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

Russell's introduction of types (see [Russell 1906]) has been motivated by the need to avoid paradoxes arising due to violating the 'vicious circle principle'. Russell's solution of this problem is connected with a strong intuition concerning our understanding such linguistic phenomena like predication. If some X is predicated of some y, then we feel that y cannot be of the same 'order' as X. Being yellow can be predicated of some particular thing ('individual') and being a colour can 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 dog can be predicated of an individual and being a property can be predicated of being a dog but not, of course, of the respective individual.

Russell's hierarchy of types has been formulated in terms of sets and relations. The limitations given by this approach have been excellently summed up and criticised in [Tichý 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 a fact corresponding to this combination. So saying that 2 is a prime we combine the predicate prime 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 odd is not clear. (problem of false sentences.) Not only that. What happens when the function + is combined with the pair <2, 3>? No referring to a fact is possible here. And such functions like + are not members of Russell's 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) 'creating' a 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 of ... being 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.)

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 objects (entities), the latter truth-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 t - represents 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 define s as a separate type.

Montague's 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 Tichý's work, e.g., in [1988]

Transparent intensional logic (TIL), first formulated by Tichý 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 a PWS (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 (Frege's error) but truth conditions (= propositions), empirical common nouns cannot denote classes but properties, etc.

iii) One among the possible worlds is the real, actual one. 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' or 'naked' 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 Tichý's [1983].)

vi) A 'Parmenides' Principle (to be found already in Frege's Grundlagen der Arithmetik):

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.

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 and falsity 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? The classical answer is: there are two of them, T(ruth) and F(alsity).

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 letter o , 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 letter i .

Distinguishing between sentences like

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 letter t .

The choice of the most 'controversial' fourth atomic type is motivated as follows:

What is the semantics of empirical expressions? Let us compare the mathematical (i.e., non-empirical) expression prime (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 contingent fact. 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 a property. The distinction between classes and properties can be illustrated as follows: Let individuals A_1,... ,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 property - being the same property - changes 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 be functions 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 thinkable state of the world with

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 of possible states of the world; the broadly accepted terminology speaks about possible 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 [Tichý 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 t is a possible world at t, and possible worlds are temporal sequences ('histories') of particular possible worlds at particular time points. -

Summing up, we have got four atomic 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., expressions) on the particular components can be best modelled as functional dependence. In order to explain this principle we have to recapitulate what is meant by the notion of function.

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 if n > 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 with just one value.

Functions defined in this way obey the principle of extensionality: for n-adic functions it is given by the valid formula (f, g are variables ranging over n-adic functions of an arbitrary type):

" x₁...x_n (f(x_1,...x_n) = g(x₁,...,x_n)) É 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) = x² - y².

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 by functions 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') of 'primitive' 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:

2. Let a ,b ₁,...,b _m be any types of order 1. Then (a b ₁...b _m), i.e., the set of partial functions with values in a and arguments (in general, tuples whose components are, respectively) in b ₁,...,b _m, is also a type of order 1
3. A type of order 1 is only what satisfies i) and ii). -

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.

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 of classes of a -objects. Indeed, (oa) is the set of functions from a to {T,F}, i.e., of characteristic functions of the classes of a -objects: Such a function takes T on those a -objects which belong to the given class, and F for the other a -objects. So (oi) is the set (type) of classes of individuals (in our table align="center" 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 (o t t): 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 and parts 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, quite independently 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", "pracovat" - it can be an adjective - see "black", "cerný" (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", "být 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: Take the type o t w (i.e., ((o t)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 ą 2", but also "MtEverest je vyšší nez Mont Blanc", "Mont Blanc je nizší nez MtEverest" 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 align="center" is not (and no such table align="center" can be) exhaustive as a list of types possibly exploitable align="center" 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 even proper 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 [Tichý 1980, 1980a]). -

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

Our definition of types shows that intensions are functions which associate possible worlds with chronologies of a -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 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 influence supposition 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):

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 'Plato's 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. T and 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 planet in them equals 9. Our ('actual' or 'real') 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.)
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 a property, 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 t that 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 (for a any type) (t (oa)). In Czech no error can arise: 'number' is 'císlo', 'number of' is 'pocet'. -

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 a_tw , 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 the functional character of the dependencies connected with using the expression. In the linguistic area the intuition connected therewith led to the rise and development of categorial grammars ( see, e.g., [Morrill 1994]). To show a most elementary 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: if t 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 (o t). (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 of objects. 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 (o i)t w (or (o a)t w for some other a). 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 if 'big' and 'small' denoted a property, we could therefrom derive the desirable type (o i)t w of the compound expressions under the following condition: the 'new' properties being a big ant and being a small elephant would have to be interpreted as being big and being an ant and 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:

