of existence of strongly complete and

sound Hilbert-style inference systems for general logics. It is proved in [AKNS,

Thm.3.2.21]. A classõ

is a quasi-variety iff it can be axiomatized with a set of

quasi-equations, i.e. equational implications. See a footnote in section 1.

THEOREM 6.2 Assume that ®

is an algebraizable general logic. Then there is a

strongly complete and sound Hilbert-style inference system for ®

iff Alg ê

®

í is a finitely axiomatizable quasi-variety.

ã

In Theorem 6.3 below we give a sufficient and necessary condition for an alge-

braizable semantic logic to have a finitely complete Hilbert-style inference system.

Its proof can be found in [AKNS, Thm.3.2.3].

THEOREM 6.3 Assume that ä

is an algebraizable semantic logic and æ ç ê ä í

is

finite2. Then there is a finitely complete and strongly sound Hilbert-style inference

system for ä

iff Alg ê ä ígenerates a finitely axiomatizable quasi-variety.

ã

The following theorem is a characterization of existence of weakly complete

and strongly sound Hilbert-style inference systems. It is [AKNS, Thm.3.2.4].

THEOREM 6.4 Assume that ä is an algebraizable semantic logic and æ ç ê ä í is

finite. Then there is a weakly complete and strongly sound Hilbert-style infer-

ence system for ä iff there is a finitely axiomatizable quasi-variety õ such that

Alg

ê ä í ï õ ï R û ýAlg

ê ä í. The same is true for algebraizable general logics.

ã

2One can eliminate the assumption of T being finite. Then the finitary character of a Hilbert-

style inference system has to be ensured in a more subtle way. Also, “finitely axiomatizable quasi-

variety” must be replaced by “finite-schema axiomatizable quasi-variety” in the second clause, cf. e.g.

Monk [M69], or Nemeti [N91].



6. EQUIVALENCE THEOREMS 87

The following theorem, due to J. Madarasz, is a characterization of existence

of weakly complete and weakly sound Hilbert-style inference systems. Such an

inference-system is sometimes called a Gabbay-style inference system. Its rules

are not necessarily valid, but the formula-schemes they derive (from the empty set¡

â ç of premises) should be valid. 3

THEOREM 6.5 Assume that ®

is an algebraizable general logic and æ ç ê

®

í is

finite. Assume that R Alg ê

®

í á

â

ê A U í

â

{ ê A U í A

â

U . Then

there is a weakly complete and weakly sound Hilbert-style inference system for ®

iff there is a finitely axiomatizable quasi-varietyõ

such that Rõ

â

R Alg ê

®

í

4.

ã

DEFINITION 6.6 (compactness)

U

ä is satisfiability compact iff a set W has a model whenever all of its finite

subsets¡

have models, i.e.

ê

¡

ï @W

í Å Æ Ç ê

¡

í æ

â ç â

Å Æ Ç ê W í æ

â ç

ò

U

ä is consequence compact if a consequence of W is always a consequence of

a finite subset ¡

already, i.e. iff

Wá

¢

â

ð

â

ê Y

¡

ï @W

í

¡

á

¢

â

ð ò

A general logic®

â

ø ä-

è is a set ù is (satisfiability, consequence) compact

if ä -

is such for allè

.ã

We note that in general, consequence compactness and satisfiability compact-

ness are independent properties, i.e. neither of them implies the other. However,

for algebraizable general logics, consequence compactness implies satisfiability3Cf. e.g. [Mi95], [MV], [Si91]. We note that many Gabbay-style rules are even more “liberal” than

not being strongly sound in that in addition their form does not satisfy Def. 4.4 (ii). (It is not known

yet which of these two liberties is responsible for their behaviour. Our feeling is that non-strongly

soundness is the more essential.) These extremely liberal (Gabbay-style) inference systems correspond

to classes `

F Alg ± such that ` is finitely axiomatized by a

b-formulas (of a certain form) and

de

`

d

Alg ± . An example for such ` is the class of rectangularly dense cylindric algebras,

the representation theorem of which ([AGMNS]) was used in [Mi95] for obtaining a weakly-sound

completeness theorem for ± f (which will be defined in section 7). Cf. also the open problem below

[Mi95, Thm.1.3.11, p.27].4Here, h

i h Å i r t and for a set u of formulas, v

u x

u denotes

v Å

xÅ r t u . We do not know whether the condition “

d

Alg ” is needed for this

theorem. It is not needed for direction “ ”. In the other direction, we can always obtain a weakly

complete and weakly sound “ ”, but this may not be completely Hilbert-style (this is more into

the “Gabbay-Venema-Simon-Mikulas” direction).



88 ALGEBRAIC LOGIC

compactness. Also, if we have some kind of negation, then the two notions of

compactness coincide.5 In all our examples in section 7, the two notions of com-

pactness coincide.

Our next theorem characterizes consequence compactness of algebraizable gen-

eral logics. For a proof see [AKNS, Thm.3.2.20], or [ANSK, Cor.3.10].

THEOREM 6.7 (characterization of compactness) Assume that ®

is an algebraiz-

able general logic. Then®

is consequence compact iff Alg ê

®

í is closed under

taking ultraproducts, i.e. iff Alg ê

®

í

â Alg ê

®

í . ã

Now we turn to characterization of some definability properties. Beth’s defin-

ability properties of logics were defined e.g. in Barwise-Feferman [BF]. Here we

give the definitions in the framework of the present paper.

DEFINITION 6.8 (implicit definition, explicit definition) Let ø ä - è is a set ù be

a general logic. Let è ³

be sets withé - æ

â ç , and let B

def â

³ è.

U

A set ¡

ï é ´ of formulas defines B implicitly in ³ iff a è -model can be

extended to a³

-model of ¡

at most one way, i.e. iff

ê £ ë ô Å Æ Ç

´

ê

¡

í í

r ç s

´

y

¹ é -

â

r ç s

´

ª

¹ é - r ç s

´

y

â

r ç s

´

ª

ò

U

¡

defines B implicitly in ³ in the strong sense iff, in addition, any è -model

that in principle can, indeed can be extended to a³

-model of ¡

, i.e. iff

¡

definesB

implicitly in³

and

ê £ ô Å Æ Ç

-

ê

´

Å Æ Ç

´

¡

é - í í ê Y ô Å Æ Ç

´

ê

¡

í í

r ç s

´

ª

¹ é-

â

r ç s-

y

.

U

¡

definesB

explicitly in³

iff any element of B

has an “explicit definition”

that works in all models of ¡

, i.e. iff

ê n ô B í ê Y ð ô é - í ê £ ô Å Æ Ç

´

ê

¡

í í r ç s

´

y

ê n í

â

r ç s

´

y

ê ð í ò

5More on this can be found in [AKNS],[ANSK].




U

¡

definesB

local-explicitly in³

iff the above definition can vary from model

to model, i.e. iff

ê £ ô Å Æ Ç

´

ê

¡

í í ê n ô B í ê Y ð

ô é-

í r ç s

´

y

ê n í

â

r ç s

´

y

ê ð

í ò

ã

DEFINITION 6.9 (Beth definability properties) Let ®

be a general logic.

U

®

has the (strong) Beth definability property iff for allè ë ³ ë B

and ¡

as in

Def. 6.8,

(¡

definesB

implicitly in³

â

¡

definesB

explicitly in³

).

U

®

has the local Beth definability property iff for allè ë ³ ë B

and ¡

as in

Def. 6.8,

(¡

definesB

implicitly in³

â

¡

definesB

local-explicitly in³

).

U

®

has the weak Beth definability property iff for allè ë ³ ë B

and ¡

as in

Def. 6.8,

(¡

definesB

implicitly in³

in the strong sense â

¡

definesB

explicitly

in³

).

ã

DEFINITION 6.10 (patchwork property of models) Let ®

be a general logic.®

has the patchwork property of models iff

for all sets è ë ³ , and models £ ô q- , ô q

´ ,

é- ´

æ

â ç and r ç s

-

y

¹ ê è ³ í

â

r ç s

´

ª

¹ ê è ³ í

â

ê Y ô q - j ´ í ê r ç s

- j ´

k

¹ é -

â

r ç s -

y

and r ç s

- j ´

k

¹ é ´

â

r ç s

´

ª

).

ã



90 ALGEBRAIC LOGIC

Recall that if õ

is a class of algebras, then by a morphism of õ

we understand

a triple ø ô ë Ì ë ö ù , where ô ë ö ô õ and Ì ô ö is a homomorphism. A

morphismø ô ë Ì ë ö ù

is an epimorphism of õ

iff for every lô õ

and every pair

Î ë ö ö l of homomorphisms we have Î

µ

Ì

â

ö

µ

Ì implies Î

â

ö . Typical

examples of epimorphisms are the surjections. But for certain choices of õ

there

are epimorphisms of õ which are not surjective. Such is the case, e.g., when õ is

the class of distributive lattices.

Letõ

0ï õ

be two classes of algebras. Letø ô ë Ì ë ö ù

be a morphism of õ

.Ì

is said to beõ

0 -extensible iff for every algebra lô õ

0 and every homomorphism

Î ô lthere exists some

ô õ0 and

s ö such that l

ï and

s

µ

Ì

â

Î . It is important to emphasize here that l is a concrete subalgebra of

and not only is embeddable into .

THEOREM 6.11 (characterization of Beth properties) 6 Let ®

be an algebraizable

general logic which has the patchwork property of models.

(i)®

has the Beth definability property iff all the epimorphisms of Alg ê

®

í are

surjective.

(ii)®

has the local Beth definability property iff all the epimorphisms of Alg

ê

®

í

are surjective.

(iii)®

has the weak Beth definability property iff every Alg

ê

®

í-extensible epi-

morphism of Alg

ê

®

íis surjective.

ã

In the formulation of Thm. 6.11 (ii),(iii) above, it was important that Alg

ê

®

í

is not an abstract class in the sense that it is not closed under isomorphisms, since

the definition of õ

-extensibility strongly differentiates isomorphic algebras.

If õ

is a class of algebras, then max õdenotes the class of all

ï-maximal ele-

ments of õ :

max õ

def â

ô ô õ ê ö ô õ í ê ô ï ö

â

ô

â

ö í " ò

We note that e.g. max o

ÿ is the class of all fullæ $

ÿ ’s.

6The proof of (i) is in Nemeti [N82] and in Hoogland [H96]. A less general version of (i) is proved

in [HMTII, Thm.5.6.10]. Part (ii) is due to J. Madarasz. An early version of (iii) is in Sain [S90],

and the full version is proved in [H96]. The finite Beth property is obtained from the Beth property

by restricting to be finite. The emphasis in [N82] and [HMTII] was on the finite Beth property. E.

Hoogland and J. Madarasz [HoM] extended the characterization of Thm.6.11(i) to the broader (than

algebraizable) class of equivalential logics.




We will use the notions of “reflective subcategory”, and “limits of diagrams of

algebras” as in Mac Lane [MacL]. We will not recall these. Throughout, by a

reflective subcategory we understand a full and isomorphism closed one.

The weak Beth property was introduced in Friedman [Fr] (cf. references of

[BF]) and has been investigated since then, cf. e.g. [BF] pp.73–76, 689–716.

THEOREM 6.12 (characterization of weak Beth property) 7 Let ®

be an algebraiz-

able general logic which has the patchwork property of models. Assume that every

element of Alg

ê

®

ícan be extended to a maximal element of Alg

ê

®

í, i.e. that

Alg

ê

®

í ï ûmaxAlg

ê

®

í. Then conditions (i)–(iii) below are equivalent.

(i)®

has the weak Beth definability property.

(ii) Alg ê

®

í is the smallest full reflective subcategory of Alg ê

®

í containing

maxAlg

ê

®

í.

(iii) maxAlg

ê

®

í generates Alg ê

®

í by taking limits of diagrams of algebras.

I.e. there is no limit-closed proper subclass separating these two classes of algebras.

ã

Now we turn to characterizing Craig’s interpolation properties.

DEFINITION 6.13 (interpolation porperties) Let ä

â


â

ùbe a logic

with connectives. For each formulað ô é

let È õ ê ð í

denote the set of atomic

formulas occurring in ð . Let be a binary connective of ä .

U

ä

has theá

â

-interpolation property iff

for allð ë ô é

such that ð á

â

there is

ô ésuch that

È õ ê ð í ï È õ ê ð í È õ ê í

and ð á

â

á

â

.

U

ähas the

-interpolation property iff

for all ð ë ô é such that á

â

ð there is ô é such that

È õ ê í ï È õ ê ð í È õ ê í

and á

â

ð and

á

â

.

ã

7This is due to I. Sain, J. Madarasz, and I. Nemeti. For the origins of this characterization of weak

Beth property see [S90, p.223 and on].



92 ALGEBRAIC LOGIC

Next we recall from the literature the amalgamation and superamalgamation

properties of classes of algebras. Let õ be a class of algebras. We say that õ has

the amalgamation property iff any two algebras inõ

can be jointly embedded into

a third element of õ such that we can require some parts of them to go to the same

place. I.e. for anyô ë ö ë l ô ß õ

withö ô ï l

, there are ô õ

and injective

homomorphisms (embeddings) Î ö ¤ , Ì l ¤ such that Î ¹ 9

â

Ì ¹ 9 .

