type theoretical analysis

7/30/2019 Type Theoretical Analysis

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Type-theoret ical analysisas a preparation of analyzingexpressions of a natural language

(A variant proposed by transparent intensional logic (TIL))

Pavel Materna

1. Why types?

2. Montague

3. Tich

a) Choice of atomic types

b) Functional approach. Molecular types

c) Intensionalism

d) Constructions

e) Parmenides principle

f) De re, de dicto

g) Higher order types

h) Other constructions

4. Type-theoretical analysis

5. Type-theoretical synthesis

6. Difficulties, problems

1. Why types?

Russells introduction of types (see [Russell 1906]) has been motivated by the need to

avoid paradoxes arising due to violating the vicious circle principle. Russells solution of

this problem is connected with a strong intuition concerning our understanding such linguistic

phenomena likepredication. If someX is predicated of somey, then we feel thaty cannot be

of the same order asX.Being yellow can be predicated of some particular thing

(individual) and being a colourcan be predicated of(being) yellow, but being a colour

cannot be predicated of an individual which is yellow. So there are some degrees of

predication. Similarly being a dogcan be predicated of an individual and being a property

can be predicated ofbeing a dogbut not, of course, of the respective individual.

Russells hierarchy of types has been formulated in terms of sets and relations. Thelimitations given by this approach have been excellently summed up and criticised in [Tich


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1988, p.68-70]. One of them consists in ignoring other than propositional functions. A

predicate can be conceived of as a function, viz. such a function whose values are truth-

values. What happens if we combine a predicate with an object (or an m-tuple of objects)? For

Russell, such a combination can be understood if there is afactcorresponding to this

combination. So saying that 2 is a prime we combine the predicateprime with the (name of

the) number 2, which corresponds with a fact: 2 is really a prime. But what happens if 2 is

combined with the predicate oddis not clear. (problem of false sentences.) Not only that.

What happens when the function + is combined with the pair ? No referring to a fact

is possible here. And such functions like + are not members of Russells hierarchy.

A very expressive tool for dealing with functions has been described by A.Church [1940].

His l-calculus, originally typeless and important for investigating the problems of

computability (l-definability!), has been connected with type-theoretical hierarchy (to avoid

paradoxes). As Manzano 1997 says:

The offshoot of this was fantastic, since added to the formalising capacity of the lambda

language was the naturalness of type representation. (226)

From our viewpoint, the most important contribution of the typed l-calculus to the functional

approach in logic consists in the following principle:

Analyses based on the functional approach need two fundamental operations: a)

creatinga function by abstracting, b) applying the given function to an argument.

Thus when we want to take into account our linguistic intuition originating in our

understanding the predication, we should be able to generalise this intuition so that it captured

not only expressions that realise predication but all expressions, including expressions which

denote functions other than those ones whose values are truth-values. Thus a type ascribed

to (the object denoted by)sin is surely another than the type ascribed to the possible

arguments of sin, i.e., numbers, and it is also distinct from the type ofbeing a periodic

function.

The hierarchy of types accepted in TIL is inspired by Church rather than by Russell in that

the types are definable in terms of functions. (See 3.)

2. Montague

R.Montague and Montagovians (see [Thomason 1974] ) make up a most influential school

of theoretical linguistic and logical analysis of language. Similarly as in TIL, the types are

construed as sets of functions. There are two basic (atomic) types: e, t; the former represents


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objects (entities), the lattertruth-values. Sets of functions over these types are the remaining

types. Thus the type (e,t) i.e., the set of functions from e to trepresents the classes of

objects (for these functions are characteristic functions of classes of objects), (t,t) is the type

of unary truth-functions, etc. etc. Since Montague was aware of the fact that the s.c.

extensions are not sufficient for analysing expressions of a natural language he wanted to take

into account also intensions but instead of introducing the new atomic type for possible

worlds he defined only the types of intensions as (s,X) (for X any type) but did not defines as

a separate type.

Montagues analyses are well-known. They differ in essential respects from TIL, sharing at

the same time some features with it, in particular being based on types and functions and

respecting in a sense the necessity of distinguishing extensions and intensions.

Some critical commentaries to Montague can be found in Tichs work, e.g., in [1988]

3. Tich

Transparent intensional logic (TIL), first formulated by Tich in the seventies and finding

its most developed form in his [1988], can be characterised by some not just standard

features. Since type-theoretical analyses which we want to practice here could be

misunderstood without knowing some general principles of TIL, we will try to briefly

reproduce them without detailed motivation.

i) TIL is an objectual, i.e., anti-formalist system. For TIL, logic is not a study of formal

languages; it studies some logical objects like truth-values, individuals, classes, functions,

properties, propositions and alike, plus the ways they can be combined to create new logical

objects. (See his [1978, p.275].) Formal, artificial languages and any symbols only serve to

this primary purpose. One of the reasons why logic (construed in this way) is important is that

the mentioned ways of combining, i.e., constructions, can be associated to particular

expressions of a natural language and help so perform logical analyses of natural languages.

ii) Semantics based on TIL is aPWS (Possible-world semantics). A most natural way

how to handle modalities is articulated in terms of possible worlds. Possible worlds, as

(maximum) consistent collections of possible facts, are indispensable not only when

modalities are to be analysed, but also when we analyse empirical expressions, since the latter

cannot denote actual, real objects but only some conditions; so empirical sentences cannot

denote truth-values (Freges error) but truth conditions (= propositions), empirical common

nouns cannot denote classes but properties, etc.


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iii) One among the possible worlds is the real, actualone. Since to know, which one it is

would mean to know all actual facts, which only an omniscient being could do, we cannot

know, which among the possible worlds is the actual one. Therefore our logical analyses of

empirical sentences cannot calculate their truth-values, they cannot discover the actual

population of a property, etc.

iv) The universe of discourse, whose inhabitants are individuals, is one and the same in

all possible worlds. Most other PWS suppose that universes are world-specific, i.e., that every

possible world owns another universe of discourse. (See, e.g., [Kripke 1979] .)

v) The preceding point is connected with the way how individuals are construed in

TIL. They are bare ornaked in the sense that they do not possess any non-trivial, property

necessarily. So we can say, e.g., that no wooden table is necessarily wooden, for no wooden

table is necessarily a table, even no table is necessarily a table. (This point is against Kripke;

see Tichs [1983].)

vi) A Parmenides Principle (to be found already in Freges Grundlagen der

Arithmetik):

An expression E cannot be about an object that is not mentioned in E.

Thus the expression the highest mountain is not about MtEverest, semantics of this expression

cannot be connected with a factual, empirical information (this holds for any semantics). See

point e).

Now concrete points necessary for type-theoretical analysis.

a) Choice of atomic types

Since one of the purposes of building up TIL was to make it a good tool for logically

analysing expressions of natural language, the choice of the basic, atomic types could not

have been entirely arbitrary. The choice it has made is surely nothing absolutely

unchangeable, but it is well founded and there are strong intuitions that support it. We will try

to articulate some of them.

First of all, a natural language is a tool for expressing our claims, convictions, hypotheses,

beliefs. A common feature of all such linguistically important entities is that they are

expressed by (declarative) sentences. But such sentences are sometimes (among semanticists

or epistemologists) called truth-bearers: the distinction between truth andfalsity is a

fundamental distinction which should be respected by all analysts of a natural language. Thus

one of the atomic types is the set of truth-values. How many truth-values do we have? Theclassical answer is: there are two of them, T(ruth) and F(alsity).


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There are, of course, some logicians who use many-valued logics. The latter are perhaps

interesting from a formal or technical viewpoint, but the notion of truth, as used in a normal

discourse, presupposes the classical standpoint. So the first atomic type, denoted by the Greek

lettero, is the set {T, F}.

Remarks:

1. In most (maybe all) natural languages there are even particular words denoting truth-

values: So the English Yes denotes T, No denotes F.

2. We cannot expect that the expressions of a natural language would denote concrete

objects: we have to be aware of the fact that T, as well as F, are abstract entities. -

The lowest level of our ascribing properties and alike to objects presupposes that there are

some such lowest level objects. These are called individuals and the respective type, i.e., the

set of individuals, is denoted by the Greek letteri.

Distinguishing between sentences like

A new age has arrived.

A new age arrives.

