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GENTZEN SYSTEMS AND DECISION PROCEDURES

FOR RELEVANT LOGICS

by

Steve Giambrone

B.A., University of Southwestern Louisiana, 1971 M.A., Louisiana State University, 1975

A dissertation submitted for the degree of

Doctor of Philosophy, in the Australian

National University, February, 1983.

The research contained herein was carried

out either by me independently or in

conjunction with Dr. R.K. Meyer. In

particular, the semantic results of §1.7

and §1.8 are based on collaborative work

performed by Dr. Meyer and myself.

~~ Steve Giambrone

iii

ABSTRACT

This dissertation is primarily a proof theoretic

investigation of the positive fragments and boolean

extensions of two of the principal relevant logics

T and R, with and without contraction, and of th.e

corresponding positive semilattice relevant logics. In

addition to motivational and syntactic preliminaries,

Chapter 1 contains some new semantic results which are Oj _0-,

useful in the later chapters. In particular, T , TW and

RWo 1 are proved to be complete with respect to their

boolean semantics, and are then shown to be conservative

extensions of T0

, TW and RW, respectively. In Chapter 2

we develop subscripted Gentzen systems for four

positive semilattice logics. Appropriate Cut Theorems

are proved, and one system is shown to be equivalent to

uR+' Decision procedures are then given for the two

contractionless systems. In Chapter 3 Gentzen systems

are given for TW+, T+, RW+ and R+, Cut Theorems and

equivalences are proved, and TW+ and RW+ are shown to

be decidable. The sequent calculi that are used are

multiply structured as required for relevant logics.

Chapter 4 begins by collecting decision procedures for

fragments of TW+ and RW+. We then discuss and make

some progress toward solving some open problems, viz.,

the decision questions for EW+, TW and RW, and the

question of equivalence between RW+ and its semilattice

counterpart uRW+.

TABLE OF CONTENTS

Abstract iii

Acknowledgements vi

Preface ix

CHAPTER 1. INTRODUCTION AND PRELIMINARIES

§1. Introduction

2. Relevant Gentzen Systems

3. Decision Questions for Relevant Logics

1

3

9

4. Contractionless Relevant Logics 14

5. Syntactic Preliminaries 19

6. Axiomatics 21

7. Semantic Completeness for 24 Boolean Relevant Logics

8. Conservative Extension 37

9. Semilattice Semantics 52 CHAPTER 2. SUBSCRIPTED GENTZEN SYSTEMS

§1. Introduction

2. Preliminaries

3. Critique of Kron 78

4. Critique of Kron 80

5. G-Systems

6. Vanishing-t

7. Cut and Modus Ponens

8 u c u . R+ - G R+ u u

9. R+ is G R+

10. Decidability

64

66

69

79

85

95

100

113

123

125

CHAPTER 3. DUNN-STYLE GENTZEN SYSTEMS

§l. Introduction 137

2. Formulation l, Definitions and 139 Facts

3. Cut Theorem 148

4. Equivalence and Represent- 155 ational Adequacy

5. Formulations 2 and 3: 165 Vanishing-t

6. Denesting 175

7. Reduction 187

8. Degree and Decidability 197

CHAPTER 4. CONCLUDING RESULTS AND OPEN QUESTIONS

§l. Introduction 204

2. Decidable Fragments 205

3. E+ and EW+ 208

4. Extensions and Decidability 211

5. RW+ = uRW+? 226

BIBLIOGRAPHY 229

vi

ACKNOWLEDGEMENTS

It has been an enormous privilege and pleasure

to have worked with my principal supervisors Dr. R.K. Meyer

and Dr. R. Routley. The influence of their writings and

of both logical and philosophical conversations with them

permeates this work. It has been with Dr. Meyer that I

have worked most closely, and to whom credit is deserved

for much of what is good to be found herein.

I am also indebted to several other scholars

who have been at one time or another members of the

Logic Group of the Philosophy Department, RSSS during my

course of study at the Australian National University,

namely, Dr. C. Mortensen, Dr. M.A. McRobbie, Dr. E.P. Martin,

Dr. J. Slaney, Mr. P. Thistlewaite, and Mr. Adrian Abraham.

I am particularly indebted to Dr. M.A. McRobbie as a proof

theoretic island in an ocean of algebraist, and to

Dr. E.P. Martin for so generously sharing his knowledge of

and insights into Ticket Entailment. I would also like to

single out Mr. P. Thistlewaite who read most of this

manuscript and suggested numerous (needed) corrections.

Above and beyond the debts owed to individual members of

the Logic Group, the Group as a whole deserves mention

for the enthusiastic atmosphere that it provides. It

seems inconceivable that an isolated researcher could

find comparable stimulation for his work, nor enjoyment

therein.

~o~uaw puB pua~~d

~alaw ·x ~~aqo~ or

One cannot write a dissertation without torturing

friends and family (largely with boredom) if, that is,

one is fortunate enough to have such who will endure it.

vii

In this respect I have been abundantly blessed. Those

individuals already mentioned must be thanked for their

tolerance and immeasurable support. In addition, special

thanks also go to Dr. C. Fahlander, Dr M. Dronjak-Fahlander,

Mr. P. Filmer-Sanke, Ms. V. Sieveking, Mr. J. Larocque,

and Ms. L. Sachs.

Ms. A. Duncanson deserves both gratitude and praise

for transforming an illegible manuscript filled with

technical notation into a fine typescript in a very short

space of time.

Finally, I want to thank Mr. Bruce Toohey, an

unsung hero of logic and the source of A-grade inspiration.

1

SECTION 1. Introduction

This thesis is primarily a contribution to the

proof theoretic investigation of sentential or zero-order

relevant logics, although some new semantic results are

contained in this chapter. Chapter II is devoted to

subscripted Gentzen systems for positive semi-lattice

relevant logics based on previous work of Aleksandar Kron.

The systems G~W+, G~RW+' GuT+ and GuR+ are formulated,

and suitable Cut Theorems are proved. Moreover, the

contractionless systems are shown to be decidable.

Although it is likely that all of the systems are equivalent

to their axiomatic namesakes, we have a proof of

equivalence only for GuR+ and u· R+.

In Chapter 3 we build primarily on the work of

J. Michael Dunn to formulate Gentzen systems which are • 0 0 0 0

proved to be equ1valent to TW+' RW+' T+ and R+' respectively.

We then build on insights from the previous Chapter to show

that these contractionless systems are likewise decidable.

Chapter 4 begins by gathering some easy results for

fragments of the logics treated in Chapters 2 and 3. The

latter sections are devoted to discussing some interesting

open questions which arose from this research, and to

contributing as much as we now can to their solution.

These problems are the decidability of EW+' TW and RW, and

the equivalence of RW+ with uRW · +

Sections 2~ of this chapter are devoted to

historical and motivational remarks, while §5 and §6 give

2

the necessary syntactic and axiomatic preliminaries. In

§ 7 semantics are presented for T01, TW

01 and RW

0

1

and completeness is proved. These systems are then shown 0

to be conservative extensions of T , TW and RW,

respectively, in § 8. In § 9, the known semantic

completeness results for semi-lattice relevant logics

are recorded, and an alternative semantics is given for

uRW+. These semantic results are useful for the work to

be done in the following chapters.

3

SECTION 2. Relevant Gentzen Systems

Sequent calculi (consecution calculi, Gentzen

system~ have been a powerful tool of formal research since

their introduction in Gentzen 35. It was recognised early

on that "normal" sequents or consecutions utilizing sequences

of formulae could be used to provide Gentzen systems for

pure implicational fragments of relevant logics. The first

of these was the system LI announced in Belnap 59, which

is equivalent to the pure implicational fragment of E.

(It is presented in detail in Belnap 60.) R was fitted +

with a similar sequent calculus in Kripke 59, in which R+

is also shown to be decidable. A similar style of

formulation was developed in Belnap and Wallace 65 for E , - +-

which is also shown to be decidable there. And Meyer 66

extends the use of simple sequences of formulae in relevant

Gentzen systems all the way to the system R-Distribution.

But to this point no one knew how to fully

accommodate conjunction and disjunction (even just along

with implication) in a relevant sequent calculus. It was

J. Michael Dunn who made the breakthrough (announced in

Dunn 73).

In a sequent calculus for, say, classical logic,

a sequence of formulae is "implicitly representing", if

you like, the conjunction of those formulae when it occurs

in the antecedent of a sequent, and the disjunction of

those formulae when it occurs in the consequent. In the

relevant Gentzen systems which had previously been

formulated, sequences of formulae were being used alternately

4

to implicitly represent intensional conjunction (fusion)

and disjunction (fission). From this point of view, what

Dunn discovered was that two different kinds of sequences

would be required to formulate Gentzen systems for the full

positive relevant logics, in particular for R:. Moreover,

such sequences must be allowed to be arbitrarily nested

within one another.

Such a simplified overview (made with a great

deal of hindsight), misrepresents these ideas as being

simple. They are far from it. We should also note

that equal credit is due to Mine 72 which develops a 0 1 sequent calculus for R .

In any event, Dunn's system LR+ was presented in

Dunn 75. A Cut Theorem is proved there and the system

is shown to be equivalent to R:t. Sequents are singular

on the right, but the antecedents in the general case are

structures, nested within one another to arbitrary depths.

The next advance came from Meyer 76a. Although

that paper deals explicitly only with systems of pure

implication, the moral is obviously more general: Since

sequences are standing in for "generalized" conjunction

of one form or another, why not explicitly introduce

structural connectives corresponding to the formulae

connectives in question. Each such structural connective

is to be governed by structural rules expressing its

particular "combinatorial" properties.

These ideas lead naturally to Gentzen systems

for other positive relevant logics via Routley and

Meyer 72.

5

This technique gives definite notational

simplification, and, we think, conceptual clarity and unity

to the complex Gentzen systems of positive relevant logics.

Indeed, we view the ideas of Meyer 76a as a bridge between

Dunn 75 and Belnap 8+.

Display Logic is presented in Belnap 8+. It is a

very general and powerful Gentzen system which can

simultaneously accommodate an indefinite number of logics -

many well-known and others yet to be "discovered". The

central idea is to conceive of connectives as coming in

"families". Each family has a (formula) "conjunction", a

"disjunction", a "negation", an "implication", and at least

one sentential constant (0-ary connective), "the true".

Some families may have other formula connectives, such as

necessity in modal cases. But all of the formula

connectives can be defined in terms of "kernel" connectives

of conjunction, disjunction, negation and truth.

Further each family has three structural connectives.

One is a negative structural connective, obviously standing

in for negation. The ''interpretations" of the other two

are context dependent, just as with Gentzen's commas and

the empty symbol. So one of these is alternately

interpreted as the conjunction (of the family) or the

disjunction, depending on whether or not it occurs as an

6

2 antecedent or consequent part of a sequent. And the final

structural connective alternately stands in for the true

or the negation of the true (the false, when such is

postulated as a primitive formula connective}. 3

Now for each family of connectives, three sets of

postulates are given. The first is a group of display

equivalences. The second is a set of structural rules.

These two together determine the character of the structural

connectives of a given family. Finally, there are logical

or (formula) connective rules, one for introducing a

given connective on the left, and another for introduction

on the right.

Different logics can now be associated with one or

more families of connectives. Classical logic, for instance,

is ·associated with th.e boolean connectives, whereas modal

and relevant logics are individually associated with the

boolean family of connectives and a family of connectives

distinctive to the particular logic in question. So it

is often convenient to think of not one Display Logic,

but of many Display Logics, each having only a definite,

given family or families of connectives governed by

particular postulates. 4

Another central feature of Display Logic is

Display Theorem. (Theorem 3-2, section 3.2 of Belnap 8+).

Each antecedent part X of a consecution S can be

displayed as the antecedent (itself) of a display-

equivalent consecution X l- W; and the consequent W is

7

determined only by the position of X in S, not by what X

looks like. Similarly for consequent parts of S.

This feature of Display Logic allows a very general (and

very pretty) Cut Theorem to be proved in Belnap 8+. It

covers an enormous range of "Display Logics". (See note 4

of this section.)

Prior to Display Logic, there were no known Gentzen

systems for relevant logics with negation and a full

complement of positive principles. The logics representable

in Display Logic include the major boolean or classical

relevant logics. The conservative extension results of 0 0

§1 and Meyer and Routley 74 show that TW, RW, T and R

are also exactly represented. But the significance of

Display Logic ·goes beyond that of providing sequent

calculi for various relevant logics.

Before closing this section, we should note that

this historical sketch is in no way intended to be complete.

Most particularly, we have not discussed the ''one-sided''

Gentzen systems, as in Belnap and Wallace 65, McRobbie

and Belnap 79 and McRobbie 79, nor the merge systems of

AB75. However, what has been presented here was intended

as stage-setting for the results of succeeding chapters, in

which such systems play no part.

8

FOOTNOTES

1 .m A similar sequent calculus for R+ · was developed

independently in BGD80.

2Antecedent and consequent parts are more or less what

one would expect. For an explanation, see sections 2.3

and 2.4 of Belnap 8+, or §4.4 of this work.

3The question of how many families are to be postulated

is left open, but some interesting examples are given.

4Although this move somewhat mars the beauty and possibly

the ultimate conception of Display Logic, it is very

convenient for particular purposes. So we take that

point of view in Chapter 4.

9

SECTION 3. Decision Questions for Relevant Logics

Decision Questions for sentential relevant logics

have been notoriously difficult to answer. Until very

+ recently only one system, Q of Meyer and Routley 73c, has

been shown to be undecidable. (But see below.) The

question for the full systems T, E and R remains open in

spite of significant efforts (by many stalwart logical

champions) to arrive at a solution, especially for R.

Indeed, since Martin slew the dragon of Belnap's conjecture

in Martin 78 (also published in Martin and Meyer 8+), the

decision problem for R has come to be known as "the big

enchillada". (Terminology is due to R. K. Meyer.)

The system T of ticket entailment has proved to be

particularly recalcitrant. Aside from PW (and the zero

degree and first degree fragments of T, which it has in

common withE and R), decision questions for even interesting

subsystems ofT have remained open (until this work);

whereas, these questions for R , E , R and E have all ~ + +~ +-

been answered in the affirmative. (See Kripke 59, Belnap

and Wallace 65 and Meyer 66.)

A decidability claim for TW+ and RW+ was made in

Kron 78 on the basis of a claimed decision procedure for

the subscripted Gentzen systems GT+-W and GR+-W. However,

we show in the next Chapter that those systems are not

equivalent to TW+ and RW+. Further, the argument for

decidability is unsound.

We then build on the work of Kron 80 to produce

10

subscripted Gentzen systems for positive semi-lattice logics,

and appropriate Cut Theorems are shown. However, at this

point, only one, GuR+' can be proved equivalent to its

namesake. We then show decidability for the two

contractionless systems, GuTW+ and GRW+.

0 0

In Chapter 3 we show that TW+ and RW+ are indeed 0 0

decidable. Dunn-style Gentzen systems LTW+ and LRW+ are

developed for this purpose, and insights gained in

Chapter 2 are used to show decidability.

The subscripted systems and the Dunn-style systems

are treated in a broadly similar fashion. We take a

rather common approach of giving different formulations

best suited for different purposes. (See Curry 63 and

Kleene62, for instance.) The initial formulations utilize

t (or its structural analogue) to remain non-empty on the

left, which facilitates a proof of Cut. The systems are

then given formulations without t which are more suitable

for showing them decidable.

The arguments for decidability are obviously proof

theoretic. First, a complete and effective proof search

tree is defined for any given formula. Konig's Lemma

(Konig 27) is then used to show that all such proof search

trees are finite. As usual the Finite Fork Property

presents little problem. But as we have said before,

Gentzen systems for full positive relevant logics and their

supersystems are more complex than Gentzen systems which

use simple sequences of formulae as structures for building

sequents, so a new technique had to be developed for showing

the Finite Branch Property.

Obviously, it suffices for the Finite Branch

Property to get control over the length or "complexity''

11

of a sequent that can occur on any branch of the proof

search tree - given Irredundancy and an appropriate

Subformula Property, that is. The technique developed in

this thesis involves first ~eparating out" the intensional

and extensional structural components of sequents, and

getting control over each separately. In the Dunn-style

systems, the different structural components are explicitly

distinguished (except at the formula level) as intensional

and extensional sequences. In the subscripted systems,

intensional and extensional components of a structure must

be unpacked from the interrelations of the subscripts

occurring in it.

Control over the extensional complexity of a sequent

is gained by showing a Reduction Lemma in the spirit of

Gentzen 35. Then we develop an appropriate notion of

degree as a measure of intensional complexity. Of course,

a Degree Lemma is proved which gives control over intensional

complexity. And then it is shown that the combined effect

yields the required control on the overall complexity of

the sequent.

It is hoped that this technique can be generalized

to handle all of TW and RW. A proposed method of doing so

is sketched in §4 of Chapter 4. However, the method does

not appear to generalise to even positive systems with

contraction. And a recent result by Urquhart would lead

12

one to believe that no argument for decidability will do in

these cases.

We mentioned before that the system Q+ has been

shown to be undecidable. However, that system was motivated

simply as an undecidable relevant logic. The system KR

(see 5. 4 of RLRI), on the other hand, was independently

motivated and of interest in its own right. In Urquhart 82

the word problem for semi-groups is encoded into KR - via

a theory of projective geometry - thus showing that the

system is undecidable.

If one can, as many suspect, encode at least an

appropriate portion of KR into R, the big enchillada will

be made a meal of. As essentially recorded in Meyer and

Giambrone 80, if such encoding can be done in R+' then the

decidability questions for (R+,) T+, E+, T and E will

likewise be answered in the negative.

ADDENDUM

We received a copy of Urquhart 8+ while we were

in the process of making final corrections to this

manuscript. That work ends an era of research into

decision questions for relevant logics by proving that

T, E and R - and many other relevant logics - are

undecidable. To this author's knowledge, T, E and R

are now the first philosophically well-motivated

sentential logics to be shown undecidable.

Although we have not yet had time to study the

paper in detail, we suspect that it dashes our hope of

showing TW and RW decidable £[ the method proposed in

§ 4. 4 .

13

14

SECTION 4. Contractionless Relevant Logics

The study of contractionless logics goes back at

least as far as the development of Zukasiewicz's three­

valued logic Z3 first published in Zukasiewicz 20. (See

§1.6 for a statement of the Contraction Axiom.) And

interest in contractionless relevant logics was present

very early in the study of relevant logics per se. §8.11

of AB75 gives Belnap's Conjecture, namely, that if A+B and

B+A are both theorems of PW, then A and B are the same

formula. The original date of the conjecture is unknown to

this author, but progress toward its solution had been

reported by Powers as early as 1968. (The results were

eventually published in Powers 76.) We have already noted

that the conjecture was proved in Martin 78.

RW+ appears in Smiley 591 and is studied in Meredith

and Prior 63. Both it and PW were given formal semantics in

Urquhart 73, as well as subscripted Gentzen systems. But

interest in full contractionless systems was first

stimulated by Meyer, Routley and Dunn 79, where it is shown

(with due acknowledgement to Curry and Feys 58) that a

non-trivial naive set theory can not be founded on R, T

nor E. The problem, of course, is that the contraction

axiom in the presence of other very minimal logical

principles will collapse any theory containing the full

Abstraction Principle.

The ability to be used in formally investigating

non-trivial but inconsistent theories, i.e. being weakly

paraconsisten~ has always been a motivating feature

15

of relevant logics. And naive set theory has always been

near the top of the list of interesting such theories -

both within and outside of the relevant program. So

contractionless relevant logics have found favor with those

who want a logic suitable for such purposes, but want to

maintain as much as possible of the full motivation of the

traditional relevant logics. 2

Another point on which the contractionless relevant

systems commend themselves is that of being more Catholic

than the Pope with respect to a central feature of relevant

logics. AB75 begins with the claim that ''the heart of

logic [lies] in the notion of 'if ... then -· •; .... ''We

take the point to be that the central task of a logic is to

separate out valid inferences from invalid ones. Contraction­

less relevant logics can be seen as taking this point further.

Distinguishing valid inferences from invalid inferences is

not simply the central task of logic, it is the task of

logic.

To be sure, logic must say something "about" truth

functions, since they are needed in the vocabulary to

express certain truths about implication, such as, if A

implies B and A implies C, then A implies B and C. But

whether or not Excluded Middle, for instance, is to be

accepted is not a matter to be determined by logic.

Accordingly, the contractionless relevant logics (we

have TW, RW and EW particularly in mind) do not record any

purely truth-functional tautologies as a logical truth.

More precisely, no formula in which an+ does not occur

is a theorem of these logics.3

16

It is not that such logics deny any of the putative

truths about truth functions - or for that matter about

quantifiers, or alethic modal operators, or what have you.

Rather, it is that such mattersare to be decided on

non-logical grounds, and recorded in theories appropriate

to those subject matters.

And with respect to taking valid inference to be

its proper subject matter, TW outshines its cousins. For

Slaney 83 establishes the following fact.

TW Implicational Fact. 4 Every theorem of TW is equivalent

to a conjunction of theorems each of which is a disjunction

one of whose disjuncts is a valid implication.

As Slaney puts it, this fact ''establishes a good sense in

which TW is fundamentally implicational 1'. Which is, we

think, as a logic ought to be.

Now one point needs clarification here. It is an

historical accident that RW, TW and EW are called

lcontractionless logics!. There are (in many ways good)

historical reasons for the name, these reasons having to

do with the usual axiomatisations of R, T and E. But

there is nothing sacrosanct about these axiomatisations.

One could add Excluded Middle, for instance, to any of

these contractionless logics and still have subsystems of

the original logics in which some instances of Contraction

were not valid- i.e., still have contractionless systems.

It is not the simple lack of Contraction as a theorem which

17

gives these systems the characteristic which we have been

commending above.

Finally, we should note that the contractionless

relevant logics are prime; that is, a disjunction is a

theorem of one of these systems just in case one of the

disjuncts is. This is a trait which is put to good use in

§1.7 and 1.8. And of course it is a point on which to

recommend these systems to those who cherish constructivism.

FOOTNOTES

1As the pure implicational fragment of a system with

negation also primitive.

18

2However, serious doubts are cast on the usefulness of RW

in this respect in Slaney So and Slaney 83.

3we have avoided the normal reference to zero degree

formulae, since a slightly different notion of degree is

employed in this work.

4The name is our own.

19

SECTION 5. Syntactic Preliminaries

A number of logics are treated in this work. For

each, a formal language£, a set of formulae, is assumed,

being built up in the usual way from (a denumerably infinite

stock of) atomic formulae - and a sentential constant t,

where appropriate - via the suitable connective(s) from

among the unary connectives ~and 1, and the binary

connectives &, v, o and~. In order of occurrence, these

connectives are De Morgan negation, boolean negation,

(extensional) conjunction, (extensional) disjunction,

fusion or intensional conjunction, and implication.

Other connectives and/or sentential constants are defined

when useful.

We take formulae to be obs, more or less in the

sense of Curry 63. We use

p,q,r,pl,ql,rl,···

as sentential parameters taking atomic formulae as values,

and

A,B,C,D,A 1 ,B 1 ,C 1 ,D 1 , •••

as formula variables. Representations of formulae are

disambiguated according to the following conventions:

the connectives bind less tightly in the order presented

above (so- binds most tightly and+ least tightly);

dots and parentheses are used in the standard way;

otherwise, disambiguate by associating to the left.

However, where there is no chance of confusion, we use

20

simple juxtaposition for o, in which case it binds more

tightly than any other connective.

Notation for various Gentzen systems is introduced

along with those systems. Local variables, for sets for

instance, are introduced when needed. c We let u, n, -,

and E serve their usual set theoretic functions; and let

c be proper subset, while u is generalised set union

(see p.lll of Kuratowski and Mostowski 68, for instance.)

21

SECTION 6. Axiomatics

Hilbert style axiomatizations of the logics

treated in this work can be formulated from the following

axiom schemata and rules. For each logic we assume the

appropriate base language£; and for the boolean systems,

it is convenient to have r~l according to

Definition. A~B =df IA¥B

Axl. A+A

Ax2. A+B+.B+C+.A+C

Ax3. B+C+.A+B+.A+C

Ax4. (A+.A+B)+.A+B

Ax5. A+.A+B+B

Ax6. A+B+.A+B+C+C

A!-7. AB+C+.A+.B+C

Ax8. (A+.B+C)+.AB+C

Ax9. A&B+A, A&B+B

AxlO. (A+B)&(A+C)+.A+B&C

Axll. (A+A+A)&(B+B+B)+.A&B+A&B+A&B

-Axl2. A+AvB, B+AvB

Axl3. (A+C)&(B+C)+.AvB+C

-Axl4. A&(BvC)+.(A&B)vC

Axl5. -A+A

Axl6. A+-B+.B+-A

Identity

Suffixing

Prefixing

Contraction

Assertion

Restricted Assertion Exportation

Importation

Simplification

Composition

Addition

V+

Distribution

-Double negation

-contraposition

Axl7. A-+~A-+~A

Axl8. t

Axl9. t-+.A-+A

Ax20. A&IA-+B

Ax21. A-+.B-+CVIC

Ax22. (A-+B)~.A~B

Rl

R2

R3

A A-+B B

A B A&B

A-+.B-+C AB+C

22

~Reductio

Ex Falsum Quod

Libet

Modus Ponens

Adjunction

Rule Importation

R4 A&(A 1&ql-+· ... +.A &a+C)+.B 1&q1+ .... -+.B &q +B Fine's Rule n n n n

A& CA1.&.q1-+ .... -+.An &<JDD )+ .B1 &q1-+ •... -+. Bn &<1n +B

A&(Al-+· ... -+. An -+CvD)-+.Bl-+ ..... -r.Bn -+B

with 0 ~ n, provided ~occurs only where indicated.

FORMULATIONS

PW. Axl-3; Rl

T. Axl-4,9,10,12-17; Rl,R2

E. T + Ax6,11

R. T + Ax5

urr. T + R4

uR. R + R4

23

Fragments and extensionscan be formulated by dropping

and adding, respectively, the appropriate axioms and rules.

Contractionless (W-) systems are obtained by dropping Ax4.

Not all of the axioms are independent in the various

sytems. Ax22, in particular, is redundant in R01 , T01

RW 01. However, it is independent in TW 01 , as the following

matrices (due to Meyer) show.

Matrices for&, v, and I can be read from the

following Hasse diagram

3

1\ 2 1

\I 0

with- identified with !,and 3 the only designated value.

The +-matrix can be specified by

1. a+b = 3, if a< b, and

2. a+b = 2, otherwise.

We now leave it to the reader to verify the claim.

In the next section we will help ourselves to

well-known and/or easily proved theorems of all of the

boolean logics, such as truth-functional tautologies

in & , v, and -l, DeMorgan Laws and Double-negation for I,

as well as (boolean) Disjunctive Syllogism, i.e.,

~L A&(IAvB)+B.

Throughout this work we use I f-1 subscripted

with the name of a logic or with L to indicate theoremhood

in a logic or logics - as we have just done above.

24 SECTION 7. Semantic Completeness for Boolean Relevant Logics

In Meyer & Routley 74a, R"~ is shown to be complete

c respect to its boolean semantics, and semantic argument 0

is given to show that it is a conservative extension of R .

In this sectjon we show that T"~, TW"~ and RW"~ are

similarly complete with respect to their expected semantics,

as a prelude to showin~ analogous conservative extension

results. So let L range over those three systems.

Now let a L model structure (L m.s.) be a

quadruple <K,O,R,*> with K a set, OEK, *a unary operation

on K, and R a ternary relation on K satisfying postulates

from the following as given below.

pl. ROab iff a = b

p2. Raaa

p3. Rabc => R2abbc

p4. R2abcd => R2a(bc)d and R2b(ac)d

p5. R2abcd => R2acbd (Pasch)

p6. a** = a

p7. Rabc => Rac*b*

p8. Raa*a

with the following definitions

Dfl. R2 abcd = df for some x (Rabx and Rxcd)

Df2. R2 a(bc)d = df for some x (Rbcx and Raxd)

Df3. a < b = ROab df

I. For TW"~: pl, p4, p6, p7.

II. 01 6 For RW : pl, p5, p, p7.

III. For T0

1: pl, p3, p4, p6, p7, p8.

IV. 01 6 For R : pl, p2, p5, p , p7.

For any (L) model structure~= <K,O,R,*> a

valuation V on M is a function from {atomic formulae

of£} x K into {True, False}. And for any such

25

valuation V, we define the interpretation I associated with

Vas a function from{£ x K} into {True, False}, satisfying

the following conditions:

Ip. I(A,a) = True iff V(A,a) = True, for any atomic

formula A.

I&. I(B&C,a) = True iff I(B,a) = True and I(C,a) = True.

Iv. I(BvC,a) = True iff I(B,a) = True or I(C,a) =True.

I+. I(B+C,a) =True iff for all b,c EK, if Rabc and

I(B,b) = True, then I(C,c) = True.

I-. I(-B,a) =True iff I(B,a*) =False.

II. I(IB,a) = True iff I(B,a) = False.

Then let us say that a formula A is true on an

interpretation I (on a model structure ~ = <K,O,R,*>) at

a point x (i.e., XEK) just in case I(A,x) =True. A is

verified by I on ~ just in case I(A,O) = True. Finally, a

formula A is L valid just in case for every L m.s. ~

and every interpretation I thereon, A is verified by I

on M -· We leave it to the reader to show that

Theorem 1.7.1. Lis consistent with respect to the L

semantics, i.e., for any formula A, A is provable in L

only if A is valid in L.

For the sake of notational convenience let us

write lA I for II(A,x) = Truel. We will then write X

lA = Fl for II(A,x) = Fl. Now we will proceed to show X

completeness along the standard lines, i.e., using L-

theories.

An L-theory is a non-null set of formulae

26

closed under adjunction and L-implication, i.e., if A is

an element of an L-theory and ~LA+B, then B is an element

of that L-theory. Taking L to be its set of theorems, a

theory S is regular iff L c S. S is prime iff AVBES

only if AES or BES. It is !-complete just in case for all

A, either AES or IAES; and it is !-consistent iff for no A

is it the case that AES and IAES. (Note that for any

L-theory S, Sis !-consistent iff Sis non-trivial.) And

S is !-maximal iff it is both !-consistent and !-complete.

Since AviA is a theorem of L, regularity and primeness

imply !-completeness.

Let us now say that an L-theory S is faithful

just in case it is closed under modus ponens, i.e., if

AES and (A+B)ES, then BES. Not all theories of the

contractionless systems are faithful. But

Fact 1.6.1. For any regular L-theory x, x is faithful.