By a partially ordered algebra we mean a structure ø ô ë ù where ô is an

algebra and

is a partial ordering on the universe9

of ô

. A classõ

of par-

tially ordered algebras has the superamalgamation property iff any two algebras

as above can be embedded in a third one such that only the necessary coinci-

dences and ordering would hold, i.e. if for any ô ¥ ô õ , § | and for any em-

beddings§ ñ ô

0 ô ñ

and§ å ô

0 ô å

there exist anô ô õ

and embed-

dings r ñ ô ñ ô and r å ô å ô such that r ñ

µ

§ ñ

â

r å

µ

§ å and for

V ë ö "

â

R ë | ",

ê A ô 9 í ê U ô 9 ó í

r ê A í r ó ê U í

â

ê Y ~ ô 90

í ê A § ê ~ íand

§ ó ê ~ í U í ò

DEFINITION 6.14 Let ä

be a compositional logic. We say that ä

has a deduction

theorem iff there is a binary derived connective such that for all¡

ï é , ð ë ô

é

¡ P

ð " á

¢

â

iff

¡

á

¢

â

ð ò

Such a is called a deduction term.ã

THEOREM 6.15 (characterization of interpolation properties) 8

Let ä

be an algebraizable semantic logic.

(i) Assume that ä

is consequence compact and usual conjunction & isinæ ç ê ä í

.

Assume that ä has a deduction theorem. Then ä has the á

â -interpolation

property iff Alg ê ä íhas the amalgamation property.

(ii) Assume that Alg ê ä íconsists of normal

ã ä

¡

$. Assume that Alg ê ä í

is al-

gebraized via the usual Boolean biconditional l , i.e. in the filter-property

is l . Let denote the usual Boolean implication term. Then ä has

the -interpolation property iff Rû ý Alg ê ä í has the superamalgamation

property.

8The proof of (i) is in Czelakowski [Cz, Thm.3]. The proof of (ii) is in Madarasz [M96], [M97].




The above is only a sample of the equivalence theorems in algebraic logic.

Other kinds of investigations are connecting deduction property of a logic ä with

Alg ê ä íhaving equationally definable principal congruences (EDPC) [BP89a]-

[BP89c], [BP97]; theorems connecting e.g. atomicity of the formula-algebra of ä

with Godel’s incompleteness property of ä

([N85], [N86]), theorems connecting

logical meanings to neat-reducts of formula-algebras ([A], [SA]), etc.



CHAPTER 7

EXAMPLES AND APPLICATIONS

In this section we give some applications for the previous theorems. Most of

the logics we use are well-known, but we recall their definitions for illustrating

how they are special cases of the concept defined herein, and also for fixing our

notation. More and also different kinds of examples are given in [AKNS], [NA],

[ANSK].

1. Classical sentential logic®

â ø ä -

â

èis a set

ù.

Below, we often will omit the index .

The set of logical connectives isæ ç

&ë "

, & is binary, is unary. Letè

be any set. Thus the set of formulas of ä -

â

isé -

â

é ê è ë æ ç í

.

A model of sentential logic ä -

â

is a function assigning 0 (false) or 1 (true) to

each atomic proposition ô è

. Thus the classq -

â

of models of ä -

â

is- |

, the set

of all functions mapping è to | . (Recall that |

ë R " .)

We can extend any model£ è |

to the seté -

â

of all formulas: for all

ð ë ô é -

â

we let

£ ê ð í

R if £ ê ð í

if £ ê ð í R ë

£ ê ð & í

Rif

£ ê ð í £ ê í R

otherwiseò

Now, the meaning of ð

in£

is£ ê ð í

, i.e.r ç s - ê ð ë £ í £ ê ð í

, andð

is

valid in£

,£

âð

, iff £ ê ð í R

.

We letä -

â

def

ø é -

â

ë q -

â

ë r ç s - ë â ùand

®

â

def

ø ä -

â

èis a set

ù.

By this, we have defined®

â . We are going to show that®

â is an algebraizable

general logic with Alg ê

®

âí ã ä

.

That®

â is compositional comes immediately from the definition. Let 2 denote

the 2-element Boolean algebra with universe 2. Then£ ê ë ê £ í 2 for all

£ ô

95



96 ALGEBRAIC LOGIC

q -,

è æ

. Thus Alg

ê

®

â í

2 ", and it can be seen from the definition that

any homomorphism Ì ú-

2 is a meaning-function of some model (namely,

that of Ì ¹ è

), thusq ç s - î È r ê ú - ë

2 " í. Thus by Proposition 5.2,

®

â

has the semantical substitution property, so®

â is structural. Then Alg ê

®

â í

û ý

2 " ã äby Theorem 5.3 (and also Alg ê ä -

â

í

Ê

ã ä’s of cardinality

r A ê ë è í ”). It can be seen that®

â has the filter-property with r ç R ,

0

ê ð ë í

def

ê ð l í

def

ê ð & í & ê ð & í,

0

ê ð í

def

ðand

{0

ê ð í

def

m B

def

ê & ífor a fixed

ô è.

Thus®

â is an algebraizable general logic with Alg ê

®

âí ã ä

, Alg

ê

®

âí

2 " .

By Theorem 6.7,®

â is compact, because ã ä

ã ä . By Theorem 6.2,®

â

has a strongly complete and sound Hilbert-style inference system, because ã ä is a

finitely axiomatizable quasi-variety. Moreover, the proof of Thm. 6.2 (in [AKNS])

constructs such a Hilbert-style inference system (given any axiomatization of ã ä

).

We give here the inference system we get from the proof.

In the next inference system, we will useð ë ë

and{

as formula-variables,

and we will use l and m B as derived connectives.

The axioms are:

ð & l & ð ,

ð & ê & í l ê ð & í & ,

ð l ê ê ð & í & ê ð & í í.

The rules are as follows:

ð l ð

ë

ð l

l ð

ë

ð l ë l

ð l

ë

ð l

ð l

ë

ð l ë l {

ê ð & í l ê & { í

ë

ð l m B

ð

ë

ð

ð l m B

ò

It is easy to check that®

â has the patchwork property.®

â has the Beth prop-

erty by Theorem 6.11, because epimorphisms are surjective inã ä

. Thus®

â has



7. EXAMPLES AND APPLICATIONS 97

the weak Beth property, so by Theorem6.12, 2 generatesã ä

by limits, since

maxAlg

ê

®

â í Alg

ê

®

â í

2 " .

A deduction term for®

â is¤

0 ¤ ñ

def

ê ¤0 &

¤ ñ í. Since

ã ähas su-

peramalgamation, Theorem 6.15 implies that®

â has the interpolation properties.

The validity problem of ®

â is decidable, the set of admissible rules of ®

â isdecidable, the set of valid formula-schemes of

®

âis decidable by Theorem 5.4

and Corollary 5.5, because the quasi-equational theory of ã ä is decidable.

2. Sentential logic in a slightly different form,®

â

¡

.

The set of connectives, thus the set of formulas are just like in the previous case.

The models are different. Letè

be a set.

q

¡

â

def

ø ë ù is a non-empty set and

è 8 ê í " ò

Thus a model£ ø ë ù

is a non-empty set together with an assignment assign-

ing a subset of to each ô è . Let £ ø ë ù be a model. We call the

set of possible situations (or states, or worlds) of £

. For any formulað

, we define

£ ë ð , which we read as “ ð is true in £ at ”, as follows:

£ ë iff

ô ê í, for

ô è.

£ ë ðiff

£ ë æ

ð,

£ ë ð & iff ( £ ë ð and £ ë ).

We say thatð

is valid in £ if £ë ð

for all ô

.

The above amounts to saying that the meaning-function r ç s y is the homo-

morphic extension of into the algebraø 8 ê í ë ë ù

, i.e.r ç s y ú -

ø 8 ê í ë ë ù , and

£ ë ðiff

ô r ç s y ê ð í,

£ ðiff

r ç s y ê ð í.

Now,®

â

¡

is defined. It is compositional, Alg

ê

®

â

¡

í ¯ ° ã ä

def

û

ø 8 ê í ë ë ù is a non-empty set

" “the class of all non-trivial set Boolean

algebras”, andq ç s - ê

®

â

¡

í î È r ê ú - ë ¯ ° ã ä í.



98 ALGEBRAIC LOGIC

Thus®

â

¡

is algebraizable, Alg ê

®

â

¡

í ã ä, Alg

ê

®

¡

â

í ¯ ° ã ä. By Theo-

rem 5.4, it has the same semantical consequence relation

¢

, same admissible

rules, same valid formula-schemes and same valid formulas as®

â .

3. Modal logic S5.

The set of connectives is

&ë ë ³ " , & binary,

ë ³ unary. The class of models

is the same as for®

â

¡

. The “meaning of ³ ” is as follows:

£ ë ³ ðiff (

£ ë

¡

ð

for some

¡

ô ).

This is the same as saying that

r ç sy

ê ³ ð í æ µ

0

ê r ç sy

ê ð í í, where for any set ¶

ï

æ

µ

0

ê ¶ í

if ¶

æ

otherwise ò

The rest of the definition of S5 goes the same way as in the case of ®

â

¡

above.

It can be checked that S5 has the filter property with the same terms as sentential

logic®

â .

Thus S5 is an algebraizable general logic with Alg

ê S5 í o ñ

û

ø 8 ê í ë ë ë æ µ

0

ù is a nonempty set " , Alg ê S5 í û ý o ñ

» o ä

ñ.

A deduction term for S5 is ³ ¤ 0 ³ ¤ ñ

.

Therefore S5 is decidable, compact, has a strongly complete and sound Hilbert-

style inference system, and has the Beth and interpolation properties by Theo-

rems 5.4, 6.7, 6.2, 6.11, 6.15, because »o ä ñ is a decidable, finitely axiomatizable

variety having the superamalgamation property, see Thm.2.3(ii) .

4. Arrow logic® ¾ ¿ À

.

The field of Arrow Logics grew out of application areas in Logic, Language

and Computation, and plays an important role there, cf. e.g. van Benthem [vB96],

[vB91a], and the proceedings of the Arrow Logic day at the conference “Logic at

Work” (Dec.1992, Amsterdam). These arrow logics go back to the investigations

in Tarski-Givant [TG]. Tarski defined in 1951 basicallyä

¾¿ À

to give the first

example of an undecidable propositional logic.




The set of connectives of ® ¾ ¿ À

is

&ë ë

µ

ë

ñ

", &

ë

µ

binary, ë

ñ

unary. The

models are as in S5, except that we require that the elements of be all pairs over

some set , i.e.

q

-

¾¿ À def

ø ë ù u for some and è 8 ê í " ò

The definition of £ ë ð is as in the previous case, and we define the meaningsof

µ

and ñ

as

£ ë ø D ë ~ ù ð

µ

iff (

£ ë ø D ë A ù ðand

£ ë ø A ë ~ ù for some

A),

£ ë ø D ë ~ ù ð

ñ

iff £ ë ø ~ ë D ù ð

.

This amounts to saying that the meaning of µ

is relation composition, and the

meaning of ñ

is relation conversion, i.e.

r ç sy

ê ð

µ

í r ç sy

ê ð í

µ

r ç sy

ê í ,

r ç s y ê ð

ñ

í ê r ç s y ê ð í í

ñ

.

Otherwise everything is the same as before, e.g.£ ð

iff r ç s y ê ð í

.

Now,®

¾¿ À

is an algebraizable general logic with Algê

®

¾¿ À

í ã » ä,

Alg

ê

® ¾ ¿ À

í ¯ ° ã » ä.

Thus, by our equivalence theorems in section 6 and by our algebraic theo-

rems onã » ä

in section 1, we obtain that®

¾¿ À

is undecidable, compact, has no

finitely complete and strongly sound Hilbert-style inference system. Since in ã » ä

epimorphisms are not surjective 1,®

¾¿ À

does not have the Beth property.

A deduction term for

® ¾ ¿ À

ism B

µ

¤ 0

µ

m B m B

µ

¤ñ

µ

m B

.Since

ã » ädoes not have the amalgamation property, by Theorem 6.15

®¾

¿ À

does not have the interpolation property.

We can add “equality” to® ¾ ¿ À

, obtaining®

¾¿ À

as follows. We add ÃÇ to the

set of connectives as a zero-ary connective, and we define its meaning as for any

model £ ø u ë ù

r ç s y ê Ã Ç í

ø D ë D ù D ô " ò

1See the methods in Sain [S90].



100 ALGEBRAIC LOGIC

Then®

¾¿ À

is an algebraizable general logic with Alg ê

®

¾¿ À

í » » ä,

Alg

ê

®

¾¿ À

í ¯ ° » » ä . Just as in the previous cases we get properties of ®

¾¿ À

by using the theorems about »» ä

stated in section 1, and the equivalencetheorems

stated in section 6.

5. First-order logic withç

variables, with substituted atomic formulas,

®

ÿ

¡

.

Let ç ô , let ÿ

def

¥ § ¨ ç " , our set of variables. Let Ä be any set (our

relation symbols). The setè

of atomic formulas of the logicä

¡

ÿ

def

ä

¡

ÿ Å

is

è

def

B ê A 0 ë ò ò ò ë A ÿ ñ

í B ô Ä ë A 0 ë ò ò ò ë A ÿ ñ

ô ÿ " ò

The set of connectives is

&ë ë

¥ ë Y

¥ § ë V ¨ ç "

, & binary, ë Y

¥ unary,

and ¥

zero-ary.2 This defines the seté

¡

ÿÅ

of formulas of ä

¡

ÿÅ

. The class of

models is the usual one,

q

Å

ÿ

def

ø q ë B

y

ù

¾

ý

Å

qis a nonempty set and

B

y

ï

ÿ

q

for allB ô Ä " ò

Let £ ø q ë B

y

ù

¾

ý

Å

be a model. Then we define as before: Let ð ô é and

Ì ô

ÿ

q. We call

Ìan evaluation of the variables. Because of the tradition, we

will write £ ð

Ì in place of £ ë Ì ð .