A new age will arrive.

we see that grammatical tenses surely possess the semantic dimension. Time points are to be

taken into account. But what are (the members of the set of) time points? One possibility

(chosen by TIL) is that time is continuum. This means that the set of time points can be

handled in the same way as the set of real numbers. Thus the next choice is the set of time

points or of real numbers, denoted by the Greek lettert.

The choice of the most controversial fourth atomic typeis motivated as follows:

What is the semantics of empirical expressions? Let us compare the mathematical (i.e., non-

empirical) expressionprime (number) with the empirical expression dog. The former

expression obviously denotes a class of numbers. The members of this class are independent

of time, of course, but moreover, if our world changed somehow (so that, e.g., it contained

other events, other natural laws or so), it would have absolutely no influence on which

numbers belong to the class of primes. So 2 would be a prime even if there were no forests in

Finland etc. The fact of class membership is independent of empirical facts. On the other

hand, the fact that some individual is a dog is a contingentfact. To determine whether an

individual is a dog we have to inspect the state of the world. Thus empirical expressions do

not denote real concrete objects: they denote something like conditions. In our example, dog

does not denote a definite class, it denotes aproperty. The distinction between classes and


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properties can be illustrated as follows: Let individuals A1, ,A5000 be the only members of a

class X. Let them share (at some time point) the property living in the village Y, so that they

are (at that time point) the only inhabitants of Y. Now imagine that some members of X

move, change their addresses. What changes is their possessing the property above. What

does not is their membership in X. So the class cannot change in any respect, whereas the

propertybeing the same propertychanges its population. This temporal variability,

characteristic of empirical expressions like names of properties, could be grasped by saying

that, e.g., properties determine classes dependently on time points, i.e., that they can be

construed to befunctions which associate time points with classes. Such functions from time

points to something will be called chronologies. Yet this temporal variability is accompanied

with what can be called modal variability: Even at the given time point it is only a contingent

fact that the individuals above share the property above. Thus this fact cannot be calculated

from our knowledge of the property, because to recognise the actual population of this

property we need experience in the sense of a contact with the world. Hence a property could

be better construed to be a function which associates every thinkablestate of the worldwith

a chronology of classes, and similar considerations can be applied in the case of any

empirical expression. The consequence of the above considerations is that a further atomic

type is necessary, the type that corresponds to our intuition ofpossible states of the world; the

broadly accepted terminology speaks aboutpossible worlds. The collection of all possible

worlds is denoted by w, which is the last atomic type.

Remark: Various theories of possible worlds are relevant for philosophical logic. Their

construal in TIL is best described in [Tich 1988]. For the purpose of analysing expressions

of natural language it is sufficient to assume that whatever consistent collection of all

thinkable states of the world (empirical facts) at the time point tis a possible world at t, and

possible worlds are temporal sequences (histories) of particular possible worlds at particular

time points.

b) Functional approach. Molecular types

Summing up, we have got fouratomic types at our disposal: o, i, t, w. We will show that a(nearly?) sufficient class of kinds of object and, therefore, of kinds of expression to be

analysed can be type-theoretically described in terms of these types.

First of all we have to define how the compound (molecular) types are created in terms of

atomic types. The main principle underlying the way of combining the latter to get the former

can be formulated as the principle of functionality: the dependence of compound entities (e.g.,


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expressions) on the particular components can be best modelled asfunctional dependence. In

order to explain this principle we have to recapitulate what is meant by the notion offunction.

The contemporary notion of function, as used in mathematics, construes functions as

mappings; they are set-theoretical entities which can be (in principle) represented as (perhaps

infinite) tables where the left n columns are arguments (they are ordered n-tuples ifn > 1) and

the right column represents values of the function on particular arguments.Partial functions

associate every argument (a line in the table) with at most one value, total functions are a

kind of partial functions such that they associate every argument withjust one value.

Functions defined in this way obey theprinciple of extensionality: forn-adic functions it is

given by the valid formula (f, gare variables ranging overn-adic functions of an arbitrary

type):

"x1xn(f(x1,xn) =g(x1,,xn)) f=g.

An earlier notion of function construed functions as rules, instructions determining particular

steps of calculations. Functions construed in this way do not obey this principle; to see this

consider two following functions-instructions F and G:

F(x,y) = (x y) * (x +y)

G(x,y) =x2y2.

F and G are distinct instructions and, therefore, distinct functions-as-instructions. We can

easily see that they are identical as mappings.

Here what we mean byfunctions are functions as mappings. Functions-as-instructions have

been rehabilitated in TIL and got the name construction (see d)).

The way of obtaining compound types from atomic types is based on functional approach in

the following sense:

Atomic types are collections (sets) ofprimitive objects (truth-values, individuals, time

points/real numbers, possible worlds). Compound types are collections (sets) of (partial)

functions. So the inductive step of the definition of types will be:

Definition 1 (Types of order 1)

i) o, i, t, w are types of order 1.ii) Let a,b1,,bm be any types of order 1. Then (ab1bm), i.e., the set of partial

functions with values in a and arguments (in general, tuples whose componentsare, respectively) in b1,,bm, is also a type of order 1

iii) A typeof order 1 is only what satisfies i) and ii). -


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Types defined in this way are types of order one. Later ( see g)) we will see that our

analyses will in some cases need types of higher orders.

c) Intensionalism

We have promised to show that objects and, therefore, expressions denoting objects

can be associated with various types derivable from the four atomic types. Further we

stated that the compound types are always sets of functions (as mappings). A motivation

of this functional approach has been suggested: dependencies are best modelled as

functions. Now we will show it via some more or less systematic examples. The

following logically relevant and distinguishable categories of objects can be type-

theoretically described as follows (types ((at)w) are written atw for any type a):

Table 1

Category of objects Type Example (object) Example (expression)

Numbers t 2 two

Classes of numbers (ot) Primes a prime

Classes of individuals (oi) {Plato, Einstein} Plato and Einstein

Binary relations of

numbers

(ott) greater than >

Properties of

individuals

(oi)tw being black black

Binary relations

between individuals

(oii)tw being a sibling of a sibling of

Binary arithmetical

operations

(ttt) subtracting minus

Propositions otw that some animals are

carnivorous

Some animals are carnivorous

Magnitudes ttw the number of planets the number of planets

Unary truth-functions (oo) negation It is not the case that

Binary truth-functions (ooo) conjunction and

Existential quantifier

over individuals

(o(oi)) non-emptiness of a class

of individuals

Some


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Comments:

1. Let a be any type whatsoever. The type (oa) is then according to the definition of

types the type of classes of objects which are members of the type a,briefly ofclasses

ofa-objects. Indeed, (oa) is the set of functions from a to {T,F}, i.e., of characteristic

functions of the classes ofa-objects: Such a function takes T on those a-objects which

belong to the given class, and F for the othera-objects. So (oi) is the set (type) of

classes of individuals (in our table one such class is {Plato, Einstein}), (ot) is the type

of classes of numbers (here such a class is the class of primes), (o(oi)) is the type of

classes of classes of individuals (so $, the existential quantifier, is the class of all non-

empty classes of individuals), etc.

By analogy, the types of (non-empirical) relations are sets of characteristic functions of

these relations, so the type of > is (ott): the members of this type are functions which

return T just to those ordered pairs of (real) numbers whose first member is greater than

the second.

2. We can immediately state that there is no systematic correspondence between types

andparts of speech (p.o.s.). We can surely assume that this holds for any natural

language, since categorization of p.o.s. originated, in any natural language, quiteindependently of type-theoretical analyses, which does not mean that a language could

not express the type-theoretical distinctions, so important for logical analyses. To

adduce some examples of discrepancy between a linguistic and a type-theoretical

classification, let us ask, which p.o.s. is connected with denoting properties of

individuals. It is easily seen that not only one: it can be a noun see dog,(in Czech:

pes) it can be an intransitive verb see to work, pracovatit can be an

adjective see black, ern (but the case of adjectives is not that simple, see later).

In the case of nouns one can a little complicate the problem, claiming, that is, that the

respective property is not denoted by the noun alone but rather by the combination

copula + noun, e.g., to be a dog, bt pes. Yet this is compatible with our claim.