27

Proof. Let x be an arbitrary regular L-theory, and

assume A+BEx and Adr.. Since 1-1

A+B+II(A+B) and x is

closed under L-implication, II(A+B)Ex. But ~L A+B~.A~B;

whence II(A+B)&(A+B~.A~B)Ex, since x is regular and

closed under adjunction. So by (boolean) Disjunctive •

Syllogism and the fact that x is closed under L-implication,

A~BEx. But on assumption AEx. So by a similar argument,

BEx as required.

The reader should take particular notice of this fact.

We will feel free to call upon it without reference in

what follows.

Now for any theorioox andy, let x0y = {CI there

is a B such that B+CEX and BEy}. Where there is no

danger of confusion, we let xy be x®y. And for any

L-theory S, an S-theory will be a non-null set of formulae

closed under adjunction and S-implication.

Now for any regular, maximal L-theory S, let

ST = {S-theories}. Note that since S is faithful, it is

itself an S-theory. As a prelude to completeness, we will

want some facts about ST. So in what follows, let S be

an arbitrary regular, maximal L-theory.

Fact 1.7.2. For all x,yEST, xyEST.

Proof. Let x andy be elements of ST. For any A,B and C,

28

(A+.B+CviC)EL. So since x is non-empty

and closed under 3-implication, whence under L-implication

since S is regular, (B+CviC)Ex, for all B. But y is

non-empty; so AviAExy by the definition®. Thus xy is non-empty

The by now standard argumenwutilizing Composition

and Prefixing, respectively, will show that xy is closed

under adjunction and S-implicatio~ to finish the proof.

~~~~--~~----

Fact 1.7.3. For all XEST, Sx = x; and in the case of

R01 and RW 01 , xS = x.

Proof. ~L A+A and ~ RWo A+.A+B+B, from which facts +

the reader can easily construct a proof.

For any XEST and any formula A, let [x,AJ =

{CI for some BEx, (B&A+C)ES}. Then

Fact 1.7 .4. For any x and A, [x,A]EST; and if Aix,

then xc[x,AJ.

Then let us define a ternary relation R on ST

as follows: Rabc iff ab £c. R2 is defined in the obvious

way. The reader can easily verify

Fact 1.7.5. R2abcd iff abc~ d, and R2a(bc)d iff

a(bc) ~ d. Further, R satisfies the

postulates from p2-p5 appropriate to L.

Of course, we are well on the way to doing the

typical canonical modelling. To accommodate * and Iv,

we will move to !-maximal S-theories. A few more facts

will put the needed machinery in good working order.

Fact 1.7.6. All maximal S-theories are prime.

Proof. Use De Morgan Laws for I.

Fact 1.7.7. Fov any !-maximal XEST, y = {AI-Aix} is a

!-maximal S-theory.

29

Proof. Choose an arbitrary !-maximal S-theory, say x, and

let y = {CI-cix}. We must show:

(1) y is non-empty. But since x is a !-maximal,

and thus !-consistent, S-theory, at least one of --p and

--lp fails to be in x. Whence y is non-empty on definition.

(2) y is closed under adjunction. Use Fact 1.7.6.

(3) y is closed under S-implication. Use

-Contraposition.

~

Next we prove a few lemmas to help us verify the

postulates when the time comes.

30

Lemma 1. 7 .1. For any x,y,zEST such that z is !-maximal,

if Rxyz, then there is a !-maximal WEST such that

Rwyz and x ~ w.

Proof. Choose arbitrary x, y and z satisfying the

conditions of the Lemma. Let X be the set of all w E ST

such that x c w and wy c z. X is non-empty (x is in it)

and partially ordered by c. Further, each chain of X is

bounded from above by the (possibly infinite) union of its

members. So by Zorn's Lemma, there is a maximal element of

X (maximal in terms of c, that is.) Let x ' be one such

maximal element. It will now suffice to show that x' is

!-maximal. Since z is !-maximal and hence !-consistent,

x' is !-consistent, since x'y c z. So it will suffice to ,

show that x is !-complete.

So assume for reductio that Aix' and IAix'.

By Fact 1. 7 .4 [x' ,AJ and [x' ,lA] are members of ST,

x'C[x',AJ and x'C[x',IAJ. So by the maximality of X~

[x',A]y ~ z and [x',IA]y ~ z. So let B,B',C, D,E,F be

such that BEx~ B'Ex: EEy, FEy, Ciz, Diz, (B&A+.E+C)ES

and (B'&IA+,F+D)ES. Then note that since S is regular,

((B&B')&(AviA)+.E&F+CvD)ES. Further AviAEx', whence so

is (B&B')&(AviA). But then E&F+CVDEx' and E&FEy. So

CvDEZ. But z is maximal, and hence prime by Fact 1. 7 .6.

So either CEZ or DEz which contradicts our assumptions .

• So x is !-maximal, which suffices for the proof.

31

In a similar fashion, one can prove

Lemma l. 7 . 2 . For all x,y,zEST such that y is !-consistent,

z is !-maximal and Rxyz, there is a !-maximal wEST such that

RXWZ and y C W.

Lemma l. 7 . 3 . For all w,x,y,zEST such that z is !-maximal

and R2w(xy)z, xy is !-consistent.

Proof. Let w,X,Y,Z be S-theories in accordance with the

lemma. Then w(xy) c z, by Fact 1.7.5. w is not empty,

so let AEW. z is !-maximal, so assume Biz. Then assume

for reductio that C&ICExy.

Now note that I-LA+.-B+CviC1 and 1-LCviC+-(C&IC). So

1-LA+.C&IC+B, whence (A+.C&IC+B)ES. So (C&IC+B)EW and BEZ,

which is absurd. So the proof is finished.

~

We are now ready for the first crucial lemma for

completeness. So let S be a regular, !-maximal L-theory.

Then let K = {xlx is !-maximal and XEST}. Let R be c c

the restriction of R(on ST) to K , and let a* = {AI-Aia} c

for each aEKc. Then

Lemma 1.7.4. Canonical Model Structure Lemma

<K ,S,R ,*> is an L.m.s. c c

32

Proof. SeK , R is a ternary relation on K by definition, c c c and Kc is closed under* by Fact 1.7.7. So it will suffice

to show that the appropriate postulates are satisfied.

Ad.pl. Right to left is immediate by Fact 1.7.3.

For left to right, use Fact 1.7.3, the !-completeness of a

and !-consistency of b.

Ad.p2,p6,p7 and p8, where appropriate,

The arguments are straightforward using Contraction for the

first and Double Negation, Contraposition and Reductio,

respectively, for the last three.

Ad.p3,p4,p5, where appropriate.

The arguments are similar in each case. We show only that

R~abcd ~ R~b(ac)d for TW0

'.

Choose a,b,c,deK and assume that R2abcd, i.e., c c

abc c d. We must show that for some xeK , R acx and - c c

Rcbxd, i.e., ac c x and bx c d.

On assumption, R;abcd and hence R 2 abcd by definition.

So by Fact 1.7.5, R 2b(ac)d, i.e., b(ac) c d. Then

by Lemma 1.7.3, ac is !-consistent. Recalling that d

is !-maximal, we see that bw c d and ac c w for some weKc

by Lemma l. 7. 2. But that is precisely what was to be shown,

so we are finished.

The other cases can now be left to the reader. (Use

Lemma 1. 7 .1 in place of Lemma 1. 7. 2 for the. other half of

p4, and Contraction for p3.) So the proof is completed.

33

Again let S be a regular, maximal L-theory, and

let M = <K ,S,R ,#> be as in the previous lemma. Now define c c

a canonical valuation V on M by: c -

Vc(A,x) =True iff AEX, for all atomic formulae A and all

XEK c.

Obviously, V is well-defined. So let I be the associated c c

interpretation. Then

Lemma 1.7.5. Canonical Interpretation Lemma

A iff AEx, for all formulae A and all XEK . X C

Proof. The standard argument, as in Meyer and Routley

73b, for instance, will do. (N presents no difficulties.)

The avid reader who actually wants to check the proof

will find Fact 1.7.6 useful for Iv; and Lemmas 1.7.1

and 1.7.2 will be handy for I•.

~

Given these two lemmas, the following will clinch

the completeness argument.

Lemma 1.7.6. Refutation Lemma

For any A, a non-theorem of L,there is a regular,

!-maximal L-theory of which A is not a member.

34

Proof. Let A be an arbitrary non-theorem of L, and let

X = {wlw is a regular, !-consistent L-theory such that

Aiw}. LeX, so X is not empty. By the standard argument,

X has a maximal member. Let x be one such. Then it will

suffice to show that x is !-complete.

First we show

Refutation Fact. IA•x

Proof. Assume for reductio that IAix. Then x c [x,IAJ,

whence [x,IAJ is regular and thus faithful. Further, by Fact

1.7.4 we see that [~,IAJ is an L-theory. So by the

maximality of x, either AE[x,IAJ or [x,IAJ is !-inconsistent.

But since IAE[x,IA], it is !-inconsistent in either case.

So assume ~ 1B&IA+C&IC, with BEx. But then it is

easy to show that l-1

B+.Av(C&IC), whence ~ 1B+A. But

since x is an L-theory and B•x, AEx. But this contradicts

x•X, thus proving the fact.

Then we show x is !-complete by reductio.

So assume Dix and IDix. Then let y = [x,DJ.

Again xcy, so by the Refutation Fact, IAEy. Now the argument

used for the Refutation Fact will show IDex, whis is absurd.

So the proof is finished.

~

That lemma puts us in the home stretch.

35

Theorem 1.7.2. L is complete with respect to the L

semantics, i.e., for any formula A, A is valid in the

L semantics only if ~LA.

Proof. By contraposition. So let A be a non-theorem of

L. By the Refutation Lemma, there is a regular, 1-maximal

L-theory of which A is not a member. Let S be such.

Then let M = <K ,S,R ,#> be as before. By the Canonical - c c

Model Structure Lemma, M is an L.m.s. Then let Vc and Ic

be as in the Canonical Interpetation Lemma. By that

lemma Ic(A,S) =False, since AiS on assumption. So A is

not L valid, and we are finished.

~

We sum up this section with

Theorem 1.7.3. Boolean Completeness Lemma

R0

1, Tol, RW 0 1, TW 0 1 are consistent and complete with

respect to their boolean semantics.

36

FOOTNOTES

1This is of course an instance of Ax21, which is stronger

than is needed with fusion in L, since A~BviB would do.

But without fusion, the extra strength is needed in

precisely this spot for the boolean T-systems.

37

SECTION 8. Conservative Extension

~~ 0 That R is a conservative extension of R · was

shown in Meyer and Routley 74. We can now extend this

result to T0

~, TW 0~ and RW 01 . Recent results in the study

of contractionless systems will ease the way in the latter

two cases. The technique used for the first case is an

adaptation of one used by Adrian Abraham in an independent

proof of the result of Meyer and Routley 74. So let us

first turn to T0 1.

The argument is semantic in nature. The full

semantics of T0 has yet to be published, but it turns

out to be much as one would expect from Routley and

Meyer 72. That is, a T 0

m. s. is Oj like aT m.s., except

for having

, pl . ROaa, and

pl". R2 0abc ~ Rabc

in place of pl. And a valuation must satisfy

Hereditary Condition. (HC). For any points x, y and

atomic formula p, if V(p,x) = True and x ~ y, then

V(p,y) =True. (x ~ y iff ROxy.)

All other relevant definitions are as usual. Notice that

adding - to T+ does not necessitate adding a set of

''regular worlds'' with special postulates to T+ m.s. as

was the case with E. (See Routley and Meyer 8+.)

38

The completeness proof of Routley and Meyer 1973a

can be straightforwardly adapted to show

Theorem 1.8.1. To Completeness Theorem

0

~ToA iff A is valid in the T semantics.

And let us say that a T0

m.s. is normal just in

case it satisfies the additional

pl'n. (Normality Postulate) 0 = o*.

Further let us say that A is normally valid just in case

it is valid on the normal semantics. Then we have

Lemma 1. 8 . 1. 0

T Normality Lemma

~ToA iff A is normally valid.

Proof. As in the Normality Lemma of Routley and Meyer

72b. Left to right is obvious by Theorem 1.7.1, since

every normal m.s. is an m.s. For right to left, assume 0

A is a non-theorem ofT . By Theorem 1.7.1, let

* 0 M = <K,O,R, >be a T m.s. with V a valuation thereon

such that I(A,O) = False, with I the interpretation induced , # , # * by V. Then let K = Ku{O }, let 0' = 0 and a = a ,

for all aEK, and let R' be defined by:

For all a,b,cEK,

' 1. R abc iff Rabc

2. RO'o'a iff ROOa

39

3 . RO'ab iff ROab

4. RaO 'b iff RaOb

5. R'O'aO' iff ROaO*

6. R'aO'O' iff Raoo*

7. R'abO' iff Rabo*

8. RO'O'O'

+ Now for all aEK, let a be a and a be a. And

,+ ,- * let 0 be 0 and 0 be 0 . The reader can confirm that , . + + - , R abc lff either Ra b c or a=b=c=O . Then note that

ROa-a+, and that Ro*oo* by p8. Further, one can assume

without loss of generality that SN1-SN3, below, hold for

M. (It is simple to show that they hold for all T"

canonical model structures.) With these facts in hand,

we leave it to the reader to show that~· = <K' ,0' ,R1 ,#>is a T"r

Now define a valuation v' on ~K such that for any

A, V'(A,a) = V(A,a), for all aEK; and V'(A,O') = V(A,O).

Let I' be the interpretation on~ associated with V'.

- + The reader will have already noticed in verifying ROa a

that ROo*o. The argument for Theorem 4 of section VIII of

Routley and Meyer 72b can then be adapted to show that for

all aEK and for any formula B, I'(B,a) = I(B,a) and if

I(B,O) =False, then I'(B,O') =False. (Use SNl below for

fusion.)

So! 'is a normal T"m.s. and I' (A,O') =False,

which suffices for the Lemma.

~

40

The Normality Lemma is strong enough for the

conservative extension argument for R01 . But T01 requires

something stronger. So let us say that a T0

m.s. is

super-normal if it is normal and additionally satisfies

SNl. c ~ d and Rabc * Rabd;

SN2. a~ b and Rcbd * Read; and

SN3. RaOb and Rbcd * Racd.

And let us say that A is super-normally valid iff it is

valid on the super-normal semantics. Then

Lemma 1.8.2. Super-Normality Lemma

~ToA iff A is super-normally valid.

Proof. Simply verify that the canonical model structures

satisfy SNl - SN3, and that this property is preserved

under the normalization construction.

The strategy for showing T01 is a conservative 0 0

extension of T will be to show that a normal T model

can be ''changed into" a T01 model that refutes the same

formulae in the original vocabulary.

The essence of the procedure is to add a new base

world and give it the properties that it must have.

Thinking in terms of canonical models, this amounts to

making a "copy" of the base theory, renaming it and fixing

R for it in the desired way. c

* 0 So let M = <K,O,R, > be a super-normal T m.s.

and let V be a valuation thereon with I the associated

41

interpretation. Then let !:'!--, = <K1 ,o1 ,~,#> where

(1) K1 = Ku {01 }

(2) # * a = a , for all aEK; # 01 = o1 , and

(3) (i) R1 abc iff Rabc, for all a,b,cEK;

(ii) ~b01c iff RbOc, for all a,b,EK;

(iii) R1 bc01 iff RbcO, for all b,cEK;

(iv) R1 o1 bc iff b = c, for all b,cEK1 ; and

(v) Rao1 o1 iff RaOO, for all aEK.

It is at (3)(1) and (3)(iii) that this technique

differs from the usual one. Pasch and p7 guarantee that

RabO iff RaOb* iff ROab* in normal canonical models, whence

a= b*. And of course ROab iff RaOb. But this is not the

case in general without Pasch. So we give R1 01 bc the

necessary property, and otherwise let 01 behave like the

original 0. Now to show that it works.

Lemma 1.8.3. Let ~and~~ be as above. ~~is a T 01

model structure.

Proof. As these things usually go, only p4 requires any

work to check. It obviously holds when a,b,c and d are

members of K. To finish the proof, one alternatively sets

a = 01 , b = o1 , c = 01 and d = 01 and shows that the

postulate holds in each case. We leave it to the reader

to check that supernormality gets you through.

* 0 Now let m = <K,O,R, >beaT .m.s., and let V be

a valuation thereon with I the associated interpretation.

Then let ~~ = <K1 ,o1 ,R1 ,#> be the corresponding T01 m.s.

as above. And define a valuation v1 on~~ as follows:

42

V1 (A,x) = True iff V(A 1x) = True, for all atomic formulae

A and all xEK; and

V1 (A,01 ) = True iff V(A,O) = True, for all atomic formula A.

And let r1 be the associated boolean interpretation. Then

Lemma l. 8 • 4 • For any A in the language of T0

, I1 (A,01 ) =

True iff I(A,O) = True, and for an XEK, I1 (A,x) = True

iff I(A,x) = True.

Proof. The two claims are proved simultaneously by

induction on the complexity of A. The argument follows

the familiar pattern and is left to the reader.

These two lemmas make easy work of

01 0 Theorem 1.8.1. T is a conservative extension of T ,

i.e.' f. ToA iff 0

f.ToiA, for any A in the language of T .

Proof. Left to right is immediate. Right to left is

by contraposition. So let A be a non-theorem of T0

• Then

by the Supernormality Lemma there is a super-normal T0

m.s.

~ and a valuation V and associated interpretation I on M

which falsifies A at O(of ~). Whence by Lemmas 1.8.3

and 1.8.4, A is not valid in the T01 semantics. So by

the Boolean Completeness Theorem of the previous section,

A is a non-theorem of T01 , which completes the proof.

~-

43

This strategy for showing conservative extension

will have to be amended for the contractionless systems.

The Normality Theorem for T depends upon the

fact that prime, regular T-theories are

-- complete since ~T Av-A. But of course (De Morgan)

Excluded Middle is not a thorem of RW and TW. Hence,

we do not have the liberty to set 01 = o1#, and will be

forced to introduce both a new 0 and 0*.

However, this task can be simplified if we take

into account some recent results from Slaney 8+a. There

the techniques of Meyer 76 are extended to contractionless

relevant logics showing them to be metacomplete. Slaney

utilizes a double metavaluation and notes that a more

''standard'' three-valued metavaluation could be used. 1

We shall not recapitulate the argument here, but merely

restate a useful corollary noted in that paper.

Slaney Lemma 1. RW and TW are prime.

As Slaney points out, his results easily extend

to RW0

, since o can be defined therein in the standard

way. Of course, this is not the case for TW, and all 0

of our attempts to extend these results to TW have

been barren to date. 2 So we abandon oat this point. 3

The RW and TW semantics can be specified from

the RW 01 and TW 01 in a similar fashion to that of To

from T01. That is, pl is exchanged for pl' and plN, the

Hereditary Condition is put in force, and, of course, Io

and II are dropped from the specification of an interpretation.

44

But given Slaney Lemma 1, RW and TW can be shown complete

with respect to their semantics by a straightforward

adaptation of the argument of Routley and Meyer 72, simply

defining * on theories as was done earlier in this section.

The reader can easily check this claim, so we move on to

reap the benefits.

Let K be the set of prime TW(RW) theories. Define c

a ternary relation Rc on Kc by Rabc iff a~ b ~ c, for

all a,b,cEKc. And for all aEKc' let a*= {AI~Aia}. '!'hen

define I : £xK ~{True,False} by I (A,x) = True iff AEx, c c c for all AE£ and XEK . Then c

Fact 1.8.1. <K ,TW,R ,*>is a TW model structure, and Ic c c

is an interpretation (associated with the obvious valuation)

thereon. Similarly for RW.

And of course

Lemma 1. 8 . 5 . Single Canonical Model Lemma

~TWA (~RWA) iff Ic(A,TW) =True (Ic(A,RW) =True).

We call <K ,TW,R ,*>the TW canonical model structure, and c c

Ic the canonical interpretation. Similarly, for RW.

We can now build a TW 01 (RW 01 ) model which will

refute all non-theorems of TW(RW). So let K1 * = Kcu{o1 ,o1 },

# * # '* and let a = a , for aEKc and set o1 = o1 and *# o, = o,. Then let R1 be the result of adding the following triples

to R : c

45

(i) <o1 ,a,a>, for all aEK1 ;

and in the case of RW 01,

( ii) <a,o1 ,a>, for all aEK1 ; and

(iii) # # <a,a ,o1 >, for all aEK,·

# And let ~~ = <K1 ,o1 ,R1 , >. Then

Lemma 1.8.6. M is a TW 01 model structure or a RW01

model structure, as the case may be.

Proof. Left to the reader. (As in Lemma, 1.7.3, but easier)

Now we will want two more results from Slaney 8+a.

Slaney Lemma 2. For all A ,B, ~ RW-( A+B) iff ~ RWA and

~ RW-B; and

Slaney Lemma 3. For all A,B, -(A+B) is not a theorem of TW.

For given these, we can define an interpretation on~~

in the expected way which will refute all non~heorems of

TW or RW, as the case may be.

So define a valuation v1 on ~~ as follows, for all

atomic formulae A (with 0 being TW(RW)):

l. v1 (A,x) = IC(A,x), for all XEKC;

2. v1 (A,D1 ) = Ic(A,O); and

3. V1 (A,o;) = Ic(A,O*).

And let I 1 be the associated interpretation on~~· We

now show

46

Lemma 1.8.7. For any formula A in the language of TW(RW),

(i) T (A,x) = I (A,x), for all XEK ; -, c c

(ii) I 1 (A,o1 ) = Ic(A,O); and

(iii) I1 (A,O~) = Ic(A,O*),

with I1 as above and Ic the canonical interpretation,

Proof. By induction on the complexity of A. The lemma

is guaranteed by definition of I 1 when A is an atomic

formula, and the cases for -,& and v are straightforward.

So assume A= B~c. We now proceed to show (i), (ii) and

(iii) by cases.

Case 1. Assume for (i) that XEK • It is then c

straightforward to show that I 1 (A,x) = Ic(A,x).

Case 2. It is straightforward on inductive hypothesis

that if Ic(B+C,O) = True, then I1 (B~c,o1 ) = True.

So to prove (ii), it will suffice to show the converse.

So assume that I (B~c,o) = False, which justifies the c

assumption that R Oxy and I (B,x) = True and I (C,y) = c c c

False, for specific x and y in K . Then by the Hereditary c

Condition, Ic(C,x) = False. So on inductive hypothesis,

I1

(B,x) = True and r1 (C,x) =False, which suffices to show

that I1

(B+C,0 1 ) = False, as required.

Case 3. It is again straightforward that if

Ic(B+C,O*) = True, then I9 (B+C,O~) = True. So to prove

(iii), we again show the converse.

Assume I (B+C,O~) = True. For TW, it is always

* the case that Ic(B+C,O ) = True, by Slaney Lemma 3 and

47

the Single Canonical Model Lemma. For RW, assume for

reductio that I (B+C,O*) =False. Then by I-, I (-(B+C),O) = c c

True. And by Slaney Lemma 2 and the Single Canonical Model

Lemma, I (B,O) = True and I (-G,O) = True. But then c c

Ic(c,o*) =False. So on inductive hypothesis I1

(B,01 ) = # # # # True and ~(c,o1 ) = False. But R0

1o

1o1 . So I

1(B+c,o1 ) =

False, contradicting the initial case assumption and

finishing the proof of the case and lemma.

------------------------

Given Lemmas 1.8.6 and 1.8.7 and the completeness

proofs of the previous section, we present without further prooJ

Theorem 1.8.2. TW0

1 and RW01 are conservative extensions

of TW and RW, respectively.

Notice that to this point neither t nor T have

been in the language. We shall actually want them, but

only as "placeholders'' in Gentzen systems for the Classical

or Boolean Relevant Logics. So we do want to know that

they can be added conservatively to these systems.

That question has already been answered in the

affirmative for R01 in Meyer 1979a. T is no problem in

48

any of the systems we are considering, since it can simply

be defined as pvlp. We now use the results of the previous

section to take care oft.

In the boolean semantics, t is added by the

interpretation clause:

It. t iff X = 0. X

And it is a straightforward matter to show that the semantic

extension is conservative, i.e.,

Lemma 1.8.8. For

L semantics iff it

Equally simple is

any t-free formula

is valid in the Lt

A, A is valid in the

4 semantics.

Lemma 1.8. 9. Lt is consistent with respect to the Lt

semantics.

Finally, we have

Theorem 1.8.3. Lt is a conservative extension of its

non-boolean counterpart without t.

Proof. t Let A be a non-boolean, t-free theorem of L .

By Lemma 1.8.9, A is valid in Lt, whence it is valid in

L by Lemma 1.8.8. So by Theorems 1.8.1 and 1.8.2,

it is a theorem of the non-boolean counterpart of L,

49

which suffices.

~

For ease of future reference, we sum up this

section with

Theorem l. 8. 4. Tolt, Ro1 t, TWolt and RWolt are

conservative extensions of To, R0

, TW and RW.

Note, however, that we have not claimed that the

boolean extensions with t are conservative extensions of

corresponding non-boolean systems. The following non-theorem

of R0

is valid in the TW 01 semantics: -tv(-t&t+A). So

Theorem l. 8 . 5 . TWo It •

o1t o1t o1t RW , T and R are not

conservative extensions of TWt, RWt, Tt and Rt, respectively.

Obviously, they are not conservative extensions of those

systems with fusion either.

It is also worth noting that the rule y

!-A 1--AvB 1-B

is not admissible in TW1 and RW1 . For -(p+p) v 1-(p+p)

is a theorem as an instance of boolean Excluded Middle.

And, of course, p+p is an axiom. But the following matrices

can be used to show that 1-(p+p) is a non-theorem of RW1

.

-+

3 2

1

0

3 J

3

3

3

3

2 - 1 -0 1

2 1

1 3

3 3

3

/'""' 1 2

"""/ 0

0 -0

0

1

3

3 2

1

0

50

The &, v and I matrices can be read from the Hasse

diagram above in the usual way. 3 and 2 are designated.

We leave it to the reader to verify that the axioms of RW1

are valid and that modus ponens and adjunction are

admissible rules. And 1-(p-+p) can be falsified by assigning

p the value of two, which suffices to substantiate the claim.

51

FOOTNOTES

l The three-valued metacompleteness argument for TW was

independently discovered also by Meyer and Martin.

2 Slaney does not discuss adding o to TW. We assume that

his attempts met a similar fate.

3of course this is no loss for RW, since o can be

reinstated by definition. Although o has turned out to

be technically important in the study of all relevant

logics, we do not think that this connective has a very

plausible, intuitive interpretation in systems of Ticket

Entailment. So we feel no great loss in not dealing 0

with TW .

4 of of of L still ranges over T , TW and RW .

52

SECTION 9. Semilattice Semantics

As a final preliminary, we present the semilattice

semantics for uR+ which will be used in the next chapter. u Formally, a R+ model structure is a triple <K,¢,v>

where K is a set, ¢EK (not necessarily the null set) and

v is a binary operation on K satisfying the following

postulates for all x,~ZEK:

pl. ¢vx = x

p2. XYX = X

p3. xv(yvz) = (xvy)vz

p4. xvy = yvx.

A valuation is defined on such a model structure

in the usual way. Given a valuation V on a model

structure, the associated interpretation I is a function

from £xK into {True,False} satisfying

Ip. I(A,x) = True iff V(A,x) = True, for all atomic A;

I&. I(A&B,x) = True iff I(A,x) = I(B,x) = True;

Iv. I(AvB,x) =True iff either I(A,x) =True or

I(B,x) = True; and

I+. I(A+B,x) = True iff for all y, if I(A,y) = True,

then I(B,xvy) = True.

Note that we assume £ to be without o and t, although

they can be added straightforwardly. We shall not be

53

concerned with them here. The interested reader is

referred particularly to Urquhart 73 where they are

dealt with in some detail.

Other semantic terminology can be brought forward

from the previous sections. Then we record from

Charlwood 81:

Charlwood Theorem 1. Semilattice Completeness

For any formula A, ~ A iff A is valid in the uR+ UR+

semantics.

Some variations on the uR+ semantics are discussed

in Urquhart 72, but Urquhart 73 contains a more in~depth

exploration. In particular, semantics are offered for a

positive semi-lattice system of Ticket Entailment, and

for corresponding contractionless systems. We name and

record them as follows.

uT+ semantics. A uT+ model structure is a

quadruple <K,¢,v,<;;>with K,¢ and vas before . .;; is a

binary relation on K satisfying

p6. .;; is transitive;

p7 . .;; is montane, i.e., if x.;; y, then xvz.;; yvz; and

p8. for all XEK, ¢ .;; x.

Valuations, interpetations, etc. are defined as before

except that I+ is replaced by

54

I +, I(A+B,x) = True iff for all y ~ x, if I(A,y) = True

then I(B,xvy) = True.

The semantics for uTW+ and uRW+ are specified by simply

dropping the idempotence condition p2 from the definition of

a model structure in the corresponding semantics with

contraction. And the semantics for pure implication and

implication-conjunction fragments are obviously specified

by dropping the irrelevant condition(s) on an interpretation.

With respect to these, Urquhart 73 gives the following

results:

Urquhart Theorem 1. The pure implication and conjunction­

implication fragments of TW, T, RW and R are complete

with respect to their corresponding semi-lattice semantics.

Of course, since R4 is not a rule of the corresponding

u u fragments of R+ and T+' the above result applies to

them as well.

u u u Completeness results for TW+, RW+ and T+ are

unfortunately still lacking. However, it is interesting

to note that semi-lattices can be regained as model

structures for uRW+ by putting a disjointness condition

on I+, 1 as will now be shown.

u u Let d RW+ m.s. be as R+ m.s., i.e., they are

triples whose base set K is a semilattide with an identity

¢. Valuations will be as before. To define an associated

interpretation, we need to define a notion of disjointness:

for all x,yEK: XAy = ¢ (x is disjoint from y) iff

either (1) x = ¢ or y = ¢, or otherwise (2) there is no

ZEK such that z ~ ¢ and either zvx = y or zvy = x. We

should also note that xAy = ¢ iff yAx = ¢, and

XA(yvw) = ¢ iff XAy = ¢ and XAW = ¢.

55

Given a valuation V, the associated interpretation

I is a function from £xK into {True,False} satisfying the

conditions for atoms, & and v as before, and also

di~. I(A+B,x) = True iff for all yEK, if yAx = ¢ and

I(A,y) = True, then I(B,xvy) = True.

Other relevant definitions are assumed as before. And

we write rAx1 for ri(A,x) = Truel and r, Ai for I(A,x) = False.

We now use techniques similar to those of Meyer 77

to show

Theorem 1. 9 .1. For any formula A, A is valid in the

d-uRW+ semantics iff it is valid in the uRW+ semantics.

Proof. The remainder of this section is devoted to proving

the theorem. Naturally, it will be done in two stages,

left to right and right to left. The strategy will be

similar in both cases, using contraposition.

STAGE I. Left to right.

The proof proceeds by contraposition. So choose an

arbitrary formula, A, an arbitrary uRW+ m.s. <K,0,0>, and

an arbitrarary valuation V thereon with associated

interpretation I. Then assume as

Stage I Hypothesis (SIH) IA0

, i.e. I(A,O) = False.

It will suffice to show that there is a countermodel to

u A in the d RW+ semantics. Let O,a 1 ,a2 , ••• be the

(distinct) elements of K. 2 We now proceed to build a u

56

d- RW+ m.s. So let 1 2 n 1 2 n ai,al, ... ,al, ... ,az,az, ... ,az,···,···

be distinct entities, and let Kd be the closure under

binary set union of {¢}U{a~/l~i and l~j}. J

Then define a function f:Kd+K as follows:

l. f(¢) = 0;

2. f({am}) =a , 1 ~ m and 1 ~ n; n n

3. f({x 1 u ... uxn}) = f(x 1 )® ... ®f(xn)' for x1' ... ,xn

distinct singleton elements of Kd.

To see that f is well-defined, simply note that

each element of Kd is either ¢, a singleton or the

union of a finite number of such singletons, and that

® is a commutative and associative operation on K.

Then note that f is an onto function (a surjection),

and further

Fact 1. 9 .1. For all x and y elements of Kd such that

xny = ¢, f(xuy) = f(x)®f(y).

57

The point of the construction is as follows. In

the commutative monoid semantics, a (multiple) fusion of

a point x need not be x; and further, it may be doing

some work, specifically in refuting some A+B. So in

converting • into a semilattice join, we will want as

many ''copies'' of x as will be needed to give us disjoint

points whose union can do the work of the multiple fusion

of x. That the construction works as we want is

essentially recorded in

Fact 1.9.2. For all x,yEK and all zdEKd such that

f(zd) = x, there is a WdEKd such that f(wd) = y,

zdnwd = ¢ and f(zduwd) = x®y.

The proof of the fact is straightforward and left to

the reader.

So let ~d = <Kd,¢,v>. Obviously, ~dis a

d-uRW+ m.s. To complete the proof of Stage I, it will

now suffice to define an interpretation on ~d that

"mimics" I, so that A is false at ¢. But this is easy

to do. Simply define Vd on ~d thus, for all atomic

formulae Band all XEKd:

Vd(B,x) = V(B,f(x)).

Then let Id be the associated interpretation on ~d' And

let us agree to write ldBxl for 1 Id(B,x) = True1 and

rdiBxl for Id(B,x) =False. Then note

58

Fact 1. 9. 3. For all w,zEKd and all formulae B, if

f(w) = f(z), then dB iff dB . w z

Proof. By induction on complexity of B. The base step

is guaranteed by definition of Vd. Only the case for+

in the inductive step is shown since the others are

straightforward. So choose arbitrary D and C, and assume

as

Inductive hypothesis (IH). The fact holds for all formula

D' of complexity less than that of D+C.

Next choose arbitrary w and ZEKd such that f(w) =

f(z). It will now suffice to show·d~(D+C) iff d~(D+C) . w z

The argument for left to right is ana,logous to

that for right to left. So assume

Case Assumption (C). d~(D+C) , whence by I~+ assume z

yEKd' dDY, d~C and ynz = 0. yuz

And recalling that f(w) = f(z), by Fact 1. 9.2, let

y'EKd be such that f(yF) = f(y), wny' =¢and f(wuy') =

f(w)®f(y), i.e., f(wuy') = f(z)®f(y). But by C and

Fact 1. 9.1, f(z)®f(y) = f(zuy). But by C and IH, for

all u,vEKd,

dD and d~C u v

such that f(u) = f(y) and f(v) = f(zuy),

So dD , and d~C ,, whence ~(D+C) -which y wuy w

finishes the proo~

Now we can prove

Lemma 1. 9 • 1. For all XEK and all formulae B, (Bx

iff for all yEKd such that f(y) = x, dBx).

Proof. The proof of the lemma proceeds by induction

on the complexity of B. The base step is immediate on

59

the definition of Vd. As usual, only the case for+ in

the inductive step requires serious checking. So choose

arbitrary D and C, and assume as

Inductive Hypothesis (IH). The lemma holds for all E of

complexity less than that of D+C.

Then choose arbitrary XEK. For left to right, we

proceed by contraposition. So choose yEKd and assume

Case assumption 1 (Cl). f(y) = x and di(D+C) . y

By Cl, let WEKd' wny = ¢, dD and diG w yuw Then Fact

1.9.3 allows us to use IH to conclude that Df(w) and

ICf(yuw)" But by Fact 1.9.1, f(yuw) = f(y)0f(w).

So Df(w) and ICf(y)®f(w)' i.e., ICx®f(w)" Whence by

I+, I(D+C)x, as required.

Right to left also proceeds by contraposition.

So assume

Case assumption 2 (C2). I(D+C)x

Then by C2 and I+, let ZEK, D and IC = . Then choose Z X'"'Z

arbitrary yEKd such that f(y) = x. By Fact 1. 9 .2, let