£ B ê ¥ Ç ò ò ò ¥ É Ê

`

í

Ì iff ø Ì ê § 0 í ë ò ò ò ë Ì ê § ÿ ñ í ù ô B

y

,

£ ¥

Ì iff

Ì ¥ Ì ,

£

Y ¥ ð

Ì iff ( £

ð

Ì

¡

for some Ì

¡

ô

ÿ

q such that Ì and Ì

¡

differ at most at§.)

£ ê ð & í

Ì and £ ð

Ì are as before.

£ ÿ ð

iff (£ ð

Ì

for allÌ ô

ÿ

q

).

We define3

r ç sy

ê ð í

def

Ì ô

ÿ

q £ ð

Ì " ò

We defineä

¡

ÿ Å

def

ø é

¡

ÿ Å

ë q

Å

ÿ

ë r ç s ë ÿ ù, and

®

ÿ

¡ def

ø ä

¡

ÿ Å

Äis a set

ù.

Now,®

ÿ

¡

is compositional, and has the filter property. But it is not structural

(and then it is not semantically structural either) as the following example shows.

Let

def

B ê 0 ë ñ

í and 0

def

B ê 0 ë 0 í ò

2Notice that ÍÎ Í Ï

is not an atomic formula but rather a zero-ary logical connective.3In the literature,

j is called the relation defined by

in the model

. Thus

Ñ Ò

is the algebra of

-variable definable relations in

.




Then it can be checked that

ÿ 0l

Y ñ ê ñ

0 & í ë

but

æ

ÿ lY

ñ ê ñ

0 & í ò

Another example is the following. Let

def

B ê 0 ë ñ

í and 0

def

B ê ñ

ë 0 í ò

Then it can be checked that

ÿ Y ñ l

Y ñ ê ñ

0 &Y

0 0í ë

but

æ ÿ Y

ñ 0 lY

ñ ê ñ 0 &

Y 0 0

í ò

The logicä

¡

ÿ

is not structural because the meanings of the atomic formulas are

not independent of each other: as soon as we know the meaning of B ê 0 ò ò ò ÿ ñ í

,

this will determine the meanings of B ê A 0 ò ò ò A ÿ ñ

í where A 0 ë ò ò ò ë A ÿ ñ

are arbi-

trary variables. It would be natural to treat onlyB ê 0 ò ò ò ÿ

ñí

as an atomic for-

mula. Then we would like to obtain the substituted atomic formulas B ê A 0 ò ò ò A ÿ ñ

í

as “complex”, built-up formulas. We will achieve this in two different ways. In the

first case we will use Tarski’s observation that substitution can be expressed with

quantifiers and equality4 and in the second case we will introduce substitutions

explicitly as logical connectives 5.

6. First-order logic with Ó variables, structural version 6 ®

ÿ , for Ó¨ and

for Ó any ordinal.

This is exactly like the previous example, except that we keep as atomic formu-

las onlyB ê

0ë ò ò ò ë

ÿ ñ í ë B ô Ä

. Since the order of the variables is fixed in our

atomic formulas, we will simply writeB

in place of B ê

0ë ò ò ò ë

ÿ ñ í

. The set of

connectives, the class of models, and the meaning function are exactly as before,

the only difference is that now the set of atomic formulas is Ä itself. When Ó is aninfinite ordinal, everything is analogous (then

Bstands for

B ê 0

ò ò ò ¥

ò ò ò í¥ Ô ÿ ).

Notation:®

ÿ ø Õ

Å

ÿ

Ä is a set ù , Õ

Å

ÿ

ø é

Å

ÿ

ë ò ò ò ù .

Then,®

ÿ is compositional, and q Ó s

÷ Ö

É

î È r ê ú

Å

ë o ÿ í , so®

ÿ has the se-

mantic substitution property by Proposition 5.2.®

ÿhas the filter property (with

4C.f. Tarski [T51], [T65].5For more detail see [BP89, Appendix C], and [HMTII, × 4.3].6This is called a full restricted first-order language in [HMTII, × 4.3]. “Restricted” refers to the fact

that we keep atomic formulas only with a fixed sequence of variables, and “full” refers to the fact that

the arity (or rank) of each relation symbol is . This logic is investigated in [BP89, Appendix C], too.

Most of what we say in this example, can be generalized to the infinitary version±

fØ

H of ± f

studied

in finite model theory, see Example 11 herein.



102 ALGEBRAIC LOGIC

¤ 0 ë m B ë ¤ 0 l ¤ ñas

ë { ë ), so

®

ÿis an algebraizable general logic. It is

easy to check that Alg

ê Õ

Å

ÿ

í is the class of Ä -generated o ÿ ’s, so Alg

ê

®

ÿ í

o ÿ

. Thus Alg ê

®

ÿ í » o ä ÿby Thm.5.3(ii), since »

o ä ÿ û ý o ÿ. Using

Alg

ê

®

ÿ í o ÿ and Alg ê

®

í » o ä ÿ , we begin to apply the theorems in sec-

tions 2,6 to®

ÿ .

®

ÿ is compact for all Ó by Thm.6.7, because »o ä ÿ

»o ä ÿ by Thm.2.3.

For finite Ó , a deduction term for®

ÿ is Y

0ò ò ò Y

ÿ ñ ¤

0 Y

0ò ò ò Y

ÿ ñ ¤ ñ

.

THEOREM 7.1 Let ÓÜ |

. There is no weakly complete and strongly sound

Hilbert-style inference system for ®

ÿ . As a contrast, there are strongly complete

and sound Hilbert-style inference systems for ®

å,

®

ñ,

®

0 .

Proof. For ÓÜ |

, this follows from Thm.6.4, because »o ä

ÿ is a nonfinitely ax-

iomatizable variety (by Thms.2.3, 2.4, 2.5). For Ó |

, this follows from Thm.6.2,

because »o ä

ÿ , Ó |

is a finitely axiomatizable quasi-variety (by Thm.2.3).

Soon we will give a strongly sound and complete inference system å Þ

for®

å.

The above negative result can be meaningfully generalized to most known vari-

ants 7 of ®

ÿ ,®

ÿ without equality, and the infinitary version®

ÿß

of

®

ÿ studied

e.g. in finite model theory (e.g. [EF], [O]). See Example 11 herein.

The proof of Thm. 7.1 above is a typical example of applying algebraic logic

to logic. There are analogous theorems (using the same “general methodology”).

An example is provided by the positive results giving completeness theorems for

relativized versions of ®

ÿcf. e.g. [AvBN97] or [N96]. Different kinds of positive

results relevant to Theorem 7.1 above are in [S95], [SGy].

OPEN PROBLEM 7.2 Is there a weakly complete and weakly sound Hilbert-style

inference system for ®

ÿ, Ó

Ü |?

By Thm.6.5, Open Problem 7.2 above is equivalent to Problem 2.11 (i.e. whether

»o ä

ÿ R õ

for some finitely axiomatizable quasi-varietyõ

), because »o ä

ÿ

ê A l U í R A U, where

A l U ê A á U í. Actually, in the present case

a positive answer would imply the existence of a strongly complete and weakly

sound “

” for®

ÿ , because Algê

®

ÿí

is a variety.

7e.g. to±

fã of Example 5.




Next we turn to investigating inference systems suggested by the connections

between »o ä ÿ , o

ä ÿ , and û ä å ÿ o ä

, for r 6 Ó (see Thm.2.14). We will define

two provability relations, ÿ

and ÿ

a

for

®

ÿ. (Of these,

ÿis given by a Hilbert-

style inference system, while ÿ

a

is not.)

In the following we will heavily use thaté

Å

ÿ

ï é

Å

ÿ Þ

(which is so because the

atomic formulas of Õ

Å

ÿ

and Õ

Å

ÿÞ

are identified).

DEFINITION 7.3 (provability relations

ÿ and

ÿ

a

for

®

ÿ ) 8

(i) First we define

ÿ which will be given by the Hilbert-style inference system

ø 9 Aÿ

ë B Dÿ

ù. In the formula-shemes below we will use

ð ë as formula-variables

(instead of ¤

0ë ¤ ñ

), and

¥ ,

are derived connectives:

¥

ð

def

Y ¥

ð

ð

def

ê ð & í ò

Recall that Y ¥

, ¥

are logical connectives for § ë V ¨ Ó

.

9 A ÿ consists of the following formula-shemes of ®

ÿ : For all § ë V ë ö ¨ Ó

, a propositional tautology 9 , i.e. a valid formula-sheme of ®

0

¥

ê ð í ê ¥

ð ¥

í

¥

ð ð

¥

ð

¥ ð

¥

ð ¥

¥

ð

Y ¥ ð ¥

Y ¥ ð

¥ ¥

¥ ê

¥ ó ó í

Y

¥ê

¥ í

¥

ó

ê

¥

í if ö æ § ë V .

¥ ê ð

¥ê

¥ ð í

, if § æ

V

.

8

fç è , in a slightly different form, is defined in [HMTII, p. 157], and also in [N86], [N92], [N95],

[BP89, Appendix C], [CzP]. In [BP89], f is denoted by

° é

É . One could get the definition of f

by mechanically translating the êë f

-axioms, as we did with the ìë

-axioms in Example 1, and then

polishing the so obtained axioms and rules.9To keep í

h f finite for t ï , we replace the infinitely many shemes here with íh ð , where

í h ð á is a strongly complete and sound Hilbert-styleinference system for

± ð. Such systems

are known, cf. e.g. [ABNPS].



104 ALGEBRAIC LOGIC

B D ÿconsists of the rules Modus Ponens

ê q ñ íand Generalization

ê ò í ¥for

§ ¨ Ó , where ê q ñ í and ê ò í ¥ are, respectively:

ð ë ð

and ð

¥ ð

ò

Now,

ÿ

is the derivability relation given byø 9 A

ÿë B D

ÿù

(for

®

ÿ

).(ii) To define ÿ

a

, let Ó r . Let 9 A

ÿ

ï é

Å

consist of the following

formulas:

ê B í B ¥

Bif Ó

§ ¨ r ëand

B ô Ä ò

Now the inference system ÿ

a

ï 8 ê é

Å

ÿ

í u é

Å

ÿ

is defined to be “ ê 9 A

ÿ

í

restricted to®

ÿ”, i.e.

ÿ

a

def

ø

¡

ë ð ù

¡ P

9 A

ÿ

ð ë

¡ P

ð " ï é

Å

ÿ

" ò

Thus, in an

ÿ

a

-proof, in addition to the instances of

9 A , we also can use

B ¥

B , for Ó

§ ¨ r,

B ô Ä. If

ÿ

a

ð, then we say that “

ðis provable

withr

variables” , or “ð

isr

-variable provable”.

It is not hard to check that both ÿ

and ÿ

a

are strongly sound for

®

ÿ, and

ÿ

is given by a Hilbert-style inference system (if Ó is finite).

Therefore, from Thm.7.1 we can conclude that there are infinitely many valid®

ÿ -formulas which are not ÿ -provable, i.e. ÿ is incomplete for®

ÿ (in a rather

strong way). On the other hand, we will see below that

ÿ

a

@ is strongly complete

for®

ÿ .

We are going to prove that the inference system

ÿ is the logical equivalent of

the algebraic axiom system defining the variety oä

ÿ .

Any formulað ô Õ

Å

ÿ

can be identified with a term in the algebraic language of

oä ÿ such that the elements of Ä are considered as (algebraic) variables, assuming

that we identify the operations of oä

ÿ with the connectives of ®

ÿ . Henceð R

is

an equation in the language of oä ÿ

(forð ô Õ

Å

ÿ

). We writeð ô Õ

Å

ÿ

forð ô é

Å

ÿ

.

Also,ð ô

®

ÿ means ( Y set Ä )ð ô Õ

Å

ÿ

.

THEOREM 7.4 Let ð ô Õ

Å

ÿ

, Ó any ordinal, Ä any set. Then (i)–(iii) below hold

for all Ó r .

(i)

ÿð

iff oä

ÿ ð R

.

(ii) ÿ

a

ð iff û ä å ÿ o ä

ð R .

(iii) ÿð

iff »o ä

ÿ ð R

.




Proof. (i)–(iii) are proved in [HMTII] as Cor. 4.3.26, Thm. 4.3.25, and Thm.

4.3.17, respectively. See also 4.3.57, 4.3.59 therein. To check that of [HMTII]

is the same as our ÿ

a

, it suffices to check that in the proof of 4.3.22, only the

axioms of our 9 A ÿ are used.

Now we are ready to state the logical corollaries of Thm.2.14(ii).

COROLLARY 7.5

(i) ÿ

a

ÿÞ

@is strongly complete for

®

ÿ, for all Ó .

(ii) ÿ

a

ÿ Þ

is not even weakly complete for ®

ÿ , if r ¨ .

(iii) Let ð ô

®

ÿ . Then ÿð iff

ÿ

a

ÿ Þ ð for some r ¨ .

(iv) For all Óë r ¨

there are valid Ó -variable formulas which cannot be

proved withr

variables. For each valid Ó -variable formulað

there is an

r ¨ such that ð is r -variable provable.

Proof. (i): Weak completeness of ÿ

a

ÿ Þ@ follows immediately from Thm. 7.4,

Thm. 2.14 ê » o ä ÿ û ä å ÿ o ä ÿ Þ@

í . For Ó¨ , then strong completeness fol-

lows, because®

ÿis compact and has a deduction theorem. For Ó

6 ,

®

ÿis still

compact, but it does not have a deduction theorem. However, ÿ

a

ÿ Þ@

is strongly

complete for®

ÿby [HMTII, 4.3.23(ii)].