3. Types unlike expressions are coarse-grained: they are just sets (of primitive

objects or of functions-mappings). Thus determining the type of an (abstract!) object

(and, derivatively, of the respective expression) is only a first, most simple step in the

logical analysis. If semantics stopped after a type-theoretical analysis has been made,

then we would need no semantics. This simplicity of types can be illustrated as follows:


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Take the type otw (i.e., ((ot)w) ),the type of propositions, i.e., of functions that associate

every possible world with a chronology of truth-values. It is not only so that all possible

declarative empirical sentences share this type but there are theoretically infinitely many

such sentences which denote one and the same proposition: So the sentences

MtEverest is higher than Mont Blanc, Mont Blanc is lower than MtEverest, If

1+1 = 2, then Mont Blanc is lower than MtEverest, If MtEverest is not higher than

Mont Blanc, then 1+1 2, but also MtEverest je vyneMont Blanc, Mont

Blanc je nineMtEverest etc.etc. not only share the propositional type but all of

them denote one and the same proposition. Indeed, take any possible world W at any

time point T: whenever one of the sentences above is true (false) in W at T, all the other

sentences are true (false) in W at T. The truth-conditions are the same (= the proposition

is the same).

4. Our table is not (and no such table can be) exhaustive as a list of types possibly

exploitable in a language. Neither is it exhaustive as a list of representatives of

expressions to which types can be ascribed. Finally, some important categories of

expressions pose some interesting (but still open) problems as concerns their type-

theoretical description. We can mention interrogative sentences, imperative sentences,

indexicals, demonstratives and evenproper names. Some other linguistic units (for

example tenses, episodic verbs) have been already successfully analyzed but these

analyses are rather complicated and whoever wants to be acquainted with them can use

some literature (see [Tich 1980, 1980a] ).

There are two groups of types in our table. First, types of the form atw.

Definition 2 (Intensions, extensions)

The members of the types atware intensions. The members of the other types areextensions.

Our definition of types shows that intensions are functions which associate possible worlds

with chronologies ofa-objects. Their type indicates the temporal and modal variability we

mentioned above. Slightly simplifying we can suggest a criterion of ascribing intensional

types to expressions: Unless an expression is a mathematical/logical expression we can

classify it with expressions denoting intensions. (Some exceptions can be expected, one of


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them is the expression colour: we would say that colour is simply the set of particular colours

like white, blue, yellow etc.. But then since these particular rs are properties of individuals,

i.e., (oi)tw-objects the type of this set is (o(oi)tw), so no intensional type. This presupposes

though that the population of the object colour is fixed, independent of empirical facts. If we

can admit that what is called colour can change, or that it is only a contingent fact that there

are just those colours we know, then the object colour would be a property of particular

colours, rather then a set, and its type would be indeed (o(oi)tw)tw, i.e., an intensional type.)

The second group contains the other types. Their members are called extensions. They are

members of atomic types, classes, relations(-in-extension), mathematical functions etc.

Remark: Montagovians use the idea of distinguishing intensions and extensions in another

way. For them any expression possesses its intension and its extension. Thus the word dog

can be handled in some contexts as a name of a class, in other contexts as a name of a

property. TIL is an intensionalistic theory: every empirical (non-mathematical, if you like)

expression denotes an intension, the other expressions denote extensions, both independently

of a context. This anti-contextualism is justified, for wherever it seems that the context

makes the given empirical expression concern extension, it can be shown that distinct

contexts influencesupposition rather than meaning or denotation. See f).

Intensions can be in general characterised as conditions. Let us compare some intensions

with corresponding extensions (examples are objects in the sense of entities independent of

a language; the respective expressions can be easily added):

Table 2


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Extension/type Example Intension/type Example

individual / i Plato individual role/ itw the most famous teacher of Aristotle

truth-value / o T proposition / otw that there are nine major planets

number / t 9 magnitude / ttw the number of major planetsclass of

individuals /

(oi)

{Mozart,

Homer}

property of

individuals / (oi)tw

(being a) composer

relation-in-

extension of

numbers /

(ott)

> relation-in-intension

of individuals /

(oii)tw

(being) taller than

Our intuitive criteria of ascribing types in general and of recognising intensionality in

particular can be supported by following considerations concerning the examples in the table:

Compare

1. Plato and the most famous teacher of Aristotle. Ignoring for the time being some special

problems with semantics of proper names we can say that Plato is something like a

label put on an individual. In this case there is nothing what could change Platos

identity: the numerical identity of an individual cannot be influenced by what just

happens to be the state of the world. On the other hand the most famous teacher of

Aristotle could have been any other individual. Just who was the most famous teacher

of Aristotle is of course dependent on the state of the world: it is world-time dependent.

This is why we do not a priori determine a definite individual when we use a definite

description.

2. Tand that there are nine major planets: The extension T is simply the abstract truth-

value. True mathematical sentences can be said to denote it. On the other hand that there

are nine major planets can be true only if the given world and time satisfy certain

conditions. In other words, the proposition will be true just in those worlds-times where

the number of those individuals which have the property being a major planetin them

equals 9. Our (actual orreal) world is now among those world-times, which is only a

contingent fact which cannot be deduced a priori, it has to be empirically discovered. (It

is not a mathematical or logical fact.)


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3. 9 and the number of major planets. The number 9 is entirely immune to any influence of

empirical facts. Which is the number of major planets is again given dependently on the

given world-time. If it were mathematically, a priori derivable, its type would be t.

4. {Mozart, Homer} and (being) a composer. Again, construing proper names as labels, or,

better, taking Mozart, Homer simply as particular individuals independent of those

properties which we are used to connect with them we can safely claim that no empirical

facts can influence that a priori kind of identity which is characteristic of individuals. So

we can speak about a class of individuals here. On the other hand, whether somebody is

or is not a composer is a fact which is a part of a given world-time, and therefore there is

no class of composers: only aproperty, which generates various classes of individuals

in particular worlds-times. Indeed, we are able to recognise that an individual is a

composer, but our knowledge does not stem from the knowledge that there is a fixed

class of individuals and that the respective individual is a member of it this would be a

mathematical knowledge, like if you know that A belongs to the class, say, {A,B,}.

5. > and (being) taller than. The fact that a number is greater than another number is an a

priori piece of knowledge. Nothing such can happen in the world what could change the

fact that, e.g., 2 is greater than 1. On the other hand, whether an individual A is taller

than an individual B, is something which cannot be deducible from the meaning of

taller (modal variability, we can rationally admit that A could be not taller than B),

and as concerns the temporal variability, everybody knows that if it holds at the time

point tthat A is taller than B then it can often happen that at some time point t A will

be no more taller than B.

Remark: In 3. we meet a frequently occurring problem. The expression number can be a t-

object but the expression number of denotes another object: it is a function which associates

a class of some a-objects with that number which is the cardinal number of that class. So it

type is (fora any type) (t(oa)). In Czech no error can arise: number is slo, number of

is poet. -

The criterion that enables us to decide whether the given expression is intensional, i.e.,

whether the type ascribed to it is of the form atw, consists in deciding whether this expression

is empirical. It is, however, only one of the criteria that help us to determine the type. Another

criterion consists in making clear thefunctional characterof the dependencies connected with

using the expression. In the linguistic area the intuition connected therewith led to the rise and

development ofcategorial grammars ( see, e.g., [Morrill 1994] ). To show a most elementary


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example, consider a common noun like prime or its combination with copula is a prime.

Using this expression together with a name of a number we get a truth-value. So the type of

prime can be derived from this fact: ift will be the type ascribed to the names of numbers

and o the type of (an expression denoting) a truth-value, then to get o after combining (is a)

prime with a name we have to construe this expression as representing a function with t-

objects as arguments and truth-values as values; the resulting type is (ot). (In Montague,

where individuals and, e.g., numbers are not type-theoretically distinguished, belonging both

to the type e, the resulting type would be (e,t) (see 2.). )

This functional criterion holds primarily in the area ofobjects. We have just shown,

however, that we can ascribe types to expressions, assuming that there is a semantic relation

connecting expressions with objects, let it be denoting relation, so that prime denotes (in

English) the set of primes, and the latter can be said to be the denotation of the former (see

[Church 1956]).

Now we will apply such a functional analysis to a more difficult case. We have adduced

black as an example of an expression denoting a property. An easy generalisation could

suggest that the type ascribable to adjectives should be (oi)tw (or (oa)tw for some othera).

