1 THE PROCEDURAL CONCEPTION OF LANGUAGE AND FACT ...

THE PROCEDURAL CONCEPTION OF LANGUAGE AND FACT Jiří Raclavský Masaryk University, Czech Republic Abstract: Within the intensional logic a proposition was defined as a function from possible worlds to truthvalues and a possible world is defined as a certain collection of state-of-affairs, (possible) facts and fact as true proposition. However, propositions are not structured, they do not consist of in certain individual instantiating certain property what is a fact in the pre-theoretical sense. Tichý’s concept of propositional construction is such a structured item (language was conceived by him as procedural). I argue for the necessity to adopt constructions as explicates for the notion of state-of-affair (fact).

The notion of possible world is used for the intensional explication of language as well as for the explication of fact (state-of-affairs1). Recently the possible world explication of language meanings was found insufficient and it has started the quest for suitable hyperintensional entity. Tichý’s concept of s.c. construction is such a procedural entity. However, Tichý himself did not adopt a procedural explication of facts. My aim is to show the inevitable need to do it, therefore both language and fact would be explained in a unified way.

Intensional Explication of Language The possible-worlds semantics (modal and intensional logics) explains language, language meanings, functional way. Not every theoretic entity may then be a function; some atomic entities must be accepted. According to common-view, there are individuals which instantiate various properties. The universe of discourse (sign it ι), a collection of individuals, will be reasonably our first category of atomic objects. To form classes and functions we have to represent two logical qualities, affirmative and negative; collection ο (whose members are the truth-values T and F) will be our second atomic category. The plentitude of combinatorically give functions over {ι,ο} does not help us suitably explicate the language meanings as well as the notion of fact. Individual’s belonging tocertain class is a necessary matter, but individual’s possession of a property is typically a contingent thing. It is usual to adopt then a set of thinkable alternatives, possible worlds (ω).2 Facts depend also on temporal 1

One can use the following terminology: a state-of-affairs that obtains is an actual fact; a state-of-affairs that does not obtain is a possible fact. I will use the conjoint term “fact”. 2 The most widespread notion of possible world conceives possible worlds as containing individuals (for a sharp criticism mainly of D. Lewis’ account see [7], 177-180); because we do not accept this conception, we do not discuss it below.

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factor, so let us append a collection τ of real numbers representing time-moments too. In Tichý’s transparent intensional logic (TIL, for brief), the logic I prefer, the set {ι,ο,ω,τ} gives rise to many intensional or non-intensional (extensional) functions. Among intensions propositions are functions from possible worlds and time-moments to truth-values, properties are functions from possible worlds and time-moments to classes of individuals (or other entities), m-ary relations are functions to classes of m-tuples of individuals, etc. According to Tichý, these functions serve as explications of pre-theoretically given attributes. In the eighties of the 20th century Tichý clearly realized that the typed lambda-calculi, he adopted for handling with functions, may be understood purely objectually and the respective λ-terms, he called constructions, became to be viewed as those procedural, hyperintensional entities. For example, there is only one proposition which is true in all possible worlds (and times), however, there are many mathematical equations (2+3=5, 25=52, etc.) or logical tautologies that construct it. The expressions which denote that proposition differ; they differ in that they express distinct procedures-constructions which are structured nearly the isomorphous way to the expressions themselves. Constants of λ-calculi are understood as s.c. trivializations (written as 0X), they serve for immediate graspings of objects; also variables are understood not as letters but as genuine constructions. Application of function on argument ([F A1...An]) is renamed to composition; the corresponding term stands for that procedure, not for the value the function has at the respective argument. Similarly λ-abstraction (λx [...x...]), renamed to closure, produces a function but the corresponding term stands for that closure not for the constructed function. Meanings of sentences are closures of the form (in a bit shortened notation) λwλt [...wt...], i.e., constructions constructing propositions, functions from possible worlds (“λw”) and timemoments (“λt”). Constructions are irreducible structured entities which typically construct non-structured set-theoretical objects like classes, properties, propositions etc. (such procedural conception of language enables us to avoid the paradox of omniscience which is fatal for the intensional conception of language). Argumentation for and the respective definitions are in [7].