WEKd,f(w) = z, xnw =¢and f(xuw) = f(x)0f(w). It is

then straightforward by IH that I(D+C) • Whence by y

Fact 1. 9 .3, for all yEKd such that f(y) = x, I(D+C)Y,

which completes the proof

Now by SIH, lA . 0

By definition off, f(¢) = 0.

60

So by Lemma 1. 9 .1, diA¢. u So A is invalid in the d- RW+

semantics. Which completes Stage I of the proof of

Theorem 1. 9 . 1 .

STAGE II. Right to left.

We again proceed by contraposition. So choose an

arbitrary A, an arbitrary d-uRW+ m.s., ~=<Kd,¢,v>, and an

arbitrary valuation Vd thereon with associated

interpretation rd. Then assume

Stage II hypothesis (SIIH). diA¢.

The construction this time is the dual of the previous

one. Two non-disjoint points in the semi-lattice are

doing no work in determining the values assigned by Id

to any formula at either of those points. So in defining

®on ''copies" of such points, say x and y, we want to be

sure that x®y likewise does no work. If one thinks of a

point of a model as the theory determined by the

interpretation and thinks of ® as the theory fusion of

the semantics of traditional relevant logics, the question

becomes that of what theory we can assign to the fusion

61

of theories which were non-disjoint points that will not

change any of the theories we already have (since they

already do what we want) - preferably one which can be

added singly to the set of theories and leave it closed

under ®. In these terms, the answer is obvious. The

trivial or absurd theory exactly fits the bill.

So let ¢,a 1 ,a2 , ••• be the distinct elements of Kd.

Then let 1, ¢', a 1 ',a2 ', ... be distinct entities, and

let K = {1, ¢', a 1 ', a 2 ', ... }. Then define® on K by: 3

1. 1®z = z6¢1 = 1, for all ZEK;

2. for x,yEKd'

(a) x'®y' = (xvy)', if XAY = ¢;

(b) otherwise, x'®y' = 1.

It is clear that ® is a well-defined binary operation on

K, and that M = <K,¢' ,®> is a uR+m.s.

So define a valuation V on Mas follows, for all

atomic formulae Band all XEKd:

1. V(B,l) = True;

2. V(B,x') = Vd(A,x).

and let I be the associated interpretation.

It is then easy to check that 1 behaves as desired,

i.e., for all formulae B,

Fact 1. 9. 4. I(B,1) = True.

And by a straightforward induction on the complexity of

A, one can then show

Lemma 1. 9 .2. For any formula B and any XEKd,

I(B,x') = Id(B,x).

From this lemma and SIIH, we get immediately that

I(A,~') =False, which completes the proof of Stage II,

and thus of Theorem 1. 9 .1.

~

62

63

FOOTNOTES

1This idea, originally due to Meyer, arises quite

naturally from consideration of the subscripted Gentzen

systems of Chapter 2.

2The notation builds in an assumption that K is

denumerable, but this is mere notational convenience.

The construction to come would work as well on the

contrary assumption. The same applies for the

construction at Stage II.

3where convenient, as in this definition, we treat '

as a function from Kd into K.

SWITlSXS NITZlNIT9 GITldiHJSHOS . Z HITldVHJ

64

SECTION 1. Introduction

This Chapter is devoted to the study of subscripted

Gentzen systems for the positive semi-lattice logics

~H+' ~T+' ~RW+ and ~TW+. This work is based on that of

Aleksandar Kron. Kron 78 and 80 present six (cut-free)

subscripted Gentzen systems GT+-W, GR+-W' G2T+' G2R+'

G1T+ and G1R+. Unlike the systems of Urquhart 73, they

are proof theoretic in character being based on earlier

work by Kron (Kron 73 and 76) on deduction theorems for

relevant logics.

The first four of these systems were claimed to

be equivalent to their axiomatic counterparts. An argument

for the decidability of the first two is presented in

Kron 78, and TW+ and RW+ are claimed to be decidable on

this basis. Kron's work was broadly well-conceived, we

think. However, it is seriously flawed.

These systems are set up in §2. Section 3 and 4

are then devoted to a critique of Kron 78 and 80,

"respectively. Most significantly we show there that the

above cited equivalence claims are false. In the process

we show that the proffered proofs that Cut and/or

modus ponens is admissible (for all six of the systems)

are unsound. And we show that the decidability argument

of Kron 78 is likewise faulty.

So in §5 we make a fresh start, presenting the

systems GuT+' GuR+' GuTW+ and GuRW+ and gathering

preliminary facts about them. The systems are fitted

65

with a ''place-holder'' to facilitate the proofs of Cut and

modus ponens in §7. We then show in §6 that the place­

holders can be suitably eliminated, and that GuT+ and

GuR+ are in fact equivalent in an appropriate sense to

G1T+ and G1R+ of Kron SO.

developed independently.)

(However, our G-systems were

In §S and 9, we show that GuR+ is equivalent to

uR+. To do so, we utilize results from Charlwood So.

Although we believe that the other G-systems are equivalent

to the corresponding axiom systems, proofs (at least in

the style of §S and 9) must await proofs of semantic

completeness and of equivalence of the matching natural

deduction systems.

Finally, in §10 we show that GuTW+ and GuRW+

are decidable. As a matter of history, it was Kron 7S

which stimulated our interest in the decision questions

for contractionless relevant logics. And it was in

reflecting on the argument for decidability given in §10

of this chapter that we discovered the decidability

argument for LTW+ and LRW+ as presented in Chapter 3.

The problem of developing "natural" cut-free,

subscripted Gentzen systems for T+' R+' TW+ and RW+

remains open. This is a shame. For the subscripted

Gentzen systems provide, we think, a simpler proof theory

than do the Dunn-style systems of Chapter 3. Although the

complexity of nested extensional and intensional sequences

(or structural connectives) is simply traded in for the

complication of using subscripted formulae instead of

formulae, subscripts are just finite sets of integers.

And what could be more simple?

SECTION 2. Preliminaries.

A subscripted formula (sf) is an ordered pair,

the first member of which is a formula, and the second

member (the subscript) of which is a finite subset of

the natural numbers ( {1, 2, 3, ... } ) . (The Language L is

assumed without any propositional constants - for the

66

time being.) We use a,b,c,d,e,w,x,y,z with or without

subscripts (in the ordinary syntactic sense) and/or super­

scripts as variables ranging over subscripts. In practice

we write rAal for <A,a>. Max(a) is the numerically largest

member of a, if a is not empty; otherwisei it is 0. A

structure is a possibly empty sequence of sfs, and W,X,Y and Z

(with or without scripts) are used as variables ranging thereove

Then a sequent is anything of the form X 1- A a, provided that ¢

does not occur as a subscript on an sf in X. Let us call X the

<ultecedent and Aa the consequent of such a sequent. (We speak of

an occurrence of a sequent, a structure, an sf, a formula

or of a subscript in the obvious way.) And we often

write rJ-AI instead of r~A~I. In context we often use x

as the union of all subscripts occurring in X. And we

use E with or without scripting as a variable ranging

over sequents.

Kron's systems can then be formulated from the

following set of axioms and rules. (Standard set theoretic

notation is assumed throughout.)

AXIOMS

Aa f-Aa (a f. ¢) •

STRUCTURAL RULES

c f-

K 1 1-

X,Aa,Bb,Y }- Cc

X,Bb,Aa,Y }- Cc

Xf-C c

X, A }- C a c

X,A ,Y }-C Kz 1- a c

X,Aa,Y,Ba f-Cc

LOGICAL RULES

& f-X,A f- C a c

W}-X,A ,A ,Y }-C a a c

X,Aa,Y f-Cc

provided (1) a f. ¢; and

(2)aS.c.

67

provided (1) a is a singleton

X,Ba f-Cc I-&

Xf-Aa XI-Ba

X,(A&B) }-C a c X,(A&B)ai-Cc XI-(A&B)a

v 1-X,A ,Y,Z f-C a c X,B ,Y,Z f-C a ~ c provided

X,AvB ,Y ,z f- C a c (1) xny = x.nz = ynz = ¢;

(2) If Y is non-empty, a is the only prefix occurring

therein;

(3) Z is idemdis

(4) max(x) ~ max(a) ~ max(z).

1- v

+ 1-

X 1- A

X I- A vB a

XI-B a X I-AvBa

XI-Aa Y,Baubi-Cc

X, Y, (A+B)b t- Cc provided ( 1) b 1' ¢;

(2) anb = ¢;

68

(3) max(b) ~ max(a).

1-+ X,Aa 1- 8aub

X I- (A+B)b provided (1) anx 1' ¢;

(2) max(b) ~ max(a).

C 1- anq W 1- are rules of permutation and contraction,

respectively. K1 l-and K2 1- are weakening or thinning

rules. So in applications of such rules we refer to the

permuted sfs, the contracted sf and to the sf weakened in

the obvious ways.

Formulations of various systems are given in

different sections to come. Provisos that certain

subscripts not be empty are not given by Kron, but it

is apparent that they do no harm. Where provisos for

disjointness and max restrictions are assumed together,

~ is effectively < . Idemdis is defined in the next

section. Also, provisos (2) and (3) of v 1- differ from

those of Kron, but they suffice. Indeed, Fact 2.3.2,

below, shows that (1), (2) and (3) are unnecessary in

Kron 78. Finally, where neither (1), (2), (3) nor (4)

is in effect on v 1- , it can be taken as

X,A 1-C X,B 1-C a c a c

X, (AvB)a 1- Cc

69

SECTION 3. Critique of Kron 78

The systems GT+-W and GR+-W presented in Kron 781

can be specified as follows. GT+-W has the axioms given

in the previous section and the following rules:

W f-, K2 f- and all of the logical rules as stated.

c f- '

GR -W + comes from the former simply by dropping the provisos on

max from the rules that have such.

A derivation of a sequent E in GT+-W (GR+-W)

is a finite tree, branching upward such that

(1) each node of the tree is a sequent (occurrence);

(2) the bottom node is (an occurrence of) E; and

(3) each node is either an axiom or follows from the

node(s) immediately above it by one of the rules

of GT+-W(GR+-W).

The notion of immediately above (below) is taken as

primitive. The notion of above (below) is its transitive

closure. And we say that A is provable iff f-A is

derivable.

Where Der is a derivation and o is a particular

occurrence of some sequent therein, the subderivation

determined by o is the derivation that one would get by

deleting from Der all sequent occurrences except o and

those above it. A sequent occurrence o (immediately)

precedes a sequent occurrence o' in a derivation just in

case o is (immediately) above o'; similarly for (immediately)

succeeds. And predecessor and successor are used in the

70

obvious way. Then a branch of a derivation is a sequence

ol, ... ,on of sequent occurrences such that o1 has no pre­

decessors and on has no successors, and for all 1 < i < n, xi

immediately precedes oi+l" A branch segment is a subsequence

of a branch.

The weight of a derivation, say Der, is the length

of a longest branch, and the weight of a sequent occurrence

o in Der is the weight of the subderivation determined by

o. The conclusion (bottom node) of a derivation that has

weight n is said to be derivable with weight n.

Finally, the height of a sequent occurrence, say

o, in a derivation Der is the length of the branch segment

consisting of o and all sequent occurrences below it.

Kron makes the following major claims for the

systems given above.

Claim 1. Cut is admissible in an appropriate form.

(Theorems 3.1 and 4.1) 2

Claim 2. Modus ponens is admissible, i.e., if A and A+B

are provable, so is B.

Claim 3. GT+-W (GR+-W) is equivalent (in terms of provable

formulae) to TW+(RW+).

Claim 4. The systems are decidable. (Theorem 6.10)

We will show that his proofs of all of these claims

are unsound, and indeed that at least half of 1 is false,

and 2 and 3 are entirely false. To do so, we begin by

collecting a few facts about these systems.

In the first place

Fact 2.3.1. If X f-C is derivable, then c = x, i.e., c

c is the union of subscripts occurring in X.

Proof. On inspection, the axioms are such and the rules

clearly preserve the property.

The fact is elementary and will be taken for granted

hereafter.

Now let us say that a sequent l: is idemdis

(identical or disjoint) just in case for any subscripts

x and y having distinct occurrences in the antecedent

of E, either x = y or xny = ¢. A derivation is idemdis

iff all sequents occurring therein are. And a system

is idemdis just in case all of its derivations are.

Then let us call a sequent l: singular iff for

any subscripts x and y with distinct occurrences in the

antecedent of l:, if x = y, then xis a singleton. And

extend this terminology to derivations and systems as

before. Finally, a sequent (derivation, system) is

singularly idemdis just in case it is both singular and

idemdis.

Fact 2.3.2. GT+-W and GR+-W are singularly idemdis.

Proof. The axioms are singularly idemdis, and the rules

7l

72

preserve this property. The only rule that really requires

checking is+~. So choose an arbitrary_instance thereof,

say

E1 = X~A X Y,Bbux~Cc = E2

E = X,Y,(A+B)b!-Cc

and assume

Fact Assumption. E1 and E2 are singularly idemdis.

It will suffice to show that E is singularly idemdis.

If X is empty, b = bux and we are finished on the

fact assumption. If X is not empty, then bux is not a

singleton by provisos (1} and (2) of + !-. So by the fact

assumption, yn(bux) = ¢. But then ynb = ¢ and ynx = ¢.

Further, bnx = ¢ by proviso (2) of + ~. So we are finished

on the fact assumption.

~~

Note that provisos (1), (2) and (3) on vj-are

not needed to prove the above fact. So one can easily show

that they are not needed simply using the fact itself and

c ~ .

Then note that Kron's Theorem 2.1 is correct 3,

which is recorded formally as

Fact 2.3.3. If A is provable in GT+-W (GR+-W), it is

provable in TW+(RW+).

73

With these facts in hand, let us turn to Claim 1

given above. To show a Cut Theorem, one would have to show

that the subscripts occurring in a derivable sequent can

be re-written in certain ways, preserving derivability.

Kron attempts to accomplish this in his Theorems 2.3 and

2.4. However, if Theorem 2.4 were true from right to left,

other things being_equal, one could trade in singleton

subscripts occurring in the antecedent of a derivable

13equent for non-singletons. Proviso (1) on K2 1- leads

one to rightly suspect the contrary. For

Fact 2.3.4. It is straightforward to show that

(1) (A&(BvC)) { 1

} ~ ( (A&B)vC) { J}

is derivable in both GT+-W and GR+-W. But

(2) (A&(BvC)){ 1 , 2 } ~ ( (A&B)vC) { 1 , 2 }

is not.

Proof. Assume for reductio that (2) is derivable. It

is not an axiom, and by examination of the rules it could

only be the conclusion of an instance of & ~, f-v or

W ~ . And Fact 2. 3. 3 rules out & ~ and ~ v. So

(3) (A&(BvC)){I, 2 },(A&(BvC)){l,z} ~ ((A&B)vC){!, 2 }

must be derivable. But this is impossible by Fact 2.3.2,

ending the proof.

74

To finish the matter off formally, Kron's Theorem

2.4 from right to left can be stated as follows:

Kl. For any derivable sequent, say <A 1,(al-b)>, ... ,

<An,(an-b) > ~ <A,(a-b) >, such that for all l.;;; i.;;; n,

either ain b

derivable.

= ¢ or bcai, <A1 ,al >, ... ,<A ,a > is also n n

But of course, the previous fact presents a

clear counterexample to it. And we now know where to look

for trouble in the argument(s) for Claim 1, namely cases

which require subscript rewriting. Consider the following

case which arises in the putative proofs of Kron's Theorems

3.1 and 4.1. Assume that there is a derivation ending as

follows

(l) X,A ~B a aux

(2) X 1-A+Bx

and another ending thus

(3) y' ~ Ay Z B I- C ' xuy c

( 5) Y,Z,A+Bx I-Cc

It is required that

(6) y ,z,x 1- cc

be derivable. His argument for (6) runs as follows:

On the basis of (l) and the subscript re~writing theorems,

(l ) X,A ~ B y yux

is derivable- but we needn't go any further.

75

We have already seen that the suggested rewriting could

not always be done when a is a singleton but y is not.

Now if Theorem 4.1 were true in spite of its

proof being unsound, Cut would be admissible in GR+-W'

as in Claim 2. And if Claim 2 were true, Claim 3 would be

also. So producing a TW+ theorem (and thus RW+ theorem)

which is not provable in GR+-W (thus, nor in GT+-W)

will suffice to substantiate most of our own negative claims.

And the previous arguments and discussion tell us how to

find such: Look for a TW+ theorem for which any GR+-W

derivation would require that a suitable version of

Distribution be used as the right premise of + ~ .

Meyer suggested (p+qvr)&(p+p)+.(p&q+r)&(r+r)+.p+r;

and it works. However, p+q&(rvs)+.p+(q&r)vs

is simpler and will do as well, as the diligent reader can

easily verify.

Before turning to Claim 4, we should point out

briefly what is wrong with

the admissibility of modus

His claim that

the independent argument 4

ponens (pp. 72-73) .

(1) A+BEK iff A+BEJ and either AiK or BEK

for

is false and does not follow from his definition of Del.

If A+BEK, then A+BEJ and either A is not retained in J

(and thus inK) orB is not deleted from J. But that B

is not deleted from J doffinot imply that BEK, since it

could be the case that B is not deleted from J because it

wasn't in J to begin with.

76

With respect to Claim 4, the decidability argument

appeals to his Theorem 6.5. This theorem claims that if

A and B occur in the antecedent of a sequent of some X y

derivation with xny = ¢ and have descendants (in the

usual sense) in the antecedent of a succeeding sequent,

the prefixes of those descendantsare likewise disjoint.

But this is false, since such descendants may be one and

the same sf occurrence, as in the following (foreshortened)

derivation.

l: = (A+B) {2}' (B+C) { 3 }'A{4} 1- C{2' 3,4} 1-+

( A+B) { 2} ,(B+C) { 3} 1- (A+C) { 2 , 3} 1-+

(A+B) {2} 1- (B+C+.A+C) {2 } D{l,2} I-D{l,2}

l: , = (A+B) {2}, ((B+C+.A+C)+D) { 1 } 1- D{l , 2}

B+C{ 3} and A{ 4 } together satisfy the initial conditions

of the theorem in l:, but have the same descendant in l:'.

So Theorem 6.5 is false and the decidability

argument is undone, as we claimed. The argument of §2.10

below can be adapted to show that GT+-W and GR+-W are

decidable. But these systems are too weak to make that

fact interesting.

Finally, there are several other mistakes in the

article, most of which are minor (possibly typographical)

+I-

and can be easily corrected. We now point out and correct

one of them, since our claim for Fact 2.3.3 relies upon

it. The restriction on modus ponens for T+-W in the

definition of propf from hypotheses (p.62) should read:

77

if S = T+-W and bj I¢, then max(bi)>max(bj). His

Theorem 2.1 can then be proved along the lines suggested

there.

FOOTNOTES

1unless otherwise specified, referenc~in this section

are to Kron 78.

2The statement of Theorem 4.1 is slightly garbled

(probably typographical errors); but the intention is

reasonably clear in context and can be sorted out by

referring to Theorem 6.1 of Kron 80, in any event.

78

3However, his proof requires the minor correction supplied

at the end of this section.

4Kron's argument is an adaptation of the techniques of

Harrop 56, which are similar to the metavaluational

techniques developed independently in Meyer 76a. It was

Meyer who first pointed out the problem in Kron's

argument.

SECTION 4. Critique of Kron 80.

G2T+ and G2R+ (of Kron 80, of course) can be

formulated by dropping the contraction restriction

(proviso (2) anb = ¢) on+ ~and restrictions (1), (2)

and (3) on v ~from GT+-W and GR+-W' respectively.

Appropriate definitions carry over from the previous

sections. We refer to G2-systems in the obvious way.

Kron claims

79

Claim 5. Cut is admissible in an appropriate form in the

Gz-systems. (Theorems 4.1 and 6.1).

Claim 6. Modus ponens is admissible in the G2-systems.

(Theorems 5.1 and 6.3).

Claim 7. The G2-systems are equivalent to their

axiomatic counterparts. (Theorem 7. 4).

But given the form of weakening, namely K2 ~ again, one

would expect these claims to go the way of Claims 1, 2

and 3 of the previous section. And this is indeed the

case. For some of the subscript re-writing theorems

(Theorems 3.3 and 3.4) are false, just as their analogue

from Kron 78 was, as will presently be shown.

First we need a few facts. Obviously

Fact 2.4.1. Let G' be the result of deleting+~ from

G2 R+. The distribution sequents (1) and (2) of p. 73

of the previous section are derivable in G' iff they

so

are derivable in G2 R+.

And recalling that provisos ( l), ( 2) and ( 3) on v f- were

redundant in GR+-W, it is clear on inspection that

Fact 2.4.2. G' is a subsystem of GR+-W, i.e., for any

sequent E, E is G' derivable only if it is GR+-W derivable.

Sequent (l) referred to in Fact 2.4.1 is easily

derived in G2T+' hence also in G2 R+. So if either of

Kron's Theorems 3.3 or 3.4 were true for the G2 -systems,

sequent (2) referred to in Fact 2.4.1 would be derivable

in G2R+' and thus by that fact also in G'. But it is

not so derivable, by Fact 2.3.4 and Fact 2.4.2. So those

theorems are false (for the G2-systems), as was previously

claimed.

Note that Theorems 3.3 and 3.4 are not corollaries

of Theorem 3.2 as claimed in the paper. But they are

corollaries of the corrected Theorem 3.2 given in

Kron 81. So the argument above suffices to show that

the corrected theorem is itself false.

As one would expect, the arguments for Claim 5

fail in the same way as the arguments for the Cut Theorem

of Kron 78. Claims 6 and 7 are false. Again, it will

suffice to show that Claim 7 is false; and the same

counterexample will do. But since the G2 -systems are not

known to be decidable, the argument requires a bit more

work. So let us gather a few facts.

It is relatively straightforward to show

Fact 2.4.3. p +q&(rvs)+. p+ (q&r)vs is provable (in a

G2 -system) iff (p +q&(rvs)){l}' p{2

} r ((q&r)vs){1 2

} is . '

derivable.

Now define a function T from sequents to formulae

(of the classical sentential logic K) as follows:

(i) T(rA)=A, and

(ii) T(<A1 ,al), ... ,(An,an) r(A,a)) = A1& ... &An +A.

Then by a straightforward induction on the weight of

derivation of ~. we have

Fact 2.4.4. For any sequent ~. E is derivable in G2~ only

if T(I) is a theorem of K.

Next, let r be the set of all finite sequences

(including the empty sequence) that can be built up from

p +q&(rvs){l}'p{ 2 }' and q&(rvs){l, 2 }. Using Fact 2.4.4

above and Theorem 3.1, it is simple to show

Fact 2.4.5. For all X E r, neither rp nor X rp{l} is

derivable in G2R+.

Further, Facts 2.4.4 and 2.4.5 can be used in a

straightforward induction on weight of derivation to prove

81

let Y E 6 with Der a G2 R+ derivation of

Y,q{l, 2 } ~ ((q&r)vs){l, 2 }. Then let S be the rightmost

branch of Der. By inspection of the rules, either

r{l, 2 } ~r{l, 2 } or s{l, 2 } ~s{l, 2 } occurs inS. But again

by inspection of the rules (using Fact 2.4.7 for~~ and

noting the singleton restriction on K2 ) q{l 2 } occurs in '

the antecedent of every member of S - which is absurd.

Hence (i) cannot be the case. And a similar argument

will show that (ii) cannot be the case. So the proof

is finished.

~

Slb

83

with combined weight w iff for each l ~ i ~ n Xi ~<Ai,xi>

is derivable with weight wi and w = w1+ ... +wn. Let X be

<AI,aJ>, ... ,<An,an> and let hl,···•hn be the complexities

of AJ,···•An, respectively. The complexity of X is

hJ+ ... +hn.

We shall write rx I instead of rxl if X is not a

empty and each prefix occurring in X is a. And we say

that ~Xa is derivable just in case each formula occurring

in Xa is provable. And for any two structures X andY,

we write rx;y;l to indicate that each sf occurring in Y

also occurs in X.

Now let Y be <B 1 ,b 1>, ... ,<Bm,bm> with the following

conditions holding for any l ~ j ~ m:

(l) either aS bj or a n bj = ~; and for G1T+

(2) if an bj =~'then max(a) < max(bj);

0

and let Y be the result of deleting all sfs with the

subscript¢ from <B 1 ,b 1-a>, ... ,<B ,b -a>. Finally we state m m

Kron's Theorem 5.1. If U is of degree h, ~U and a a

Y!Ual ~Bb are derivable with combined weight w, (l) and

(2) are satisfied and all sfs occurring in Y with the

subscript a are members of U , then Yo ~<B,b-a> is derivable. a

The proffered proof of this theorem by double

induction breaks down in the following sort of case. Assume

there are derivations ending as follows:

84

l:1 = 1-A

E 2 = 1- AvB

and

1: 3 = X,AvB ,A I- C a a c X,AvBa,Ba 1- Cc = E4

E 5 = X,AvB ,AvB I- C a a c

with xna = ¢, just to make things simple. Note that the

complexity of AvB ,A is greater than that of AvB ; and, a a a indeed, the combined weight of derivation of E1 and E2

is greater than that of 1: 2 • So neither inductive hypothesis

(p.395) can be used to apply cut to E1,E 2 and 1: 3 as would

be required. Thus the argument fails.

Note that if 'x/Y/ 1 is taken to indicate that all

sfs occurring in Y occur in X the same number of times as

they occur in Y, the argument will break on the obvious

cases involving contraction. Indeed, all variations on

Kron's argument which this author has attempted have met

with similar fates. So it is best to start from the

beginning to investigate systems similar to the G2-systems.

85

SECTION 5. G-Systems

We will now give a first formulation of subscripted

Gentzen systems, GuTW+, GuT+' G ~W+ and G ~+'

So we let L range over the obvious systems, and refer to

the G-systems, GT-systems, etc. in the straightforward way.

Contractionless systems will be referred to as GLW, or simply

as the W-systems. With an eye to proving a Cut Theorem, we

introduce a structural analogue of t, namely I, as in

Belnap 8+. So let <I,a> be an sf for any non-empty subscript

a, and note that I is a structural constant, not a formula.

Otherwise, appropriate definitions are brought forward

from previous sections.

The G-systems can be formulated from the following:

AXIOMS

A a ~A a, for any formula A and non-empty subscript a.

RULES

Structural Rules

c 1-

K 1-

X,Z,W,Y ~ Cc

X,W,Z,Y ~Cc

X ~ Cc

X, Y ~ C c

w ~ X,Y,Y ~Cc

x,Y ~cc

provided (l) y ~ ¢ and ¢ does

not occur in Y; and

(2)ys_c.

The G-systems can be formulated as follows:

u G TW+ has all of the axioms and rules as stated,

but without proviso (2) on j- ,..,. ;

G0RW+ comes from G~W+ by dropping proviso (3)

from + 1- and 1- +, and adding proviso ( 2) on 1-+.

The systems with contraction come from their

W-counterparts by dropping proviso (2) from + 1- and

1- +, and adding I 1- with proviso (3) for GuT+ and

without proviso (3) for G0 R+.

Note that GW does not have I 1- as a rule, not

even with the restriction that anb = ¢, as would be

87

required. The reason for this is simple. The subscript

manipulation that seems to be required to handle I l-

in the proof of Cutdo not preserve derivability in those

systems. We can think of no reason why this should be.

So we suspect that there is some "book-keeping" fact

which has so far gone unnoticed.

The lack of I 1- forces G0RW to remain non-empty

on the left for the proof of Cut, and blocks the proof

of modus ponens for both W-systems. This situation does

no harm for the time being, since we are not yet in a

position to prove an equivalence for those systems even

with modus ponens. But it is inelegant and ought to be

rectified.

Without I 1- there is no point in having sfs with

I as their first member. So we banish them from the

86

Logical Rules

& f-

v f-

..,. f-

f--+

I f-

X,Aa f-Cc X,B f- C a c f-&

X,(A&B)a f-Cc X,(A&B) f- C a c

Xf-Aa Xf-Ba

Xf-(A&B)a

X,A f-C X,B f-C a c a c

X, ( AvB) a f-Cc

X f-A Y ,<B,aub> f- C a c

X,Y, (A-+B)b f-Cc

X,Aa f-<B, x ua>

X f- (A-+B) X

X,<A,aub> f- C c

X, Ib , A a f- C c

f- v X f-Aa

X f- ( AvB) a

X f- Ba

X f- ( AvB) a

provided (1) b i ¢;

(2) bna = ¢;

(3) max(a) ~ max(b).

provided (1) xna = ¢;

(2) X f ¢

(3) max(a) ~ max(x)

provided (1) a i ¢ and b i ¢;

(2) max(a) ~ max(b);

In an application of I f- , we say that the displayed

occurrences of <A,aub> and <A,a> have been weakened onto.

88

W-systems.

Now the following simple facts can oe established

by straightforward inductions on weight, which are left

to the reader.

Fact 2.5.1. Let E be a derivable sequent in GL.

Then

( 1) The antecedent of l: is not empty when L is

GURW

(2) The subscript of the consequent of l: is equal

to the union of the subscripts occurring in the

antecedent of E.

(3) The null set does not occur in the antecedent

of E.

(For the remainder of this Chapter, references to

sequents of a G-system are to sequents satisfying the

conditions of the above fact.)

Fact 2.5.2. ~ & is invertible in the sense of Curry 63,

89 i.e, if X ~(A&B) is derivable, so are X f-A and X ~B

X X X

Fact 2.5.3. !;-+is invertible,

i.e., if X f-A+B is derivable, so is X,A r <B,xua>, for x a

flome a satisfying the appropriate proviso(s) of f--...

Proving Cut and Equivalance will require the

ability to rewrite subscripts in certain, sometimes

peculiar, ways. The following strong rewriting Lemma

will help us prove the facts that are needed. For the

sake of convenience let us allow formula variables to

range over formulae and I for the rest of this section -

except where I would obviously not be permitted.

For any subscripts al•···•an,ai, ... ,a~ let

a= {a 1, ... ,a} and a' = {aj, ... ,a'}. Let a= Ua and n n

a'= Ua'. And let o andy range over the various unions of

the ai, i.e.,, over elements of {Ufll fl ~a and fl I </>}.

And . ,

where o lS a 1 u ... ua., for example, let o be J ,

similarly for y and y . Note that in a , ,

aw ... uaj;

degenerate case, o might simply be a 1 , for instance.

Then

Lemma 2.5.1. Rewriting Lemma

, , For any subscripts a 1, ... ,ah,a 1, ... ,an and any formulae

A,Al, ... ,A , if l: = <A1 ,a1>, ... ,<A ,a > r<A,a> is derivable n n n

. u u , , , ,. ln G R+(G T+), then so is l: = <A 1 ,a 1>, ... ,<Anp.n> f-<A,a >

provided, for any o and y,

(1) , ,

if o c y, then o c y , and

(2) for the Gl\r+ for alll <;; i,j

then max(ail > max(a~),

90

.;; n, if max Ca i l > max (a j )

Proof. By induction on weight of derivation of E. The

base step is straightforward, and only four cases of the

inductive step require checking, since only four

rules alter subscripts in any way.

Case 1. For K f- , assume that

E = <A 1,a 1 >, ... ,<A ,a >,<B 1,b 1 >, ... ,<B ,b >f-< A,a> n n m m

is derivable following from

E 1 = <A 1 , a 1 >, ... , <An , ar{> f- < A, a > .

, ' , , And let a 1 , ... ,an,b 1, ... ,bm satisfy the provisos of the

, , , lemma, whence b 1u ... ubm Sa. On inductive hypothesis,

, E 1

, , , = <A 1 ,a 1>, ... ,<A 1 ,a > f-<A,a >

n

is derivable. And by the previous observation, it is , ,

clear that E follows from E1 by K f-, which suffices.

Case 2. For f-+, assume there is a derivation

ending as follows

E1 = <A 1,a 1>, ... ,<A ,a >,<B,b>f-<C,aub> n n

E = <A 1 , a 1 >, ... , <~ , an> f- <B+C , a>

with boa=¢, of course.

the proviso of the lemma.

, , Then choose a 1, ... ,a satisfying

n , p

Obviously, a1, .. ,an,

9l

{max(a')+l} also satisfies the proviso. So on inductive

hypothesis,

, , , , , , <A 1 ,a 1>, ... ,<A ,a >,<B, max (a ) + 1> 1- <c, a u {max( a ) +1}> n n E I =

is derivable, from which r' follows by ~+. (The proviso's

on disjointness and maxima are obviously met.)

Case 3. For + ~ , assume that

E = <AI,aJ>, ... ,<A ,a >,<B 1 ,b 1>, ... ,<B ,b >,<A+C,c>I-<D,aubuc> n n m m

is derivable, following from

r 1 =<A 1 ,a 1>, ... ,<An,an>~Aa' and

l:2 = <BI,bl>, ... ,<B ,a >,<C,auc> ~<D,aubuc> . m m , , , , ,

Then choose a 1 , ••• ,a ,b 1 , ••• ,b ,c satisfying the proviso. n m

Obviously, r{ is derivable on inductive hypothesis.

Further, r' follows from it and

, , , , , , , . , E2 = <Bl>b 1>, ... ,<B ,b >,<C,a uc > 1-<D,a ub u c > m m

p

by + 1-. So it will suffice to show that 1: 2 is derivable,

for which it will suffice to show that , , , ,

bJ, ... ,b ,(auc),b 1 , ••• ,b ,(a uc) satisfy the proviso. m m

But this is clearly the case, since auc = a 1 u ... uanuc and

a'uc' = aju ... ua'uc', and since obviously max(a') ~ max(c') n

in the appropriate case; so we are finished.

Case 4. The argument for I ~is similar to that

of Case 3 and is left to the reader.

-~

This is quite a powerful rewriting lemma, which

has the useful

Corollary 2.5.1. For any structure X, for any

formulae B and C, for any subscript b such that bnx = ¢

and for any subscript d whatsoever, X,Bb ~<C,xub> is

derivable in GuR+(GuT+) only if X,Bd ~<C,xud> is

provided, in the case of GuT+' that for all c occurring

in X, max(c) ~ max(b) iff max(c) ~ max(d).

92

The corresponding rewriting lemma for the W-systems

is more complex.

Lemma 2.5.2. W-Rewriting Lemma

, , For any subscripts a 1, ... ,a ,a1, ... ,a and any formulae n n

A1 , ••• ,An,A' if

z: = <A 1 ,a 1>, ... ,<A ,a > i-<A,a> is derivable in a particular n n

, , , W-system, then so is <A 1 ,a 1>, ... ,<A a > ~<A,a >

n' n .

provided

(l) for any o, y, if o S y, then o' c y';

(2) for all 1 ~ i, j ~ n, if ainaj

and

93

= ¢, then a:na: = ¢; l J

(3) for GuTW+' for all 1 ~ i,j ~ n, max(ai) ~ max(aj)

only if max(a~) ~ max(aj)

Proof. The proof proceeds in a fashion similar to that of

Lemma 2.5.1 and is left to the reader.

It has the slightly weaker

Corollary 2.5.2. For any structure X, for any formulae

B and C, and for any subscripts b and d such that bnx = ¢

and dnx = ¢, X, Bb ~ <C,xub> is derivable in a particular

W-system only if X, Bd ~<C,xu:i>

for GuTW+ that max(b) ~ max(c)

all c occurring in X.

is - provided, of course,

iff max(d) ~ max(c) for

Some particular little facts, all corollaries of

the rewriting lemmas, will be useful - the first for the

Vanishing-t Theorem to come and the second for handling

I ~ in the proof of Cut.

Fact. 2.5.4. Let a 1 , ••• ,a ,b be subscripts such that for n

all 1 ~ i ~ n, either bna. = ¢ or b c a.. Then for any l l

formulae A,A 1 , ••• ,A , <A 1 ,a 1>, ... ,<A ,a > ~ <A,a> is n n n derivable in G only if <A 1 ,a 1-b>, ... ,<A ,a -b>