But take such expressions which combine an adjective with a noun where the latter denotes a

property; now compare, e.g.,

(being a) big ant

and

(being a) small elephant

ant, as well as elephant, denote properties of individuals. big ant and small elephant

obviously denote properties of individuals too. Now ifbig and small denoted a property,

we could therefrom derive the desirable type (oi)tw of the compound expressions under the

following condition: the new properties being a big antand being a small elephantwould

have to be interpreted as being big and being an antand being small and being an elephant,

respectively. But then our linguistic intuition would be lost: a logical consequence of this

reading would be that if an individual is a big ant, then it is big, and if an individual is a small

elephant, then it is small. Looking then at a big ant and, at the same time at a small elephant

we would have to state that the former is big whereas the latter is small.

This absurd consequence can be avoided if the type of big, small and of adjectives in

general is changed. Montagovians as well as TIL have realised this change. TIL proceeds as

follows:


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Empirical adjectives (where this problem arises) being empirical denote intensions. So

their type has the form atw, i.e., ((at)w). Let A be such an adjective.

They are combined with other expressions, which denote properties, i.e., (oi)tw-objects. Let

E be such an expression.

In every world W at every time (point) T we apply the adjective first to W (getting the type

(at) ) and then to W (getting the type a). What happens, if this result is applied to (the object

denoted by) E? We know that the resulting type of doing this procedure in all worlds-times

should be (oi)tw. So it should be (oi) in one particular world-time. Summing up: We reduced

our problem to the following one: What type must a be if applying A to E with E/ (oi)tw gives

the type (oi) ? To answer this question is already easy: we apply A to E (in the given W and

T) and the result is an (oi)-object. If the a-object is a function (it has to be, since a is surelya compound type), then its application to an (oi)tw-object results in an (oi)-object only ifa is

of the type ((oi)(oitw)). And since this procedure repeats in all worlds and times, the resulting

type of an empirical adjective will be

((oi)(oi)tw)tw.

Now we avoid the absurd consequence mentioned above. A new property arising from the

combination of an adjective and an expression denoting a property is not necessarily the same

property as that one which would be expected if the adjective denoted simply a property: sofrom the fact thatx is a big something it does not follow thatx is big, etc.

Exercise: Prove that the same result will be obtained when expressions with cumulated

adjectives (a big pink elephant and alike) are analysed.

The proper analysis of expressions in TIL presupposes that a type-theoretical analysis has

been performed for (most) simple expressions. Since theprinciple of compositionality is

assumed to hold we must be able to unambiguously derive the type of a compound expression

from the types of its components. It should be clear

that the way the particular expressions are combined to create more compound

expressions is given by the grammar of the given language and

that from the specific, language dependent character of grammars and universal,

language independent character of logic it follows that there are two branches of

what can be called logical analysis of natural language:

a) a general theory that formulates some principles independent of what is

specific for particular languages,


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b) application of this theory to particular languages.

The tasks to be solved by b) are dependent on the branch a). To solve b) for English tried

Tich in his posthumous work (to appear in some future). It is, for every language, a gigantic

work to do. The result of such a work done for a language L should be a set of L-specific

rules which make it possible to create for any (however complex) expression of L pairs , where e is the expression and m is the construction corresponding to e.

Here we only suggest some points relevant for a) with an emphasis on type-theoretical

analysis and synthesis. The key notion ofconstruction is introduced in the next section.

Remark: We adduce types of some important extensions.

a) Truth functions: Unary truth functions (in particular negation)(oo). Binary truth

functions (ooo).

b) Quantifiers.The universal (") and existential ($) quantifiers are type-theoretically

polymorph, which means that their type depends on the area over which quantification

goes on. In general, the schema of their types is (o(oa)). The universal quantifier is the

class whose only member is the class of all a-members. The existential quantifier is the

class of all non-empty subclasses ofa.

c) Identity. Identity ( = ) too is type-theoretically polymorph. The schema is (oaa). It is the

class of all such pairs a, b ofa-members where a is the same object as b.d) Singulariser. Singulariser (whose linguistic counterpart is commonly called descriptive

operatorand the symbol of which is the iota inversum, but because of the fact that this

symbol is often missing, it is replaced by simple i) is also type-theoretically polymorph; it

is a partial function of the type (a (oa)) which is defined on classes containing just one a-

member and returns this very member, and is undefined on all other classes. -

d) ConstructionsWhat the standard analyses (like Montague see [Thomason 1974], [Cresswell 1985] )

mostly do is that they define some artificial language (l-language or so) and define some

rules of translation from the natural language to this artificial language, to show then that we

can in many cases demonstrate equivalence of expressions of natural language with their

translations.

For TIL this way is not viable. There is, first of all, a fundamental theoretical flaw in the

mentioned approach: IfA, an expression of a language L, is correctly translated asAinto a

language L, then there must be something what is common toA andA, evidently what is


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called meaning (ofA andA). Now we can hardly claim that an expression of an artificial

language is the meaningof the expression whose translation it is supposed to be. LetAbe an

expression of a natural language, and A its translation into an artificial language. Suppose

for a while thatAis really the meaningofA. We can ask: what is the meaning ofA? Is it

perhapsA ? But this is an obvious nonsense, forA andAare distinct, so that the meaning of

A would be distinct from the meaning ofA.

The way out which has been chosen by TIL is theoretically very important; but also in

practising our analyses it makes it possible to avoid some problems that jeopardise the other

approach (see Cresswells [1975], where the author thinks that variables of possible worlds

cannot be used in the object language, since the so-called principle of proximity would be

lost). It is the notion ofconstruction which offers the solution.

Clearly, if meanings of expressions were construed as expressions (say, of another

language), then regressus infinitus is what we get. The only solution can be that meanings are

extra-linguistic entities.

Constructions, as defined below, are such extra-linguistic entities. TIL assumes that what

makes expressions of a natural language meaningful is a system ofabstract procedures,

which are encoded by the given language and which can be imagined as series of intellectual

steps. They are abstract (like, e.g., numbers, functions etc.) and structured, i.e., they contain

particular components plus the way they are combined (without this plus they would be

mere lists of components). Already by now we can see that infinitely many such procedures

can result in constructing one and the same object. To illustrate this fact consider a very

simple example: A simple way to the number 2 is simply posing 2. Another such way

consists in a calculation 1 + 1, and in infinitely many such calculations, still another way is to

say the only even prime, etc. With the exception of the simple posing the procedures we

have in our minds consist in making some steps: identifying the number 1, the function

addingand applyingthe latter to the pair , or identifying the classes of even numbers,

of prime numbers, the intersection, applyingthe intersection to the pair , identifying the partial function the only x such thatand applying it to the

result of the intersection, etc. Italicizing the expression applying suggests that there is some

way of handling the particular components of the procedures, in this case the operation of

applying a function to its arguments.

Accepting the general idea of constructions (as characterized but not defined above) we

have to ask some questions.


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First, we could be interested unless we avoid philosophical questions in knowing what

kind ofstatus the constructions enjoy: are they mental, or objective entities? For TIL they

are objective, since mental character does not guarantee intersubjective agreement. For

anybody who is not interested in such problems this question can seem to be irrelevant. Let it

be so.

Second, if constructions are extra-linguistic, then a rather practical question arises: Well, let

constructions be abstract entities, but how do we communicate with such entities? Without

linguistic means it is obviously impossible. Thus a following objection to the notion of

constructions can be formulated: Let constructions be something like meaningof

expressions. The only possibility how to handle such constructions seems to be to define an

artificial language, which is just what has been criticised above.

Yes, defining constructions we will define a sort ofartificial language, let it be called CL.

What is, however, important, is that it will be not the expressions ofCL what will be taken to

be meanings: The expressions ofCL will be only a kind of fixing particular kinds of

construction; by the way, if the way of fixation were changed, i.e., if another artificial

language CL were chosen, the constructions themselves would remain the same. This is very

similar to the case of various distinct notations of truth-functional logic. Whether the language

of truth-functional logic is Russellian, Hilbertian, or perhaps a language of Polish notation,

the logic itself is the same, it is always the same truth-functional ( propositional) logic.

Another analogy: Using the English expression elephant, say, in the sentence Some

elephants live in Asia we do not speak about the English word elephant: we speak about

elephants, or, more precisely, about the property (being an) elephant.