(ii)–(iv) follow from Thm. 7.4 and Thm. 2.14. E.g., assume ð ô

®

ÿ . Then

ÿ ð

iff »o ä ÿ ð R

iff (by »o ä ÿ û ä å ÿ o ä ÿ

Þ@

)û ä å ÿ o ä ÿ

Þ¹ @ ð R

iff ÿ

a

ÿÞ

@ ð iff (by the definition of ÿ

a

) ÿ

a

ÿÞ

ð for some r ¨ .

We note that Corollary 7.5 speaks also about usual first-order logic, because an

Ó -variable formula is valid in®

ÿ iff it is valid in usual first-order logic (and every

first order formulað

has a normal formð

¡

which is in®

ÿ , for some Óô

).

To investigate further the provability relations ÿ , ÿ

a

, now we compare their“deductive powers”. Results in cylindric algebra theory yield the following.

THEOREM 7.6 (the deductive powers of ÿ

, ÿ

a

)

(i) For any R ¨ Ó ¨ , ÿ

æ

ê ÿ Þ

ñ ¹

®

ÿí , i.e. there is an Ó -variable formula

ðsuch that

æ ÿ ð but ÿÞ

ñ ð ò

(ii) If Ó6

, then ÿ

a

æ

ÿ

a

Þ ñfor all r ¨ , i.e. there is a

ð ô

®

@ such

that

æ @

a

ðand

@

a

Þ ñ ð ò



106 ALGEBRAIC LOGIC

(iii) å æ

å

aó

å

a

for all

r 6 õ.

(iv) If Ó R or Ó

6 , then for any Ó -variable ð , ÿ ð iff ð , for all

r 6 Ó. If Ó

R, then

ÿ

ÿ

a

for all

r 6 Ó.

Proof. (i) follows from Thm 7.4 and [HMT, 2.6.14], as follows. Let | Ó ¨

. Then there is an equationo

in the language of o

äÿ

such thato

äÿ

æ

o

ando

ä ÿ Þñ

o , by [HMT, 2.6.14]. We may assume that o is of form ð R , and then

Thm. 7.4(i) finishes the proof.

Similarly, (iv) follows from Thm. 7.4 and [HMT, 2.6.8, 2.6.9], and from oä

ÿ

»o

ä ÿ ûä å

ÿ o ä ÿ Þ if Ó R .

(ii) follows from (Thm. 7.4(ii) and) û ä å ÿ o ä ÿ Þ

æ

û ä å ÿ Þ Þ ñ

for all Ó6 ,

r ¨ which is an unpublished result of Don Pigozzi.

(iii) follows from oä å æ

û ä å å o ä

ó

» o ä å û ä å å o ä @

, see [HMT, 2.6.42,

3.2.65]

OPEN PROBLEM 7.7 Is it true that ÿ

a

æ

ÿ

a

Þ ñwhenever õ

Ó

r ¨ ? I.e. is there a õ -variable formula ð for all õ r ¨ such that ð is not

provable with r variables but it is provable with r ÷ R variables?

The above is Problem 2.12 in [HMT]. Results in this direction are proved e.g.

in Maddux [Ma83], [Ma91a], Andreka [A97]. The present status of this problem

is summarized in [A97, Remark 3].

REMARK

ÿ

a

is a (structural) derivability relation for

®

ÿ in the sense of

[BP89], i.e. for anyð ô Õ

Å

ÿ

and $ Ä Õ

Å

ÿ

, if

ÿ

a

ðthen

ÿ

a

$ ê ð í.

This follows from Thm. 7.4(ii). Thm. 7.4(ii) also implies that

ÿ

a

ÿ Þñ

for Ó¨

can be given by some Hilbert-style inference systemø 9 A

¡

ÿ

ë B D

¡

ÿ

ù; while

ÿ

a

ÿÞ

with r 6 | cannot be given with such. The latter is so because û ä å ÿ o ä ÿ Þñ

is a finitely axiomatizable variety while û ä å ÿ o ä ÿ Þ

, r 6 | is not finitely ax-

iomatizable (see [A97, Thm. 2.3]), and then one can use the presently discussed

“methodogy of algebraization”, cf. Thm.6.3, to infer the above information.

ÿ

a

ÿ Þ@

is strongly complete for®

ÿ , but the ÿ

a

ÿ Þ@

-proofs use formulas that

are not in®

ÿ . Different kinds of complete inference systems for®

ÿ , where the

proofs use only®

ÿ -formulas, are in [Si91], [V91], [MV, ú 5.5.1], [Mi95], [Mi96].

A common feature of the latter inference systems is that they are not strongly

sound. (This is natural to expect because by Thm. 7.1 there cannot exist strongly

sound and complete Hilbert-style inference systems for®

ÿ if ÓÜ |

.)




Now we turn to®

å. If Ó

R, then

ÿis strongly complete for

®

ÿby Thm. 7.4,

because then oä ÿ » o ä ÿ .

åis not complete for

®

å(but

å

aó is). We will

show that if we add a rule or axiom expressing

ê í Dom

ê B í R

B Dom

ê B í uRng

ê B í ë

(and the same for Rngê B í

), then we get a strongly complete Hilbert-style inferencesystem for®

å. Namely, consider the following formula-shema and rule for

§ æ

V,

§ ë V ¨ |:

ê 9 íY

0

ð & Y ñ ð & ð Y ¥

ê Y ê 0

ñ & Y ¥

ð í & 0

æ

ñ í

ê B í

Y

¥ð & Y ê

0 ñ & Y

¥ð í

0 ñ

ð l ê Y 0 ð & Y ñ

ð í

ò

THEOREM 7.8 Bothø 9 A å

P

ê 9 í ë B D å ùand

ø 9 A å ë B D å

P

ê B í ùare strongly

complete for ®

å.

Proof. This follows from Thm. 7.4 and Thm. 2.3(iii).

Now we turn to checking what Theorems 6.11 and 6.15 say about definability

and interpolation properties of ®

ÿ.

®

ÿhas the patchwork property of models.

So®

ÿ for Ó6 | does not have the local Beth definability property by Thm.6.11,

because epimorphisms are not surjective in o ÿ

(see [KMPT]+[M97a, T.7.4.(i)],

for | Ó ¨ see [ACN], for Ó

6 see [N88], [S90, Thm. 10]).

It is proved in [KSS] that®

ó does not have the weak Beth property, and we

conjecture that this extends to õ¨ Ó ¨ , while

®

å has the weak Beth property.

It is proved in [KSS] that for Ó¨

,®

ÿ has the weak local Beth property. By

Thm.6.15,®

ÿ does not have the interpolation property for all ÓÜ R , since »

o ä ÿ

does not have the amalgamation property (see [KMPT], this is a result of Comer

[Co69]).Further definability and interpolation results for

®

ÿ ,

ÿ (both Ó¨

and

Ó6

) are in Madarasz [M97b]. That paper is devoted to solving problems from

Pigozzi [P72].

Summing up:

®

0 is equivalent to sentential logic®

â

¡

.

®

ñis equivalent to S5.

®

å: Our characterization theorems in section 6 and the corresponding algebraic

theorems in section 2 give the following properties for®

å:

®

åis decidable, it has a



108 ALGEBRAIC LOGIC

strongly complete and sound Hilbert-style inference system, which can be obtained

from the equational axiomatization of »o ä

å.

®

åhas the finite model property.

The algebraic version of this is stated in [HMTII, 3.2.66]. It does not have the

Beth (definability) property , and it does not have the interpolation property. We

conjecture that®

åhas the weak Beth property.

®

ÿ for õ Ó ¨

: The characterization theorems and the corresponding

algebraic theorems give the following properties of ®

ÿ:

®

ÿis undecidable,

®

ÿ

does not have a strongly sound and complete Hilbert-style inference system. It is

open whether it has a weakly sound and weakly complete Hilbert-style inference

system, cf. Problem 2.11 and Theorem 5.4.®

ÿ has neither the Beth property nor

the interpolation property.

®

@ : This is called “Finitary logic of infinitary relations”. Model theoretic re-

sults (using AL) are in Nemeti [N90].

7. First-order logic with Ó variables with substitutions, with and without

equality,®

ÿ

»

,®

ÿ

»

, ( Ó ).

First we define®

»

ÿ

. The set of connectives is

&ë ë Y ¥ ë

¥ ,

¥ ë ë ¥

§ ë V ¨ Ó "

, & binary, ¥

zero-ary, and the rest unary. Everything is as in

the previous example, we only have to give the meanings of the logical connectives

¥ ë

¥ë

. Let£

be a model, and recall 10 the operations

§ V ë

§ ë V mapping

Ó to Ó . Now

r Ó s y ê

¥ ð í

Ì ô

ÿ

q Ì

µ

§ V ô r Ó s y ê ð í " ,

r Ó s y ê

¥ ë ð í

Ì ô

ÿ

q Ì

µ

§ ë V ô r Ó s y ê ð í ".

By this, we have defined®

»

ÿ . Itis not hard tocheckthat®

»

ÿ is an algebraizablegeneral logic.

The theory of quasi-polyadic algebras û

¦

ä ’s is analogous with that of cylindric

algebras. Exactly as cylindric algebras are the algebraic counterparts of quanti-

fier logics with equality, û

¦

ä ’s are the algebraic counterparts of quantifier logics

without equality, cf. section 3 herein. »û

¦

ä

ÿ

and û

¦

ÿ

denote the classes of rep-

resentable û

¦

ä ’s of dimension Ó and quasi-polyadic set algebras (of dimension

Ó ) respectively as introduced e.g. in [N91] and in section 3 herein. Analogously,

û

¦ü

ä

ÿ

and û

¦

¯

ÿ

denote the same classes but with equality.

Now, Alg

ê

®

»

ÿ

í û

¦

¯

ÿ

, Algê

®

»

ÿ

í » û

¦ü

ä

ÿ

.

10 ý

rþ ÿ

sends r to ÿ and leaves everything else fixed, and ý

r ÿ

interchanges r and ÿ and leaves

everything else fixed.




If we omit equality from the set of connectives, then we get the equality-free

version®

»

ÿ

of the logic. This is also an algebraizable general logic with Alg

ê

®

»

ÿ

í

û

¦

ÿ

, Alg ê

®

»

ÿ

í » û

¦

ä

ÿ

.

We turn to showing how to retrieve substituted atomic formulasB ê

¥ Çò ò ò

¥ É Ê

`

í

in®

ÿ ,®

»

ÿ

. Here we assume Ó¨ .

First we treat the case®

»

ÿ

. Since a finite mapping can always be written as

a product of

§ ë V ’s and

§ V ’s, we obtain that for any sequence

A0

ë ò ò ò ë Aÿ

ñof

variables there is a sequence

§ ñ ë V ñ ë ò ò ò ë

§ % V % of “substitutions” such that for all

models£

and relation symbolsB

,

r Ó sy

ê B ê A 0 ò ò ò A ÿ ñ

í í r Ó sy

ê

¥

`

ë

`

ò ò ò

¥

!

ë

!

B í .

(Here the first meaning-function is taken from Õ

¡

ÿ

, while the second one from

Õ

»

ÿ

.) This shows that in®

»

ÿ

we do have our substituted atomic formulas back as

“complex” formulas. (On the other hand, the expressive power of ®

»

ÿ

is not bigger

than that of ®

¡

ÿ

, because of the following. It can be proved with a simple induction

that the meaning of the formula

¥ë ð

is the same as that of the formula we get

from ð by interchanging ¥ and

in it everywhere (in the connectives Y ¥ also),

and the meaning of the formula

¥ ð

coincides with that of the formula we get

fromð

by replacing ¥ everywhere it it with

. Hereð

is a formula of Õ

¡

ÿ

.)

Now we show how to get substituted atomic formulas back in®

ÿ by using

Tarski’s observation that substitution can be expressed with quantifiers and equal-

ity. By the above, it is enough to express the meaning of the formulas

¥ ð

and

¥ ë ð , for § æ

V . So let £ be a model and Ì an evaluation of the variables in

q. Then it can be checked that

£ ÿ ê

¥ ð l Y ¥ ê ¥

&ð í í

Ì ,

and if ö æ

§ ë V

,£ ÿ ð l Y

óð

, then

£ ÿ ê

¥ ë ð l

¥ ó

ó

¥ ð í

Ì .

Thus to express

¥ë

we need one extra free variable. We can get this e.g. by

treating Õ

¡

ÿÅ

as the following theory of Õ

Å

ÿ Þñ

:

ê Y ÿ B í l B B ô Ä "

and then treat the atomic formulaB ê

0ë ò ò ò ë

ÿ ñ í

of Õ

¡

ÿ

as the atomic formulaB

of Õ ÿ Þñ

. For more on this see [HMTII, ú 4.3], [BP89].



110 ALGEBRAIC LOGIC

8. First-order logic, ranked 11 version,®

£

ÿó ¥

½

¦ §

À

.

The set of connectives is©

Ó

&ë ë Y

¥ë

¥ § ë V ¨ "

, & binary, ë Y

¥

unary, and ¥

zero-ary. (This is the same as that of ®

@ .)

Let Ä be a set (the set of relation-symbols), and let ) Ä be a function

(the rank-function, )ê B í

is the rank of B

). First we define the logic Õ

¦ §

À

.

Our atomic formulas will beB ê 0 ë ò ò ò ë

¾

ñ

ífor

B ô Ä. We do not include

B ê ¥

Ç

ë ò ò ò ë ¥

Ê

`

í into the set of atomic formulas for the same reason as in our

previous examples: because they would immediately make our logic unstructural.