n n

J <A,a-b> is - provided in the case of the GT-systems that

for all l ~ i ~ n, max(a.) > max(b). l

Fact 2.5.5. Let a 1, ... ,am,b 1, ... ,bn,c be subscripts

satisfying the following conditions

(l) for all l ~ i, j ~ n and for all l ~ k ~ m,

max(bi) = max(bj) and max(bi) ~ max(ak);

(2) in the case of the W-systems, for all

l ~ i ~ n, for all l ~ j ~ m, bi nc =

ajnc = ¢.

(3) in the case for the GT-systems,

max ( b 1 ) ;;;. max ( c ) .

Then for all AJ, ••• ,~,BJ,.,.,Bn,D' if

<A 1 , a 1 >, ... , <A , a >, <B 1 , b 1>, .•• , <B , b > f- <D, au b> m m n n ·

is derivable in GL, so is <AJ,al>, ... ,<Am,am>'

<B1 ,b1uc>, ... ,<Bn,bnu c> f-< D,aubuc>.

We now have sufficient control over subscripts

to prove the desired Cut and Equivalence Theorems. But

first we will want to show that our placeholder, I, can

be done away with under the appropriate conditions.

94

95

SECTION 6. Vanishing-t

Technically, we will show a vanishing - I theorem,

but somehow it doesn't have the same ring to it. And I

"really" is t, just with a different name. Of course, we are

concerned here with only GuT+ and GuR+. So let L range only

over them in this section. And for the sake of notational

convenience, we will identify sequents which differ only in

the order of the constituents of their antecedents. 1

Lemma 2.6.1. Vanishing-t Lemma

Let X and Y be arbitrary structures such that for some

subscript Y

(1) all sfs occurring in Y have y as their subscripts; and

(2) for each subscript a occurring in X, either

any = ¢ or yea;

(3) for G0 T+' if X is non-empty, then max(a) > max(y),

for all a occurring in X,

Then if X,Y rCc is GL derivable, so is X- r<C,c-y> ,

where x- is the result of replacing a by a-y in X, for

each a occurring in X.

Proof. By strong induction on weight of derivation.

Note that if Y is empty, the lemma holds by Fact 2.5.5.

Thus the base step is straightforward, since I does not

occur in axioms. So choose an arbitrary j ~ 1 and assume

Inductive Hypothesis (H). For any X', Y', y' satisfying

the conditions of the lemma and for any c , if x', y' ~c c c

is derivable with weight j'<j, then x'- r <C,c-y'> is

derivable with some weight k ~ j'.

96

Next choose arbitrary X,Y,y and arbitrary Cc' and assum

Conditional Hypothesis CCl. l: = X, Y J- C c is derivable with

weight j and X,Y and y satisfy the conditions of the lemma.

(Let Derc be the derivation of E.)

It will then suffice to show that E- = x- i-<C, c-y > is

derivable with some weight k ~ j. The inductive step

proceeds by cases.

Case l. Derc ends with an application of I 1-,

say

E 1 = W, <A, au y >, Z 1- C c

E =W,A ,I ,ZI-C a y c

with X = W,A andY = I ,z. 2 Note that by choice of y in a Y

accordance with (2) of the lemma, yea or yna ~ ¢. So

a-y = (auy)-y = a, and ye(auy). Further, the subscripts

in z 1 satisfy condition (3) for the lemma when applicable.

So apply (H) to z 1 to finish the case.

Case 2. Derc ends with an application of + 1- :

E1 = W1 ,W2 i-<A,w 1 uw 2 > Z1,Z 2 ,<B,buw 1uw 2 > I-Cc = E2

E = W1 ,z 1, (A+B\,W 2 ,Z 2 1- Cc

with X= W1,Z 1,(A+B)b andY= W2 ,z 2 , thus y = w2 = z 2 .

Note that either w1ny =¢or yew 1 , and bny =¢or ycb.

It is clear by (C) that E1 satisfies the conditions of

the lemma. So by (H), Ej = W1 I- <A,w 1-w 2> is derivable

with appropriate weight. It is equally clear by (C) that

E? satisfies the required conditions, and that

97

(w 1nw2 =¢or w2cw 1) and (bnw 2 =¢or w2cb). So again

by (H), Ez = z1,.q,,(b-w2)u(w1-w2)>1-<c,c-w2> is derivable

with appropriate weight. Then note that in the case for

GuT+' max(w 1uw2);;;. max(b) by proviso (3) on+ 1-; but by

(C) -condition (3), in particular- max(b) > max(w2 ).

Whence, max(w 1);;;. max(b). So E- follows from E! and Ez by + ~ , and hence is derivable with appropriate weight -

which finishes the case.

Case 3. If Derc ends with an application of

any other rule, the argument is straightforward and

left to the reader.

~

Now let us say that a sequent is I-free just

in case I does not occur in it, and that a derivation

is I-free just in case each sequent occurring in it is

I-free. The Vanishing-t Lemma then makes short work of

Theorem 2.6.1. Vanishing-t Theorem

For any subscript a and any formula A, Ia ~ Aa is derivable

in GL iff there is an I-free derivation of 1- A.

Proof. Left to right is immediate by the Vanishing-t

Lemma and inspection of the rules. Right to left is by

induction on the weight of derivation of 1- A. If A is,

say, B+C, then Bbl- Cb is derivable for some b, by Fact

2.5.3. But by Corollary 2.5.1 or 2.5.2, as the case

may be, .q,,au{max(a)+U>I-<c, a'u{max(a)+l}> is also

derivable. Whence by I 1-, I ,.q,,{max(a)+l}> ~ a

<c,au{max(a)+l}> is derivable. So by 1-+, Ia ~ (B+C)a

98

is derivable, as desired.

Other cases are simple on inductive hypothesis.

So the proof is finished. ~

Before moving on, we should note that by inspection

of the rules

Theorem 2.6.2. &R+ and GuT+ are equivalent to Kron's

G1R+ and G1T+' respectively; i.e., for any sequent E

in which I does not occur, E is derivable in &R+(GuT+) iff

it is derivable in G1R+(G 1T+).

Hence the proofs to come of Cut and modus ponens for the

G-systems will vindicate Kron's analogous claims for his

G1-systems.

99

FOOTNOTE

1That is, we treat antecedents as firesets or multisets.

(See Meyer and McRobbie 1979.) Obviously, the rules are

taken to be likewise specified, so in essence there is

no rule C ~'or at least no need thereof.

2Technically, we are treating only one subcase of this

case. The other subcase is when the sf introduced by

I 1- is not a member of Y. But this subcase is

straightforward on (H) and is left to the reader.

100

SECTION 7. Cut and Modus Ponens

A Cut Theorem in the style of Theorem 4.1 of

Kron 80 could now be shown. But we prefer one along the

lines of Dunn 75. So we begin with an analysis of the

rules.

First, an inference is an ordered pair consisting

of a finite (non-null) sequence of sequents - the premises -

as left member and a sequent - the conclusion - as right

member. A rule is a set of inferences, and its members

are called instances thereof. A calculus or system is a

set of sequents - the axioms - together with a set of

rules.

Let o be a sf occurrence in a premise of an

inference Inf. The immediate descendant of o is the sf

occurrence in the conclusion of Inf which "matches" o in

the sense which is obvious from the statement of the rule

of which Inf is an instance, with the following exception.

If the conclusion of an inference is the same sequent as

one of the premises, then the immediate descendant relation

is determined by similarity of position. 1 An sf occurrence

in the premise of an inference is the immediate ancestor

of its immediate descendant. This terminology is taken

over in the obvious way to derivations. The relation of

ancestor is the transitive closure of immediate ancestor.

An sf occurrence in the conclusion of an inference

which is an instance of a logical rule other than I ~is a

101

principal constituent thereof just in case it is the "newly

introduced" sf occurrence. The immediate ancestor(s) of a

principal constituent are subaltern(s). All other sf

occurrences in a premise or in the conclusion of an

inference are parametric constituents, either premise

parameters or conclusion parameters, as the case may be.

Note that all immediate ancestors of a conclusion parameter

have the same subscriptas it unless it was weakened onto.

Now we shall say that a rule Ru is closed under

parametric substitution if it satisfies the following

conditions. Let Inf be an arbitrary instance of Ru. Let

a be a set containing some conclusion parametersin Inf and

all of their immediate ancestors, such that each conclusion

parameter in a has the same subscript, say x, and is not of

the form Ix nor is it an sf that has been weakened onto. And

for an arbitrary structure X, the union of the subscripts

occurring in which is of course x, let Inf[X/a] be the result

of substituting X (in the premise(s) and conclusion of Inf)

for each member of a. Then either InfiX/a] is an instance

of Ru or its "conclusion~ is the same sequent as its

"premise".

This definition is a minor modification of the

expected adaptation of the analogous definition of left

regularity in Dunn 75. It differs from the exf)ected only in

having the"· .. is the same sequent as ... " clause.

Now the following lemma can be verified by

inspection of the rules.

102

Lemma 2.6.1. Closure Under Parametric Substitution

The rules of GL are closed under parametric substitution.

Next we say that a Rule Ru is antecedent expandable

if it satisfies the following. Assume that for

1 .;;; i .;;; n, l:i is Xi~ <ci'ci > and l:n+l = X ~Cc, and that

Y,Bc,Z ~Dd is a sequent. Then suppose that

( 1 ) l: 1 , • • • , l: n

l: n+l

is an instance of Ru, with the displayed occurrence of Cc

in l:n+l parametric. Then

(2) l:l, · ·. ,1:' n l: , n+l

is an instance of Ru, where l:n+l'

1.;;; i.;;; n l:~ is either Y,Xi,Z ~Dd

whether or not C. is an immediate l

is Y ,X,Z ~ Dd and for all

or l:. depending on l

ancestor of C in (2).

This definition is the appropriate analogue of right

regularity in Dunn 75. It is not so general as the

definition to come in §3.3, but will suffice for our

purposes. For

Lemma 2.6.2. Antecedent Expandability

All of the rules of GL are antecedent expandable.

Now for the needed notion of rank in a derivation.

Let Der be a GL derivation of l:. Unless l: is the top

node of a branch of Der, let Inf be the inference (in Der)

of which I: is the conclusion, and let a be a set of

constituents of E. Then define the rank of a in Der as

follows. If a is empty, its rank in Der is 0. If a is

103

non-empty but contains no premise parameters, then the

rank of a in Der is 1. (In this case, a is in fact a

singleton. Otherwise let inf be

(1) El•···•E n

I:

with Der. the subderivation determined by E. for each ~ 1

1 ~ i ~ n, and let ai be the set containing all and only

immediate ancestors in Ei of members of a. (Note that

if all members of a were weakened in, then ai = 0.) Let k

be the maximum rank of any a. in its corresponding Der .. 1 ~

Then the rank in Der of a is k+l. And following BGD 80

we talk of the consequent rank of Der as the rank of

a in Der when a is the singleton containing the consequent

of the conclusion of Der.

Then, where a is a set of sf occurrences in Y(E),

let Y[X/aJ (E[X/aJ) be the result of substituting X in

Y(E) for each member of a. (When a is a singleton, say

{o}, we let E[X/oJ = E[X/a].)

We are finally ready for the Cut Theorem which can be

stated as follows:

Theorem 3.3.1. Cut Theorem

Let a be a set of occurrences of any sf Ax in a

structure Y. If XI-A and YI-C are GL derivable, then X C

so is YCX/aJ ~c .3 c

Proof. The proof proceeds as in Dunn 75 by a double

induction. So choose arbitrary m>O, j and k such that

j+k>O, and assume

104

Outer inductive hypothesis (OH). For all X,Y,C ,A and a C X

(a set of occurrences of A in Y), if the complexity of A X

is less than m, then if X 1- A and Y ~ C are derivable, so X C

is YCX/aJ ~c ; and c

Inner inductive hypothesis (IH). For all X,Y,C ,A of C X

complexity m, and a (a set of occurrences of A in Y), if X~ A X X

is derivable with consequent rank j', and there is a derivation

of Y ~C in which the rank of a is k' and j'+k'<j+k, then c

YCX/aJ ~C is derivable. c

Next choose arbitrary A with A of complexity m X

and arbitrary X,Y,C and a (a set of occurrences of Ax in Y),

and assume

Conditional Hypothesis (CH). DerL is a GL derivation of

X ~Ax with consequent rank j and DerR is a GL derivation

of Y ~ Cc in which the rank of a is k.

It will suffice to show that Y[X/a] 1-C is GL derivable. c

For the sake of notational convenience, let

L-premise = X ~A , R-premise = Y ~ C and Conclusion = X C

Y[X/a] 1-C. c

105

We now proceed by cases.

Case 1. k = 0, whence ~ is empty, CNote that j is

never 0.) Then Conclusion is L-premise and we are finished

by CH.

Case 2. k = l. There are three subcases.

Case 2.1. R-premise is an axiom.

Then Conclusion is L-premise, and we are finished

by CH.

Case 2.2. R-premise follows from a sequent, call

it E, by K ~. Then each member of a was weakened in and

is parametric, (but has no immediate ancestor;) whence by

closure under parametric substitution, either Conclusion

is E or follows therefrom by K ~ .

Case 2.3. R-premise follows by a logical rule

on the left, call it Ru. Then a is a singleton containing

a principal constituent of R-premise. There are two

subcases.

Case 2.3.1. j = l. Then L-premise is either

an axiom or follows by a logical rule on the right "matching"

Ru. In the first instance, Conclusion is R-premise,

whence it is derivable by CH. So assume L-premise follows

by a logical rule, call it Ru'. There are three subcases

one for each pair of matching logical rules.

(Note that A cannot be I.) In each case one applies OH

to the appropriate premises to get

Conclusion. The subcases for v ~ and & ~ are

straightforward and left to the reader. So we now show

that Kron's difficulties with~ ~ do not afflict us.

106

But since there are now no singleton requirements on any

of the rules, we easily got Corollaries 2.5.1 and 2.5.2,

which see us through as follows:

Suppose Der1 ends thus

El

L-premise

=

=

X,Aa l- <l3,xua>

X 1-A~B X

and that DerR ends thus:

E2 = Z f-A Y,<B,zu x>f- C z c

R-premise = Y ,A+B f- C X C

= E3

We must show that Conclusion = X,Y f- C is derivable. c

Now, in the GT-systems, note that by proviso (3)

on f- + and + f- , max (a) ;;;. max ( x) and max ( z) ;;;. max ( x) .

Whence, max(a);;;. max(b) and max(z) ;;;. max(b), for any b

occurring in X. (Note that z ~ ¢) And for GuTW+' by

107

proviso ( 1) of ~ + and proviso ( 2) of + f- , anx = znx = ¢.

So by Corollary 2.5.1 or 2.5.2, as the case may be

Ej = X,A f-B is derivable, since E1 is. Then, applying z ~z

OH to E; and E3 , it follows that Conclusion is derivable,

as desired.

We proceed in a similar fashion in the GR-systems

when z I-¢. So for GR+ assume z = ¢. Then by the

Vanishing-t Theorem and z2 , we have that

(3) I f-A a a

is derivable. Then by (OH) and (3) and Et, we get

(4) X,Ia 1-<B,xua>.

From which by the Vanishing-t Theorem, we get

(5) Xf-B . X

Now, since z =¢on assumption, x = xuz. So by (OH),

(5) and E3, we have X,Y ~ C as required. c

108

Case 2.3.2. j>l, whence the consequent of L-premise

is parametric. For the sake of simplicity, assume that

DerL ends with an instance of a single premised rule

Ru' as follows:

E

L-premise

=

=

W ~A X

X 1- Ax

Then E has consequent rank (in the subderivation

determined by it) of j-1. So by IH we find that

E' = W,Y ~C is derivable. But since A is parametric C X

and a is a singleton, according to Antecedent Expandability,

Conclusion= X,W 1-C follows from E' by Ru', as required. c

Cases where L-premise follows from a two-premised rule are

handled in a similar fa.shion, and are left to the reader.

Case 3. k>l. Then suppose DerR ends with the

following instance of some rule Ru other than I 1- :

E 1 ' • • • ' l:n '

(l .;;; n .;;; 2 ) R-premise = y 1- c

and let for 1 .;;; i .;;; n Deri be the subderivation determined

by Ei. Then let a' be the set of all conclusion parameters

in a, and for 1 .;;; i .;;; n let ai be the set containing all

immediate ancesters in Ei of members of a'. Then note

that for each such ai' the rank of ai in Deri is less

thank, whence j+(rank of a~ is less than j+k. So by IH

we see that l:.[X/a.J is derivable, l.;;; i.;;; n. Whence by l l

closure under parametric substitution, E = Y[X/a'J }C

is derivable by Ru, or r 1 is E. (Call the derivation of , ) , E Der . If a = a, we are finished. Otherwise r 1 I E

and l: and Conclusion are of the form W ,A a, Z 1--C c and

W,X,Z ]-Cc' respectively, with the displayed occurrence

of Aa being principal in the application of Ru ending

109

, Der . Whence we advert to case 2.3 to complete the proof.

Finally, assume that DerR ends with the following

instance of I } :

l:l = <D 1 ,d1>, ... ,<Dh,dh.>,A ... ,A ,<A,xuy>,z 1-C X X C

R-premise <D 1 ,d 1>, ... ,<Dh,dh>'Ax'''''Ax,Iy,Ax,z tCc

with a containing all and only the displayed occurrences

of A in R-premise, with max(x)nz = ¢ and with X

max(x) = max(di)' for each 1 ~ i ~h. And assume

L- premise = W,<l31 ,b1>, ... ,<13 ,b > 1-A n n x

with max(bi) = max(x) and max(x) nw = ¢, for each 1 ~ i ~ n.

Then note that this is not a case for the

W-systems, and that for GIJT+' max(x) ;;;. max(y).

by proviso (2) of I 1-. By CH r 1 and L-premise are

derivable, whence by Fact 2.5.5 so are

, E 1 = <D1 ,d 1uy>, ... ,<Dh,dhuy>,<A,xuy>, ... ,<A,xuy>,z l-Cc

and

L-premise' = W,<l3 1,b 1uy>, ... ,<l3 ,buy> ~<A,xuy> n n

llO

Then note that they are derivable in such a fashion

that the consequent rank of L-premise' = j and the consequent

rank of a'= k-1, where a 1 contains only the occurrences of ,

<A,xuy> displayed in E{. 4 So Conclusion follows from E1

and L-premise' by an application of Cut licensed by IH,

followed by applications of I ~ to restore the original

subscripts, followed by applications of W ~ to remove

excess occurrences of <l,y>.

The proof of the Cut Theorem is now completed.

~

It is worth noting that Vanishing-t was not needed

in this proof for the GT-systems. This is because proviso

(3) on + ~ guarantees that no premise of an instance of

that rule will have an empty antecedent.

The reader can now use the invertibility of ~+,

the Vanishing-t Theorem and the Cut Theorem ·to show

Theorem 3.3.2. Modus ponens is admissible in

u u G T+ and G R+.

We will now proceed to show that G~+ is equivalent

in an appropriate sense to uR+. The proof will use both

the semilattice semantics and Urquhart's natural deduction

system. The question of equivalence between the other

G-systems and their namesakes must be left open.

111

For neither the appropriate natural deduction systems nor

semantics are known to be equivalent to the other logics.

112

FOOTNOTES

1This convention is slightly broader than that of BGDBO,

p.35l; but the effect is the same. Strictly speaking,

we should either stop treating sequents as multisets, or

define a function to determine the relation of immediate

descendent, since otherwise "matching" really makes no

sense. But to do so would cloud what is essentially clear

terminology, and needlessly complicate the statement of

rather simple facts.

2 Note that although some instances of I ~ are also instances

of K ~' when structures are treated as multisets, this

causes no problems.

3Note that no sequent of the form X ~I is derivable. X

Further, x ~ ¢ in the statement of the Theorem when a

is not empty.

4This is a simple matter to check in the proofs of the

Rewriting Lemmas. Indeed, given E and E1 with E

derivable and E1 being the result of rewriting subscripts

in E according to one of the Lemmas, one can obtain a

derivation of E 1 by simply re-writing the subscripts

occurring in any derivation of E in an appropriate fasnion.

SECTION 8. UR c G UR + - +

We want to show that

for each theorem A of uR+.

u ~A is derivable in G R+

As a matter of notational

convenience, we will refer to G and to N, rather than

GuR+ and N~1R+ - in this section and the next, only,

The strategy of the proof will be in essence

to translate normal proofs in N into derivations in G.

Charlwood Theorems 2 and 3 (given below) will then

finish the work for us. This method neatly by-passes

the cumbersome rule R4. So let us introduce Urquhart's

N.

ASSUMPTIONS

113

Ax may be entered as a top node, for any x and for any A.

RULES

&I

vi

A B X X

A

A&B X

X

AvB X

&E

B vE X

AVB X

AvB X

A&B X

Ax

[A J X

. c xuy --c xuy

A&Bx

B X

[Bx]

. c xuy

+I [A{k}]

Bxu{k}

A+Bx-{k}

+E A+B X Ay_

A+Bxuy

With reference to these schemata, the principal premise

of +E is A+B ; of &E is A&B ; and of vE is AvB . All X X X

other premises are lesser. We refer to the lesser

premises of vE as AC and BC in the obvious way. x uy xuy

We stipulate that all assumptions of the form A{k}

above A+Bx-{k}' the conclusion of an application of +I,

are discharged at this application of +I. In the same

context, we refer to {k} as the deleted subscript. At

an application of vE with principal premise AvB , all X

undischarged assumptions of the form Ax (Bx) above and

including AC (BC ) are discharged at once. 1 And xuy xuy

let us say that the premises, where there is more than

one, of any application of a rule are side-connected.

A derivation in N is any finite tree whose nodes are sf

occurrences such that:

1. the tree branches upward and has the usual

properties;

2. no subscript is deleted more than once; and

3. if

of

Ay is a premise of +I discharging assumptions

the form B{k}' then every undischarged +I

assumption above A with subscript {k} is of y

the form B{k} ·

114

115

A proo! in N is a derivation in which every

assumption is discharged. And we use the normal notion

of the sub-derivation determined by a particular sf

occurrence in a given derivation.

Let us record a few facts established in Charlwood

80.

Charlwood Theorem 2. A is a theorem of u R+ iff A is

provable in N.

Charlwood Theorem 3. A¢ is provable in N iff it has a

normal proof in N; and a normal proof of A¢ has the

following properties:

(l) Subformula Property

Every formula occuring in it is a subformula of A; and

(2) Normality

No principal premise of vE is subscripted with ¢. 2

It is useful to think of derivations in N as

branching downward below a principal premise of vE,

until the vE assumptions are discharged. That is, we

want to think of the minor premises of vE, and all nodes

above them, as being ''below" the principal premise of

the application of vE discharging those assumptions.

So we introduce a notion of (immediately) precedes,

which is like that of (immediately) above, except that

the principal premise of an application of vE immediately

precedes the vE assumptions (discharged by that application

116

of vE), but does not immediately precede the conclusion

of that application of vE. The conclusion is the

immediate successor of the minor premises only. For the

sake of simplicity, we will have it that each node precedes

itself.

Then for any derivation der and for any sf occurrence

s therein, define the weight of 6 (wt(o)) in der as

follows:

(1) wt(a) = 1, if a is an +I assumption; otherwise

(2) wt(o) = j+l, where j is the maximum weight of the

immediate predecessor(s) of s.

And a derivation has weight m, if its conclusion has

weight m therein.

For any occurrence a of any sf, say A , in a X

normal proof, say of B, let X0 be the sequence, in some

assumed order, of undischarged +I assumptions preceding

a. And let X0 be the sequence, in some assumed order,

of undischarged vE assumptions. Then let g(o) be

X0,X0 ~Ax. We will want to show that g(a) is derivable

in G, for each such o. The following facts will be needed

to do so.

In the first place, the definition of proof

guarantees

Fact 2.8.1. Let o be an occurrence in a proof of an sf,

say A , and let y be an occurrence in the same proof of X

some By' such that x £ y. Then x is the union of the

117

subscripts occurring in X and each subscript occurring in 0

X0

is a subset of x. Further, each member of X0

is a

member of Y . y

Similarly, the reader can easily establish

Fact 2.8.2. Let a be the principal premise, 8 and y

the lesser premises and o the conclusion of an application

of vE in a proof. Let A and B be the vE assumptions X X

discharged by that application, whence X is the sequence a

of undischarged vE assumptions preceding a. Let Y8

,

YY and Y0 be the corresponding sequences for a, y and o.

Then there are W1 and W2 such that Y0 is X ,w 1 ,A andY ~ a x y

is X ,W2 ,B . a x . Further, an sf occurs in Y

0 iff it

occurs in either X , W1 or Wz. a

Finally, we can show

Lemma 2.8.1. For any A, for any normal proof P of A¢

in N and for any sf occurrence o in P, g(o) is derivable

in G.

Proof. By induction on weight of o in P. The base step

is straightforward. So choose an arbitrary k>l, and assume

Inductive Hypothesis. (H) For any A and any normal proof

P of A¢ in N, and for any sf occurrence y of weight

ll8

< k in P, g(y) is derivable in G.

Next choose arbitrary A and an arbitrary P, a normal proof

of A~ in N, and an arbitrary o of weight k+l in P. It

will suffice to show that g(o) is derivable in G. The

proof proceeds by cases.

Case 1. o is a vE assumption, say Bx. Note that

by Normality of Charlwood Theorem 3, x is not null.

Obviously, B ~B is derivable in G, from which X,, X X u

X0 ~Bx' i.e. g(o), follows by K~(if necessary) in light

of Fact 2.7.1. So the case is finished.

Case 2. o is the conclusion of +I. Let the

case be as follows:

[Bk]

y = <c(u{k}> o = B+C)

X

assuming without lose of generality that k I x. For the

sake of simplicity, let z = xu{k}. Using Fact 2.8.1 and

the definition of a proof, it is clear that Zy is X0,B{k}"

Further, any occurrence that is a vE assumption and

precedes o, likewise precedes y; so Zy = X0 . So by (H),

g(y) = x5,B{k}, x8

.f-<(B+C), xu{k}> is derivable in G.

And again by Fact 2.8.1, each subscript in X0 is a subset

of x. So we see that X0 , X0 ~ (B+C)x, i.e., g(o), follows

from g(y) by f- +, finishing the case.

Case 3. 6 follows by +E. Then let the case be

as follows:

By (H)

(1)

( 2)

a = (B+C)

6 =

B = y z

c yuz

y Y ~(B+C) , and a' a Y

Z ,z t- B y y z

are derivable in G. Then by (1) and Fact 2.5.3,

( 1,) Y ,Y ,Bb i-<c,yub> a a

is derivable, for some b such that bny = ¢ and b ~ ¢.

There are now two subcases.

Case 2.1. z = ¢, whence (2) is l-B. Then by

the Vanishing-t theorem,

( 2' ) Ib 1- Bb

is derivable in G. So by (1'), (2•) and Cut,

( 3) Ib 'y a 'y a f- <c' y ub>

119

is derivable in G. Recalling that bny = ¢, we see by the

Vanishing-t Lemma that g(o) is derivable as required.

Case 2.2 z I¢. Then by (1') ahd the rewriting

Corollary 2.5.1,

( 1 N ) Y , Y N , B 1- <.c, y uz> a ~ z

is derivable in G. Whence by (2), (1") and Cut, g(o)

is derivable as required.

Case 3. o follows by vE. Let the case be

a = (BvC) z

0 =

[B J z

<D,zuy> = y

<D,zuy>

[C J z

<D,zuy> = A

120

and for notational convenience let x = zuy. By Fact 2.8.1,

X = X = X = y A 0 Z ,WJ, for some structure W1. Cl

And by

Fact 2.8.2, Xy = z ,w2 ,B Cl z and XA = za,w3,cz' for some

W2 and w3 . So by (H)

(1) Z , W 1 , Z , W 2 , B l- D , and a · Cl Z X

(2) z ,w 1 ,z ,w 3 ,c 1- D a a Z X

are derivable in G. Then by K 1-

( 1,) z ,w1,Z ,w 2 ,W3,B 1- D , and a a Z X

( 2,) Z ,WI ,Z ,W2 ,W3 ,C 1- D a a Z X

are also derivable. So by v 1- ,

(3) Z ,W 1,Z ,W2 ,W3, (BvC) l- D a a Z X

is derivable in G.

But by (H) again,

( 4 ) Z ,z 1- (BvC) Cl Cl z

121

is also derivable. So by (4), (3), Cut and W 1-,

(5) Z ,W1,Z ,Wz,W3} Dx., i.e", X_,.,.Z ,Wz,W3 r D a u v a X

is derivable in G. But by Fact 2.8.2, g(o) either is (5)

or follows therefrom by W 1-, which finishes the case.

Case 4. If 6 follows by any other rule, the

argument is straightforward and can be left to the reader.

So the proof is finished.

We did use I in the proof of the Lemma, but Theorem

2.6.1 tells us that its use was inessential. So henceforward

we take all of the G-systems to be formulated without I.

The above lemma and Charlwood Theorems 2 and 3 now finish

the business of this section.

Theorem 2. 8 .1. A is a theorem of uR+ only if 1- A is

derivable in G.

FOOTNOTES

1Note that an application of vE discharging assumptions

of the form B which occur above a particular lesser X

premise, say y, does not discharge any assumptions

above y which are not (occurrences of) Bx.

122

2The definition of normal is given in Charlwood 80 on p.8.

However, only those properties of normal proofs just

listed are of interest to us.

3rn order not to confuse some relatively simple matters,

we are a bit lax here and below with the distinction

between an sf and an sf occurrence.

123

SECTION 9. UR . GUR + 15 +

As before, let us say that a formula A is provable

in G just in case ~A¢ is derivable. We need to show that

all formulae provable in G are theorems of u R+. Given

Charlwood Theorem 1, it will suffice to show that all

theorems of G are valid in the uR+ semantics laid out in

Chapter 1. A technique similar to that of Urquhart 73 is used.

The concept of the technique is very simple.

Sequents will be interpreted as metasemantic statements.

Then we will show that the interpretation of each derivable

sequent is true, with the desired conclusion then

being immediate. Notational conventions will be adopted

to make the actual technique match the concept in simplicity.

So on interpretation, we will let integers be

variables ranging over points of a uR+ model structure.

So let n1

, ••• ,nk be the elements of a· non-empty subscript,

say x. For any formula A, we let [A J be the statement X

that I(A,n 1 u ... unk) =True on m, where I is a variable

ranging over interpretations and m is a variable ranging

u + over R model structures. And let [A0 J be the statement

that I(A,¢) =True on m. Then let us agree to abbreviate

I[A J1 to 1 A 1 , for any subscript x. Then for any X X

non-empty structure X, say <A 1 ,a 1 >, ... ,<A ,a >, and for any m m

sf Aa' let [X ~AaJ be the statement that if

[<A 1 ,a! >J, ... , C<A ,a >J then [A J. And let [ ~ AJ be [A_,~ J, m m a "'

for any formula A. Finally, for any sequent X ~A a, let

124

t(X !-A) be the statement u that for all R+ model structures

m, for all interpretations I and for all points nJ,···>nk'

[X!- AaJ - where n 1 , ••• ,nk are all of the positive integers

occurring in X!- A . (With the obvious understanding when a a = ¢)

With these definitions, we can conveniently state

Theorem 2.9.1. For any sequent E, E is derivable in G

only if t(E).

Proof. By a straightforward induction on weight of

derivation of E.

Putting this Theorem and Charlwood Theorem l

together we get

Theorem 2.9.2. A is provable in G only if A is provable u

in R+.

And this theorem with Theorem 2.8.1 finishes the

equivalence:

Theorem 2.9.3. Equivalence Theorem

G is equivalent to u R+.

125

SECTION 10. Decidability

Now we turn to showing that the GW-systems are

decidable. The simplicity of structures - they are,

after all, just sequences - makes at least part of the

job straightforward. The overall strategy of the proof

will be to define a complete and effective proof search

procedure which builds proof search trees, then show that

such trees are finite via Konig's Lemma (see below).

For once, we actually have to worry (a little)

about the Finite Fork Property. W;i. th the rule 1- + as it

stands, a search for a proof of ~A+B, for instance, is

immediately infinite, since on the face of it, we must

check A{l} f-B{l}' A{z} ~B{z}'···· But this is easily

remedied.

Let G'LW come from GLW by changing proviso (1) to

(1') a = {max(x) + 1}.

Using the re-writing Corollary 2.5.2, it is simple to

show that GLW and G'LW are equivalent in terms of derivable

sequents. So we will not bother to distinguish between them;

we simply take GLW to be formulated with proviso (1')

instead of (1) on~+. And let us now say that max(x) + 1

is discharged by an application of~+.

To show that any branch of a proof search tree

is finite, we will take the simple approach of showing

that only finitely many distinct sequents can occur only a

finite number of times on a given branch. However, with

126

W ~as a rule, it is clear that we must reduce sequents

that can occur, i.e., put an upper bound on the number of

times that an sf can occur in a sequent.

But, again, the simplicity of our structures makes

this an easy task. We have already been taking structures

to be mul tisets. Given K ~and W 1- , nothing stands in the

way of going to sets simpliciter.

So let an s-structure be a (possibly empty) set

of sfs, and bring forward other appropriate definitions

in the obvious way. So structure variables now range over

s-structures. And let us simplify notation by dropping

parentheses from singletons when we wish and allowing commas

to stand in for the set union sign. We now officially

formulate G2 uTW+ and G2uRW+ by taking their axioms and

rules to be specified by the statement of the same for

GuTW+ and GuRW+, respectively. It is then straightforward

that

Lemma 2.10.1. Reduction Lemma

GLW and G2 LW are equivalent in the obvious sense.

Notice, however, that the premise and conclusion of an

instance of W 1- in G2LW are the same sequent. Likewise for

C 1-. So we take G2LW to be formulated without these rules.

If the elements of our structures were simply

formulae, the Finite Branch Property would be guaranteed

by the Reduction Lemma and the Subformula Property, below.

(Given Irredundancy, that is). For it would then be the

127

case that all sequents occurring on a branch of a proof

search tree would be built out of a finite number of

formulae, each of which could occur only a finite number

of times in a given sequent. Obviously, only a finite

number of distinct sequents could be so constructed. But

the constituents of our sequents are sfs. The Reduction

Lemma puts a definite, finite upper bound on the number of

times a formula can occur with the same subscript in

any sequent. But it gives no information about the number

of times that a formula can occur with a different subscript.

So the problem that remains for decidability is to

get an upper bound on the number of distinct subscripts