Thus handling abstract extra-linguistic entities need not be a piece of a pseudo-

philosophical speculation: it can be as well precise as if we pretend to handle chains of

symbols in a formal language.

We will return to this question after constructions are defined.

Before we proceed to the definition proper let us explain some motivation for choosing just

those kinds of construction which will be defined.

First of all, we have defined an objectual environment consisting of the objects whose

types are types of order 1. These 1storder objects (intensions and extensions) are, of course,

distinct from the way they are constructed. Expressions of a language can denote them only

due to some in general structured way that Frege had in his mind when he introduced the

termsense (Sinn). Thus the role of the first two kinds of construction consists in mediating


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between the world of 1st order objects and the world of constructions. This role is played

by variables and trivialisation.

Further, our functional approach (see section b) ) needs two other kinds, well-known from

l-calculi: constructions that apply functions to arguments, here called composition, and

constructions thatproduce functions by abstraction, here called closure.

Finally, in section h) we will consider the possibility + usefulness of defining still other

kinds of construction.

Now we willpre-theoretically characteriseparticular kinds of construction as mentioned

above and afterwards we will formulate the core definition of this chapter.

1) Variables. We have shown that there are infinitely many types (see sections a) and b) ).

Now for each of them we have at our disposal infinitely many incomplete constructions: these

are variables for the given type, say, a-variables for the type a. They are constructions in

that they always construct exactly one object of the given type, and they are incomplete in that

they do their job dependently on a total function called valuation: valuations are functions

which associate every variable with just one object of the respective type. They can be

imagined to work as follows: For every type there are infinitely many infinite sequences of the

members of the type. A valuation always selects one such sequence for every type, and then

the i-th variable of that type constructs the i-th member of that sequence.

Example: Imagine that the (infinite) sequence of the members ofo begins as follows:

T, T, T, F, T, F, F, .

Now letp5be the 5tho-variable. If the sequence above has been selected by the valuation v,

then we say thatp5 v-constructsT (since T is the 5th member of that sequence).

We have explained the character of variables as constructions without resorting to any

reference to a linguistic object. Thus we see that variableslike all constructions are no

linguistic objects. A standard conception of variables has it that variables are letters,

characters, i.e., linguistic objects. For us, the usually used letters likex,y,p,q,f,q,etc.,

are not variables but only arbitrarily chosen names of variables.

To make clear that variables construct objects dependently on valuations we will say that

variables (and constructions that contain free variables) v-construct objects, where v is a

parameter of valuations.


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2) Trivialisation. Trivialisations make constructions from objects: Let X be any object

whatsoever (but X may be also a construction, which is important for building up the higher

order objects, see section g) ). Then 0X is a construction called trivialization, which constructs

just X without any change. From the present viewpoint trivialization is necessary, since the

remaining constructions consist of constructions only, not of objects.

3) Composition. The precise definition of composition follows in the definition below.

Here an approximate description of its role can be formulated as follows: If X (v-)constructs

an m-ary function, and X1,,Xm (v-)construct arguments (i.e., an m-tuple of the components

of the argument) of this function, then the composition (v-)constructs the value (if any) of this

function on the argument(s).

4) Closure. Again, here only characteristics: If X (v-)constructs a-objects andx1,,xmv-

construct b1-,,bm-objects, respectively, then the closure (v-)constructs a function determined

by what object is constructed by X when the respective objects replace the (possibly

occurring) variablesx1,,xm. The result is an (ab1bm)-object.

So we have interface with objects( 1), 2) ), applying function to arguments (3) and

creatinga function (4). We will have some thoughts about whether this is sufficient later

(section h).

Definition 3 (Constructions)

i) Variables are constructions.

ii) Let X be any object or even construction. Then0X is a construction called

trivialisation. It constructs X without any change.

iii) Let X, X1, , Xm be constructions v-constructing (ab1bm)-, b1-, , bm-objects,respectively. Then [XX1Xm] is a construction called composition. If the function

v-constructed by X is not defined on the objects v-constructed by X1,,Xm, then

[XX1Xm] is v-improper, i.e., it does not v-construct anything. Otherwise, it v-

constructs the value of the function on the tuple v-constructed by X1,,Xm.

iv) Letx1,,xmbe pairwise distinct variablesthat v-construct b1-,,bm-objects, andlet X be a constructionv-constructinga-objects. Then [lx1xm X] is aconstruction called closure. It v-constructs the following function F: Let b1,,bm


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be b1-,,bm-objects, respectively; let vassociatexiwith biand be otherwiseidentical with v; then the value of F on b1,,bm is the object v-constructed by X;

if X is v-improper (see iii) ), then F is not defined on b1,,bm.

v) A construction is only what satisfies points i) iv).

Remark: As for the point v), see, however, section h).

Examples. A sufficient number of examples will be given in the following sections. Some

most simple illustrations follow:

Letx1 be a variable v-constructing real numbers (i.e.: ranging overt). Let a particular

valuation v associatex1 with the number 2. Thenx1v-constructs 2, whereas0x1 constructs

(i.e., v-constructs for every valuation v) just the variablex1. Notice thatx1 is a t-variable, for 2

is a t-object. On the other hand, 0x1 is nota variable, it is a construction which constructs the

variable, i.e., another construction. Our definitions do not make it possible to ascribe a type to

constructions: the latter are not 1st order objects. Ascribing types to constructions will be

enabled by a ramified hierarchy of types, section g).

Let > be the greater_than relation between real numbers, so >/ (ott). The construction

[0>x00]

(x ranging overt) v-constructs truth-values: T ifxv-constructs a positive number, F

otherwise.

Notice that the type of the object constructed by the construction is unambiguously given by

the types of the objects constructed by its components. We have

[0> x 00]

(ott) t t

o

(see Definition 3 iii) ).

Now consider the construction

[lx [0>x00]].

According to the Definition 3 iv) this construction constructs a function which takes the value

T on those numbers for which the following composition takes T, and F on the other

numbers. Again, we can see that the type of this function, i.e., (ot), is unambiguously

derivable from the types of the objects constructed by the components of the construction,


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here it suffices to derive it from the type v-constructed by the composition and the type t

whichx ranges over (we omit the outermost brackets):

lx [0>x00]

t o

(ot)

Remark: We recapitulate the distinction between a construction and the artificial expression

that guarantees its fixation. In our last example we can illustrate this claim as follows. This

construction does not contain brackets, does not contain l, and whereas the artificial

expression contains two occurrences ofx, the construction itself contains one occurrence ofxonlywriting lx we only fix what the construction does, we do not add a new occurrence

ofx to the construction itself.

Let div be the dividing function, so div/ (ttt). Letx, y be again numerical variables.

Consider the construction

[0divx y].

It obviously v-constructs the result of dividing the numberv-constructed byx by the number

v-constructed byy. Now we will bindthe variablexvia the following closure:

lx [0divx y].

Let v associatex with 6 andyby 4. The closure ignores v as concernsx (see Definition 3 iv)),

so that it v-constructs the function which takes any numberkto the result of dividing kby 4,

i.e., by the numberv-constructed by [0divx y]. Thus we can see that the valuation influences

only those variables which are not in the scope ofl. (It does not influence the variables which

are inside a trivialization either, but before the section g) we cannot formulate the general

definition of free and bound variables.)

Let us consider the construction

[0divx 00].

Clearly, for every valuation v this construction v-constructs nothing, i.e., it is (v-)improper.

The closure

lx [0divx 00],

however, is not improper: it constructs a function which is on each argument undefined.

Some examples of simple closures:

Letx range over a type a.


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The closure

lxx

constructs the identity function overa.

The closure

lx 02

constructs the constant function which associates every member ofa with the number 2.

Letx, y range overt. Compare the following closures:

a) lxy [0>x y],

b) lyx [0>x y],

c) lx[ly [0>x y]]

The closure a) constructs the function that takes T on such pairs of numbers where m

is greater than n, and F otherwise. It constructs, therefore, the same relation as does0>.

The closure b) constructs the function that takes T on such pairs of numbers where n

is greater than m, and F otherwise. So it constructs the same relation as does0x y]] v-

constructs, i.e.,) the function which takes every numbern to T iffm is greater than n. So it

takes, e.g., the number 2 to the class of all numbers n such that 2 is greater than n, in other

words, to the class of all numbers less than 2. (We can see that c) constructs a function which

models the same dependency as does in another way the function constructed by a). )

The definition of constructions will be extended in section g) (and perhaps in h) ). Before,

however, we will exemplify using constructions of intensions by analysing the already

mentionedParmenidesPrinciple and the problem ofde re vs. de dicto suppositions.

e) Parmenidesprinciple

We already know that this principle can be formulated as follows:An expression E cannot be about an object that is not mentioned in E.