However, these substituted atomic formulas will be present in our logic as (com-

plex) formulas, because they can be expressed by quantifiers and equality (see our

previous remark on this). Since the sequence ê 0 ë ò ò ò ë

¾

ñ

í of variables is de-

termined by ) , we will just writeB

in place of B ê

0ë ò ò ò ë

¾

ñ

í. (This will

be convenient also when we will compare our present logic with®

@ .) Thus the

set of atomic formulas is Ä . Then the formula-algebraú

of Õ

¦ §

À

has universe

é ê Ä ë ©Ó

í.

The models are£ ø q ë B

y

ù

¾

ý

Å

whereB

y

is a )ê B í

-ary relation onq

for allB ô Ä

. I.e.

q

ø q ë B

y

ù

¾

ý

Å

B ï

¾

qfor all

B ô Ä " ò

Validity and the meaning function are practically the same as those of ®

@ , therefore

we only give here the concise algebraic definition: Let£

be a model.

r Ó s y ê B í

Ì ô

@

q Ì ¹ ) ê B í ô B

y

", and

r Ó s y ú

ø ê

@

q í ë © #

¥

ë % #

¥

ù¥

a

Ô@ is a homomorphism.

£ ðiff

r Ó sy

ê ð í

@

q.

Now®

£

ÿó ¥

½

¦ §

À

ø Õ

¦ §

À

)

is a function into ù ò

Let®

®

£

ÿó ¥

½

¦ §

À

. Then®

is compositional, and has the filter-property. Also

we have that

Alg

ê

®

í o ( @ and Alg ê

®

í ) ( @ ò

The second statement is proved in [HMTII, 4.3.28(iii)].

But®

is not substitutional, even Õ

¦ §

À

is not substitutional if )æ

ç . An

example is: LetB

be an Ó -ary relation symbol in ) and letð

denote the formula

0

ñ & ò ò ò & 0

ÿ . Then

B ÿ B

whileæ

ð ÿ ð ò

11These are called ordinary languages in [HMTII, × 4.3].




It is easy to see that®

is compact. Since

) (@

æ

) (@ , this logic shows that

the condition of structurality in Theorem 6.7 is necessary.

We can extend our inference system @ of the non-ranked logic®

@ to get a

complete one for®

£

ÿó ¥

½

¦ §

À

, as follows.

For any rank-function ) Ä , let 9 A

denote the set of the followingformulas:

ê B

¡

í B ¥

B ëif )

ê B í § ¨ ëand

B ô Ä ò

( 9 A

is a straightforward modification of 9 A

ÿ

.) Then

is defined as

def

ø

¡

ë ð ù

¡ P

9 A

@ ð ë

¡ P

ð " ï Õ

¦ §

À

" ò

Now,

provides a complete inference system for the ranked version of first-order

logic®

£ ÿ ó ¥ ½

¦ §

À

. I.e.:

THEOREM 7.9 (Godel’s completeness theorem) For every formula ð of Õ

¦ §

À

we have

ð iff

ð ò

Proof. This is a corollary of Thm. 7.4(ii) and) ( @ ï » o ä @ R û ý o ( @

(Thm’s

2.14(i), 2.4(iii)), as follows. Let5

def

ê ð ë í

ð l ". Then

ú

5 ô ) ( @

by Thm. 7.4(i) and Rngê ) í ï

. Assumeæ

ð. Then

ú

5 æ

ð

R, hence

o ( @ æ

ð R

by) ( @ ï R û ý o ( @

, i.e.æ

ð

.

At this point we should emphasize that there are valid schemes 12& of Õ

¦ §

À

such that although & , we have æ

& . This is so because there is no difference

between the schema languages of Õ

¦ §

À

and®

@, and also the valid schemes of

®

£

ÿó ¥

½

¦ §

À

and®

@ coincide by Cor.5.5, because Eq ê ) ( @ í Eq ê » o ä @ í, and the

-provable and @ -provable schemas coincide 13.

How it is possible that there is an

-unprovable valid formula-scheme&

?

This means that though each instance of & in Õ

¦ §

À

is

-provable (because of

Thm. 7.9), these

-proofs vary form instance to instance. We cannot give a “uni-

form”

–proof for these instances, in spite of there being a uniform “cause” & of

their validity.

12If @ is a formula schema of (cf. Def.4.4), then by -derivability A of @ we understand the

natural extension of Def.4.4 to a mixed language consisting of both -formulas and schemes. I.e. in a

derivation @ D

@ f

of @

,@ Î

is built up from atomic formulasG Ï

of and formula-variablesH

Ï Ç I

(using the connectives T

of

).13This not quite trivial, but can be proved with ê

ë-theoretic methods, e.g. one can use [HMT,

2.5.26].



112 ALGEBRAIC LOGIC

Theorem 2.14(i) statingQ R

@ï » o ä

@ can be used to overcome schema-incom-

pleteness of

. Using this theorem, one can obtain enriched inference systems

Þ

by adding brand new variables ¥ ê § ¨ í

to the language and new ax-

ioms postulating the effects of the fact that ¥ does not occur in the old formulas.

Roughly, these axioms say that

B ¥

B

if )

ê B í § ¨

and¤

¥¤

if § ¨ ë ¤

is a formula-variable.

These inference systems are strongly complete for the formula-schemas of ®

£

ÿó ¥

½

¦ §

À

.

These completeness theorems (based on theQ R

-representation result) are proved

in [AGN77], [HMTII, ú 4.3].

The reason for®

£

ÿó ¥

½

¦ §

À

not being substitutional is that the atomic formulas

cannot take the meanings of any formula, because an atomic formula has a fixed

finite rank, while formulas can have meanings of arbitrarily large finite ranks. This

will be repared in our next example.

9. First-order logic, rank-free

14

(or type-less) version,

®

¦ §

À

.

The set of connectives are as in the previous case. Let Ä be a set (of relation

symbols). Then the set of atomic formulas of Õ

Å

¦ §

À

is Ä , as before.

The models will be different (as the information ) Ä

is missing): We

only know thatB

denotes a finitary relation, we do not know what its arity is. The

actual arity will be given by the model. I.e., the models are£ ø q ë B

y

ù

¾

ý

Åwhere B

y

is an arbitrary finitary relation on q for all B ô Ä ,

q

Å

ø q ë B

y

ù

¾

ý

Å

ê B ô Ä í ê Y Ó ô í B ï

ÿ

q " ò

Validity and the meaning function are the same as in the previous case, the only

difference is that

r Ó s y ê B í

Ì ô

@

q Ì ¹ Ó ô B

y

for some Ó"

.

Let®

¦ §

À

denote the system of these logics. Now this general logic is struc-

tural, since

q Ó s

Å

î È r ê ú

Å

ë o ( @ í ò

Thus®

¦ §

À

is an algebraizable general logic with

Alg

ê

®

¦ §

À

í o ( @ and Alg

ê

®

¦ §

À

í û ý o ( @ ò

14Rank-free first-order logic was introduced in Henkin-Tarski [HT], and elaborated in more detail in

[A73], [AGN77, sec. IV]. See also [HMTII, section 4.3]. A nice proof system for this logic is given in

Simon [Si91].




Thus Theorem 6.7 says that®

¦ §

À

is not compact becauseû ý o (

@æ

û ý o (@

by Thm. 2.4(iii). Or vice versa, one can prove the algebraic theorem û ý o ( @ æ

û ý o (@ by showing that

®

¦ §

À

is not compact, as follows:®

¦ §

À

is not com-

pact because the set¡

ê B l Y ¥ B í § ¨ " of formulas, where B is any

relation symbol, is not satisfiable while all of its finite subsets are. Thus Theo-

rem 6.7 says that û ý o ( @ æ

û ý o ( @ because Alg ê

®

¦ §

À

í û ý o ( @ .

Thm.6.4 admits a generalization to logics like®

¦ §

À

above. Then we obtain the

following corollary of this generalized result and of Thm.2.5 (saying that »o ä ÿ is

not finite schema axiomatizable).

COROLLARY 7.10 Assume that ø 9 A ë B D ù

defines a strongly sound and weakly

complete inference system for ®

¦ §

À

. Then ø 9 A ë B D ù must involve an infinite

set of formula-variables. I.e.®

¦ §

À

is not finite-schema axiomatizable. The same

applies for ®

@ in place of ®

¦ §

À

.

Improved versions of this negative result are in [A97] where it is proved that

ø 9 A ë B D ù has to be extremely complex, too, besides involving infinitely many

formula-variables. Positive results kind of side-stepping Corollary 7.10 above arein [S95], [S97], [SGy]. These present expansions of

®

¦ §

À

with further logical

connectives, such that the new®

Þ

¦ §

À

becomes finite schema axiomatizable.

At this point the reader might have the impression that Corollary 7.10 seems to

contradict Godels completeness theorem. However, Godels theorem holds for the

ranked version®

£

ÿó ¥

½

¦ §

À

of first order logic but not for®

¦ §

À

. The essential differ-

ence between these two logics is that®

£

ÿó ¥

½

¦ §

À

is not structural (substitutional). No

structural version of first order logic is known for which Godel’s completeness the-

orem would hold. More precisely, the only such versions are the logics presented

in [SGy] etc. cited above. The presently discussed issue is highly relevant to the

propositional modal versions of first order logic, cf. e.g. van Benthem [vB97],

[vB96], [vBtM], [V95a], [MV].

Now we briefly compare our three versions of FOL: non-ranked®

@ , the ranked

version®

£

ÿó ¥

½

¦ §

À

, and the rank-free one,®

¦ §

À

. The same formula-schemes are

valid in them, and they have the same admissible rules by Theorem 5.4, because

the same quasi-equations are true in their algebraized forms byû ý

o ( @

»o ä @

Alg ê

®

@ í . Also, this set of admissible formula-schemes is recursively

enumerable, and the validity problem in these logics is not decidable, by Theo-

rem 2.4, Theorem 5.4, Corollary 5.6.

As a contrast, here we will give a logic which has a decidable validity prob-

lem and at the same time the set of valid formula-schemes is not even recursively

enumerable.



114 ALGEBRAIC LOGIC

10. Equality logic, monadic logic.

First we treat equality logic Õ

¥

. This is the same as first-order logic with

variables and with no atomic formulas, i.e. Õ

¥

def

Õ T

@

. Therefore, this is not a

general logic. Notice that the set of formulas is non-empty because ¥

is

a zero-ary logical connective. A model is just a setq

and the meaning-algebra

£ ê ë ê q í of this model is the subalgebra of ø ê

@

q í ë © ¥ ë % ¥

ù ¥

a

Ô@ generated by

%¥

§ ë V ¨ ". These are called the minimal cylindric set algebras, and their

class is denoted by ¯ ° Å U @ , while Å U @

def

ß ¯ ° Å U @ .

Õ

¥

is an algebraizable semantic logic with Alg

ê Õ

¥

í ¯ ° Å U @and Alg ê Õ

¥

í

û ý Å U @. ( Õ

¥

is substitutional because its set of atomic formulas is empty.)

It is well known that the validity problem of Õ

¥

is decidable, it has the finite

model property, and it admits an elimination-of-quantifiers theorem. (See e.g.

[M64a].)

However, the set of valid formula-schemes of Õ

¥

is not even recursively enu-

merable. This is so by Corollary 5.5, because Eq ê Å U @ í is not recursively enu-

merable.15

More generally, consider now ranked first-order logics Õ

@

. Ranked first-order

logic Õ

ÿ

with Ó variables, Ó¨

can be defined analogously for ) Ä Ó

. Let

Ó

. If every relation symbol is unary, i.e. if B

Ós ) ï

R ", then Õ

ÿ

is called

a monadic logic. Let Õ

Å

ÿ

denote monadic logic with relation symbols Ä , i.e.

Õ

Å

ÿ

def

Õ

ÿ

where ) Ä u

R ".

Õ

Å

ÿ

is a compositional logic with filter-property. It is not substitutional.

It is known that the validity problem of monadic logics is also decidable, they

have the finite model property, and they admit elimination of quantifiers. (See also

[M64a].)

Let ÓÜ |

. If Ó is infinite, then the valid schemes of Õ

Å

ÿ

are not recursively

enumerable. If ) is not monadic, then the valid schemes of Õ

ÿ

are recursively

enumerable (and the validities become undecidable). (If Ó |

, then the set of

valid schemes of Õ

ÿ

is decidable.) These are proved in [N87] by showing that

the equational theories of the corresponding classes of algebras are not recursively

enumerable (and using Theorem 5.4). The logical implications and the reasons for

this behaviour are also explained carefully in [N87].

11. Infinitary version®

ÿß

@

of the finite variable fragments®

ÿ .

LetW

be an infinite cardinal.®

ÿX

@

is obtained from®

ÿby adding

W-ary con-

junction to the logical connectives. More formally, let é ÿ be the set of formulas of

Õ

Å

ÿ

ø é ÿ ë ò ò ò ù

. Leté

ÿX

be the smallest set satisfyingê § í

–ê § § § í

below.

15This was proved by M. Rubin, and independently by I. Nemeti, see [N87].




ê § í é

ÿ

X

é ÿ

,

ê § § í é

ÿX

is closed under the connectives of Õ ÿ ,

ê § § § í î ¨ W

ê & î í ô é

ÿX

, for anyî ï é

ÿX

.