Empirical adjectives (where this problem arises) - being empirical - denote intensions. So their type has the form a_tw , 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 surely a compound type), then its application to an (oi)_tw -object results in an (oi)-object only if a is of the type ((oi)(oi_tw)). 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: so from the fact that x is a big something it does not follow that x 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 the principle 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

1. a general theory that formulates some principles independent of what is specific for particular languages,
2. 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 Tichý 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 <e, m>, 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 of construction is introduced in the next section.

Remark: We adduce types of some important extensions.

1. Truth functions: Unary truth functions (in particular negation) - (oo). Binary truth functions - (ooo).
2. 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 of a .
3. Identity. Identity ( =) too is type-theoretically polymorph. The schema is (oaa). It is the class of all such pairs a, b of a -members where a is the same object as b.
4. Singulariser. Singulariser (whose linguistic counterpart is commonly called descriptive operator and 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. -

What 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: If A, an expression of a language L, is correctly translated as A' into a language L', then there must be something what is common to A and A', evidently what is called meaning (of A and A'). Now we can hardly claim that an expression of an artificial language is the meaning of the expression whose translation it is supposed to be. Let A be an expression of a natural language, and A' its translation into an artificial language. Suppose for a while that A' is really the meaning of A. We can ask: what is the meaning of A' ? Is it perhaps A ? But this is an obvious nonsense, for A and A' are distinct, so that the meaning of A would be distinct from the meaning of A' .

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 Cresswell's [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 of construction 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 of abstract 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 adding and applying the latter to the pair <1,1> , or identifying the classes of even numbers, of prime numbers, the intersection, applying the intersection to the pair <even numbers, prime numbers>, identifying the partial function the only x such that and 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.

First, we could be interested - unless we avoid philosophical questions - in knowing what kind of 'status' 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 meaning of 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 of 'artificial language', let it be called CL. What is, however, important, is that it will be not the expressions of CL what will be taken to be meanings: The expressions of CL 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 '1^st order 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 term sense ('Sinn'). Thus the role of the first two kinds of construction consists in mediating between the 'world' of 1^st 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 that produce 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 will pre-theoretically characterise particular 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 of o begins as follows:

Now let p₅ be the 5^th o -variable. If the sequence above has been selected by the valuation v, then we say that p₅ v-constructs T (since T is the 5^th 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 variables - like 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 like x,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.

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,...,X_m (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 and x₁,...,x_m v-construct b₁-,...,b_m-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) variables x₁,...,x_m. The result is an (a b₁...b_m)-object.

So we have 'interface with objects' ( 1), 2)), applying function to arguments (3) and 'creating' a function (4). We will have some thoughts about whether this is sufficient later (section h).

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:

Let x₁ be a variable v-constructing real numbers (i.e.: 'ranging over t '). Let a particular valuation v associate x₁ with the number 2. Then x₁ v-constructs 2, whereas ⁰x₁ constructs

(i.e., v-constructs for every valuation v) just the variable x₁. Notice that x₁ is a t -variable, for 2 is a t -object. On the other hand, ⁰x₁ is not a 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 1^st 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 >/ (o t t). The construction

[⁰> x ⁰0]

(x ranging over t) v-constructs truth-values: T if x v-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   

(see Definition 3 iii)).

Now consider the construction

[l x [⁰> x ⁰0]].

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., (o t), is unambiguously derivable from the types of the objects constructed by the components of the construction, here it suffices to derive it from the type v-constructed by the composition and the type t which x ranges over (we omit the outermost brackets):

l x [⁰> x ⁰0]

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 of x, the construction itself contains one occurrence of x only - writing 'l x' we only fix what the construction does, we do not 'add' a new occurrence of x to the construction itself. -

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

Consider the construction

[⁰div x y].

It obviously v-constructs the result of dividing the number v-constructed by x by the number v-constructed by y. Now we will bind the variable x via the following closure:

l x [⁰div x y].

Let v associate x with 6 and y by 4. The closure ignores v as concerns x (see Definition 3 iv)), so that it v-constructs the function which takes any number k to the result of dividing k by 4, i.e., by the number v'-constructed by [⁰div x y]. Thus we can see that the valuation influences only those variables which are not in the scope of l . (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

[⁰div x ⁰0].