It could seem that it is an entirely obvious principle. Yet violating it is often very tempting. A

semantic analysis of an expression should result in showing a structure (construction) that

identifies (constructs) just what the expression is about. A consequence thereof is that

empirical expressions should be analysed so that the respective construction constructed an

intension rather than its value in the actual world-time (see section c) ). This is very important

because of one of the main purposes of analysis: to make it possible to make just the correct

inferences. To illustrate this claim observe the following premise:


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properties themselves are functions from possible worlds and times to classes, so that

comparing heights is always possible only among the members of some class, i.e., of the

value of the given property in some world-time. Thus claiming that some object is the highest

one means claiming that it is the highest member of a class. But if which individual is the

highest one is dependent on a given class, then we can say that it is afunction from classes of

individuals to i. Not forgetting that all this is dependent on worlds and times we come to the

result that a in our case is (i (oi)). So we have

(the)H(ighest) / (i (oi))tw.

B. Type-theoretical synthesis.

We recapitulate: The three objects whose constructions should combine to make up a

construction underlying the whole expression are

H / (i (oi))tw, M / (oi)tw, A / (oi)tw .

The whole expression should denote as an empirical expressiona proposition, i.e., an otw-

object rather than a truth-value: not taking this into account means not obeying Parmenides

principle: no truth-value is mentionedin the sentence. Hence our task is now to find such a

construction which would construct an otw-object and whose subconstructions would be the

constructions constructing the three objects above. (This task is an instance of the tasktype-

theoretical synthesis.)

A useful definition will determinesubconstructions:

Definition 4 (Subconstructions)

Let X be a construction.

i) X is a subconstruction ofX.

ii) X is a subconstruction of0X.

iii) If X is [YY1Ym], then Y, Y1, , Ym are subconstructions ofX.

iv) If X is lx1xm Y, then Y is a subconstruction ofX.v) If X is a subconstruction ofY and Y is a subconstruction ofZ, then X is a

subconstruction ofZ (transitivity).

As we already suggested the rules of the transition from an expression to the underlying

construction, in particular the rules governing the combining of subconstructions to get the

required construction, are language dependent, i.e., they will be distinct for distinct languages.

Having a linguistic input plus the rules we should get the desired construction practically


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automatically. We will solve our task without possessing such rules (for English), we have to

suppose their existence and give some general hints, the only advantage thereof being that

such general hints may support solving similar tasks for other languages as well.

The linguistic input should make it clear that in our case the being in Asia is predicated

about some object which in turn is determined by the expression the highest mountain. Now a

temptation we mentioned above consists in our confusing some piece of empirical knowledge

(the highest mountain is surely MtEverest) with what the expression really says; it does not

mention MtEverest, and the information contained in it is not an information about

MtEverest: it informs us that whatever mountain is the highest one it is in Asia. The sentence

would say the same thing even if MtEverest were not the highest mountain. (In that case, what

could change would be at most the truth-value, the meaning would remain the same.) Now

what is certain is that being in Asia is predicated about some individual; but no definite

individual can occur (via its trivialization) in the resulting construction, for no definite

individual is mentioned in the sentence (Parmenides principle). The problem has to be solved

with H / (i (oi))tw and M / (oi)tw at our disposal.

Let us begin from the other end: The resulting construction should construct a proposition.

To construct an otw-object (i.e., an ((ot)w)-object!) means to construct a function from

possible worlds to chronologies of truth-values. The basic procedure of constructing functions

is a closure. Thus let us choose once for all two variables: w ranging overw, and tranging

overt. The resulting construction will obviously possess the following form:

lw[ltX],

where X (v-)constructs truth-values. (We will abbreviate this notation by lwltX.) Now the

truth-values v-constructed by X have to be mediated by possible worlds and times, i.e., the

variables w and tmust be present in X, otherwise the resulting construction would produce a

trivialproposition, i.e., a constant function associating every world-time with one and the

same truth-value, which contradicts the fact that the sentence is an empirical sentence. So the

predicating of A to that indefinite individual is in fact applying, for every world-time, the

class of those individuals that happen to possess the property being in Asia in the given

world-time to that individual. We have therefore (abbreviating [[Xw]t] by Xwt)

lwlt[0AwtY],

where Y is a construction v-constructing that indefinite individual, or better, that individual

which is the result of applying H in the given world-time to the class which is the value of the


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property M in the given world-time. But this is already the solution of our problem, for we

have

lwlt[0Awt[0Hwt

0Mwt]].

An easy algorithm checks the type-theoretical adequacy as follows:

lw lt [[[0A w] t] [[[0H w] t] [[0M w] t]]]

w t (oi)tw w t (i (oi))tw w t (oi)tw w t

((oi)t) t ((i (oi))t) t ((oi)t) t

(oi) (i (oi)) (oi)

(oi) i

o

t (ot)

w ((ot)w)

The principle of this algorithm is clear:

Let X (v-)construct an (ab1bm)-object and let X1,,Xm (v-)construct respectively b1-,,bm-objects. Then [XX1Xm] (v-)constructs an a-object.Let X (v-)construct an a-object and letx1,,xmv-construct b1-,,bm-objects,respectively. Then lx1xm X (v-)constructs an (ab1bm)-object.Notice that the composition itselfv-constructs a truth-value. Yet we cannot calculate the

truth-value of the sentence, since the composition contains free variables w, t. And this is as

it should be: in the empirical case the truth-value of a sentence depends on possible worlds

(modal variability) and on time points (temporal variability). Therefore the resulting type

must be an intensional type, here the type of propositions, i.e., truth conditions. These truth

conditions can be read off from the construction: in our example we can see that the

respective proposition is true in such worlds-times W, T where the class of those individualsthat are in Asia in W at T contains the individual which happens to be in W at T the highest

one in the class of those individuals which are mountains in W at T. No other information is

given in particular we are not informed about MtEverest.

f) De re, de dicto

The TIL-like analysis is anti-contextualistic: it presupposes that any disambiguated

expression has the same meaning (and, therefore, denotes the same object) in any context. It


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could seem, however, that there are strong arguments against. We will argue that at least one

kind of them can be refuted in the following way.

Compare the sentences

Some politicians are honest.

Honesty is a virtue.

The expression honesty is in English the substantialised form of the adjective honest(in

Czech, for example, we can observe the similar caseestnost, estn). For TIL the meaning

of both grammatically distinct expressions is the same. It is not as if the second sentence

gavehonesty another meaning and/or type than the first sentence w.r.t. honest. In both cases

honest(y) denotes a property of individuals, so its type is (oi)tw. The following analysis will

demonstrate that the distinction does not concern meaning or type, but so-calledsupposition

only.

Analysis: Some $ / (o (oi)). P(olitician) / (oi)tw, H / (oi)tw, V(irtue) / (o (oi)tw) (let virtue

be a class of properties; an alternative analysis would consider virtue to be aproperty of

properties). A standard conception needs also / (ooo). (One can construe quantifiers in

such a more natural way that no conjunction is necessary in existentially quantified

constructions.)

The first sentence is empirical. So we have (x ranging overi)

lwlt[0$lx [0 [0Pwtx] [0Hwtx]]].

The second sentence is not empirical due to our choice of the type for V. We have

[0V 0H].

Now let us ask: Can we see any distinction between H in the former construction and H in the

latter? Well, let us accept that in our simplified analysis the meaning ofhonest(y) is 0H.

(Nothing essential would change if the construction underlying honest(y) were a complex

construction inspired by some definition.) In this respect no distinction can be observed. Yet

one distinction is easily seen. In the first construction the property H is applied to w and (then)

to t. No such application happens in the second construction. It means that the truth-value v-

constructed by the respective composition in the first construction does and in the second

composition does not depend on value of H in a given world-time.

Definition 5 (de r e, de d i c t o )

Let X be a composition and Y a subconstruction of X. Y is said to be in de re supposition

in X iff the value of what is v-constructed by X in a world W at the time point T


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depends on the value of what is v-constructed by Y in W at T. Otherwise Y is in de dicto

supposition in X.