that can occur on a branch of a proof search tree. To solve

this problem, it will be helpful to think of relevant logics

as a mixture of the intensional and extensional, as Meyer

has often urged (in Meyer and McRobbie 79, for instance) -

or as being ''hybrid'', as it is put in Belnap 8+. 1 In our

context, & and v are extensional connectives, and + is

an intensional connective.

Now let us think of GLW along these lines. 2 Given

& ~, K ~and W ~, it is clear that two sfs with the same

subscript occurring in the antecedent of a sequent are

being structurally represented as being conjoined, that

is as being extensionally related. So the Subformula

Property and Reduction Lemma give us control over the

extensional complexity, if you like, of the antecedent of

sequents that must be considered for a proof search for a

given sequent.3

128

Now in terms of intensional complexity, f- + and

+ !- obviously indicate that sfs in the antecedent with

disjoint subscripts are being represented as being

intensionally related. GLW is not idemdis, and sfs with

"overlapping" subscripts muddy the water, but we now have

a fair hint for a measure of the intensional complexity

of a sequent.

First, for any formula A and subscript x, define

the degree of A (deg(A )) as the number of + 1 s occurring X X

in A. Then for any structures X, Y, let Y be an intensional

barometer of X, if

(l) Y ~ X; and

(2) all subscripts occurring in Yare pairwise disjoint.

And for any structure Y satisfying (2), define the indicator

of Y ind(Y) as the sum of the degrees of its elements. And

for any structure X, define deg(X), as the maximum of the

indicatorsof its intensional barometers. Obviously, if X

is empty, the degree of X is 0. Finally, for any sequent

E, let deg(E) be the sum of the degrees of its antecedent

and consequent.

Two questions remain to be answered. Can we get

control over the intensional complexity of sequents

occurring in a GLW derivation of a given sequent? If so,

will the combined control over extensional and intensional

complexity, as we choose to put it, yield a decision

procedure? The answers to these questions are ''Yes'' and

''Yes'' - respectively.

129

First, note that

Lemma 2.10.2. Degree Lemma

The rules of G2LW are degree preserving. That is, the

conclusion of an instance of any rule is at least as great

as that of any of its premises.

Proof. By cases.

Case l. Let the following be an arbitrary instance

of 1- +:

E1 = X,A f-<B,xua > a E = X 1- ( A+B)

X

Let Y be an intensional barometer of X with maximum

indicator. Then deg(E) = ind(Y) + deg(A) + deg(B) + l.

But since anx = ¢, deg(E 1 ) = ind(Y) + deg(A) + deg(B).

So deg(E) is greater than deg(E 1)

Case 2. Let the following be an arbitrary instance

of + }- :

E1 =Xf-A X

Z ,<B, 8!W X> f- C c

E = X,Z, (A+B)a 1- Cc

First note

= Ez

(l) Every intensional barometer of the antecedent

of E1 or of the antecedent of E2 is an intensional

barometer of the antecedent of E;

130

and

(2) anx = ¢, by proviso (2) of + 1- •

To show deg(E) ~ deg(E 1 ), let Y be an intensional

barometer of E1 with maximum indicator. Then

deg(E 1 ) = ind(Y) + deg(A ). But by (1) and (2), Y u {(A+B) } x a

is an intensional barometer of the antecedent of E, and its

indicator is ind(Y) + deg(A) + deg(B) + 1, which suffices.

Then let W be an intensional barometer of E2 with

maximum indicator.

(W- {<B, awx>}) u

If<B,aux>E W, then by (1) and (2),

{(A+B)a} is an intensional barometer

of the antecedent of E. So an argument similar to the one

immediately above will suffice. And if <B, aux > i W, we

are finished by (1). So the case is completed.

Case 3. All of the other rules are straightforward

on inspection, which finishes the proof.

~

It is important to note from the above proof that ~+

is degree increasing.

The Degree Lemma will yield a finite upper bound

on the number of subscripts that can occur in a branch of

a proof search tre~ which will work together with the

Reduction and Irredundancy Lemmas and Subformula Property

to yield the Finite Branch Property. So let us get the

decidability argument properly underway.

131

First say that a tree is irredundant provided that

no sequent occurs more than once on any branch of it.

Clearly

Lemma 2.10.3. Irredundancy Lemma

Any sequent E is G2LW derivable iff it has an irredundant

derivation.

Next, let us specify as follows a proof search

procedure which produces the LRW:(LTW:) proof search tree

of E for any sequent E:

(l) Enter E as the bottom node;

(2) above each sequent E' occurring with height k

(in the tree so far constructed) (a) enter

nothing, if E' is an axiom, (b) otherwise enter

(in some assumed order) all sequents EN such

that EN is a premise of some G2LW inference of

which E' is the conclusion and such that the tree

remains irredundant.

Obviously

Lemma 2.10.4. Effectiveness Lemma

The proof search procedure thus specified is effective.

Now let us say that a (possibly null) tree T'

is a subtree of a tree T iff it is the result of deleting

132

some (possibly no) sequent occurrences in T and all sequent

occurrences above them. Then by the Irredundancy Lemma

and the above specification:

Lemma 2.10.5. Completeness Lemma

The proof search procedure is complete, i.e., E is G2LW

derivable iff some subtree of the proof search tree of

E is a G2 LW derivation of E.

As usual, a tree has the finite fork property

iff it has at most finitely many nodes of any given height;

and a tree has the finite branch property iff each of its

branches contains at most finitely many nodes. And recall

Konigs Lemma. A tree is finite iff it has the finite

fork and finite branch property. (Konig 27)

Now, by inspection of the rules

Lemma 2.10.6. The proof search tree of any sequent E

has the finite fork property.

To show the finite branch property, we need a few more

facts and lemmas. Of course G2LW has the Subformula

Property, which we state as follows:

Lemma 3.8.6. For any inference of G2LW, every formula

occurring in a premise thereof is a subformula of a

formula occurring in the conclusion.

133

As was indicated in the earlier discussion, what

is needed now is control over the number of distinct

subscripts that can occur in the sequents of a branch of

a proof search tree. For clearly the move to sets in

the Reduction Lemma gives us

Fact 2.10.1, For any sequent E and subscript x, there

are at most finitely many sequents in which

(1) all formulae that occur are subformulae of formulae

occurring in E; and

(2) all subscripts that occur are subsets of x.

So for any branch 8 of a proof search tree, let

cs(8) (the conclusion subscripts of 8) be the union of all

subscripts occurring in the bottom node of 8. And let

ds(8) be the set of all positive integers discharged by

~+on 8. Then by the definition of a proof search tree

and inspection of the rules

Fact 2.10.2. Subscript Fact

All subscripts occurring in 8, a branch of a proof search

tree, are subsets of cs(8) u ds(8).

Now it is time for the Degree Lemma to do its job.

Lemma 2.10.5. Subscript Lemma

For any sequent " and branch s of a proof search tree for "'

134

cs(S) u ds(S) is finite.

Proof. Choose arbitrary E and S in accordance with the

lemma. Obviously cs(S) is finite, whence it will suffice

to show that ds(S) is finite. So assume for reductio

that ds(S) is infinite. Then clearly there are infinitely

many distinct applications of ~+ on B. Now recall that

~+is degree increasing. Then by the Degree Lemma,

deg(E) is infinite - which is absurd. So the lemma is

proved.

~~-

So straightaway we have

Lemma 2.10.6. Finite Branch Property

All G2 LW proof search treeshave the finite branch property.

Proof. By the subformula property, every formula occurring

on a branch is a subformula of that branch's bottom node,

call it r. And by the Subscript Lemma, cs(r)u ds(E) is

finite. So by the Subscript Fact and Fact 2.10.1, only

finitely many distinct sequences occur on any branch. Since

the proof search procedure guarantees that a proof search

tree is irredundant, every branch of such is finite,

as was required.

135

This Chapter can now be concluded with

Theorem 2.10.1. G2LW, and hence GLW, is decidable.

Proof. By the Effectiveness and Completeness Lemmas and

Konig's Lemm~along with the Finite Fork and Finite

Branch Properties.

136

FOOTNOTES

1we do not intend to suggest that there is any underlying

philosophical agreement among the authors cited.

2Although we have no proof, it seems very reasonable to

believe that the GW-systems are formulations of relevant

logics, namely, of uRW+ and uTW+' respectively.

3This much is a drastic oversimplification, but is close

enough to be of heuristic value.

137

SECTION 1. Introduction

The major result of this work, namely that TW+

and RW+ are decidable, is contained in this Chapter which

is a study of Dunn style Gentzen systems for TW+, RW+'

T+ and R+. The L-systems are presented in §2 where we also

spend some time to gather some basic facts. These systems

are actually ''hybrid", utilising extensional sequences as

in Dunn 75 and an intensional, binary structural connective

as in Meyer 76b and Belnap 8+. The reason for this is that

although the use of binary structural connectives has

definite notational advantages, sequences are much simpler

to deal with for the sort of extensional reduction needed

for the decidability argument.

In §3 the systems are fitted with appropriate

Cut Theorems. The desired equivalences are then proved

in §4, where we also develop a notion of representational

adequacy and show that the L-systems meet the criterion

thereof.

Next we begin the business of showing that the

contractionless systems are decidable. The strategy is

an appropriate modification of the one used for the

subscripted systems of Chapter 2.

In §5 we reformulate the systems to be empty on

the left, ridding ourselves of t via a Vanishing-t theorem

analogous to that of the previous chapter. Doing so is a

first step toward a suitable formulation to show

decidability for the contractionless systems, since the

138

rule t- ~is not degree-preserving. And in §6 we give

a final formulation which "denests" extensional sequences.

(The terminology is explained in §6.) This move facilitates

the extensional reduction required for the decidability

argument.

The proof of an appropriate extensional reduction

lemma is given in §7. But rather than moving to sets,

i.e., trading in denested E-sequences for sets, we simply

show that there is a finite upper bound on the number of

repetitions of a structure as an immediate constituent

of an extensional sequence. The reason for this is

purely practical. We will basically adopt the notational

conventions of Dunn 75, than which we can find no better.

And our own experience has been that with such notation,

it is far easier to check case-ridden arguments, as are

common in proof theoretic investigations.

Finally, in §8 we formulate an appropriate notion of

degree and then give a decision procedure for LTW+ and

LRW+. Given the equivalence results of §4, this suffices

to show that TW+ and RW+ are decidable. (The difficulties

involved in extending the argument to cover EW+ are

discussed in §3 of the final chapter.)

139

SECTION 2. Formulation 1, Definitions and Facts

The primary task of this section is to present

Dunn-type Gentzen Systems for R:t and T:t with and without

contraction, introduce vocabulary and gather a few facts.

Accordingly, unless otherwise specified L ranges over ot o~ ot ot

TW+' RW+ , T+ and R+ \'i'e will call

these Gentzen systems 'L-systems'. Since other formulations

will be presented later, superscipting on 'L' will be used

to distinguish the different formulations, e.g., L 1 T~ 0 ,

L2RW~ 0 , etc. Further, 'LT-systems', 'L 2 R-systems',

'LW-systems', etc. are used in the obvious way.

Of course, a base language built out of atomic

formulae in the usual manner using the connectives and

constants appropriate to the system(s) under consideration

is assumed throughout. From the wffs, nested

structures will be built as was indicated in 13.1.

We will have extensional sequencffias in Dunn 75, for which

our notation will be the same as there. And we will have

a binary intensional structural connective as in Meyer 76b

and Belnap 8+, which will be represented by a semicolon.

Then letting 'X', 'Y', 'Z' and 'W' with or without

subscripts and/or superscripts be structural variables,

a structure is defined recursively:

(1) A is a structure, for any wff A;

(2) if X and Yare structures, so is X;Y; and

( 3) if X1 , ••. •X"n are structures and n ;;;>- 2, then so is

E (X 1 , ... , Xn) .

140

Note that there are no null structures nor any

structures of the form E(X). With respect to the latter,

we say that our structures are denuded. This is the first

move toward simplifying the counting of structures, as we

will of course want to do for decidability. There is no

conceptual or technical loss involved, since structures

of that form carried no representational load. Unless

otherwise indicated, semicolons are taken to be associated

to the left. Parentheses are used to disambiguate notation

as necessary.

Structures of the form of (3) are called

extensional structures, extensional sequences, or

e-sequences. Those of the form of (2) are intensional

structures or i-structures. And we say that a structure

X occurs in a structure Y just in case

(1) X is Y; or

(2) Y is W;Z and X occurs in W or in Z; or

(3) Y is E(W1 , ... ,Wn) and X occurs in some Wi.

Of course, if X occurs in Y, then X is a substructure of

Y, and the appropriate occurrence(s) of X is/are a

constituent(s) of Y. (The notion of a particular occurrence

of a structure is taken as primitive. However, the

distinction between a structure and a particular occurrence

thereof is often ignored when it is not likely to cause

confusion.) And for 1 ~ i ~ n, the "displayed" occurrence

of Xi in E(X 1, ... ;Xn) is an immediate constituent thereof.

And for an intensional structure, say X;Y, we refer to X

as the left constituent and to Y as the right constituent.

141

A sequent is an entity of the form X ~A. &, with

or without scripting, is used as a variable ranging over

sequents. X is the antecedent and A is the consequent of

X ~ A. And Y occurs in the sequent l:, just in case it

occurs in its antecedent or consequent. The use of

constituent is similarly extended. And we say that

structures and sequents are built up from or built up

out of the wffs that occur in them.

The following structural analogue to the notion

of the length or complexity of formulae will be very

useful. So define the structural complexity (sc) of

a structure and of a sequent as follows

(1) sc(A) = 1, for any formula A;

(2) sc(X;Y) = sc(X) + sc(Y) + 1, for any structures

X and Y;

(3) sc(E(Xl•···•Xn)) = sc(X1 ) + ... + sc(Xn) + 1, for

any structures X1, ... ,Xn; and

(4) sc(X ~A) = sc(X) + sc(A), for any X and A.

Upper case Greek letters (except 'E') are used to

range over (possibly empty) sequences of symbols drawn from:

formula variables and parameters, 'E', left and right

parenthesis, the comma and the semicolon. For example

r 1 X r 2 ~A

represents a sequent. Further, a particular occurrence

of X is taken to have been displayed.

142

Now Formulation 1 of the L-systems can be given

from the following set of axioms and rules. Two-sided

rules are indicated by •~•

AXIOMS

A 1- A

RULES

Structural Rules

r 1x r 2 1- c r 1 E ( x1

, ••• , Y , w, ... , x ) r 2 1- c Ce 1- n , N ;;. 0 !

r 1 E(X1

, •.• ,W,Y, ... ,Xn)r 2 1- C Ke 1-

r 1E(X,Y) r 2 1- C

r 1E(X,X)r 2 1- c We 1-

r 1 X r 2 1- c

ee 1- r 1E(X 1 , •.• ,E(Y 1 , ••• ,Ym), ... ,Xn)rz 1- C""

r 1E(X 1 , ••• ,Y 1 , ••• ,Ym, ... ,Xn)r 2 1- c, n;;. 1 and m;;. 2

r 1 (X;Y)r 2 1- c B- i I-

r 1 ( (X;Y) ;Z)r 2 1- c CIH

r 1 (Y;X)r 2 1- c r 1 (X;(Y;Z))r 2 1- c

r 1 (X;(Y;Z))r 2 1- c r 1 (X;(Y;Z))r 2 1- c Bi 1- B, . 1-

r 1 (X;Y;Z)r 2 1- c l r 1 (Y;X;Z)r 2 ~ c

& ~

v ~

_, ~

0 ~

t ~

Logical Rules

r 1A r 2 ~c

r 1A&B r 2 ~ c

r 1B r 2 ~c

r 1 A&B r 2 ~ c

r 1A r 2 ~c r 1 B r 2 ~c

r 1AvB r 2 ~ c

Y~Ar 1Br 2 ~c

r 1 (A+B;Y) r 2 ~c

r 1 (A;B) r 2 ~ c

r 1 AoB r 2 f-C

r 1x r 2 ~c

r 1 (t;X) r 2 ~c

~v

~-+

~ 0

t- f-

X f-A X f-B f-&

X f- A&B

X ~A X ~B

X~AvB X~AvB

X; A ~B

X ~ A-+B

X ~A Y ~B

X;Y ~AoB

r 1 (t;X) r 2 ~c

r 1 x r 2 ~c

The axioms, all of the logical rules and Ke ~,

143

We ~ , to

LTW+

Cie ~ and ee f- are common to the L-systems. To get

add Bi f- and B' i ~ . to For LRW+ replace B'i f- by Cii f-

and B-il-instead. And for t~ to , . LT t and LR+ add Wl 1- to the

appropriate contractionless system.

The reader familiar with Dunn 75, Meyer 76b and/or

Belnap 8+ will feel at homewith this formulation, even though

144

some of the rules are slightly different. This form of

We ~ is demanded for preserving well-formedness, since our

structures are denuded. The choice of logical rules for &

is discussed later in this section, as well as the superfluity

of ee ~. And the role oft-~ is discussed in §4. Now

some definitions are wanted, before a few facts are collected.

In the first place we should re-establish some

familiar terminology. Ke ~ is of course an extensional rule

of weakening, and we speak in the obvious way of a structure

having been weakened in. Ce ~ and CTi ~ are Permutation

rules, and We~ and Wi ~ are Contraction rules. Again we

speak of a permuted structure and of a contracted structure.

A derivation in LL is a finite tree branching

upward with the normal sorts of properties, and a proof of

A is a derivation of t ~A. We take the notion of a sequent

(occurrence) being immediately above (below) another sequent

(occurrence) as primitive. Being above (below) is the

transitive closure of immediately above (below). So where

Der is a derivation and x is a particular occurrence of some

sequent therein, the subderivation determined by x iS the

derivation that one would get by deleting from Der all

sequent occurrencffiexcept x and those above it. A sequent

occurrence x (immediately) precedes a sequent occurrence

y in a derivation just in case x is (immediately) above y;

similarly for (immediately) succeeds. And predecessor and

successor are used in the obvious way. Then a branch of

a derivation is a sequence x 1 , .•. ,x of sequent occurrences n

such that x 1 has no predecessors and xn has no successors,

145

and for all 1 < i<n, xi is immediately above xi+l' A branch

segment is a subsequence of a branch.

The weight of a derivation, say Der, is the length

of a longest branch, and the weight of a sequent occurrence

x in Der is the w.eight of the subderi vat ion determined by x.

The conclusion (bottom node) of a derivation that has weight

n is said to be derivable with weight n. The concept of

weight is an important one, since many arguments to follow

will be by induction on weight.

Finally, the height of a sequent occurrence, say s,

in a derivation Der is the length of the branch segment

consisting of s and all sequent occurrences below it.

Now to gather a few facts. In particular we want

to show that various logical rules are

invertible (cf. Curry 63). In each case, the

proof is by a straightforward induction on weight of

derivation, which is generally left to the reader. Some of

these facts will be useful in what follows. Others are

given just out of interest.

Fact 3.2.1. o ~ is invertible, i.e.,

r 1 AoBfz 1- C is L1L derivable only if r 1 (A;B)r 2 1- C is.

Fact 3.2.2. 1- + is invertible.

Fact 3.2.3. vf- is invertible, i.e.,

r 1AvB r 2 I-C is derivable only if both r 1 A r 2 1-C and

r 1 B r 2 1- c are .

146

Fact 3.2.4. ~ & is invertible.

Similar claims for ~ v, f- o, -+ f- and & f- will not be

forthcoming, mainly because they would be false. But

there is an analogous fact for & f- in its Ketonin style

(see Curry 63, p.20l). Dunn 75 uses Ketonin & ~ and non­

Ketonin f- & , but they do not fit so nicely with the sort

of ''canonical forming" to be done later on e-sequences.

However, they are admissible.

Fact 3.2.5. If X f-A and YI-B are derivable, so is

E(X,Y) f-A&B.

Proof. Use Ke 1- twice, then f- & .

Fact 3.2.6. If r 1E(A,B)r 2 f-C is derivable, so is

r 1 A&Br 2 ]-C.

Proof. Use & f- twice, then We f- •

Fact 3.2.7. The inverse of the Ketonin form of & 1-

is admissible.

And while we are at it, the two following facts

are worth explicit note.

Fact 3.2.8. B'i ~ is admissible in the

LR-systems.

So by inspection of the rules and Fact 3.2.8,

it follows that

Fact 3.2.9 The LT-syste!Jl3·are subsystems of the • ot ot • LR-systems, i.e., E lS derivable in LT+ (LTW+) only lf

. ot ot it is derivable ln LR+ (LRW+ ).

Before moving on to more serious business,

it should be noted that L 1L does not have a strict

subformula property as is common for Gentzen systems.

Obviously, t- ~is the culprit. It has instead

Theorem 3.2.1. Approximate Subformula Property.

Let E' be a sequent occurring in a LlL derivation of

some sequent E. If A occurs in E', then either A is

tor A is a subformula of some formula occurring in E.

But we defer a discussion of this fact until §5,

in order to deal with more immediately pressing matters.

147

148

SECTION 3. Cut Theorem

As we said before, Formulation l was chosen for

relative ease in proving Cut and the desired equivalences.

The business of this section will be to give definitions

and an analysis of the rules that allow us to state and

prove an appropriate Cut Theorem. We begin with an analysis

of the rules as in §2.7 . So of course, the analysis is

along the lines of Dunn 75 and BGD80.

First, an inference is an ordered pair consisting

of a finite (non-null) sequence of sequents - the premises -

as left member and a sequent - the conclusion - as right

member. A rule is a set of inferences, and its members are

called instances thereof. A calculus or system is a set

of sequents - the axioms - together with a set of rules.

Again, we define the immediate ancestor(s) of a

formula constituent of the conclusion of an instance of

any rule of L1L by reference to the statement of the rule, as

in §2. Recall from §2.7 our ad ho~ convention for instances of

a rule in which the conclusion is the same sequent as the

premise, and that this convention is somewhat broader than

that of BGD80, p.351. The converse notion of immediate

descendant is also taken as defined.

Premise and conclusion parameters, principal and

subaltern are defined in the obvious ways analogous to the

definition of Chapter 2. But note that the occurrence of t

in the premise of the instance <t ;p )- p, p f- p> of t- !-is

a subaltern.

Now we will say, as in Chapter 2, that a rule Ru

149

is closed under parametric substitution just in case the

following conditions are met. Let Inf be an arbitrary

instance of Ru, and let a be a set containing only some

conclusion pararnetersof Inf and all of their immediate

ancestors. Then for an arbitrary structure X, let

Inf[X/aJ be the result of substituting (in the premise(s)

and conclusion of Inf) X for each member of a. Then

either Inf[X/aJ is an instance of Ru or its ''conclusion''

is the same sequent as its "premise". Note again that

the definition is a minor modification of those of AB75

and BGD80. We see straightaway

Lemma 3.3.1. Closure under parametric substitution.

The rules of LL are closed under parametric substitution.

Proof. Verification is relatively easy on inspection of

the rules.

Next, we say that a rule Ru is antecedent

expandable if it satisfies the following. Assume that

for 1 .:;; i .:;; n Ei is Xi f- Ci and En+l is X 1- C, and that

r 1 Ar 2 is a structure. Then suppose that

( 2) l:l, ... ,En

l:n+l

is an instance of Ru, Then

( 3) ' E' El,···, n

En+l ,

is an instance of Ru, where En+l' is r 1 xr 2 f- C and for

l ~ i ~ n E~ is either r 1xir 2 ~Ci or Ei depending on

whether or not Ci is an immediate ancestor of C in (3).

Again, this definition is a modification of BGD80.

And we reap

Lemma 3. 3. 2. All of the L1L rules except f- + and f- o

are antecedent expandable.

Proof. By inspection of the rules.

150

Now for the needed notion of rank in a derivation.

Let Der be a LlL derivation of E. U.nless E is an axiom

(the top node of a branch of Der), let Inf be the inference

(in Der) of which E is the conclusion, and let a be a set

of formula constituents of E. Then define the rank of a

in Der as follows. If a is empty, its rank in Der is 0.

If a is non-empty but contains no conclusion parameters,

then the rank of a in Der is l. (In this case, a is in

fact a singleton.) Otherwise, let inf be

(l) E 1 ' •.• 'En

E

with Der. the subderivation determined by E. for each ~ l

l ~ i ~ n, and let ai be the set containing all and only

immediate ancestors in E. of members of a. (Note that l

if all members of a were weakened in by Ke f- , then

ai = ¢.) Let k be the maximum rank of any ai in its

corresponding Der .. Then the rank in Der of a is k+l. ~

And following BGD80 we talk of the consequent rank of Der

151

as the rank of a in Der when a is the singleton containing

the consequent of the conclusion of Der.

Then, where a is a set of formula occurrences in

Y(E), let Y[X/aJ (E[X/a] be the result of substituting

X in Y(E) for each member of a. (When a is a singleton,

say {y}, we let E[X/y] = E[X/aJ.)

We are finally ready for the Cut 'l'heorem which can be

stated as follows:

Theorem 3.3.1. Cut Theorem.

Let a be a set of occurrences of any formula A

in a structure Y. If X f-A and Y f- C are L1L derivable,

then so is Y[X/a] f- C.

Proof. The proof proceeds as in Dunn 75 by a double

induction. Since the base steps of the inductions are trivial,

choose arbitrary m>O,.i and k such that j+k>O, and assume

Outer inductive hypotheses (OH). For all X,Y,C,A and a

(a set of occurrences of A in Y), if the complexity of A

is less than m, then if X 1- A and Y 1- C are derivable, so

is Y[X/aJ f- C; and

Inner inductive hypotheses (IH). For all X,Y,C,A of

complexity m, and a (a set of occurrences of A in Y), if

X 1-A is derivable with rank j' and there is a derivation

of Y f-C in which the rank of a is k' and j'+k'<j+k, then

Y[X/aJ f-C is derivable.

Next choose arbitrary A of complexity m and

arbitrary X,Y,C and a (a set of occurrences of A in a),

and assume

Conditional Hypothesis (CH). DerL is an LlL derivation

of X ~A with consequent rank j and DerR is a L1L

derivation of Y ~C in which the rank of a is k.

It will suffice to show that Y[X/aJ ~C is L1L derivable.

For the sake of notational convenience, let

L-premise = X ~A, R-premise = Y ~ C, and Conclusion =

Y[X/aJ ~C.

We now proceed by cases.

Case l. k = 0, so a is empty. (Note that j is

never 0.) Then Conclusion is L-premise and we are

finished by CH.

Case 2. k = 1. There are three subcases.

Case 2.1. R-premise is an axiom.

152

Then Conclusion is L-premise, and we are finished

by CH.

Case 2.2. R-premise follows from a sequent,

call it E, by Ke}. Then each member of a is parametric,

(but has no immediate ancesto~) whence by closure under

parametric substitution, either Conclusion is E or

153

follows therefrom by Ke }- .

Case 2.3. R-premise follows by a logical rule

on the left,call it Ru. Then a is a singleton containing

the principal constituent of R-premise. There are two

subcases.

Case 2.3.1. j = l. Then L-premise is either

an axiom or follows by a logical rule "matching" Ru on

the right. In the former instance, Conclusion is R-premise,

whence it is derivable by CH. So assume L-premise follows

by a logical rule, call it Ru '. There are four subcases

matching the four different logical rules on the right.

(Note that A cannot bet.) In each case, one applies OH

to the appropriate premises (possibly twice) to get

Conclusion. One never uses a structural rule as is

sometimes required in the analogous case in AB 75 and

BGD 80, We show one case as an example and leave the

rest to the reader.

Suppose Der1 ends as follows:

I 1 = X~B X~D = I2

L-premise = X ~ B&D

and DerR ends thus (without loss of generality):

I3

R-premise

=

=

r 1Dr 2 ~c

r 1B&Dr 2 ~ c

We want to show that Conclusion = r 1xr 2 ~C is derivable.

Let d be the set containing the displayed occurrence of D

154

in r 3, and note that on assumption B&D has complexity m,

whence D is of complexity less than m. Since E2 and E3

are derivable on the case assumption, so is

E4 = r 1xr 2 ~c by OH, as required.

Case 2.3.2. j>l, whence the consequent of L-premise

is parametric. This case is left to the reader to be

handled in a similar fashion to case 2.3.2 of the Cut

Theorem of §2.7.

Case 3. k>l. Again as in §2.7, but without the

special case for I ]- .

~

With the Cut Theorem in hand, we proceed to the

desired equivalences.

SECTION 4. Equivalence and Representational Adequacy 155

We are in a much happier position for showing

that the L-systems are equivalent to their Hilbert-style

analogues than was the case with the G-systems. The two

types of complex structures of the L-systems are directly

representing formula connectives, namely o and &. So define

a function t from the set of structures and sequents into £

by the following recursive specifications:

(1)

( 2)

( 3)

( 4)

t(A) = A, for every formula A;

t(X;Y) = t(X)ot(Y);

t(E(X 1 , ••• ,Xn)) = t(X 1 )& ... &t(Xn); and

t(X ~A) = t(X)->A.

We will want to show that t tA is L1L derivable

iff I-LA. Left to right is straightforward and simply

recorded as

Lemma 3.4.1. If X tA is L 1L derivable, then tLt(X tA).

This lemma shows that the translation is a good

one. However, the following fact is also significant in

that respect - and will be useful for other purposes.

Facts 3.4.1. r 1xr 2 1-A is L1L derivable iff r 1 t(X)r 2 1-C

is.

Proof. By induction on the structural complexity of X.

Base step. sc(X) = 1. Then X is a formula and t(X) = X,

156

so we are finished.

Now choose an arbitrary m>l and assume

Inductive hypothesis. ll 1 X'll 2 f-C is L1L derivable iff

61 t(X' )6 2 ~ C is, for all X' such that sc(X' )<m.

Inductive step. sc(X) = m. There are two cases.

Case l. X is, say, E(W 1 , ••• ,Wn). Then for all

l ~ i ~ n, sc(Wi)<m. So using the inductive hypothesis

n times, we have r 1E(W 1 , •.• ,Wn)r 2 f-C is L1L derivable

iff r 1E(t(W1 ), ••• ,t(W ))r 2 f-C is. But by Fact 3.2.6 n

and Fact 3.2.9, r 1E(t(W 1 ), ••• ,t(W ))r 2 f-C is derivable n iff r 1 t(W 1 )& ... &t(W )r 2 f-C is. Then by transitivity and

n

the definition oft, r 1E(Wl•· .. ,W )r 2 rC is derivable n

iff r 1 t(E(W 1 , ••• ,w ))r 2 f-C is, which finishes the case. . n

Case 2. X is, say, Y;Z. Proceed as in case l,

using the inductive hypothesis (twice), • f- and Fact 3.2.1

Returning to the matter of equivalence, for the

right to left half, it will be necessary to show that

the rules of L are admissible (in appropriate form) in LL.

Lemma 3.4.2. Rl, R2, and R3 are admissible, i.e.,

(l) If t f-A and t f- A+B are L1L derivable, so is t f- B;

(2) If t f-A and t f-B are L1L derivable, so is

t f-A&B; and

( 3) t 1- A+ .B+C is derivable in an LT-system only if

t I- AB+C is.

157

Proof. (2) is straightforward by 1-&. For (l), assume that

t ~A and t ~ A+B are L 1L derivable, and note that in any

event A+B;A f-B is LlL derivable. So using Cut twice,

t; t 1- B is derivable. Whence by t- ~ , t 1- B is derivable

as desired. Finally, for (3) from left to right, assume

that t f-A+.B+C is L1L-derivable. Again using the converse

of ~+,(t;A);B f-C is derivable. So by t- f-, A;B f-C is

derivable. Then by o ~ , AB f- C is derivable; whence by

t f- and 1- +, t f- AB+C is derivable to complete the proof.

~

These two lemmas make the proof of the Equivalence

Theorem straightforward.

Theorem 3.lJ.l. L 1L Equivalence Theorem

t j-A is derivable in L 1L iff ~LA.

Proof. For left to right, assume t f-A is L 1L derivable.

Then by Lemma 3. 4 .l, 1-L t+A, whence ~ LA.

Right to left proceeds by induction on length

of proof of A, for which it is necessary and sufficient

to show that t 1-A is LL derivable for each axiom A of L

and that Rl-R3 are admissible in the requisite sense.

The latter was accomplished in Lemma 3.4.2. For the

158

former, we show one example and leave the rest to the

reader. Since this is the first time for any LT-system

to be discussed in detail in print, we choose suffixing

0 t , l-in LTW+ as the example and thereby show that B i

does its job.

+ 1- A j- A Bj-B

A+B;A j-B c 1- c + 1-

B+C; (A+B; A) ]- C B'i 1-

A+B;B+C;A J-C J-+

A+B ;B+C I- A+C 1-+

A+B I- B+C+. A+C t 1-

t; A+B ]- B+C+. A+C

t ]- A+B+. B+C+. A+C.

~

The theorem is much as would be expected. The

only reason for bothering with its proof in so much

detail is that it provides an appropriate context for the

promised discussion of t- 1- •

First note that the use oft- 1- fuinessential in

the proof of (l) of Lemma 3. 4. 2 for systems with Wi ]- .

But matters are different for the contractionless systems.

If one wants a definite equivalence between them and their

axiomatic counterparts, there seems to be no alternative

to having t- ]--at least as an admissible rule. Without

such, one can (except as indicated below) show an

159

equivalence in the form of X ~A being L1LW derivable iff

~LWA, where X is a structure built up from t. But such

an equivalence will not suffice to give decidability of

the axiomatic systems from the decidability of their

corresponding 1-systems, and as such is inadequate for

our purposes.

Further, t- 1- is not needed in the proof of Ax 8