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

l x [⁰div x ⁰0],

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

Some examples of simple closures:

Let x range over a type a .

The closure

l x x

constructs the identity function over a .

The closure

l x ⁰2

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

Let x, y range over t . Compare the following closures:

1. l xy [⁰> x y],

2. l yx [⁰> x y],

3. l x[l y [⁰> x y]]

The closure a) constructs the function that takes T on such pairs <m, n> of numbers where m is greater than n, and F otherwise. It constructs, therefore, the same relation as does ⁰> .

The closure b) constructs the function that takes T on such pairs <m, n> of numbers where n is greater than m, and F otherwise. So it constructs the same relation as does ⁰< .

The closure c) constructs the function that takes every number m to (what [l y [⁰> x y]] v'-constructs, i.e.,) the function which takes every number n to T iff m 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 mentioned Parmenides' Principle and the problem of de re vs. de dicto suppositions.

We already know that this principle can be formulated as follows:

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:

If its analysis did not obey Parmenides' principle, then it could easily enable us to draw the conclusion

• MtEverest is in Asia.

which, of course, does not follow.

Now we will derive a construction underlying our premise to show that Parmenides' principle blocks the incorrect inference above. It will be a particular illustration of the procedure that will be more thoroughly described in chapters 4 and 5.

Remark: One could wonder why we speak about a construction... . Why not the construction...? We will return to this question in the beginning of chapter 4. -

1. Type-theoretical analysis.

The easiest expression to be type-theoretically analyzed is obviously mountain. It is an empirical common noun, so it is about a property of individuals. So we have

M(ountain) / (oi)_tw

Simplifying a little, we can take the expression is in Asia as denoting a property of individuals too. So we have

(being in) A(sia) / (oi)_tw

Now the expression the highest is more difficult to submit to a type-theoretical analysis. Let us try.

First, whether something is the highest object of some kind is dependent on empirical facts. So we can say that whatever object is denoted by the highest, its type will be a _{t w} for some type a .

Second, let us suppose (for simplicity's sake) that the superlative (the ...est, in Czech nej...) indicates that the object which is the highest one is always at most one. And since high (more or less) can be only individuals (from the viewpoint of our type-theoretical classification, and ignoring possible metaphoric usage), we could consider the possibility of the type i t w . Yet this hypothesis is counterintuitive. Individuals according to our stipulation are mere 'pegs' of properties. The expression the highest individual is highly problematic. We can see that the highest is in any rational context accompanied with a common noun; this syntactic fact is justified by the semantic fact that being the highest always means being the highest among some objects (individuals) possessing some property: being the highest tower is not the same as being the highest mountain. Thus the individual which is the highest one with respect to one property is not automatically the highest individual with respect to another property. But 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 a function 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 .

2. 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 expression - a proposition, i.e., an o_tw -object rather than a truth-value: not taking this into account means not obeying Parmenides' principle: no truth-value is mentioned in the sentence. Hence our task is now to find such a construction which would construct an o t w -object and whose subconstructions would be the constructions constructing the three objects above. (This task is an instance of the task type-theoretical synthesis.)

A useful definition will determine subconstructions:

1. X is a subconstruction of X.
2. X is a subconstruction of ⁰X.
3. If X is [YY₁......Y_m], then Y, Y₁, ... , Y_m are subconstructions of X.
4. If X is l x₁...x_m Y, then Y is a subconstruction of X.
5. If X is a subconstruction of Y and Y is a subconstruction of Z, then X is a subconstruction of Z (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 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 o_tw -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 over w , and t ranging over t . The resulting construction will obviously possess the following form:

l w[l t X],

where X (v-)constructs truth-values. (We will abbreviate this notation by l wl t X.) Now the truth-values v-constructed by X have to be mediated by possible worlds and times, i.e., the variables w and t must be present in X, otherwise the resulting construction would produce a trivial proposition, 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 X_wt)

l w l t [⁰A_wt Y],

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

l w l t [⁰A_wt [⁰H_wt ⁰M_wt]].

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

The principle of this algorithm is clear:

Notice that the composition itself v-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 individuals that 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.

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 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

The expression honesty is in English the substantialised form of the adjective honest (in Czech, for example, we can observe the similar case - cestnost, cestný). For TIL the meaning of both grammatically distinct expressions is the same. It is not as if the second sentence 'gave' honesty 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 (o i)t w . The following analysis will demonstrate that the distinction does not concern meaning or type, but so-called supposition 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 a property of properties). A 'standard' conception needs also Ů / (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 over i)

l w l t [⁰$ l x [⁰Ů [⁰P_wt x] [⁰H_wtx]]].