The models of Õ

ÿ

Å

X

@

are the same as those of Õ

Å

ÿ

, and r Ó s

X

, é

X

are the obvious

generalizations of the definition given for Õ

Å

ÿ . Then

Õ

ÿ

Å

X

@

def

ø é

ÿ

X

ë q ÿ ë r Ó s

X

ë é

X

ù and®

ÿ

X

@

def

ø Õ

ÿ

Å

X

@

Ä is a set ù ò

®

ÿß

@

is obtained from®

ÿX

@

by removing all conditions of the form “ò ò ò ¨ W

”.

That is,é

ÿ

ß

@

`

é

ÿ

X

@

Wis a cardinal

", etc.

®

ÿ

ß

@

and®

ÿ

X

@

(withW Ü

)

are not logics in the sense of Def .4.1 because they involve infinitely long strings

of symbols. All the same,®

ÿß

@

is an interesting mathematical structure whose

study is motivated by studying logics. Most properties of logics make sense for

the “pseudo-logic”®

ÿ ß

@

, too. Studying mathematical structures like®

ÿ ß

@

,®

ÿ X

@

seems to be useful for obtaining a better understanding of logics (in the sense of

Def. 4.1).

Most of the results obtained for®

ÿ via the methods of algebraic logic can be

pushed through for®

ÿ

ß

@

by the same kinds of algebraic methods. In particular, by

stretching the algebraic methods which lead to Theorem 7.1 , one can obtain the

following. The notions of formula schema, inference system, axiom schema, rule

schema can be generalized to®

ÿ

X

@

the natural way. Herein we do not go into the

details of this.

COROLLARY 7.11 Assume is a strongly sound and weakly complete provabil-

ity relation for ®

ÿß

@

orfor ®

ÿX

@

( W 6 ). Then is not definable by a Hilbert-style

inference system. Moreover, any schemaø 9 A ë B D ù

axiomatizing

must involve

infinitely many formula variables (cf. Def.4.4 for ø 9 A ë B D ù

axiomatizing

.)

The next table summarizes the algebraic counterparts of some of the distin-guished logics.



substitutional,

Logic Õ Alg ê Õ í Alg

ê Õ ícompact

®

»

sentential logicã ä

2" ÷ ÷

S5

modal logic »o ä ñ

o ñ ÷ ÷

®¾

¿ À

arrow logicã » ä

¯ ° ã » ä ÷ ÷

®

¡

ÿ

Ó -var. FOLc d

ü£ c d

ü£

with substituted »

¦ü

ä ÿ

¦

¯

ÿ ÷

atomic fmlas®

ÿ

structural »o ä ÿ

o ÿ ÷ ÷

Ó -var. FOL®

@

finitary FOL of »o ä @ o @ ÷ ÷

-ary rel.’s

®

£ ÿ ó ¥ ½

¦ §

À

ranked FOL ) ( @o

( @ ÷

®

¦ §

À

rank-free FOLû ý o ( @ o (« @ ÷

Table 7.1.



BIBLIOGRAPHY

*

[Ac] Aczel, P., Schematic consequence. In [Ga94, pp. 261–272].

[AHS] Adamek,J. Herrlich,H. and Strecker,G., Abstract and concrete cate-

gories, or the joy of cats. John Wiley and Sons (1990).

[Al] Allen, S., (ed.) Possible Worlds in Humanities, Arts and Sciences (Proc.

of Nobel Symposium 65) W. de Gruyter, Berlin, 1989.

[A] Amer,M., Cylindric algebras of sentences (abstract). J. of Symbolic

Logic 58,2 (1993) 743.

[A73] Andreka,H., Algebraic investigation of first-order logic. (in Hungarian)

Dissertation with E otv¨ os Lor and Univ. Budapest (1973) ix+162 pp.

[A77] Andreka,H., Universal Algebraic Logic. (in Hungarian) Dissertationwith Hung. Acad. Sci. Budapest (1977) 199 pp.

[A94] Andreka,H., Weakly representable but not representable relation alge-

bras. Algebra Universalis 32 (1994) 31–43.

[A97] Andreka,H., Complexity of the equations valid in algebras of relations.

Annals of Pure and Applied Logic, to appear.

[ABNPS] Andreka,H. Blok,W. Nemeti,I. Pigozzi,D. and Sain,I., Abstract Alge-

braic Logic. Chapter for the Handbook of Algebraic Logic, in prepara-

tion.

[AvBN96] Andreka,H. van Benthem,J. and Nemeti,I., Finite model property of

the bounded fragment of first-order logic and cylindric-relativized set

algebras. Manuscript (1996).

[AvBN97] Andreka,H. van Benthem,J. and Nemeti,I., Modal languages and

bounded fragments of predicate logic. Journal of Philosophical Logic,

to appear.

[ACN] Andreka,H. Comer,S.D. Nemeti,I. Epimorphisms in cylindric algebras.

Manuscript, 1983.

[AGN73] Andreka,H. Gergely,T. and Nemeti,I., Purely algebraical construction

of first order logics. Publications of Central Res. Inst. Physics Budapest

KFKI-73-71 (1973) 46pp.

[AGN77] Andreka,H. Gergely,T. and Nemeti,I., On universal algebraic construc-

tion of logics. Studia Logica Vol 36 (1977) 9–47.

[AGMNS] Andreka,H. Givant,S. Mikulas.Sz. Nemeti,I. and Simon,A., Some no-

tions of density that imply representability in algebraic logic. Annals of

Pure and Applied Logic, to appear.

117



118 ALGEBRAIC LOGIC

[AGiN94] Andreka,H. Givant,S. and Nemeti,I., The lattice of varieties of repre-

sentable relation algebras. Journal of Symbolic Logic 59.2 (1994)

631–661.

[AGiN95] Andreka,H. Givant,S. and Nemeti,I., Perfect extensions and derived

algebras. Journal of Symbolic Logic 60,3 (1995) 775–796.

[AGiN97] Andreka,H. Givant,S. and Nemeti,I., Decision problems for equational

theories of relation algebras. Memoirs of Amer. Math. Soc. 126,604

(1997) xiv+126pp.

[AGoN] Andreka,H. Goldblatt,R. and Nemeti,I., Relativized quantification:

some canonical varieties of sequence-set algebras. J. Symbolic Logic,

to appear.

[AHN] Andreka,H. Hodkinson,I. and Nemeti,I., Finite algebras of relations are

representable on finite sets. J. Symbolic Logic, to appear.

[AJN] Andreka,H. Jonsson,B. and Nemeti,I., Free algebras in discriminator

varieties. Algebra Universalis 28 (1991) 401–447 Abstracted in

[BMP], 1–14.

[AKNS] Andreka,H., Kurucz,A., Nemeti,I. and Sain,I. Methodology of apply-

ing algebraic logic to logic. Math. Inst. Budapest, Preprint, 1993.

Shortened version (“Applying Algebraic Logic to Logic”) in: AlgebraicMethodology and Software Technology (AMAST’93), eds. M.Nivat,

C.Rattray, T.Rus and G.Scollo, in Series “Workshops in Computing”,

Springer-Verlag, 1994. pp.7-28. Also: Material for Summer School on

Algebraic Logic and the Methodology of Applying it, Budapest (1994).

http://circle.math-inst.hu/pub/algebraic-logic/meth.dvi

[AKNSS] Andreka,H., Kurucz,A., Nemeti,I. Sain,I. and Simon, A., Causes and

remedies for undecidability in arrow logics and in multi-modal logics.

In: Arrow logic and Multi-Modal Logic, M. Marx, L. Polos, M. Masuch

eds, CSLI Publications and FOLLI, Stanford, 1996. pp. 63–99.

[AMdN] Andreka,H. Maddux,R.D. and Nemeti,I., Splitting in relation algebras.

Proc. Amer. Math. Soc. (1991) 111,4 1085–1093.

[AMik] Andreka,H. and Mikulas,Sz., Lambek calculus and its relational seman-

tics: completeness and incompleteness. Journal of Logic, Language and

Information (JoLLI) 3 (1994) 1–37.

[AMN] Andreka,H. Monk,J.D. and Nemeti,I. (ed. s), Algebraic Logic. (Proc.

Conf. Budapest, 1988) Colloq. Math. Soc. J. Bolyai Vol 54, North-

Holland, Amsterdam (1991) v+746 pp.

[AN75] Andreka,H. and Nemeti,I., A simple purely algebraic proof of the com-

pleteness of some first order logics. Algebra Universalis Vol 5 (1975)

8–15.

[AN90] Andreka,H. and Nemeti,I., Constructing unusual relation set algebras.

Manuscript, Mathematical Institute, Budapest. August (1990).

[AN93] Andreka,H. and Nemeti,I., Axiomatization of identity-free equations

valid in relation algebras. Algebra Universalis 35 (1996) 256–264.



BIBLIOGRAPHY 119

[ANSK] Andreka,H. Nemeti,I. Sain,I. and Kurucz,A., General algebraic logic in-

cluding algebraic model theory: an overview. In: Logic Colloquium’92

(Proc. 1992 Summer Meeting of Assoc. Symb. Log. ), eds. L. Csirmaz,

D.M. Gabbay and M. de Rijke, CSLI Publications, Stanford (1995), 1–

60.

[ANT] Andreka,H. Nemeti,I. and Thompson,R.J., Weak cylindric set algebras

and weak subdirect indecomposability. J. Symbolic Logic Vol 55, No 2June 1990, 577–588.

[AS] Andreka,H. and Sain,I., Connections between initial algebra semantics

of CF languages and algebraic logic. Mathematical Logic in Computer

Science (Proc. Conf. Salgotarjan Hungary 1978) (Ed. s: Domolki,B.

and Gergely,T.) Colloq. Math. Soc. J. Bolyai Vol 26, North-Holland,

Amsterdam (1981) 25–83.

[AT] Andreka,H. and Thompson,R.J., A Stone-type representation theorem

for algebras of relations of higher rank. Trans. Amer. Math. Soc. Vol

309,2 Oct. 1988 671–682.

[AH] Anellis,I.H. and Houser,N., The nineteenth century roots of univer-

sal algebra and algebraic logic: A critical-bibliographical guide for the

contemporary logician. In [AMN] 1–36.

[BF] Barwise,J. and Feferman,S. (ed. s), Model-Theoretic Logics. Springer-

Verlag, Berlin, (1985).

[BU] Bednarek,A.R. and Ulam,S.M., Some remarks on relational composi-

tion in computational theory and practice. Fundamentals of Computa-

tion Theory’77, Lecture Notes in Computer Science Vol 56, Springer-

Verlag, Berlin (1977) 33–38.

[vB89] van Benthem,J., Modal logic and relational algebra. Preprint, Institute

for Language, Logic and Information, University of Amsterdam (1989).

[vB91] van Benthem,J., General dynamics. In: Theoretical Linguistics. (ed.

Ph. A. Luelsdorff) (1991), pp. 151–201.

[vB91a] van Benthem,J., Language in Action. (Categories, Lambdas and Dy-

namic Logic) North–Holland (Studies in Logic . . . Vol. 130) (1991)

x+350 pp.[vB94] van Benthem,J., A note on dynamic arrow logics. In “Logic and Infor-

mation Flow”, (J. van Eijck and A. Visser eds. ) MIT Press, Cambridge,

Mass. (1994), pp.15-29.

[vB96] van Benthem,J., Exploring logical dynamics. Studies in Logic, Lan-

guage and Information, CSLI Publications, Stanford (1996).

[vB97] van Benthem,J., Modal foundations for predicate logics. Logic Journal

of the IGPL, 5,2 (1997), 259–286.

[vBtM] van Benthem,J. and ter Meulen,A. (eds), Handbook of Logic and Lan-

guage. North-Holland, Amsterdam (1997).

[BMP] Bergman,C.H. Maddux,R.D. and Pigozzi,D.L. (eds.), Algebraic logic

and universal algebra in computer science. Lecture Notes in Computer

Science, Springer Verlag Vol 425 (1990) xi+292 p.



120 ALGEBRAIC LOGIC

[Bi92] Biro,B., Non–finite–axiomatizability results in algebraic logic. J. of

Symbolic Logic Vol 57, 3 (1992) 832–843.

[BiSh] Biro,B. and Shelah,S., Isomorphic but not lower-base-isomorphic cylin-

dric set algebras. J. of Symbolic Logic 53,3 (Sept. 1988) 846–853.

[Bl] Blok,W.J., The lattice of modal logics: An algebraic investigation. J.

Symbolic Logic Vol 45 (1980) 221–238.

[BP86] Blok,W.J. and Pigozzi,D., Protoalgebraizable logics. Studia Logica 45(1986) 337–369.

[BP89] Blok,W.J. and Pigozzi,D., Algebraizable logics. Memoirs Amer. Math.

Soc. Vol 77,396 (1989) vi+78 pp.

[BP89a] Blok,W.J. and Pigozzi,D., On the structure of varieties with equation-

ally definable principal congruences, Part III. Preprint (1989).

[BP89b] Blok,W.J. and Pigozzi,D., On the structure of varieties with equation-

ally definable principal congruences, Part IV. Preprint (1989).

[BP89c] Blok,W.J. and Pigozzi,D., The deduction theorem in algebraic logic.

Preprint, (1989).

[BP91] Blok,W.J. and Pigozzi,D., Local Deduction Theorems in Algebraic

Logic. In [AMN], 75–109.

[BP91a] Blok,W.J. and Pigozzi,D., Introduction to Studia Logica Vol 50, No 3/4

(1991).

[BP97] Blok,W.J. and Pigozzi,D., Abstract Algebraic Logic and the Deduction

Theorem. Submitted.

[BB] Bloom,S.L. and Brown,D.J., Classical abstract logics. Dissertationes

Mathematicae (Rozprawy Matematyczne) Vol CII (1973) 43–56.