in the LR-systems, but the author knows of no way to

prove (3) of Lemma 3.4.2 without the use oft-~. In

fact there appears to be no way to show the even weaker

form of admissibility of R3 without it - the weaker

form being that if Y 1-A+.B+C is derivable, for some Y

built up from t, then for some X also built up from t,

so is X ~ AB+C. (Hence the qualification on the equivalence

claim of the previous paragraph.)

Aside from its practical necessity as just

indicated, two further considerations commend t- ~ ,

at least as an admissible rule. For the first one,

note that t is an identity with respect to o in

R:t and in RW:t, and that it is a left identity with

respect to o in the corresponding T-systems. So, since

the intensional structural connective (at least on this

point of view) is representing o, t- ~ ought to hold on

the grounds of representational adequacy. (Note, however,

that this point is irrelevant to the question of

equivalence between the L-systems and their axiomatic

counterparts. For t ~A+tA and t ~tA+A are derivable

without t- ~.)

The second reason is again representational

adequacy, but from a different point of view. If one

takes L to be indicating entailment in the sense of

Quine 66, then LL can be taken as a theory of L-

entailment (or L-deducibility). To put matters a bit

more formally, let us say that A L-entails B (or B is

L-deducible from A) just in case ~LA+B. One can then

coherently interpret each sequent (in LL) X ~C as

saying that t(X) L-entials C. For

Theorem 3.4.2. The L-systems are deductively complete,

that is, B is L-deducible from A iff for all structures

X such that t(X) =A, X 1-A is LL derivable. 1

Proof. Right to left is immediate by Lemma 3.4.1.

For left to right, assume that B is L-deducible from

A, i.e., ~ 1A+B, and let X be an arbitrary structure

160

such that t(X)=A. By Theorem 3.4.1 t ~A+B is derivable

in L1L, whence by Lemma 3.2.2, so is t;A ~B. So by t- 1-,

A ~B is derivable, i.e., on assumption, t(X) ~B. But by

Fact 3. 4 .l, X~ B is derivable - which completes the

proof.

The use oft-~ is essential in the proof,

as we now proceed to show. Let L'L come from L 1L

by dropping t- ~ . Then

161

Theorem 3.4.3. The L'L-systems are deductively incomplete.

Dot S Indeed, LR+ of AB75 and LR+ of BGD 0 are likewise

deductively incomplete.

Proof. Obviously, for LR+ and LROot t is to be taken as the

translation of AB75 p.385 and of BGD80 p.348, respectively,

The only ''significant'' change is that in these cases

t(E(X)) = t(I(X)) = t(X). It now suffices to note that

rLp+.(p+p)•p, and to show

(l) E = p r(p+p)op is not derivable in any of the

Gentzen systems under consideration.

To show (1), assume for reductio that Der is a

derivation of E in one of those systems. E is not an

axiom. Furthe~ by inspection of the rules, there is a

sequent, say E', occurring above E in Der which is of

the form w r (p+p) op and is the conclusion of an

instance of 1- o. Let Y r p+p be the left premise of

that instance of 1- o. It is easy to see that pis the

only formula that occurs in Y.

It then suffices to show that t ( Y )+.p+p is not a

theorem of L contradicting (the appropriate analogue of)

Lemma 3.4.1. This is easy to do by using the following

matrices (with l, 2 and 3 designated) which are sound, ot

in the usual sense, for R+ , and assigning p the value

of 2. (The & and v matrices are the normal ones

obtained by defining a&b(avb) as the glb(lub) of {a,b}

on numerical ordering, and t = 1.)

+

0

1

2

3

0

3

0

0

0

1

3

1

0

0

2 3

3 3

2 3

1 3

0 3

0

0

1

2

3

162

0 1 2 3

0 0 0 0

0 1 2 3

0 2 3 3

0 3 3 3

Of course, in the case of ROot, the proof relies on the

conservative extension results of Meyer and Routley 1974.

~~

One final observation on t- ~is in order. The choice

of a fusion formula as the counter example for the theorem

is inessential for all of the systems except the L'T­

systems. The argument would work just as well with the

RW+ theorem p+p+p ~p. However, the L'T-systems without

fusion are deductively complete.

And lastly, there is an interesting observation

to be made about ee ~ . The rule was not used in the proof

of Cut except in those cases in which one or the other

of the premises followed by it. (Recall, in particular,

the remark made in case 2.3 and the particular example

given there.) Nor was it needed for the proof of the

admissibility of the rules of Lin Lemma 3.4.2. And

finally, it is not required for the proof of any of the

axioms. So.

Theorem 3.4.4. The L-systems obtained by dropping ee ~

are equivalent to their axiomatic counterparts, in the

sense of Theorem 3.4.1.

163

In spite of this, ee ~ is kept in the formulation

for its usefulness in the Denestation Fact (and thus

in the Denestation Theorem) of §6.

164

FOOTNOTES

1 This idea, in a slightly modified form, was originally

suggested by Robert K. Meyer.

165

SECTION 5 FORMULATIONS 2 AND 3: Vanishing-t

If we pause for a moment to think about our goal

of a decision procedure for the contractionless systems,

it is quite obvious that Formulation 1 is not well-suited

for the task. There are several problems, but in this

section we will deal with just one- t- ~- The problem is

not the approximate subformula property. Other things being

equal, it will do quite nicely. Rather it is that we shall

want to use a suitable version of the decidability argument

of the previous chapter, and as things will work out t- ~

is not degree preserving. Two solutions come quickly to

mind. In the first instance, we could replace t- ~ by

other rules that provide its effect but are better behaved

with respect to degree. This can be done by adding (where

appropriate) rules such as

t+ ~ ti-A r 1Br 2 J-C

r 1A+Br 2 1-C

The second option is to do away with t and all of its works.

By and large, our only interest in t was for the technical

succor it provides, particularly for the cut theorem.

The first option has at least this much to commend

it: it is the more complete approach. Indeed, it was the

path that we initially took. But in the end, the second

option proves to be somewhat simpler and clearer, so the

systems will be reformulated to allow sequents to be

empty on the left.

We keep the definition of structures as before.

There will be no null or empty structure. We simply

allow sequents to be entities either of the form X ~A or

of the form ~A. To do otherwise is to introduce the

166

ridiculous question of whether or not there are structures

of the form E(X 1, ... ,X ), for instance, where each X. is n l

empty. Of course, the adopted policy is not without its

own headache. Technically, whenever we want to say something

about sequents in general we must speak double, once about

structures of the form X ~A and once about structures of

the form ~A.

Of course, when one has a headache, the sensible

thing to do is to take aspirin. Our aspirin will be to

use double-speak rather than speak double. We now allow

structural variables to be existentialist variables, that

is, they range over structures and the dreaded Nothingness.

Otherwise, notation remains the same.

We must still occasionally restrict structural

variables to range only over structures. But with a bit

of goodwill (and commonsense) on the part of the reader

and a few conventions, this is not so cumbersome. In the

first place, we insist that structural variables never

range over Nothingness when used to represent an immediate

constituent of an E-sequence. And likewise for structural

variables that occur in the statement of structural rules.

Further conventions can be adopted as the need arises.

The simplest method for getting rid of t is to

first leave it in and make a few modifications (including

167

being empty on the left) to Formulation 1, and then show

that we no longer need t. So let 1 21 come from L 1L by

I. Adding ~ t as an axiom;

II. Leaving the structural rules as they are (but note

the conventions on Nothingness;)

III. For the LT-systems, insisting that (1) the left

premise of _,. ~ is never empty on the left, and

(2) the right premise of ~ • is empty on the left

only if the left premise is; and

IV. For the LR-systems, replacing t ~ by the more

general

t# ~ r 1xr 2 ~c r 1xr 2 ~c

r 1 (X;Y)r 2 ~c r 1 (Y;X)r 2 ~c

where Y is a t-structure, and at-structure of

course is a structure in which the only formula

that occurs is t.

Since this is the first opportunity for the reader

to display such, a quick check on his/her good will is in

order. The reader should understand that

A ~B

~A+B

is an instance of~+, just as

~A

t 1-A

is an instance of t ~ (and of t# ~) and

t ~A

~A

is an instance of t- ~. However,

E(p,q) ~p

E(t,p,q) ~P

is not an instance oft~. Nor is

E(p,t,q) ~p

E(p,q) ~P

an instance of t- ~ .

168

Now if one extends the translation t of the previous

section to include the following clause:

( 0) t( ~A) = A,

one can show

Fact 3.5.1. If X ~A is L 2L derivable, then ~Lt(X ~A).

Proof. As in previous section. Note that where Y is a

t-structure, ~L t(Y)+t, and that the restrictions on

emptiness for the LT-systems are completely necessary.

,......,..,__,..._.....,,.......,...,_,~,....., .................... ,..,,....., ...... ,.....,

Of course L2L is a supersystem of L1L. So

given Fact 3.5.1 and the L 1L equivalence Theorem, it is

immediate that

Theorem 3.5.1. t ~A is L2L derivable iff ~L A.

Then note by t- ~. t ~ and t# ~

Theorem 3.5.2. t ~A is L2L derivable iff ~A is.

And these two theorems give us

Corollary 3.5.1. ~A is L2 L derivable iff ~L A.

But note that this Corollary is not sufficient

169

for us to do away with t. As yet we have no guarantee

that when t is not a subformula of A and ~A is derivable,

there is a derivation of it in which t is not employed.

For, so far, all that can be established is an approximate

subformula property as in Theorem 3.2.1. To rectify this

situation, we first show

Lemma 3.5.1. Let der be an L2 -derivation of a sequent E

satisfying the following conditions:

(1) t is not a subformula of the consequent of E;

(2) t is not a proper subformula of any formula

occurring in the antecedent of E; and

(3) E is not of the form r 1E(Y 1 , ... ,X, ... ,Yn)rz ~c.

where X is a t-structure and for some l < i < n

Yi is not a t-structure.

Then every sequent in der satisfies conditions (1), (2),

and (3).

Proof. That every such sequent satisfies (1) and (2) is

obvious (more or less) from Theorem 3.2.1. For (3), let

170

L' be an arbitrary sequent (occurrence) in der. An

induction on the height of L' will show that it satisfies

(3). The base step holds on assumption (since L' is then

L), and the cases for the inductive step are straightforward

on inspection of the rules.

"'"'"'""'"'""'"'"'"'"'"' ....... """'-

Of course the conditions of the lemma do not come

out of thin air. They are conditions that must be met by

a subderivation of a proof of a t-free formula. (1) and

(2) are obvious. Condition (3) is the important one. Put

quite loosely, once an occurrence of t gets ''properly

inside'' of an E-sequence which is not a t-structure, if

some descendant of that occurrence of t is not in a

''similar position'', then some descendant must have been a

principal constituent of a non-t logical rule. And if a

sequent containing such an occurrence of t were to be

in a derivation of a t-free sequent, then some descendant

of that occurrence of t must get in a "dissimilar position''

in order to be vanished by t- ~. The lemma shows that this

cannot happen.

Now we can show the important

171

Lemma 3.5.2. Vanishing-t Lemma.

Let X be a t-structure and let E be a sequent

satisfying conditions (1), (2) and (3) of Lemma 3.5.1. If

E is L2L derivable with weight n, then if E is

r 1 (X;Y)r 2 ~C (or also, in the case of the L2 R-systems,

if r is r 1 (Y;X)r 2 ~C) then E' = r 1 Yr 2 ~Cis likewise

derivable with weight ~ n - where Y is possibly empty if

r1 and r2 are

Proof. By induction on n. The base step is trivial,

and the inductive step is reasonably straightforward on

examination of the rules. We will only make a few general

comments and consider a couple of the trickier subcases.

First notice that Lemma 3.5.1 virtually guarantees on its

own that the inductive hypothesis will be applicable when

needed. Further, since the weight of derivation of E' is

no greater than that of E, the inductive hypothesis can be

applied successively when needed, e.g.,

E 1 = r 1 ((X;Y);(X;Y)r 2 f-C

r = r 1 (X;Y)r 2 ~c w; ~ .

Two applications of the inductive hypothesis to the premis

yields r 1 (Y;Y)r 2 ~C with no increase in weight, from which

E' follows by Wi ~C with appropriate weight.

For the L2 T-systems, some subcases of Bi ~ and B'i f­

are tricky, but still easy, e.g.,

172

E1 = r 1 (W;(Z;Y)r 2 1-C

E = r 1 (Z;W;Y)r 2 1-C , where Z;W is a t-structure.

We want to show r 1 Yr 2 1-C. But W and Z are t-structures,

and both the occurrence of W and the occurrence of Z in

the premise are in position for the (successive) application(s)

of the inductive hypothesis. So we are finished.

Finally, for the LR-systems, consider

LJ = r 1xr 2 ~c t # 1-

E = r 1 (Y;X)r 2 t-c

where the desired E' is r 1Yr 2 ~c. (Note that this is not

a case for the LT-systems.) There are two principal

subcases. (l) The displayed occurrence of X in z1 is not

an immediate constituent of an e-sequence. So suppose

without loss of generality that z 1 is 61(W;X)b 2 ~C. Then

on inductive hypothesis b 1Wb 2 1-C is derivable with no

greater weight than z 1 • Whence by t# I-, b 1(W;Y)b 2 ~Cis

derivable as desired. (2) Otherwise. Then E1 is (say)

blE(Wl, ... ,X, ... ,Wm)b 2 1-C. By condition (3) onE, each

Wi is a t-structure. So assume without loss of generality

that the displayed e-sequence is not an immediate constituent

of an e-sequence. (If it were, the containing E-sequence

would likewise be a t-structure, and we would deal with it

in a similar fashion.) Then z 1 is actually, say,

A1(Z;E(W1, ... ,X, ... ,W ))A 2 I-C. Then on inductive hypothesis m

A1ZA 2 ~Cis derivable with no greater weight than E1 .

Whence A1 (Z;E(W 1, ... ,Y, ... ,Wm))A 2 1-C is derivable with

appropriate weight by t# 1-.

Other cases for the Lemma are now left to the

reader's inspection,

173

The Vanishing-t Lemma puts us virtually home and

hosed, So let us say that a sequent is t-free just in case

t is not a subformula of any constituent thereof, and that

a derivation is t-free just in case every sequent therein

is such.

Lemma 3.5.3. If L is a t-free sequent, then E is L2 L

derivable iff there is a t-free derivation of it.

Proof. Right to left is immediate. Left to right proceeds

by induction on the weight of derivation of E. The base

step is trivial. The inductive step is easy. For note

that E does not follow by t r . And if I follows from

any rule Ru except t- r, then on inductive hypothesis the

premise(s) of that application of Ru has (have) a t-free

derivation(s). Whence by an application of Ru, so does E.

And if E follows by t- r, then by applying the Vanishing-t

Lemma to its premise,we see that Lis derivable with less

weight. So we are finished on inductive hypothesis.

With this lemma in hand we can formally say that

Formulation 2 has served us well:

Theorem 3.5.3. If t is not a subformula of A, then ~LA

iff there is a t-free, L2L derivation of rA.

Proof. Immediate from Lemma 3.5.3 and Corollary 3.5.1.

174

Now let us drop t from our language, so that L

ranges over the appropriate logics without t and its works

- except in reference to L 1L or L 2L, obviously. The

well-known conservative extension results tell

us that this is no real loss. Then let Formulation 3 of

L come from Formulation 2 by dropping the axioms t rt and

rt and the rules t r, t# rand t- r. Henceforward, we

will say that A is provable (in LL) iff rA is derivable.

So

Theorem 3.5.4. A is provable in L 3L iff rLA.

Proof. Obviously A is provable in L3L iff there is at-free

L 2L derivation of rA. Whence the theorem follows by

Theorem 3.5.3 and the aforementioned conservative extension

results.

175

SECTION 6. Denesting.

With little t out of the way, it is time to turn

to the promised canonical form fore-sequences. The point

of the exercise is the following. Even if e-sequences were

limited to reduced form (as they will be in the next

section), there are still an infinite number of

e-sequences that can be built up from a single formula

by nesting, e.g.,

E(p,p), E(p,E(p,p)), E(p,p,E(p,p)), etc.

This is representationally fitting, since such

nesting represents the different ways in which the

conjuncts of a conjunction can be associated. But we

must be able to at least ignore the differences brought

on by association, if we want to show decidability. So

the 1-systems must be reformulated yet again.

Let L 4L come from L 3L by adding the following

weakening and contraction rules:

' K e 1- •

w' e ~

w' 1 ~

r 1 E (X 1 , ••• , xn) r 2 ~ c

r 1E(X1 , .•. ,xn,Y)r 2 1-C

r 1 E (X 1 , ••• , xn, Y, Y) r 2 1- c

r 1 E ( x 1 , ... , xn , Y ) r 2 f- c n ;;. 1

r 1E(X 1 , ••• ,Xn,(E(W 1 , ••• ,W );E(W 1 , ••• ,W )))r 2 ~C m m

r 1 E(X 1 , ••• ,xn,W 1 , ••• ,Wm)r 2 ~c

176

We call Ke 1- and K'e f- (for instance) the companion of one

another. Naturally, K 'e f- is the prime companion of Ke f- .

It is easy to see that the primed rules are

admissible in the L3 -systems (use the companion rule

and ee 1- ); and of course, L3L is a subsystem of L4L. So

Theorem 3.6.1. L4L Equivalence Theorem

X f-A is L 4L derivable just in case it is L 3L

derivable. Hence, A is provable in L4L iff it is

provable in L.

Now let us say that a structure X is denested

just in case it has no substructure of the form

E(YJ, ... ,E(WJ, ... ,Wm), ... ,Yn). Then for any structure X,

define the denestation of X (dN(X)) as follows:

(l) dN(A) = A, for any formula A;

(2) dN(X;Y) = dN(X); dN(Y);

( 3) dN(E(X 1 , ••• ,E(Y 1 , •.• ,Ym), ... ,Xn))

dN(E(X 1 , ••• ,YJ, ... ,Ym' .. · ,Xn) );

=

( 4) dN(E(X 1 , .•• ,Xn)) = E(dN(X 1 ), .•. ,dN(Xn)),

where no Xi is an e-sequence;

Strictly speaking, clause (3) will not suffice. But

this is its most convenient form, so we state it as

such from the beginning for ease of reference. The

diligent reader is advised to take (3) as having the

proviso that the displayed occurrence of E(Y 1 , ••• ,Ym)

178

denested just in case each sequent that occurs in it is

denested. And let us say that an occurrence of a

substructure X of a structure Y is a nested e-sequence

(in Y) just in case it is an occurrence of an e-sequence

as an immediate constituent of an e-sequence (in Y).

Some further facts and lemmas can now be gathered toward

proving what is required.

Fact 3.6.4. If Z = r 1xr 2 is such that the displayed

occurrence of X is not a nested e-sequence, then

dN(r 1xr 2 ) = dN(r 1aN(X)r 2 ) = ~ 1 dN(X)~ 2 for some ~ 1 and

~ 2 , with the ''displayed occurrence of dN(X)'' in ~ 1 dN(X)6 2

corresponding, in the obvious sense, to the displayed

occurrence of X in Z.

Proof. By a straightforward induction on complexity of

Z which is left to the reader.

Fact 3.6.5. dN(r 1E(X 1 , ••• ,E(Y 1 , ••• ,Ym), ... ,Xn)r2) =

dN(f 1E(X 1 , ••• ,Y 1 , ••• ,Ym' ... ,Xn)r 2 ) •

Proof. Again by a straightforward induction on complexity.

Loosely speaking, what lies behind these two facts (and

the proof of the upcoming lemma) is this: for any

r 1 ,r 2 ,X 1 , ... ,X, there are some tq,A 2 ,Y 1 , ... ,Y such that n m

dN(r 1 x 1 , ... ,xnr 2 ) = A1Y1 , ... ,YmA 2 • And further, A1 and

J\ 2 are functions of r 1 and r 2 only. That is, if

dN(r 1 X 1 , ••. ,xnr 2) =

dN(r 1Z1, ... ,z1/2l =

A1Y1, ... ,Y A2 , then m

A1W1, ... ,W.A 2 , for some W1, ... ,W .. J J

177

is the first immediate constituent (in order of occurrence)

of the containing e-sequence which is itself an e-sequence.

The reader can then show that (3) is a fact as stated.

And let us say that a sequent is denested just

in case its antecedent is (or is empty); and for any

sequent X ~A, define dN(X ~A) as dN(X) ~dN(A), i.e.,

dN(X) ~A. (Of course, dN( ~A) = ~ dN(A) = ~A.) Then it

is clear that

Fact 3.6.1. For any structure X, dN(X) is a denested

structure. Thus, for any sequent E, dN(E) is a denested

sequent. And

Fact 3.6.2. For any denested structure X, dN(X) = X.

Hence for any denested sequent E, dN(E) = E.

The reader will no doubt have noticed that for

any sequent E, dN(E) either is E or follows from it by

one or more applications of ee ~ , in which case E also

follows from dN(E) by a sequence of applications ofee ~

So the following important fact is immediate.

Fact 3.6.3. Denestation Fact.

For any sequent E, E is L4L derivable iff dN(E) is.

This fact shows that every sequent has an equivalent

extensional canonical form. But the decidability

argument will require that derivations have an extensional

canonical form. So let us say that a derivation is

Lemma 3.6.1. dN-Substitution Lemma.

Let Z be a structure containing an occurrence y of some

structure Y such that y is not a nested e-sequence.

By Fact 3.6.5, let dN(Z) = ~ 1 dN(Y)~ 2 • Let y' be the

displayed occurrence of dN(Y). Then for all structures

X such that the substituted occurrence of X in Z[X/yJ

179

is not a nested e-sequence, dN(Z[X/yJ) = (dN(Z))[dN(X)/y'].

Proof. The proof proceeds by induction on the complexity

of Z. Since the base case is trivial, choose an arbitrary

m>O, and assume

Inductive hypothesis (IH). The lemma holds for a Z'

satisfying the conditions of the lemma.

Then choose an arbitrary Z of complexity m, y (an occurrence

in Z of some arbitrarily chosen Y) and X, all satisfying

the appropriate conditions of the lemma; and let

~ 1 dN(Y)~ 2 and y' be as in the lemma. It will suffice to

show that dN(Z[X/yJ) = (dN(Z))[dN(X)/y'J. There are

two cases.

Case 1. Z = Y. Straightforward.

Case 2. Y is a proper substructure of Z.

Again there are two cases.

Case 2.1. Z is an e-sequence, say E(X 1 , ••• ,

r 1Yr 2 , ••• ,Xn) withy the displayed occurrence of Y.

(Keep in mind that y is not a nested e-sequence.) The

180 proof branches into two subcases.

Case 2.1.1. An immediate constituent of Z is

an e-sequence. Then assume without loss of generality

that X1 is E(WJ, ... ,Wm).

Let Z1 = E(W 1, ... ,wm•·· .,r 1Yr 2 , ... ,XJ and let y 1

be the displayed occurrence of Y therein. By Fact 3.6.4,

and the definition of denestation, let dN(Z)=dN(Z 1) =

AJdN(Y)A 2 , withy' obviously corresponding to both y and

Y!· Clearly, dN(Z)[dN(X)/y'] = dN(Z 1)[dN(X)/y'].

And by Fact 3.6.5 dN(Z[X/y]) = dN(Z 1[X/y 1J). But by

IH, dN(Zl[X/ylJ) = dN(Z 1)[dN(X)/y'],so we are finished.

Case 2.1.2. No immediate constituent of Z is

an e-sequence. Of course, Z[X/y] = E(X 1, ... ,(r 1Yr 2 )

[X/y], ... ,Xn). Then dN(Z[X/y]) = E(dN(X 1), ... ,

dN((r 1Yr 2 )[X/y]), ... ,dN(X )). So by IH, n

dN(Z[X/y]) = E(dN(X 1), ... ,(dN(r 1Yr 2 ))[dN(X)/y'J, ... ,

dN(X )), which is (E(dN(X 1), ... ,dN(r 1Yr 2 ), ... ,dN(X )) n n

[dN(X)/y'J, i.e., (dN(Z))[dN(X)/y'J as desired.

Case 2.2. Z is an intensional structure. Without

loss of generality, let Z = A1YA 2;W. Then the argument

of case 2.1.2 applies mutatis mutandis to finish the

proof.

~

The reader should become familiar with the pattern of

argument above, since it is typical of many arguments to come.

The dN-Substitution Lemma facilitates the proof

of the

Lemma 3.6.2. Denestation Lemma.

If" follows from E1(E 2) by an application of any L4L

rule Ru, then either dN(E) = dN(E 1) or it follows from

181

dN(E 1 )(dN(E 2 )) by a sequence of applications of Ru and/or

its companion and possibly Ce ~ , such that the conclusion

of each such inference is denested.

Proof. By cases

Case 1. The lemma holds for logical rules on

the right by inspection. Further, if E follows by ee ~

from E1 , then dN(E) = dN(E 1 ) by Fact 3.6.5. And in the

case of Ce ~, it is clear that dN(E) follows from dN(E 1 )

by one or more applications of Ce ~ .

Case 2. If Ru is an intensional structural rule

or a logical rule on the left, then dN(E) follows from

dN(E 1 )(dN(E 2 )) by Ru. The cases are similar and reasonably

straightforward using the dN-Substitution Lemma. The

details of the proof of one case is provided as an

example.

Case 2.1. Ru is~·~

E 1

E

=

=

y ~A r 1Br 2 ~c

r 1 (A~B;YJr 2 ~ C

= E2

Let b be the displayed occurrence of B in E2 , and by

Fact 3.6.4 let dN(r 1Br 2 ) = A1 dN(B)A 2 = A1 BA 2 with b'

the displayed occurrence of B. (Assume without loss of

generality that Y is not Nothingness). But then by the

dN-Substitution Lemma, dN(E) = A 1 dN(A~B;Y)A 2 ~C, i.e.,

A 1 (A~B;dN(Y))A 2 ~C- which obviously follows from

dN(E 1 ) and dN(E 2 ) by~~.

182

Case 3. The remaining rules are companioned and

should be treated in pairs, first showing that the lemma

holds for the primed rule and then using that result as

needed for its companion. Ce ~ is often required due

to the positioning demanded for weakened and contracted

structures. However, these can be safely ignored since

the reader will have already noticed in checking case l

that the conclusion of an instance of Ce r is denested

iff the premise is.

Case 3.1 Ru is K'e r

E1

E

=

=

f1ECX1, ... ,XN)r2 r C

r 1 E ( X 1 , ••• , XN , Y ) r 2 r C

The proof proceeds by induction on the complexity of E. The

base step is trivial, so choose arbitrary m>O and assume

Inductive hypothesis (IH). For all E' of complexity

less than m, if E' follows from some l: 1' by K' e r , dN( E')

follows from dN(E;) in accordance with the conditions of

the lemma.

Then choose an arbitrary E of complexity m and assume

Conditional hypothes~~ (C). l: follows from some

E1 (call it E1 ) by K'e ~

It now suffices to show that dN(E) follows from dN(E1)

in accordance with the conditions of the lemma.

Let E1 and E be as displayed above, and let w be the

displayed occurrence of E(X 1 , ... ,Xn) in E1 . There are

two cases.

183

Case 3.1.1. Y is an e-sequence, say E(Wl•···•Wk).

Then E' = flE(X 1 , ... ,Xn,Wl,···•Wk)r 2 rC follows from

E1 by k applications of K'E r. So one can use (IH)

k times to show that dN(E') follows from dN(E 1 ) in

accordance with the conditions of the lemma. But by

Fact 3.6.5, dN(E) = dN(E') which finishes the case.

Case 3.1.2. Y is not an e-sequence. There are

two subcases.

Case 3.1.2.1. w is a nested e-sequence. Then

E' = f1X1, ... ,Xn,Yf2 J- C follows from E{ = f1X1, ... ,Xnf2 r C

by K 'E f-. Since sc(E ') < sc(E), by IH dN(E ') follows

• from dN(E 1) in accordance with the conditions of the lemma.

But by Fact 3.6.5, dN(E 1 ) = dN(E;) and dN(E) = dN(E')

which finishes the case.

Case 3.1.2.2. w is not a nested e-sequence; and

on case assumption 3.1.2, y (the displayed occurrence

of Y in E) is not either. Now if some X. is an e-sequence, l

proceed as in similar cases that have come before. So

assume no Xi is an e-sequence. In the first place, by

Fact 3.6.4, dN(E 1 ) = dN(f 1dN(E(X 1 , ... ,Xn))r 2 ) f-C, Now

let dN(EJ) = i\ldN(E(XJ,···•Xn))i\ 2 rC, and let w' be the

displayed occurrence of dN(E(X 1 , ... ,Xnll. Since

E = E1[E(X 1 , ... ,~,Y)/wJ, by the dN-Substitution Lemma,

dN(l.) = i\ldN(E(Xl, ... ,Xo,Y)i\ 2 rC. But since neither Y

nor any Xi is an e-sequence, dN(E) = A1E(dN(X1), ... ,

dN(Xn),dN(Y))A 2 ~C and dN(E 1) ~ A1E(dN(X1 ), ... ,dN(Xn))

A 2 ~ C. So dN (E) follows from dN ( l: 1 ) by K 'e ~ as

desired -which finishes Case 3.1.

Case 3.2. Ru is Ke~:

r 1 = r 1 xr 2 ~c

E = r 1E(X,Y)r 2 ~ C

Proof. By induction on complexity of E. The base step

184

is trivial. The inductive step separates conveniently into

two cases.

Case 3.2.1. X is an e-sequence, say E(W 1 , ... ,Wm).

Then E1 = r 1E(W 1, ... ,Wm)r 2 ~C. Let E' = r 1E(W 1 , ... ,Wm,Y)

r 2 ~ C, and note that l:' follows from E 1 by K 'e ~ . So by

case 3.1, dN(l:') follows from dN(E 1) in accordance with

the conditions of the lemma. But by Fact 3.6.5

dN(E) = dN(l:'), which completes the case.

Case 3.2.2. X is not an e-sequence. If X is

an immediate constituent of an e-sequence, an argument

similar to the one above will suffice. So assume otherwise.

There are two subcases.

Case 3.2.2.1. Y is an e-sequence, say E(WJ, ... ,Wm).

Then note that E2 = r 1E(X,W 1)r 2 ~C follows from r 1 by

Ke ~ and sc(l:~<sc(r 1 ). So on inductive hypothesis

dN(E 2) follows from dN(E 1) in accordance with the lemma.

But r' = r 1E(X,W 1, ... ,Wm)r 2 ~ C follows from E2 by a sequence

of applications of K'e ~. Whence by case 3.1, dN(r')

185

follows from dN(E 2 ) in accordance with the lemma.

Obviously, dN(E') then follows from dN(E 1 ) in accordance

with the lemma. And by Fact 3.6.5, dN(E) = dN(E') to

complete the case.

Case 3.2.2.2. Y is not an e-sequence. There are

two subcases.

Case 3.2.2.2.1. The displayed occurrence of

E(X,Y) in E is nested. Then r' = r 1X,Yr 2 ~C follows

from r 1 by K'e ~ and dN(E') = dN(E), so we are finished

on inductive hypothesis.

Case 3.2.2.2.2. The displayed occurrence of

E(X,Y) is not nested. Then use the dN-Substitution

Lemma as in previous cases.

This completes case 3.2. The other cases are

handled in a similar fashion and are left to the reader.

With the Denestation Lemma, we can make short

work of the proof of the following theorem to finish

the business of this section.

Theorem 6.2. Denestation Theorem.

For any sequent E, E is L 4L derivable iff

dN(E) has a denested derivation.

186

Proof. Right to left is obvious by the Denestation Fact.

Left to right proceeds by induction on the weight of

derivation of E. The base step is simple using Fact

3.6.2, and the cases for the inductive step are

straightforward using the Denestation Fact and the

Denestation Lemma.

187

SECTION 7 Reduction.

Now that e-sequences have effectively been given a

canonical form, a decidability argument analogous to the

one of Chapter 2 can be given. The first step is to get

the obvious analogue of reduction.

So let us say that a structure is reduced just in

case no constituent thereof is an e-sequence three or more

immediate constituents of which are occurrences of the same

structure. So no structure occurs more than twice as an

immediate constituent of any given extensional substructure

of a reduced structure. Then a structure is e-reduced iff

it is denested and reduced. Of course, a sequent is

reduced (e-reduced) just in case its antecedent and

consequent are (or its antecedent is empty); and a

derivation is reduced (e-reduced) iff each sequent occurring

therein is.

Next let us say that a structure is super reduced

just in case it contains no e-sequence with two distinct

immediate constituents that are occurrences of the same

structure. Again, the definition is extended to sequents

in the obvious way. (Obviously a super reduced structure

or sequent is reduced.) Then define the super reduct of

any denested structure as follows:

(1) sr(A) = A, for any formula A;

(2) sr(Y;Z) = sr(Y); sr(Z), for any structures Y and

Z; and

188

(3) for any structures Y1 , ... ,Yn, sr(E(YJ, ... ,Yn)) =

sr(Y 1 ), if for all 1 ~ i ~ n, Y. = Y1 ; otherwise, l

sr(E(YJ, ... ,Yn)) = E(WJ, ... ,Wm)' where E(WJ, ... ,Wm)

is as follows: For each Y., let k. be the number l l

of occurrences of sr(Y.) as an immediate constituent l

of E(sr(Y 1 ), •.• ,sr(Yn)). Then E(W 1 , ... ,Wm) is

the result of deleting the first ki-1 occurrences

of sr(Yi) from E(sr(Y1

), •.• ,sr(Ym)).

Naturally, for any formula A and denested structure X,

s r (X 1- A) = s r (X) 1- s r (A) = s r (X) 1- A, and s r ( 1- A) = 1- s r (A) =

1-A. And for the sake of notational convenience, let

X '(l:') be sr(X) (sr(E)) for the remainder of this section

only.

Then note straightaway that

Fact 3.7.1. For any denested structure X and any denested

sequent l:, X' is a super reduced structure and l:' is a

super reduced sequent.

And although it is not the case that the denestation of

any super reduced structure is super reduced, it is the

case that

Fact 3.7.2. For any denested structure X and any denested

sequent l:, X' and l:' are denested, and hence, by the

previousfact, e-reduced.

Given Ce ~ and the extensional contraction and

weakening rules, it is clear from the above and the

definition of super reduction that

Fact 3.7.3. The Super Reduction Fact

189

For any denested sequent I, I is L4L derivable iff I' is.

But the fact gives no reduction control over entire

derivations. What we want to show is that I' as above

has an e-reduced derivation if it has one at all. The

following facts and lemma will clear the way for it. But

first a definition.

For any denested e-sequence X, let con(X') (the

immediate constituents of X') be {X'}, if all the super

reductions of the immediate constituents of X are the same

structure, and otherwise let con(X') be the set of

structures occurring as immediate constituents of X'.

Then obviously

Fact 3.7.4. Let Z be a denested e-sequence. Then for all W,

WE con(Z') iff there exists a Y such that Y occurs as an

immediate constituent of Z and W = Y'.

The following simple fact will be important for

the main lemma.

Fact 3.7.5. For any structures X1 , ••• ,Xn,Y, and W which

are note-sequences, if Z1 = E(X 1 , ••• ,Y, ... ,Xn) and

Z2 = E(X 1 , ... ,W, ... ,XN)' then either

190

(l) con(z;) = con(Z~),

( 2) con(z;) - {Y '} = con(Z 2'),

(3) con(z;) - {W '} = con(Z{), or

( 4) con(Z~) - {Y'} = con(Z~) - {W'}

Proof. By Fact 3.7.4, con(Z{l and con(Zz) differ at most

in Y' being an element of the former but r.ot of the latter

and/or w' being an element of the latter but not of the

former - which is exactly what the fact says.