(The terminology is borrowed from medieval philosophy, where a somewhat distinct meaning

was connected with both terms.)

From the definition we can infer that a subconstruction which is in the de re supposition is

always applied to w and t. Now a question arises: Does it hold that a subconstruction in the de

dicto supposition cannot be applied to w, t?

Surprisingly, the answer is negative. To show it we need to explain a special kind of contexts

called attitudes.

More about attitudes will be said in section g) and then in chapters 4, 5. Already now we

can, however, explain one kind of analysis ofpropositional attitudes.

Propositional attitudes are relations which are denoted by such expressions like believe, know,

doubt, thinketc. . Let B be such an attitudinal verb. The schema of the attitudinal sentences is

X Bs that Y.

What type can be ascribed to B? There are two possibilities, one of which will be referred to

in section g). Here we will choose such an interpretation of attitudinal verbs which seems at

first sight to be the most natural one: construing these relations as empirical relations linking

individuals with propositions we get

B / (oiotw)tw.

Let us analyze from this viewpoint the sentence

Charles believes that the richest man is happy.

We are here not involved in solving the difficult problem of semantics of proper names, so let

Charles be an individual;

Ch(arles) / i.

The type ofthe richestis obviously the same as that ofthe highest, so

R(ichest) / (i (oi))tw.

Being a man and being happy are properties of individuals, thus

M(an) / (oi)tw, H(appy) / (oi)tw.

According to the type of B and the linguistic intuition, believing is an empirical binary

relation between individuals and propositions. The individual is here Charles, the proposition

that the richest man is happy

is analyzed in the same way as the proposition denoted by The highest mountain is in Asia:


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lwlt[0Hwt[0Rwt

0Mwt]].

The sentence above that refers to Charles belief is thus analyzed as follows:

lwlt[0Bwt0Ch [lwlt[0Hwt[

0Rwt0Mwt]]]].

Now the subconstruction which constructs the proposition that the richest man is happy is in

the de dicto supposition, and this fact corresponds to the fact that Charles believing is

independent of the values of H, R, M in the worlds-times where his believing takes place. Yet

each of the constructions constructing these intensions is applied to w, tin the resulting

construction. So we have to state that the de dicto context makes of subconstructions

containing de re suppositions subconstructions with de dicto suppositions w.r.t. the context. In

this sense a de dicto occurrence can contain applications to w, t.

Two features of such believing etc. de dicto are characteristic and important for logical

inference:

a) Let a premise

Bill_Gates is the richest man.

be added to the premise


Can we deduce from these two premises the conclusion

Charles believes that Bill Gates is happy. ?

Surely not. Charles may believe what he believes without knowing Bill Gates, without

knowing who is the richest man. The proposition

that the richest man is happy

is distinct from the proposition

that Bill Gates is happy,

and no logical connection obtains between them.

b) Does from


follow that

there is some individual such that Charles believes of him/her/it that he/she/it is happy ?

(The respective construction being

lwlt[0$lx [0Bwt0Ch [lwlt[0Hwtx]]]] ).

Imagine that there is no richest man. (you can admit it at least when there are more rich

people with the same amount of money at their disposal, or when there are no rich people at

all). In this case the conclusion would be false: Charles simply cannot believe of some non-


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existent entity that it is happy, whereas his de dicto belief is immune to existence or non-

existence of the richest man.

These two features ( a) and b) ) distinguish believing as referred to in de dicto supposition

from believing as referred to in de re supposition. The latter can have one of the following

linguistic forms:

Charles believes of the richest man that he is happy.

or, equivalently,

The richest man is such that Charles believes of him that he is happy.

If we had at our disposal the rules that transform particular syntactic structures of English to

our constructions (i.e., properly speaking, to operations of creating functions and applying

them to arguments) we could now automatically fix the resulting construction. Here we have

to resort to linguistic intuitions or perhaps use some linguistic structure as defined in some

linguistic school (the most adequate structure would be probably offered by a categorial

grammar). Let us try without rules: Both sentences speak about believing of (somebody).

Especially the second sentence makes it clear that this time the class of those individuals of

which Charles believes them to be happy is applied (in each world-time) to the richest man.

So the form of the construction will be (x ranging overi):

lwlt[[lx ] [0Rwt0Mwt]].

Now [lx] will, for every world-time, construct the class of those individuals ofwhich

Charles believes them to be happy, i.e.,

[lx [0Bwt0Ch [lwlt[

0Hwtx]]]].

The resulting construction is

lwlt[[lx [0Bwt0Ch [lwlt[0Hwtx]]]] [

0Rwt0Mwt]].

Now unlike in the case ofde dicto supposition the value of what is v-constructed by the

composition

[lx [0Bwt0Ch [lwlt[0Hwtx]]]] [

0Rwt0Mwt]

is dependent on what is v-constructed by [0Rwt0Mwt]. Therefore, the situation sub a) above,

when the premise


is added, has to be evaluated distinctly: Suppose that the proposition denoted by this premise

is true in all worlds-times W,T, where the proposition denoted by the first premise is true.

Then the conclusion

Charles believes of Bill Gates that he is happy.


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will be true as well: that Bill Gates is a member of the class of those individuals who are

believed by Charles to be happy is surely fulfilled in W at T. Notice that the possible fact that

Charles does not know that Bill Gates is the richest man is irrelevant here, because this time

the construction

[0Rwt0Mwt]

is outside the scope of Charles believing, it is ratherwe as those who refer to this believing

who know that the richest man is Bill Gates.

Formally, the premise


expresses the construction

lwlt[0= 0Bill_Gates [0Rwt0Mwt]].

Thus we can substitute 0Bill_Gates for [0Rwt0Mwt] in the first premise without any problems

and get the conclusion.

Similarly, as for b), if there is no occupant of the office the richest man in W at T, the

premise will not be true in W at T, and otherwise such an individual existsit is just the

occupant of the role of the richest man in W at T.

One of the consequences of our distinction between de re and de dicto suppositions is:

The de dicto case does not follow from the de recase, and the de re case does not follow

from the de dicto case.

This claim is demonstrated by our analysis of the features a) and b). Besides, counterexamples

can be easily given:

The non-deducibility of de dicto from de re: Imagine that Charles belief (in W at T) concerns

the man who is the richest man (in W at T) but that Charles does not know that he is the

richest man (in W at T). Then de re holds unlike de dicto.

The non-deducibility of de re from de dicto: Imagine that Charles believes that the richest

man is happy but at the same time he would be surprised if informed that such and such

individual actually is the richest man. Maybe Charles would say: Well, if I knew that this man

is the richest man, I would not claim that he is happy. In this case Charles belief was de dicto

without being de re.


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g) Higher order types

Having defined types of order 1 (Definition 1) and trivialization (Definition 3) we have made

it possible to construct such objects whose type cannot be determined; we can say that these

object do not possess types of order 1. For a most simple example see any variable, say, a

numerical (overt ranging) variablex. We could speak about a t-variable, but t is the type of

the objects v-constructed byx, not of the variable itself. This could be an innocuous fact,

becausex, as well as any variable, does not construct itself, but a s soon as we begin to use

trivialization we are able to construct variables and any constructions. Consider0x.

According to Definition 3, ii), this construction constructsx. So what is the type of the object

constructed by 0x?

This will be possible to say after our hierarchy of types has been extended to a ramified

hierarchy. (How far is this hierarchy similar to Russells ramified hierarchy can be seen

from a comparison made by Tich in [1988, p.68-70].)

The Definition 6 below is structured as follows: First, types of order 1 are defined (referring

to Definition 1), second, a connecting link defines constructions of order n, third, types of

order n + 1 are defined. No vicious circle arises.

Definition 6 (Types of order n)

T1 Types of order 1 : see Definition 1.

Cn Let a be a type of order n.i) Ifx is a variable that v-constructs a-objects, then x is a

construction of order n.

ii) If the type of X is a, then 0X is a construction of order n.iii) If X, X1,,Xm are constructions of order n, then the composition

[XX1Xm] is a construction of order n.

iv) Ifx1,,xm, X are constructions of order n, then lx1xm X is aconstruction of order n.

v) A construction of order n is only what satisfies i) through iv).