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 of honest(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.

(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 of propositional attitudes.

Propositional attitudes are relations which are denoted by such expressions like believe, know, doubt, think etc. . Let B be such an attitudinal verb. The schema of the attitudinal sentences is

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 / (o i o _{t w})_{t w} .

Let us analyze from this viewpoint the sentence

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 of the richest is obviously the same as that of the 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:

l w l t [⁰H_wt [⁰R_wt ⁰M_wt]].

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

l w l t [⁰B_wt ⁰Ch [l w l t [⁰H_wt [⁰R_wt ⁰M_wt]]]].

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, t in 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:

1. Let a premise

be added to the premise

Can we deduce from these two premises the conclusion

Surely not. Charles may believe what he believes without knowing Bill Gates, without knowing who is the richest man. The proposition

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

l w l t [⁰$ l x [⁰B_wt ⁰Ch [l w l t [⁰H_wt x]]]]).

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-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:

or, equivalently,

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 over i):

l w l t [[l x ... ] [⁰R_wt⁰M_wt]].

Now ...[l x...] will, for every world-time, construct the class of those individuals of which Charles believes them to be happy, i.e.,

[l x [⁰B_wt ⁰Ch [l w l t [⁰H_wt x]]]].

The resulting construction is

l w l t [[l x [⁰B_wt ⁰Ch [l w l t [⁰H_wt x]]]] [⁰R_wt ⁰M_wt]].

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

[l x [⁰B_wt ⁰Ch [l w l t [⁰H_wt x]]]] [⁰R_wt ⁰M_wt]

is dependent on what is v-constructed by [⁰R_wt ⁰M_wt]. 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

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

[⁰R_wt ⁰M_wt]

is outside the scope of Charles' believing, it is rather we as those who refer to this believing who know that the richest man is Bill Gates.

Formally, the premise

expresses the construction

l w l t [⁰= ⁰Bill_Gates [⁰R_wt ⁰M_wt]].

Thus we can substitute ⁰Bill_Gates for [⁰R_wt ⁰M_wt] 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 exists - it 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 re case, 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.

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 (over t ranging) variable x. We could speak about a t -variable, but t is the type of the objects v-constructed by x, not of the variable itself. This could be an innocuous fact, because x, 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. Consider ⁰x. According to Definition 3, ii), this construction constructs x. 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 Russell's ramified hierarchy can be seen from a comparison made by Tichý 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.

Remark: "and every type of order n" in Tn+1 means that in the case when the 'original' types in (a b ₁...b _m) 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 variable x. Since x v-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)). So ⁰x 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.

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 1^st 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.

Thus HOOs are constructions and such (a b₁...b_m)-objects where at least one type from among a , b₁,...b_m is a type of order n > 1.

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.

1. If X is a variable x , then x has a free occurrence in X.
2. If X is ⁰Y, then every occurrence of x in Y is a o-bound occurrence in X.
3. If X is a composition [YY₁......Y_m], then every occurrence of x in X which is a free, o-bound, l-bound occurrence in Y, Y₁, ..., Y_m, is a free, o-bound, l-bound occurrence in X.
4. If X is l x₁...x_m Y, then if x is one of x_1,...,x_m and its occurrence in Y is not a o-bound occurrence in Y, then its occurrence is l-bound in X. If x is not one of x₁,...,x_m, then every occurrence of x which is not l- or o-bound in Y is free in X. -

The specific form of boundness, the o-boundness ("boundness by trivialization") has a very important 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-equivalent construction.

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

Example:

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

a) l x [⁰> x ⁰0], l y [⁰> y ⁰0]

1. 0[l x [⁰> x ⁰0]], ⁰[l y [⁰> y ⁰0]]

Both constructions under a) construct one and the same function, viz. the class of positive numbers; they are equivalent; the occurrence of x (y) in them is l -bound. The constructions under b) construct distinct, even if equivalent constructions: so they are not equivalent; the occurrence of x (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 constructed by a construction X, then we use X as our 'way' to the object. If we are interested in the construction X itself, then we have to construct X, 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.

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 Tichý's articles before 1988, considered 1^st order objects to be constructions sui 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 trivialization 'came into being'. Trivialization also made it possible to reject the stipulation according which objects were a kind of construction. Tichý, 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:

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> x ⁰0] and x with the number 2. Then