~~~

The fact not only provides a bit of useful information, but

also some structuring for cases in the proof of:

Lemma 3.7.1. Reduction Lemma

Let Inf

z1 (Z 2 )

Z3

be an instance of some L4L rule Ru, such that z 1 (z 2 ) and , , , , ,

Z3 are denested. Then either z3 = z 1 or z 3 = z2 , or E3 , ,

follows from z 1 (and/or z 2 ) by a sequence of applications

of Ce ~ , Ke ~ , ,

K e ~ , We 1- , w' e ~ , and/or at most one

application of Ru (if Ru be distinct from the aforementioned

rules), the conclusion of each of which is e-reduced.

Further, if no antecedent of a premise nor of the conclusion

191

of Inf is an e-sequence, then neither is the antecedent of the

conelusion of any inference in the above-mentioned ·sequence

Proof. By induction on the complexity of E. Since the

base case is trivial, choose an arbitrary m>O and assume

Inductive hypothesis (IH). The lemma holds for any rule

Ru' and any instance Inf' thereof such that the complexity

of the conclusion of Inf' is less than m (and of course

the premise(s) and conclusion are denested.)

Then choose an arbitrary rule Ru and an instance Inf

thereof with premise(s) E1 (E 2 ) and conclusion E3 , all of

which are denested and such that sc(E 3 ) = m. It will

suffice to show that the lemma holds for Inf. The proof

proceeds by cases.

Case l. Ru is ~o, ~+, ~v or ~&. Then E'3

follows from E~(E;) by Ru, by inspection.

Cases for the remaining rules all proceed in a

similar fashion. Significant details of the most difficult

case are presented below. The other cases are left to the

reader.

Case 2. Ru is v ~. There are three subcases.

Case 2.1. The antecedent of E3 is a single formula.

The case is simple, and left to the reader.

Case 2.2. The antecedent of I 3 is an intensional

structure. Assume without loss of generality that Inf is

1: 1 = x;r 1Ar 2 ~c X;r 1Br 2 f-C = l:z

l: 3 = x;r 1AvBr 2 f- c

with X a structure. By IH, (r 1AvBr 2 )' ~C either

(l) is (r 1Ar 2 )' f- C or (r 1Br 2 ) '~ C, in which case

x';(r 1AvBr 2)' f-C is x';(r 1Ar 2 )' ~cor x';(r 1Br 2 )' ~c,

. , , , , l.e., 1: 3 = 1: 1 or 1: 3 = 1: 2 ; or

192

(2) follows from (r 1Ar 2 )' f-C and/or (r 1Br 2 )' f-C in accordance

with the lemma. _Then by antecedent expandability,

t3 = X';(f1AvBr2)' ~C follows from t] = X';(r1Ar2)' f-C

and/or t'2 = X ; ( r 1Br 2 )' 1- C by a matching sequence of

applications of the same rules. And it is clear that the

conclusion of each is e-reduced, and that the furthermore

clause is fulfilled.

Case 2.3. The antecedent of l: is an extensional

structure. Technically, there are subcases according as

the super reducts of the immediate constituents of t 1, t 2

or t 3 are the same sequent or not. But the former case

is simple and is left to the reader. So assume

the latter and let Inf be

1: 1 = E(X 1, ... ,r 2Ar 2 , ... ,Xn) f-C E(X 1 , ... ,r 1Br 2 , ... ,Xn) f-C = <2

l: = E(X 1 , ... ,r 1AvBr 2 , ••• ,Xn) f-C

Now let z1 , Z2 and Z3 be the antecedents of t 1 , t 2 , and l:3

respectively. One can use Facts 3.7.5 to show that the

following cases are exhaustive.

193

Case 2.3.1. con(z3•) = con(zl') or con(Z 3') =

con(Z 2 '). Then either I; is E{ orE~, or it follows from

one of them by permuting the immediate constituents of its

antecedent. Obviously such applications of Ce ~ preserve

being e-reduced, and the furthermore clause is vacuously

fulfilled; so we are finished.

Case 2.3.2. con(Z 1 ')or con(Z2') is

con(Z 3 ')-{(r 1AvBr 2 )'}. Then Ii f~llows from ~i orr; by an

application of K'e ~, possibly followed by a sequence

of permutations of the immediate constituents of its

antecedent. Again it is obvious that the applications

of such rules preserve being e-reduced, and the

furthermore clause is vacuously fulfilled.

Case 2.3.3. Either (1) con(Z 3') =

con(Z 1 ')-{(r 1 Ar 2 )'} = con(Z 2 ')-{(r!Br 2 )'}, or

(2) con(Z 3 ')-{(r1 AvBr 2 )'} = con(Z 1 ')-{(.r 1Ar 2 1'} = con(Z 2 ')-{(r 1 Br 2 )'}. Noting·the previous remarks about

premuting, let 1:; = E(Y 1 , ... ,Y , (r 1 Ar 2 )') ~ C and let m

E'2 = E(Y 1 , ... ,Y ,(r 1Br 2 )') ~c. Then in the case of (2) m

above, 1:; is E(Y 1 , ... ,Y ,(r 1AvBr 2 ) ') ~C. In the case m

of (1), some Y. is (r 1 AvBr 2 )'. Without loss of generality, l

assume in that case that it is Y . Then 1: 3' is E(Y 1 , ... ,Y ) ~C. m m

Now note that by IH, (r 1 AvBr 2 )'~c follows from

(r 1Ar 2 )' ~ C and/or (r 1Br 2 )' ~C in accordance with the

lemma. Strictly speaking, there are two subcases according as

to whether or not v ~is used. But a proof for the latter

case is easy to construct out of a proof for the former.

So assume the former, and assume that the ''quasi-derivation''

promised by IH is as follows:

(riArz)'f-C (riBr 2 )' 1-C

xi f-C yi 1-C

xi 1-C yk 1-C

wi 1- c

wh 1- c

(riAvBr 2 ) 'f-C

with i,k and h greater than or equal to 0, respectively.

Call this quasi-derivation Deri. Then by antecedent

expandability

E(YI, ... ,Ym,(riAr 2 )') 1-C

E(YI,. .. ,Ym, XI) f-C

i E(YI, ... ,Ym, X )1-C

E(YI, ... ,Y ,(riBr 2)') 1-C m

E(YI, ... ,Y, yi) f-C m

k E(YI, ... ,Y, Y) ~C m

E(YI, ... ,Ym, wi) ~c

h E(YI, ... ,Ym, W) f-C

E(YI, ... ,Y ,(riAvBr 2 )') ~C m

is also a quasi-derivation, with each sequent following

from its predecessor(s) by an application of the same

rule as that by which its corresponding sequent followed

194

195

in Der 1. Call this new quasi-derivation Der 2 .

, , Of course the top nodes of Der 2 are E1 and E2 ,

respectively, and thus are super reduced and e-reduced.

It is simple to verify that each sequent of Der 2 is

e-reduced after noting that

(a) each sequent in Der 1 is e-reduced;

(b) since E!, E2 and E3 are e-reduced and thus denested,

neither (r 1 Ar 2 )' ,(r 1 Br 2 )' nor (r 1 AvBr 2 )' is an

e-sequence; whence (by the furthermore clause) none

X l i 1 k 1 h . d of , ... ,X ,Y , ... ,Y ,W , ... ,W are e-sequences, an

(c) each application of a rule changes the only structure

(occurrence) which could be a second occurrence of a

structure as an immediate constituent of the

antecedent of a premise of the application of that

rule.