Tn+1 Let *n be the collection of all constructions of order n.i) *n and every type of order n is a type of order n+1.ii) If a, b1 ,,bm are types of order n+1, then (ab1bm) is a type of order

n+1.

iii) A type of order n+1 is only what satisfies i), ii).


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Remark:and every type of ordern in Tn+1 means that in the case when the original

types in (ab1bm) are not of the same order, then we can and have to shift the orders of

them so that all of them are of the same (the highest) order. (Similarly for applying points iii)

and iv) in Cn.

We now illustrate Definition 6 by some simple examples.

1) Let us return to our numerical variablex. Sincexv-constructs numbers, i.e.,t-objects,

and since t is a type of order 1,x is a construction of order 1 (Cn, i) ). It is, therefore, a

member of *1, and its type is of order 2 ( Tn+1, i) ). So0x is construction of order 2 and

its type is of order 3.

2) Let Pr be the class of all proper (i.e., for all v v-proper) constructions of order 1, so Pr /

(o*1). We will determine the types of following two constructions.

[0+x01]

[0Pr0[0+ x 01]].

+ is a function of type of order 1 ( (ttt) ). Hence 0+ is a construction of order 1 and its

type is of order 2. x, asa numerical variable, is a construction of order 1 and its type is

of order 2. 1 is a t-object, type of order 1, so 01 is a construction of order 1 and its type is

of order 2. So the first construction is of order 1 (Cn iii) ) and its type is of order 2.

Now Pr is an (o*1)-object and its type is according to Tn+1, ii), of order 2. Therefore,0Pr

is a construction of order 2, and its type is of order 3. Finally, 0[0+x01] is a construction

of order 2 and its type is of order 3, so that [0Pr0[0+ x01]] is a construction of order 3

and its type is of order 4.

This hierarchy is very intuitive. Types of order 1 are types of objects over the basis o, i, t, w;

no construction can possess such a type. To mention the constructions which (v-)construct the

1st order objects we use constructions that construct the former: these are constructions of

second order. To mention the constructions of second order we use constructions of order 3,

etc. etc.

Definition 7 (Higher order objects)

Higher order objects (HOOs) are objects of a type of order n for n> 1.

Thus HOOs are constructions and such (ab1bm)-objects where at least one type from

among a, b1,bm is a type of ordern> 1.


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Now it could seem that there is no need in HOOs when expressions of a natural language are

to be analysed. It could seem that such artificial expressions like the class of proper

constructions are highly artificial and professional and that we never meet really natural

expressions which would have to be analysed in terms of HOOs. We must state that this

impression is wrong: One of the difficult problems with analyses is the problem of analyzing

attitudes. We have already mentioned this problem and have shown that one of the

possibilities is to interpret attitudes as relations between individuals and propositions (in

general, between individuals and intensions). We will see in chapters 4 and 5 that a very

important variant of these analyses consists in taking attitudes to be relations linking

individuals with constructions. Then, of course, HOOs are indispensable components of our

analyses.

Now we can define free, l-bound and o-bound occurrences of variables.

Definition 8 (free, l-bound, o-bound occurrences of variables)Let X be a construction.

i) If X is a variable x, then x has afree occurrence in X.ii) If X is

0Y, then every occurrence ofx in Y is a o-bound occurrence in X.

iii) If X is a composition [YY1Ym], then every occurrence ofx in X which is afree, o-bound, l-bound occurrence in Y, Y1, , Ym, is afree, o-bound, l-boundoccurrence inX.

iv) If X is lx1xm Y, then ifx is one ofx1,,xmand its occurrence in Y is not a o-bound occurrence in Y, then its occurrence is l-bound in X. Ifx is not one of

x1,,xm, then every occurrence ofx which is not l- or o-bound in Y isfree inX.

The specific form of boundness, the o-boundness (boundness by trivialization) has a veryimportant feature: Having a construction X which is the result of correctly substituting in X

l-bound occurrences of a variable by occurrences of another variable we can state that both

constructions are equivalent. (See the a-rule in l-calculi.) Substituting in X o-bound

occurrences of a variable by another variable always results in a non-equivalentconstruction.

Definition 9 (equivalent constructions)

A construction X is equivalentto the construction Y iff either X v-constructs for allvaluations v the same object as Y, or X and Y are for all valuations v v-improper.


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Example:

Compare following two pairs of constructions (x, y range overt):

a) lx [0>x00], ly [0>y 00]

b) 0[lx [0>x 00]], 0[ly [0>y00]]

Both constructions under a) construct one and the same function, viz. the class of positive

numbers; they are equivalent; the occurrence ofx (y) in them is l-bound. The constructions

under b) construct distinct, even if equivalent constructions: so they are not equivalent; the

occurrence ofx (y) in them is o-bound (see Definition 8, ii), iv) ).

The role of trivialization is now well visible. If what we are interested in is what is

constructedby a construction X, then we use X as ourway to the object. If we are interestedin the construction X itself, then we have to constructX, which in the easiest case happens via

trivializing X. Now we can return to a recent question: can we say that in a really natural

language we are sometimes interested in a construction itself ? And our answer is again: an

attitude can be interpreted as being sensitive to the way the proposition is given, i.e., just to

the respective construction. This will be seen in chapters 4, 5.

h) Other constructions

TIL is an open ended theory. This means that we are invited, among other things, to

consider possibilities of enlarging a) the class of atomic types, b) the class of constructions.

Here we refer to two points in this respect.

1. Constructions not mentioned in the preceding text.

The original form of TIL, as used in Tichs articles before 1988, considered 1st order

objects to be constructionssui generis; objects were constructions that constructed

themselves. The transition to the ramified hierarchy (see section g) ) showed that it was

necessary to introduce a new construction which would shift the order of the given entity.

So trivializationcame into being. Trivialization also made it possible to reject the stipulation

according which objects were a kind of construction. Tich, however, has introduced in

[1988] two other constructions: execution and double execution. Here they were not

mentioned because in our opinion they are not indispensable in most contexts. So only

briefly:


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Execution, 1X, v-constructs what X v-constructs; so if X is no construction, then 1X is

improper, otherwise it is identical with X. (Thus it is a sort of a filter, rejecting non-

constructions and doing the same job as the respective construction.)

Double execution, 2X, works as follows: if X is a v-proper construction which v-constructs a

v-proper construction Y, then 2X v-constructs the object v-constructed by Y; otherwise it is v-

improper.

An example:

Let c be a variable which ranges over numerical constructions (i.e., over (o*1) ). Let a

valuation v associate c with the construction [0>x00] andx with the number 2. Then

0c constructs just the variable c,

1c v-constructs [0>x 00],

2c v-constructs what [0>x00] v-constructs, i.e., the truth-value T.

(We have a theoretical possibility to inductively define executions of any degree, of

course.)

Execution, as well as double execution, may have a certain importance in highly theoretical

contexts. Our analyses can do without them.

2. Other types and constructions.

Applying TIL analyses in the area of conceptual (database) modeling (see [Du 2000] ),

resulting in building up a data model HIT, has shown that what is badly needed in the practice

of answering users queries is to have at ones disposal functions whose values are m-tuples

form >1. Since the type schema for functions in TIL is (ab1bm), where a is a type

according to Definition 1, no function can satisfy this requirement. Thus a new type, a tuple

type has been introduced in the HIT model. This type is, properly speaking, the Cartesian

product of the types b1,,bm, and has been denoted by (b1,,bm). Then the functional type

has been defined as (ab) for any type a; b could be, of course a tuple type. This made it

possible to have all functions monadic without the Schoenfinkel reduction (which does not

work in the case of partial functions, as has been demonstrated in [Tich 1982] ).

Having, however, a new type is only a necessary prerequisite of the analyses proper. We

have to be able to constructthe objects of the new type. For this reason two new kinds of

construction have been defined. First, a tuple construction: Let X1,,Xm be constructions v-

constructing b1-,,bm-objects b1,,bm respectively. Then (X1,Xm) is a construction v-

constructing the m-tuple . And for 1 im, theprojection constructions are


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defined, v-constructing the object v-constructed by the ith member of the tuple construction

Y. Projections are denoted, say, by Yi.

Thus, e.g., the construction ( 0Day, 0Month, 0Year)2 constructs the class {January,

February,,December}.

4. Type-theoretical analysis

a) General remark

type theoretical analysis

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