[BrS] Brown,D.J. and Suszko,R., Abstract logics. Dissertationes Mathemati-

cae (Rozprawy Matematyczne) Vol CII (1973) 9–41.

[BS] Burris,S. and Sankappanavar,H.P., A course in universal algebra. Grad-

uate Texts in Mathematics, Springer-Verlag, New York (1981).

[C65] Cohn,P.M., Universal Algebra. Harper and Row, New York (1965).

[Co69] Comer,S.D., Classes without the amalgamation property. Pacific J.

Math. 28 (1969), 309–318.

[Co72] Comer,S.D., A sheaf-theoretic duality for cylindric algebras. Trans. Amer. Math. Soc. Vol 169 (1972) 75–87.

[Cr65] Craig,W., Boolean notions extended to higher dimensions. In: The the-

ory of models (eds: J.W.Addison, L.Henkin, A.Tarski), North–Holland

(1965) 55–69.

[Cr74] Craig,W., Logic in algebraic form. North-Holland, Amsterdam (1974)

viii+204pp.

[Cr74a] Craig,W., Unification and abstraction in algebraic logic. Studies in

algebraic logic, Math. Assoc. Amer. (1974) 6–57.

[CM] Crvenkovic,S. and Madarasz,R., Relation Algebras. (In Serbian),

Beograd, 1992.

[Cz] Czelakowski,J., Logical matrices and the amalgamation property. Stu-

dia Logica 51,4 (1982) 329–342.



BIBLIOGRAPHY 121

[Cz93] Czelakowski,J., Logic, algebras, and consequence operators. Preprint,

(1992). 79pp.

[Cz97] Czelakowski,J., Protoalgebraic Logic. Kluwer, 1997 (to appear).

[CzP] Czelakowski,J. and Pigozzi,D., Amalgamation and interpolation in ab-

stract algebraic logic. Preprint, Iowa State University, Ames, (1995).

[D69] Daigneault,A., Lawvere’s elementary theories and polyadic and cylin-

dric algebras. Fund. Math. Vol 66 (1969) 307–328.[DM63] Daigneault.A. and Monk,J.D., Representation theory for polyadic alge-

bras. Fund. Math. Vol 52 (1963) 151–176.

[D1864] De Morgan,A., On the syllogism, no. IV, and on the logic of relations.

Transactions of the Cambridge Philosophical Society Vol 10 (1964)

331–358.

[Dip] Dipert,R., Peirce, Frege, the logic of relations, and Church’s theorem.

History and Philosophy of Logic Vol 5 (1984) 49–66.

[Dis91] Diskin,Z.B., Polyadic algebras for non–classical logics. Parts I–IV.

Latv. Mat. Ezhegodnik Vols 35, 36 (1991).

[Dis96] Diskin,Z.B., Abstract Universal Algebraic Logic. Part I: A unified

framework of structural hyperlogics for integrating the deductive and

model-theoretical approaches. Part II: Algebraizable logics and alge-

braic semantics (Galois connections, compactness, constructivisability)

In: Proceedings of the Latvian Academy of Sciences Vol 50, No 1

(1996), pp.13–21, 22–30.

[Dis97] Diskin,Z.B., When is sematntically defined logic is algebraizable? Acta

Universitatis Latviensis 595 (1997), 57–82.

[Du] Duntsch,I., A note on cylindric lattices. In: Algebraic Methods in Logic

and in Computer Science, ed. C. Rauszer, Banach Center Publications

Vol 28, Polish Academy of Sciences, Warszawa, (1993). pp.231–238.

[EF] Ebbinghaus,H-D. and Flum,J., Finite Model Theory. Springer Verlag,

Berlin, 1995.

[E] Epstein,R.L., The semantic foundations of logic. Vol 1: Propositional

logics. Kluwer, Dordrecht, 1990.

[F91] Ferenczi,M., Measures defined on free products of formula algebrasand analogies with cylindric homomorphisms. In [AMN] 173–181.

[FJ94] Font,J.M. and Jansana,R., On the sentential logics associated with

strongly nice and semi-nice logics. Bulletin of the IGPL 2 (1994)

55–76.

[FJ97] Font,J.M. and Jansana,R., A general algebraic semantics for sentential

logics. Lecture Notes in Logic Vol. 7, Springer, Berlin, 1997.

[Fr] Friedman,H.M., Beth’s theorem in cardinality logics. Israel Journal of

Mathematics 14 (1973) 205–212.

[Ga94] Gabbay,D.M. (ed), What is a logical system? Oxford Science Publica-

tions, Clarendron Press, Oxford, 1994.

[Gab] Gabbay,D.M., Labelled deductive systems, Part I. Oxford University

Press, (1996).



122 ALGEBRAIC LOGIC

[Gab2] Gabbay,D.M., Self fibring in predicate logics. Submitted.

[Gab3] Gabbay, D.M., Fibred semantics and the weaving of logics. Part I:

modal and intuitionistic logics. The Journal of Symbolic Logic 61,

4 (1996), 1057–1120.

[Gi91] Givant,S., Tarski’s development of logic and mathematics based on the

calculus of relations. In [AMN] 189–215.

[Gi94] Givant,S., The structure of relation algebras generated by relativiza-

tions. Contemporary Mathematics Vol 156, American Mathematical

Society,Providence, (1994). xv+134pp.

[Gi97] Givant, S, Universal classes of simple relation algebras. Journal of

Symbolic Logic, to appear.

[G90] Goldblatt,R.I., Varieties of complex algebras. Annals of Pure and Appl.

Logic Vol 44 (1990) 173–242.

[G91] Goldblatt,R.I., On closure under canonical embedding algebras. In

[AMN] 217–229.

[G95] Goldblatt,R.I., Elementary generation and canonicity for varieties of

Boolean algebras with operators. Algebra Universalis 34 (1995) 551–

607.

[Gr] Gratzer,G., Universal Algebra. Second edition. Springer Verlag, New

York, (1979).

[Ha54] Halmos,P.R., Polyadic Boolean algebras. Proc. Nat. Acad. Sci. U.S. A.

Vol 40 (1954) 296–301.

[Ha60] Halmos,P.R., Polyadic algebras. Summaries of talks presented at the

Summer Institute for Symbolic Logic, Cornell University, (1957), sec-

ond edition, Communications Research Division, Institute for Defense

Analyses (1960) 252–255.

[Ha62] Halmos,P.R., Algebraic logic. Chelsea Publ. Co. , New York (1962)

271pp.

[H55] Henkin,L., The representation theorem for cylindric algebras. In: Math-

ematical interpretation of formal systems, North-Holland, Amsterdam,

1955, pp.85-97.[He70] Henkin,L., Extending Boolean operations. Pac. J. Math. Vol 22 (1970)

723–752.

[HM74] Henkin,L. and Monk,J.D., Cylindric algebras and related structures.

Proc. Tarski Symp. , Amer. Math. Soc. Vol 25 (1974) 105–121.

[HMT] Henkin,L. Monk,J.D. and Tarski,A., Cylindric Algebras. Parts I,II

North-Holland, Amsterdam 1971 and 1985.

[HMTII] Henkin,L. Monk,J.D. and Tarski,A., Cylindric Algebras. Part II North-

Holland, Amsterdam (1985).

[HMTAN] Henkin,L. Monk,J.D. Tarski,A. Andreka,H. and Nemeti,I., Cylindric

Set Algebras. Lecture Notes in Mathematics Vol 883, Springer-Verlag,

Berlin (1981) vi+323pp.



BIBLIOGRAPHY 123

[HT] Henkin,L. and Tarski,A., Cylindric algebras. In: Lattice theory, Pro-

ceedings of symposia in pure mathematics, vol.2, ed. R. P. Dilworth,

American Mathematical Society, Providence, (1961), pp.83–113.

[HH] Hirsch,R.D. and Hodkinson,I., Step by step — building representations

in algebraic logic. J. Symbolic Logic, to appear.

[HH97] Hirsch,R.D. and Hodkinson,I., Relation algebras with Ó -dimensional

bases. Preprint, London, May 1997. Imperial College and UniversityCollege.

[H97] Hodkinson,I., Atom structures of cylindric algebras and relation alge-

bras. Annals of Pure and Applied Logic, to appear.

[H96] Hoogland,E., Algebraic characterizations of two Beth definability prop-

erties. Master Thesis, University of Amsterdam, (1996).

[Ho97] Hoogland, E., Beth definability in almost algebraizable logics. In prepa-

ration.

[HoM] Hoogland, E., and Madarasz, J., Definability properties beyond alge-

braizable logics. In preparation.

[Ja] Janssen, T.M.V., Compositionality. In [vBtM, pp. 417–474].

[JKE] Janossy,A. Kurucz,A. and Eiben,A.E., Combining algebraizable logics.

Notre Dame J. of Formal Logic 37,2 (1996) 366–380.

[Ji92] Jipsen, P., Computer aided investigations of relation algebras. Ph.D.

Dissertation, Vanderbilt University, Tennessee, 1992.

[J69] Johnson,J.S., Nonfinitizability of classes of representable polyadic al-

gebras. J. Symbolic Logic Vol 34 (1969) 344–352.

[J73] Johnson,J.S., Axiom systems for logic with finitely many variables. J.

Symbolic Logic Vol 38 (1973) 576–578.

[J82] Jonsson,B., Varieties of relation algebras. Algebra Universalis Vol 15

(1982) 273–298.

[J91] Jonsson,B., The theory of binary relations. In [AMN] 245–292.

[J95] Jonsson,B., The preservation theorem for canonical extensions of

Boolean Algebras with Operators. In: Lattice Theory and its Appli-

cations, K.A. Baker and R. Wille eds., Heldermann Verlag, (1995).

pp.121–130.[JT51] Jonsson,B. and Tarski,A., Boolean algebras with operators. Parts I-II

1951, 1952 Vol 73, 74 Amer. J. Math. 891–939, 127–162.

[JT91] Jonsson,B. and Tsinakis,C., Relation algebras as residuated Boolean

algebras. Algebra Universalis 30 (1993) 469–478.

[KSS] Kearnes,K. Sain,I. and Simon,A., Beth properties of finite variable frag-

ments of first-order logic. Preprint, Math. Inst., Budapest, (1996).

[KMPT] Kiss,E.W. Marki,L. Pr˝ ohle,P. and Tholen,W., Categorial algebraic prop-

erties. A compendium on amalgamation, congruence extension, epimor-

phisms, residual smallness and injectivity. Studia Sci. Math. Hungar. 18

(1983), 79–141.

[K97] Kurucz, A., Decidability in algebraic logic. Ph.D. Dissertation, Mathe-

matical Institute Budapest, 1997.



124 ALGEBRAIC LOGIC

[KNSS] Kurucz,A. Nemeti,I. Sain,I. and Simon,A., Decidable and undecidable

modal logics with a binary modality. J. of Logic, Language and Infor-

mation 4 (1995), 191–206.

[KNSS2] Kurucz,A. Nemeti,I. Sain,I. and Simon,A., Undecidable varieties of

semilattice-ordered semigroups, of Boolean algebras with operators,

and logics extending Lambek calculus. Bulletin of IGPL Vol 1 (1993)

91–98.

[Ly] Lyndon,R.C., The representation of relation algebras. II Ann. of Math.

series 2, Vol. 63 (1956) 294–307.

[MacL] Mac Lane, S., Categories for the working mathematician. Springer

Verlag, (1971).

[M95] Madarasz,J.X., The Craig interpolation theorem in multi-modal logics.

Bulletin of Section of Logic 24,3 (1995) 147–151.

[M96] Madarasz,J.X., Interpolation in algebraizable logics; semantics for non-

normal multi-modal logic. J. of Applied Nonclassical Logic, to appear.

[M97] Madarasz,J.X., Craig interpolation in algebraizable logics; meaningful

generalization of modal logics. J. of IGPL, to appear. Also Preprint of

Math. Inst. Hungar. Acad. Sci.

[M97a] Madarasz,J.X., Interpolation and amalgamation, pushing the limits.Studia Logica, to appear.

[M97b] Madarasz,J.X., A sequel to “Amalgamation, congruence extension, and

interpolation properties in algebras. In preparation.

[MNS97] Madarasz,J.X. Nemeti,I. and Sagi,G. On the finitization problem of

relation algebras (completeness problems for the finite variable frag-

ments). Extended abstract. Submitted.

[Ma78a] Maddux,R.D., Topics in relation algebras. Ph.D. Thesis. University of

California, Berkeley, 1978.

[Ma78] Maddux,R.D., Sufficient conditions for representability of relation al-

gebras. Algebra Universalis Vol 8 (1978) 162–172.

[Ma80] Maddux,R.D., The equational theory of oä

ó is undecidable. J. of Sym-

bolic Logic 45 (1980) 311–316.[Ma83] Maddux,R.D. A sequent calculus for relation algebras. Ann. Pure. Appl.

Logic 25 (1983) 73-101.

[Ma89] Maddux,R.D., Nonfinite axiomatizability results for cylindric and rela-

tion algebras. J. of Symbolic Logic 54, 3 (1989) 951–974.

[Ma91] Maddux,R.D., The origin of relation algebras in the development and

axiomatization of the calculus of relations. Studia Logica 50,3–4 (1991)

421–456.

[Ma91a] Maddux,R.D., The neat embedding problem and the number of vari-

ables required in proofs. Proc. Amer. Math. Soc. 112 (1991), 195–202.

[Mk87] Makkai,M., Stone duality for first order logic. Advances in Mathematics

(Academic Press, New York and London) Vol 65, No 2 August (1987).