Again, the furthermore clause is vacuously

fulfilled. Finally, for (2) above, the bottom node of

Derz is E3, and for (1) E3 follows from the bottom node

by w'e f- This completes the proof.

The Reduction Lemma makes quick work of

Theorem 3.7.1. Reduction Theorem

For any denested sequent E, E is L4L derivable iff E'

has an e-reduced derivation.

Proof. Right to left is straightforward by the Super

Reduction Fact. Left to right proceeds by induction on

the weight of derivation of E. The base step is simple,

and the inductive step is strightforward by the Super

Reduction Fact and the Reduction Lemma.

-~~

196

197

SECTION 8. Degree and Decidability

The Reduction Theorem produces a situation somewhat

familiar from Chapter 2. It will provide a finite upper bound

on the number of (denested) e-sequences built up from a

finite number of formulae that need to be examined in a

proof search - provided that a finite upper bound can be

placed simultaneously on the number of intensional structures

that can be built up from the said formulae. An appropriate

notion of degree will fill the bill for the proviso. So

define the degree (deg) of a formula as follows:

(l) deg(A) = o, if A is an atom;

( 2) deg(B&C) = deg(BvC) = deg(B) + deg(C), for

any formulae B and C; and

( 3) deg(B+C) = deg(BoC) = deg(B) + deg(C) + l,

for any formulae B and C.

Recalling that the degree of a formula is

supposed to indicate its intensional complexity, the

definition is obviously felicitous. And given that the

structural connective is ''standing in'' for fusion, it is

clear that the degree of a structure should be defined

as follows:

(l) deg(A) is of course the degree of the formula A

as defined above, for any formula A;

(2) deg(X;Y) = deg(X) + deg(Y)+l, for any structure

X and Y; and

(3) deg(E(X1 , ... ,Xn)) = max{deg(X 1), ... ,deg(Xn)},

for any structures X1 , ... ,Xn.

Naturally, for any structure X and formula A, deg(X}A)

198

= deg(X) + deg(A), and deg( ~A)= deg(A). Note again

that the degree of a sequent is not raised by 1 in virtue

of f-.

Now note the following obvious fact.

Fact 3.8.1. For any structures X, Y and Z, if deg(X) ~

deg(Y), then for any occurrence y of Yin Z, deg(Z[X/y]) ~

deg(Z).

Up to this point, all of the L4 -systems have

travelled along together. But here, as in Chapter 2,

the systems with intensional contraction part company.

For let us say that a rule is degree preserving just in

case for any instance of the rule, the degree of the

conclusion is greater than or equal to that of any premise.

Then using the above fact, it is clear that

Lemma 3.8.1. Degree Lemma

The rules of L4Rw: and L4 TW: are degree preserving.

199

Now reduction and degree will work in tandem

to give us the needed control on the total complexity of

structures that can occur in the sorts of derivations of

a given sequent to which we can restrict our attention.

The virtual coup de grace is delivered by

Lemma 3.8.2. Counting Lemma

For any formula A and any n ~ 0, there are at most

finitely many e-reduced structures of degree ~n built

up from subformulae of A.

Proof. By induction on n. The base step is simple.

So choose an arbitrary m>O and assume

Inductive hypothesis (IH). For any formula Band any

k<m, there are at most finitely many e-reduced structures

of degree ~ k.

Now choose an arbitrary formula A. It will then suffice

to show that there are at most finitely many e-reduced

structures of degree ~ m built out of subformula of A.

But any such structure is either

(1) a subformula of A, of which there are only

finitely many;

(2) an intensional structure,whose left and right

constituents are of degree < m (by the definition

of degree ) and of course are built out of

subformulae of A. But by IH there are at most

finitely many such structures to serve as left

and right constituents. Whence there are but

finitely many intensional structures of the

required kind; or

200

(3) an e-sequence, each of whose immediate

constituents is a non-extensional structure of

degree ~ m (by the definition of e-reduced and

degree) and again built out of subformulae of A.

By IH and (1) and (2) above, there are at most

finitely many structuresto serve as immediate

constituents; and by the definition of e-reduced,

none can occur more than twice as such. So

there are at most finitely many e-sequences of

the requisite sort.

And Finitely Many + FINITELY MANY + FINITELY MANY =

FINITELY MANY. So we are finished.

~

Of course, the lemma holds equally well for e-reduced

sequents built up from subformulae of any of a finite

number of formulae.

Decidability is now clearly in sight. All that

remains to be shown are well-known and/or by now obvious

facts. First, let us say that a derivation is

irredundant just in case no sequent occurs more than once

on a branch thereof. Recalling The Denestation and

Reduction Theorems, it is clear that

Theorem 3.8.1. Irredundancy Theorem

Any sequent E is L4L derivable iff sr(dN(E)) has an

irredundant, e-reduced derivation.

201

Next, let us specify as follows a proof search

procedure which produces the LRW:(LTW:) proof search tree

of E for any sequent E:

(1) Enter sr(dN(E)) as the bottom node;

(2) above each sequent E' occurring with height k

(in the tree so far constructed) (a) enter

nothing, if E' is an axiom, (b) otherwise enter

(in some assumed order) all e-reduced sequents

E" such that E" is a premise of some L4 Rw:(L 4 TW:)

inference of which E' is the conclusion and such

that the tree remains irredundant.

Obviously

Lemma 3.8.3. Effectiveness Lemma

The proof search procedure thus specified is effective.

Now let us say that a (possibly null) tree T' is a

subtree of a tree T iff it is the result of deleting

some (possibly no) sequent occurrences in T and all

sequent occurrences above them. Then by the Irredundancy

Theorem and the above specification

Lemma 3.8.4. Completeness Lemma.

The proof search procedure is complete, i.e., E is

L4L derivable iff some subtree of the proof search tree

of Eisa L4L derivation of sr(dN(E)).

As usual, a tree has the finite fork property

202

iff it has at most finitely many nodes of any given height;

and a tree has the finite branch property iff each of its

branches contains at most finitely many nodes. And recall

Konigs Lemma. A tree is finite iff it has the finite

fork and finite branch property.

Now, by inspection of the rules

Lemma 3.8.5. The proof search tree of any sequent E

has the finite fork property.

Of course L4L has the Subformula Property. But

more important for our purposes

Lemma 3.8.6. For any inference of L4L, every formula

constituent of a premise thereof is a subformula of a

formula constituent of the conclusion.

At last we have

Lemma 3. 8. "(. The proof search tree of any sequent E

has the finite branch property.

203

Proof. Choose an arbitrary sequent, say E, and let

m = deg(sr(dN(E))). By the Counting Lemma there are at

most finitely many e-reduced structures of degree < m

built up from subformulae of formula constituents of E.

Whence by the specification of the proof search procedure,

Lemma 3.8.6 and the Degree Lemma, there can be but a finite

number of different sequents, occurring no more than once,

on any given branch of the proof search tree of E - which

completes the proof.

~~~

So we conclude straightaway

Lemma 3.8.8. Finitude Lemma

The proof search tree of any sequent E is finite

Finally, by the Effectiveness, Completeness and

Finitude Lemmas and the equivalence of Theorem 3.5.4,

we get our main result

Theorem 3.8.2. L4Rw: and L4Tw: are decidable.

And then by the L4L Equivalence Theorem

Theorem 3.8.3. 0 0

RW+ and TW+ are decidable, which completes

the business of the Chapter.

SNOI.LS3:fi0 N3:d0 GN\f Sl1fiS3:l:! 9NIGn1::JNO::J ·p l:!Hld\fHJ

204

SECTION 1. Introduction

This, the concluding chapter, looks backward and

forward. In §2 we collect some easy results for various

fragments of the logics treated in the previous chapter.

And in §3 we discuss the problems of using our techniques

to show EW+ decidable.

In §4 we formulate modified Display Logics for

TW0

1 and RWol and prove appropriate equivalences. We then

outline a proposed method of extending our basic decidability

argument to these systems, filling it in as far as we now

can. The conservative extension results of §1.8 show that

the decidability of these boolean systems would suffice

for the decidability of TW and RW.

Finally, we conclude this work with a discussion in

§5 of what we take to be one of the most interesting

questions that has arisen from our research, namely,

whether or not RW+is equivalent to uRW+.

205

SECTION 2 Decidable Fragments.

The L-systems of the previous chapter give

separation results for the logics considered there. The

1-systems for+ and+,& with or without o, as well as the

positive systems without o can be shown equivalent to

their axiomatic counterparts, thus giving simple proofs of

conservative extension. 1 Most of these results are already

known. The most extensive list of such currently in print

is to be found in Meyer and Routley 74. But a more

complete and updated report will be found in RLRI.

Of course, given the above conservative extensions

and equivalences, the decision procedure of the previous 0 0

chapter also decides the relevant fragments of TW+ and RW+.

Of these, only one has been previously published. That is

of course the duly famous PW. It was first shown to be

decidable using the "merge" Gentzen systems of AB75. The

result is essentially recorded there on p.69. A second

proof of its decidability is given in Martin 79 using

semantic techniquesdeveloped therein.

Many of these results can be duplicated using the

subscripted Gentzen systems of Chapter 2. The G-systems

without v-rules and without K ~and W 1- can be shown

equivalent to T+& and R+&' respectively, by simply proving

the axioms (Cut and modus ponens will stay in force) and

then translating into the appropriate semilattice semantics

of Chapter 1 § 9, making use of Urquhart Theorem 1 given

there. Alternatively, one could translate into the

operational semantics of Fine 74 or of RLRI. The same

goes for T+ and R+.

206

And of course, the same remarks apply mutatis

mutandis for fragments of GTW+ and GRW+. Thus the decision

procedure of Chapter 2 gives yet another proof of

decidability for the pure implication and implication

conjunction fragments of TW+ and RW+.

207

FOOTNOTES

1Fragments with only one of & and v, can be formulated

without any extensional structural rules, and hence

without e-sequences period. It is another reason for

liking the particular forms of & ~ and ~ & which we use.

SECTION 3 E+ and EW+

The straightforward way to formulate the

LE-systems is to add

Cit f- r 1.(X;tl r 2 l- c

r 1 Ct;X)r 2 J-C

to the appropriate LT-systems of the previous chapter.

The rule is admissible on translation in E:t and in ot

EW+ . In the analysis of the rules, the permuted

occurrence of t can be counted as principal, to allow

208

closure under parametric substitution as usual. The proof

of Cut then goes through as in Chapter 3.

However, the proof of Vanishing-t runs into trouble.

Ct ~forces one to argue that t can be vanished as a right

constituent as well as a left constituent of an intensional

structure, as was the case with the LR-systems. But then

trouble arises in cases such as

B, ~ (X;(Y;t) ~C

Y;X;t ~ C

One can show X;Y ~Con inductive hypothesis. But lacking

a general rule of intensional permutation, the way is

apparently blocked to the desired Y;X ~C.

A solution to this problem is to formulate LE+

and LEW+ by exchanging Ct ~ for

t ~A r 1 Br 2 ~c X~ A t ~ B t+ ~ and t ~ o

r 1 (t;A+B)r 2 ~ c t;X~AoB

The L2E-systems are then specified by allowing

emptiness on the left and adding f- t as an axiom.

209

Vanishing-t and all of its works will then go through, as

will the argument for decidability for L4Ew:t

However, the normal argument for Cut breaks down

in this formulation for the case of the left cut sequent

following by f-+ and the right cut sequent following by

the matching t+ f- . And our attempts to build into the

Cut Theorem itself the required permutation of t look far

from promising. Further, an inductive argument to the

effect that Cit is admissible breaks down on the case for

B'i ~ .. We suspect that these systems are simply too weak.

The situation can be salvaged - somewhat. In the

first place, the original formulation of LE:t with Cit~

can be shown equivalent to E:t, which we axiomatize as in

Routley and Meyer 72 by adding

E+Ax. t+A+A

ot t t to T+ . EW+, of course, comes by adding E+Ax to TW+.

Now let us formulate L'EW~ by dropping t- r and the fusion rules from our original formulation of

ot LEW+ . One can then show

Indefinite Equivalence.

some t-structure X, X r A

A is a theorem of EW~ iff

is derivable in L'EW~.

for

The reason for dropping o is that without t- f- the proof

of admissibility of R3 is blocked.

210

The rule Citr presents no new problems for denesting

and reduction. And it is obviously degree preserving.

So the argument of the previous chapter can be applied to

show LEW~ decidable.

But because of the indefiniteness of the equivalence,

this does not suffice to show EW~ decidable. This is an

altogether peculiar situation. One can hardly believe that

it is irremediable. What one would really like to do, is

to replace Cit ~ by a more general rule of restricted

permutation. But we know of no way to specify in advance

just which structures are permutable. So we leave as an

Open Question. Is EW+ decidable?

211

SECTION 4. Extensions and Decidability

We noted in Chapter 3 §8 that our decidability

technique breaks down in the presence of intensional

contraction. Specifically, Wi ~ is not degree preserving.

And there appears to be no straightforward modification of

the technique which can cope with this rule. Indeed, the e.s\.o..l,..\,.;c;.\...,.s, ~ ... u.."'<io..c,d-al.. ~L~':l.

8+ geptainly ~uts the oegs in recent

"'' T, I faveur

result of Urquhart E'" n_,._~ R ,_ or H+ Baing uneesi9aGl~. As was noted in Meyer and

l Giambrone 80, if R+ is undecidable, so is T+,E+,T,E and R.

However, there is some hope of extending our

decision procedure to TW and RW, by way of the Display 2

Logics of Belnap 8+, to a discussion of which we now turn.

In some sense, Display Logic brings the ideas of

Dunn 75 and Meyer 76 to their logical conclusion.

If we can have a structural analogue of o, why not of other

connectives? Belnap 8+ shows that we can, and reaps the

benefits by showing that Display Logics can be given

for an enormous range of logics, some well-known and

others yet undreamt of.

Here we will concern ourselves with only the Display

Logics for TW 01 and RW 01 , which we call DTW and DRW for

the sake of simplicity. We assume£ to have the appropriate

vocabulary.

Equivalence will be shown via TWo 1 t and RW01

t,

in the language of which it will be convenient to define

Df. f =df -t, and

D+. A+B =df ~A+B.

212

For the Display Logics thermselves we will want

five structural connectives, represented by III, 1;1, 1,1,

1·1, 111, respectively. - and I are structural analogues

of -and I, of course. The other connectives are context

sensitive. I alternately stands in fort and f, ; for o

and +, and , for & and v.

We again use lxl, lyl, etc. as structural variables;

and parentheses will be used to disambiguate inscriptions.

Structures are defined recursively:

l. I and A are structures, for any formula A;

2. x·, xl, (X;Y) and (X,Y) are structures, for any

structures X andY.

Of course, sequents are of the form X ~Y.

DTW and DRW can then be specified from the following

axioms and rules.

AXIOMS

A I- A

RULES

Display Equivalences

X,Y~Z<>X~YI,z

X;Y~Z "'Xl-Y-;Z

X ~Y,Z .. X,YI ~z .. X ~Z,Y

X ~Y,Z .. X;Y- ~z .. X ~Z;Y

X~Y .. yl~xl <>XTI~y

x J- Y .. x- ~ Y .. x-- ~ Y

Structural Rules

I ~ X~ y

I ;X ~ Y

B; ~ W; (X;Y) ~ Z

(W;X) ;Y ~ Z

c; ~ (W;X) ;Y ~ Z (W;Y); X~ Z

c' ~ (W,X) ,Y j- Z (W,Y),Xj-Z

w' ~ Y,YJ-Z

Y F z

Logical Rules

& j-

v j-

A,B j- Z A&B I- Z

A~X B~Y AvB!-X,Y

I- j-

B,; ~

CI; ~

CI, j-

K' ~

~ &

l- v

I ;X ~ Y X ~ y

W; (X;Y) ~ Z (X;W) ;Y ~ Z

X; Y ~ Z Y ;X ~ Z

X, Y ~ Z Y ,X ~ Z

X ~ Z X, Y ~ Z

X~A Y~B X,Y ~A&B

X j- A ,B X j- AvB

213

X~A B~Y + ~ A+B ~ x- ;Y

1- A;B ~z o Ao B ~ Z

~j-A-1-Z ~A 1- Z

AI 1- Z I 1- lA 1- Z

X;A I- B ~ + X~ A+B

~o Z~A Y~B X;Y ~ AoB

1--

I-I

Z 1- A­Z I- ~A

Z 1- AI Z I- lA

All of the axioms, logical rules, display

equivalences, I 1-, I- 1-, C, ~,

214

CI, 1-, W, 1- and K, 1- are common to both DTW and DRW. For ,

DTW add B; 1- and B ; 1- • For DRW add C; 1- and CI; 1- instead.

Derivations are as usual. And we say that a

formula A is provable in DTW (DRW) just in case I ~A is

derivable therein. We will want to show that DTW and DRW

are equivalent to TW 01 and RW 01 , respectively.

All of the axioms of TW01 (RW 01 ) are provable in

DTW (DRW). To give the flavor of these Gentzen systems

we prove two of the negation axioms (!del indicates use

of one or more display equivalences.)

Axl6. B 1- B de B- 1- B-

~t-

A 1- A --£1-B-+ 1-

A+"£ 1- A-·B-de

A+~B; B 1- A-~~

A+-B;B I- ~A 1-+ A+-B 1- B+~A

I ~ I :A+-B 1- B+-A

~-+ I 1- A +-B+. B+~A

215

Ax21. c 1- c de Cl f- Cl

f-1 Cl f- IC

K' f-Cl 1 (A; B) f- IC CI, f-

(A; B) 1 Cl f- IC de A;B f- CIC f- v A;Bf-CviC 1- +

A f- B+. C viC I f-I ; A f- B+. C viC f-+

If- A+.B+.CviC

We leave it to the reader to show that the other

axioms are provable and that R2 and R3 are admissible

rules. To show Rl admissible requires a Cut Theorem. But

we have one ready-made. In §4 of Belnap 8+, a very

general proof of Cut in Display Logic is given which covers

DTW and DRW. So

Belnap Lemma 1. Cut is admissible in DTW and DRW, i.e.,

if X f-A and A f-Y are derivable, so is X f-Y;

and the reader can now easily show that Rl is admissible.

(Note, however, that I-f- must be used.) So

Lemma 4.4.1. A is provable in TW 01 (RW 01 ) only if it is

provable in DTW (DRW).

One can show the other half of the equivalence by

translating sequents into formulae with the result that

the translate of any derivable sequent is a theorem. But

since we have an eye to decidability, we will move to a

216

new formulation and complete the equivalence for it.

Although we do not have a proof of decidability,

we can go some way toward extending our techniques to one.

Here we take them as far as we can, sketch a proposal

for completing the work and discuss the difficulties

involved.

In the first place, we will need some sort of

reduction. As we commented in Chapter 3, a binary

extensional structure is not convenient for such. Since

the notation for DTW and DRW is relatively simple, we will

trade in the extensional binary structuring for sets. But

this will not suffice, since we still have ''negative''

structural connectives. So we could still nest a

"structurally negated" extensional structure within other

extensional structures.

However, when one notes that the boolean systems

have DeMorgan Laws for both boolean and DeMorgan negation,

the solution becomes obvious. Structural negation will

be driven inside of sets automatically.3

Even so, negative structural connectives still

present a problem. For the Display Equivalences allow

these connectives to appear and disappear. If decidability

is to be shown, we must get some control over how many

negative structural connectives can ''bind'' a given structure.

The following facts about the boolean systems give the

solution:

217

Fact 4.4.1. A is provably equivalent to IIA and to --A.

Fact 4.4.2. -lA is provably equivalent to !~A.

Fact 4.4.3. The boolean systems are closed under replacement

of provable equivalents.

So we can effectively "permute'' and "contract" negative

structure connectives. The simple and obvious thing to do

is to trade in negative structuring for sets of negation

formula connectives.

We now present new formulations of DTW and DRW,

which we call lsTWI and lsRWI, whose structures (s-structures)

will be ordered pairs and sets thereof. So let us

simultaneously define quasi-structures and s-structures,

using a,S,y, etc. as variables over quasi-structures,

X,Y,Z, etc. as variables over s-structures and x,y,z, etc.

as variables ranging over subsets of {~,1}:

(l) Formulae are quasi-structures, and so is I;

(2) if a is a quasi-structure and x is a subset of

{~,1}, then <a,x> is an s-structure;

(3) if X and Y are structures, then (X;Y) is a

(4)

quasi-structure; and

if X1 , ••• ,X ares-structures which are ordered pairs n

(note, not sets), then {X 1 , ••• ,X} is an s-structure. --- ---- n

218

Naturally, s-sequents are of the form X 1- Y.

Next, let us make some notational conventions and

define some useful operations. First let us identify

(s-structures which are) singletons with their sole

element, i.e., {<a,x>} = <a,x>, for instance. And when

convenient, we will write lal for '<a,¢>1. For example,

we allow ourselves to write IAI for '<A,¢>1.

We use lui for ordinary set union, I_! for binary

set complementation, and r~l for symmetric difference

(see Kuratowski and Mostowski 68, for example) which can

be defined by (where a and b are sets)

a~ b = (a-b)u(b-a); or equivalently, a~ b = (aub)-(anb).

And let us define, for any quasi-structures a 1 , .•• ,an and

any subsets of {-,1} XJ, ... ,xn y, '

{<a 1 ,x 1>, ... ,<a ,x >} U y = {<a 1 ,(x 1-'-y)>, ... ,<a ,(xn~Y)>}. n n n

Recall for this definition that any s-structure which is

an ordered pair has been identified with the singleton set

containing it. Further, that same convention makes

expressions of the form rxuYI well-defined.

Keeping these definitions and conventions firmly

in mind it is easy to specify the axioms and rules of sTW

and sRW. Interpret commas as set union, interpret

•x-1 as Xu{~}, andlx1 1 as Xu {I} in the statement of

219

the rules for DTW and DRW. sTW and sRW can then be specied

as DTW and DRW were. However, C, f- , CI, f- and W, f- are

now redundant, so we drop them.

Given this specification, it is clear that

Lemma 4.4.2. DTW and DRW are contained on the obvious

translation in sTW and sRW, respectively.

Then by Lemmas 4.4.1 and 4.4.2, we have immediately

Lemma 4.4.3. A is provable in TW 0'(RW 01 ) only if it is

provable in sTW(sRW).

To complete the desired equivalence, we will want

to translate sequents into formulae. First, we define

simultaneously two functions c and a (consequent and

antecedent- cf., §2.3 and 2.4 of Belnap 8+) from s-structures

and quasi-structures into formulae: For any formula A,

any quasi-structure a and s-structures X andY,

1. c(A) = a(A) = A; for any formula A;

2. c (I) = f and a (I) = t;

3. c(<a,¢>) = c(a) and a(<a,¢>) = a(a); for any

quasi-structure a;

4. c(X;Y) = c(X)+c(Y), and a(X;Y) = a(X)oa(Y), for any

s-structures X andY.

5. c(<a,{~}>) = ~(a(a)), and a(<a,{~]>) = ~(c(a)), for

any quasi-structure a;

6. c(<ct,x U {"""1}>) = l(a(<a,x>)) and a(<a,x U {I)>=

I( c(<a,x>) )4; and

7 . cfX1, ... ,X} = c(X 1)v ... vc(X ),and a({X1 , ... ,X }) = n n n

220

a(x 1 )& ... &a(X ), for any distinct s-structure X1•···•X, n n n > 1. (We assume an ordering of s-structures.)

Then for any s-structuresX andY, let t(X~Y) =

a(X) + c(Y).

Now with Facts 4.4.1 - 4.4.3, the reader can

easily verify

Lemma 4.4.4. X ~y is derivable in sTW(sRW) only if

t(X ~ Y) is a theorem of TWo It (RWolt).

And it is now easy to show

Theorem 4.4.1. For any formula A in the language of

TW 01, A is provable in TW 01 (RW 01

) iff it is provable in

sTW(sRW).

Proof. Left to right is immediate from Lemma 4.4.3. For

right to left, assume I ~A is derivable in one of the olt

s-systems. By Lemma 4.4.4, t+A is a theorem of TW olt or of RW , as the case may be. But t is also a theorem

(it's an axiom), whence A is a theorem. Then by Theorem

8 , ol ol 1. . 3, A lS a theorem of TW ( RW ) , to complete the proof.

221

And by Theorem l. 8 . 2, we get

Theorem 4.4,2, For any formula A in the language of TW,

A is provable in TW(RW) iff it is provable in sTW(sRW).

We now have the desired equivalences, and good

extensional control . But two questions remain for

applying our technique for decidability.

Question l. Vanishing-t (Vanishing-I)

Can the systems be formulated without I- ~· without losing

equivalences.

The answer to this question must (should?) be

yes. Our preferred method of going empty on the left-

and now thus empty on the right - presents technical

difficulties on how to translate sequents that are empty

on one side or other. For consider an s-derivation

ending as follows;

(l) ( 2)

( 3)

( 4)

I ;A ~ B

I~ A+B

<A+B {-}>]-<I {-}>

-(A+B) ~<I {-}>

<I {-I}> ~<-(A+B) {I}>

<-(A+B) ,{1,-}>~<I,{"l}>

If the systems were allowed to be empty, we would be

forced to interpret emptiness on the left alternatively

as t inland as 1-t in (3). Emptiness on the right is

222

analogous in (2) and (4). And this is only the tip of the

iceberg.

The simpler course would seem to be to leave I

in the system, stay non-empty and argue to the effect that

I;X f-Z is derivable just in case X rZ is. But in any

event, an argument for Vanishing-t will not be a

straightforward adaptation of the argument of §3.5.

For the analogue of Lemma 3. 5.1 is not immediately

forthcoming. I's can move in and out of sets with members

other than I, as illustrated below

I r p+p

n,q} r p+p

I r { <q, fl}> ,p+p}

It may be the case that no Vanishing-t Theorem

can be had. Or that the most one could hope for would be

the minimum requirement that I;I f-Z be derivable just in

case I f-Z is. For the time being, the question remains

open.

Question 2. Degree

Is there a determinable upper bound on the degree of a

sequent that can occur in a derivation of a given sequent.

Obviously, the first step is to find an

appropriate definition of degree. The following immediately

suggests itself:

223

(1) deg(I) = 0;

(2) deg(A) is the total number of ~·sand o 1 s occurring

in A;

(3) deg(X;Y) = deg(X) + deg(Y) + 1;

(4) deg(<a,x>) = deg(a);

(5) deg({X 1, ... ,Xn}) = max({deg(X 1), ... ,deg(Xn)}); and

(6) deg(X ~ Y) = deg(X) + deg(Y).

That is, ignore negation and count degree essentially as

before.

But on this definition, certain boolean display

equivalences fail to be degree-preserving, as shown by: 5

{p,p+p} ~ A~.B~C

P ~ {<p+p,CI}>,A+.B~C}

One possible solution to this problem is to define

a normal form for sequents in such a way that no s-structures

of the form Xu <a,y u {I}> occurs in it, and then count

the degree of a sequent as the degree of its normal form.

But if I ~ and ~I are to be degree preserving, such normal

forming would almost certainly have 6 to be radical enough

to eliminate s-structures of the form Xu lA.

At this point, we would not hazard a guess as to

whether such a strategy can be effected. So for the time

being, Question 2 also remains unanswered. And we conclude

this section with the partially independent

·pueees s~~ e~ dBIISUE

6D1QBPTDDp M~ PUB

MH DJV tDTqDpf8Bp [0

MM PUE LoMffi BdV "SUOT4SDnb uedo

225

FOOTNOTES

1see addendum to §1.3.

2our discussion of Display Logics and the presentation

of DTW and DRW are strictly tailored to present purpose.

A more general discussion of Display Logic is contained

in Chapter 1 § 2 , where the use of 'Display Logics I as

opposed to IDisplay Logicl is explained.

3This point, as well as the other points on negation below,

can also be found in §5.8 of Belnap 8+.

45 and 6 embody an arbitrary decision to translate

boolean negation to the outside of De Morgan negation.

The reverse procedure would do as well.

5The example also shows a further difficulty, namely,

K,r, for going empty.

6we suspect that one would even have to change the notion

of degree for formulae.

226 SECTION 5. RW = VRW ? + +.

Before getting into the question, we will want

to make some assumptions to simplify the discussion. We

have stated before that neither the uRW+ semantics nor

NuRW+ (which comes from NuR+ by putting the obvious

disjointness restriction on +E) are known to be equivalent

to uRW+. However, it seems very likely that the proofs of

Charlwood Theorem 1 and 2 will go through mutatis

. 1 mutand2s. For the sake of the discussion to follow, we

ASSUME that the three are equivalent.

The question of the equivalence of RW+ and uRW+

arose when it was noted that the standard counter examples

to the equivalence of R+ and uR+ are invalid in the duRW+

semantics given in the first chapter. The best known of

these is

Thl. (P+qvr)&(q+r)+.p+r

Thl. is provable in uR+, but not in R+. But it and its

known mates require contraction for their proof.

To see that Thl is not a theorem of uRW+, it will

suffice to give a refuting model in the duRW+ semantics,

since it is straightforward to show that uRW+ is consistent

with respect to that semantics. So let K be the power set

of {1,2}. Obviously~= <K,0,v> is a duRW+m.s. Then let

V make p true at {2}, q true at {1,2), and otherwise make

a sentential parameter false at an element of K. The

reader can quickly check that the associated interpretation

makes (p+qvr)&(q+r) true at {1), and p+r false at {1).

227

Whence Thl is false at ¢, as required.

So let us officially pose the

Open Question. Are RW+ and uRW+ (theoremwise) equivalent?

This is a very interesting question, and one which

has been raised independently by at least one other

researcher in the field, Professor Robert Bull of New

Zealand. 2 A positive answer to the question would yield a

very simple semantics for RW+. But in any event, the

process of discovering the answer should throw more light

on the relationship between the traditional Relevant

Logics and their semilattice cousins.

We suspect that the two systems are

equivalent, but the suspicion is largely based on negative

evidence, namely failure to date in finding a counter­

example. But for the time being, this question must

remain as an interesting problem for further research.

FOOTNOTES

1It seems to us that the ''book-keeping'' of such an

adaptation of Charlwood's completeness proof could be

more easily managed using the duRW+ semantics given

in Chapter 1.

2Reported in correspondence of July 1982.

228

229

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