BIBLIOGRAPHY 125

[MkP] Makkai,M. and Pare,R., Accessible categories: The foundation of cate-

gorical model theory. Contemporary Mathematics, Amer. Math. Soc. ,

vol.104, Providence, Rhode Island, (1989).

[MkR] Makkai,M. and Reyes,G.E., First Order Categorical Logic. Lecture

Notes in Mathematics Vol 611, Springer-Verlag, Berlin (1977).

[Ms77] Maksimova,L.L., Craig’s interpolation theorem and amalgamable vari-

eties. Soviet Math. Dokl., American Mathematical Society 1978 Vol 18,No 6 (1977).

[MOM] Marti-Oliet,N. and Meseguer,J., General logics and logical frameworks.

In [Ga94, pp. 355–392].

[Ma95] Marx,M., Algebraic relativization and arrow logic. PhD Dissertation,

University of Amsterdam, (1995).

[MNS] Marx,M. Nemeti,I. and Sain,I., On conjugated Boolean algebras with

operators. Manuscript, (1996).

[MPM] Marx,M. Polos,L. and Masuch,M., Arrow Logic and Multi-Modal

Logic. Studies in Logic, Language and Information, CSLI Publication,

Stanford, (1996).

[MV] Marx,M. and Venema,Y., Multi-dimensional Modal Logic. Kluwer

Academic Press, (1997).

[MP92] Masuch,M. and Polos,L. (eds), Logic at Work. (Proc. Conf. Amsterdam

December 1992). See the Arrow Logic Day section.

[McK] McKenzie,R.N., The representation of relation algebras. Dissertation,

University of Colorado, (1966).

[MMT] McKenzie,R.N. McNulty,G.F. and Taylor,W.F., Algebras, Lattices, Va-

rieties. The Wadsworth and Brooks/Cole Mathematics Series, Monterey,

California, 1987 (Vol I).

[Mi95] Mikulas,Sz., Taming Logics. PhD Dissertation, University of Amster-

dam, (1995).

[Mi96] Mikulas,Sz., Gabbay-style calculi. In: H. Wansing (ed), Proof Theory

of Modal Logic, Kluwer, 1996. pp. 243–252.

[M61] Monk,J.D., Studies in cylindric algebras. Ph.D. Thesis, University of

California, Berkeley, 1961.[M64] Monk,J.D., On representable relation algebras. Michigan Math. J. Vol

11 (1964) 207–210.

[M64a] Monk,J.D., Singulary cylindric and polyadic equality algebras. Trans.

Amer. Math. Soc. 112 (1964) 185–205.

[M69] Monk,J.D., Nonfinitizability of classes of representable cylindric alge-

bras. J. Symbolic Logic Vol 34 (1969) 331–343.

[M70] Monk,J.D., On an algebra of sets of finite sequences. J. Symbolic Logic

Vol 35 (1970) 19–28.

[M71] Monk,J.D., Provability with finitely many variables. Proc. Amer. Math.

Soc. Vol 27 (1971) 353–358.

[M74] Monk,J.D., Connections between combinatorial theory and algebraic

logic. In: Studies in Math., vol. 9. Math. Assoc. Amer. pp. 58-91.



126 ALGEBRAIC LOGIC

[M77] Monk,J.D., Some problems in algebraic logic. Colloq. Inter. de Logic,

CNRS Vol 249 (1977) 83–88.

[M91] Monk,J.D., Structure problems for cylindric algebras. In [AMN] 413–

429.

[M93] Monk,J.D., Lectures on Cylindric Set Algebras. In: Algebraic Meth-

ods in Logic and in Computer Science, ed: C. Rauszer, Banach Center

Publications Vol 28, Polish Academy of Sciences, Warszawa (1993).

pp.253–290.

[N82] Nemeti,I., Surjectiveness of epimorphisms is equivalent to Beth de-

finability property in general algebraic logic. Preprint, Mathematical

Institute, Budapest. (1982).

[N85] Nemeti,I., Logic with three variables has Godel’s incompleteness prop-

erty — thus free cylindric algebras are not atomic. Preprint No 49/85,

Math. Inst. Hungar. Acad. Sci. , Budapest July (1985).

[N85a] Nemeti,I., Exactly which varieties of cylindric algebras are decidable?

Preprint No. 34/1985, Math. Institut Budapest, 1985.

[N86] Nemeti,I., Free algebras and decidability in algebraic logic. (In Hungar-

ian) Dissertation for D.Sc. with Hung. Academy of Sciences, Budapest

(1986) xviii+169 pp.

[N87] Nemeti,I., On varieties of cylindric algebras with applications to logic.

Annals of Pure and Applied Logic Vol 36 (1987) 235–277.

[N88] Nemeti,I., Epimorphisms and definability in relation-, polyadic-, and

related algebras. Invited lecture at the “Algebraic Logic in Computer

Science” Conference, Ames, Iowa, USA, June 1988.

[N90] Nemeti,I., On cylindric algebraic model theory. In [BMP], 37–76

(1990).

[N91] Nemeti,I., Algebraizations of Quantifier Logics, an introductory

overview, 12-th version. Math. Inst. Budapest, Preprint, No 13/1996.

Electronic address:

http://circle.math-inst.hu/pub/algebraic-logic/survey.dvi

An extended abstract of this appeared in Studia Logica vol 50 No 3/4(1991) 485–569.

[N92] Nemeti,I., Decidability of weakened versions of first-order logic. In:

Proc. Logic at Work, December (1992), Amsterdam.

[N95] Nemeti,I., Decidable versions of first order logic and cylindric-

relativized set algebras. In: Logic Colloquium’92 (Proc. Veszprem,

Hungary 1992), eds: Csirmaz,L. Gabbay,D.M. and de Rijke,M., Stud-

ies in Logic, Language and Computation, CSLI Publications, (1995).

pp.177–241.

[N96] Nemeti,I., Fine-structure analysis of first order logic. In: Arrow Logic

and Multi-Modal Logic, M. Marx, L. Polos, and M. Masuch eds, CSLI

Publications, Stanford, California, (1996). pp.221–247.



BIBLIOGRAPHY 127

[NA] Nemeti,I. and Andreka,H., General algebraic logic: a perspective on

“what is logic”. In: What is a logical system, ed: D. M. Gabbay, Claren-

dron Press, Oxford, (1994). pp.393–444.

[NS96] Nemeti,I. and Sagi,G., On the equational theory of representable

polyadic equality algebras. Preprint, Math. Inst. , Budapest, (1996).

[NS97] Nemeti,I. and Sagi,G., Geometries, relation algebras, cylindric alge-

bras. Mathematical Institute Budapest, In preparation.[NSi] Nemeti,I. and Simon,A., Relation algebras from cylindric algebras and

polyadic algebras. Logic Journal of IGPL, 5,4 (1997) 575-588.

[O] Otto,M., Bounded variable logics and counting. (A study in FiniteMod-

els.) Springer Lecture Notes in Logic, Vol. 9. 1997.

[PP] Palasinska,K. and Pigozzi,D., Implication in abstract algebraic logic.

Preprint, Cracow University of Technology and Iowa State University,

December (1995).

[Pa] Partee,B.H., Montague Grammar. Academic Press, New York, 1976.

[P72] Pigozzi,D., Amalgamation, Congruence-Extension, and Interpolation

Properties in Algebra. Algebra Universalis 1,3 (1972), 269–349.

[P91] Pigozzi,D., Fregean Algebraic Logic. In [AMN] 473–502.

[PS] Pigozzi,D. and Salibra,A., Polyadic algebras over nonclassical logics.

In: Algebraic Methods in Logic and in Computer Science, Banach Cen-

ter Publications, vol.28, Institute of Mathematics, Polish Academy of

Sciences, Warszawa, (1993). pp.51–66.

[Pl94] Plotkin,B.I., Universal algebra, Algebraic logic, and Databases.

Nauka, Fizmatlit (1991) (In Russian.) English version: Kluwer Aca-

demic Publishers, (1994).

[Pr90] Pratt,V.R., Dynamic algebras as a well behaved fragment of relation

algebras. in [BMP] (1990) 77–110.

[Pr91] Pratt,V.R., Dynamic Algebras. Studia Logica 50,3–4 (1991) 571–605.

[Pr92] Pratt,V.R., Origins of the calculus of binary relations. In: Proc. 7th

Annual IEEE Symposium on Logic in Computer Science, Santa Cruz,

CA, June 1992. pp. 248-254.

[Q36] Quine,W.V., Toward a calculus of concepts. J. Symbolic Logic Vol 1(1936) 2–25.

[RT] Resek,D. and Thompson,R.J., An equational characterization of rela-

tivized cylindric algebras. In [AMN] 519–538.

[RV] de Rijke,M. and Venema,Y., Sahlquist’s Theorem for Boolean Algebras

with Operators with an application to cylindric algebras. Studia Logica

54,1 (1995) 61–79.

[R] Rybakov, V., Admissibility of Logical Inference Rules, Elsevier, North-

Holland, New-York, Amsterdam (Studies in Logic and the Foundations

of Math. vol. 132), 1997. 617 pp.

[S79] Sain,I., There are general rules for specifying semantics: Observations

on abstract model theory. Computational Linguistics and Computer

Languages (CL& CL), 13 (1979), 195–250.



128 ALGEBRAIC LOGIC

[S90] Sain,I., Beth’s and Craig’s properties via epimorphisms and amalgama-

tion in algebraic logic. In [BMP] 209–226 (1990).

[S95] Sain,I., On the problem of finitizing first order logic and its alge-

braic counterpart. (A survey of results and methods.) In: Logic Col-

loquium’92 (Proceedings of Logic Colloquium’92, eds.: Csirmaz,L.

Gabbay,D.M. and de Rijke,M.), CSLI Publications, Stanford, (1995).

pp.243–292.

[S97] Sain,I., Searching for a finitizable algebraization of first-order logic.

Annals of Pure and Applied Logic, to appear.

[SGy] Sain,I. and Gyuris,V., Finite schema axiomatizable algebraization of

first order logic. Preprint, Math. Institute, Budapest, (1994). Journal of

IGPL, to appear.

http://circle.math-inst.hu/pub/algebraic-logic/fin-scheme.dvi.gz

[ST] Sain,I. and Thompson,R.J., Strictly finite schema axiomatization of

quasi-polyadic algebras. In [AMN] 539–571.

[SA] Sayed Ahmed,T.M., Algebras of sentences of logic. Master Thesis,

Cairo University, (1997).

[Sa95] Sagi,G., Non-computability of the consequences of the axioms of the

omega-dimensional polyadic algebras. Preprint, Math. Inst., Budapest,(1995).

[SN] Sagi,G. and Nemeti,I. Modal logic of substitutions, completeness re-

sults. Preprint, Mathematical Institute Budapest, 1997.

[Se85] Sereny,G., Compact cylindric set algebras. Bull. Section of Logic

(Warsaw-Łodz, June 1985) 57–64.

[Se97] Sereny,G., Saturatedness in cylindric algebraic model theory. Logic

Journal of the IGPL 5, 1 (1997), 25–48.

[Sh] Shelah,S., On a problem in cylindric algebra. In [AMN] 645–664.

[Si91] Simon,A., Finite Schema Completeness for Typeless Logic and Rep-

resentable Cylindric Algebras. Algebraic Logic (Proc. Conf. Budapest

(1988)) Colloq. Math. Soc. J. Bolyai Vol 54, North-Holland, Amster-

dam. 665–670.

[Si93] Simon,A., What the Finitization Problem is Not. In: Algebraic Methods

in Logic and in Computer Science (Proc. Banach Semester Fall (1991)),

Banach Center Publications, Vol 28, Warsaw, (1993). pp.95–116.

[Si96] Simon,A., A new proof of the representation theorem for Quasi-

projective Relation Algebras. Manuscript, Mathematical Institute, Bu-

dapest. Nov. (1996).

[Si97] Simon,A., Non-representable algebras of relations. Ph.D. Dissertation,

Mathematical Institute, Budapest, (1997).

[T51] Tarski,A., Remarks on the formalization of the predicate calculus. (ab-

stract) Bull. Amer. Math. Soc. 57 (1951) 81–82.

[T65] Tarski,A., A simplified formalization of predicate logic with identity.

Archiv f ur Math. Logik u. Grundl. 7 (1965) 61–79.



BIBLIOGRAPHY 129

[TG] Tarski,A. and Givant,S., A formalization of set theory without vari-

ables. AMS Colloquium publications, Providence, Rhode Island Vol 41

(1987).

[V91] Venema,Y., Many–dimensional modal logic. Doctoral dissertation,

University of Amsterdam, ITLI (Institute for Language, Logic, and In-

formation) (1991) vii+178pp.

[V95] Venema,Y., Cylindric modal logic. Journal of Symbolic Logic 60,2(1995) 591–623.

[V95a] Venema,Y., A modal logic of quantification and substitution. In: Logic

Colloquium ’92, L. Csirmaz, D,M. Gabbay, and M. de Rijke, eds., CSLI

Publications and FOLLI, Stanford, 1995. pp. 293–309.

[V96] Venema,Y., Atom structures and Sahlqvist equations. Algebra Univer-

salis, to appear.

[V97] Venema,Y., Atom structures. Manuscript, Jan. (1997). Free University,

Amsterdam.

[W] Werner,H., Discriminator algebras. Akademie Verlag, Berlin (1978).

[Z] Zlatos,P., On conceptual completeness of syntactic-semantical systems.

Periodica Math. Hungar. Vol 16,3 (1985) 145–174.

algebraic logic.handbook

Documents