Intensional Explication of Fact and Possible World One can be a bit surprised that Tichý never incorporated the procedural view into his conception of fact, that he stayed out with intensional stage of it. In many his late writings we read that the fact of the author’s of Waverley being a poet is a proposition singling out T ([7], 178), that the state-of-affair is a class (i.e. a characteristic function) of possible worlds ([8], 2

39); that possible worlds are wholes of possible facts ([7], 178) etc. Thus he used a classical definition of possible world and fact which Materna articulated ([3], 29) as follows: possible world is a maximal consistent class of facts which can obtain and ([2], 14): fact is a true proposition Problems with the definition of possible world. The first problem one can see is what a syntactical concept “consistency” does mean – it cannot be applied to propositions, only to expressions denoting propositions. One may then ask whether it probably means that two propositions are not having opposite values at certain argument or not. But propositions are conceived in TIL as (typically) partial mappings; thus “having not opposite values” is not a suitable notion here. But this problem is in only on the side of Materna’s formulation: Plantinga (cf. [5]) and others talk about mutually non-contradictory propositions; they use a semantical notion which is more relevant here. On the other hand, the best way how to use the term “non-contradictory” is to apply it on the level of propositional constructions. In the following sketchy picture we omit the proposition false in all possible worlds (and times) because it cannot be a valid fact (call it P8; W1, ...,W3 are possible worlds, we omit times and partiality here): P1

P2

P3

P4

P5

P6

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W2

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If we skip, for the simplicity, certain other problem, it would be reasonable to define the possible world the following way (cf., for example, [8], 39): each of possible worlds is a collection of all propositions (or: it is individuated by propositions) which are true in it In our example, we have three worlds W1={P1, P3, P4, P6}, W2={P1, P2, P4, P7}, W3={P1, P2, P3, P5}. Unfortunately, there is an apparent definition circle: possible worlds are made up from propositions and propositions are made up from possible worlds (for example, P2={W2, W3}). How can be such proposal accepted? The only philosopher I know who explained the circle away is Pavel Tichý. When explicating propositions and other intensions by means of possible worlds, possible worlds have to be treated as primitive, non-definable entities. They must be mere surrogates whose internal structure is completely omitted (relations between worlds are diminished to numerical distinctness; [7], 178); possible worlds are explicated within such system simply trivial way. 3

Possible worlds are then mere “indices” of sentence’s being truth or not. One may ask: what are they, if it is not reasonable to conceive them as collections of possible facts? From that “explication” of possible worlds, Tichý distinguished “interpretation” of possible worlds. It consists in the assignment of objects (of appropriate kinds) to intensions; any such assignment is a determination system ([7], 178). When we have, for instance, two individuals (say, A and B) and two properties (“being white” and “being black”) there are 16 determination systems. But not all of them count as possible worlds; individual A cannot be in the extensions of both properties, the extensions of the properties cannot be overlapping classes, thus some of determination systems cannot be real alternatives to our world, hence there are only 8 possible worlds (one of them should the actual one); cf. ([7], s. 178). Problems with the definition of fact. As we have seen, fact was explained as proposition. There are two deep troubles with such an explication: it conflicts our two fundamental pre-theoretical intuitions concerned with the notion fact. First, a fact typically consists in something being such-and-such; Xenakis’ being a composer consists in Xenakis’ possessing of the property being a composer. According to common view, facts are structured − they consist in several items combined certain way. Propositions are, thanks rigor definition, structure-less entities consisting merely in couples -truth-value. There is no trace of individual or property in them at all. Having a proposition one cannot know whether it is about Bush and his presidentship or about Xenakis and his musical activities. Facts are also said to be singular or general, disjunctive or conjunctive, relational or non-relational and so on. But one cannot see anything in the proposition which should cue whether certain proposition is relational or not, no relations (conjunctions, quantifiers) are inside propositions (this is why the term “flat entity” is applied to proposition). The pre-theoretical intuition concerned with fact is therefore annihilated in the intensional notion of fact. The second no less contra-intuitive consequence of the intensional notion of fact is that having infinitely many time moments (real numbers) and perhaps infinitely many possible worlds, propositions are infinite sequences of -truth-value-couples. It is generally accepted that to enumerate infinitely many items is not in human power. I.e. to identify the proposition amounts to identify the set of those couples, to enumerate them all. And this seems impossible for us, final beings. Therefore nobody can identify, know, any proposition. According to intensional notion of facts, facts are propositions − hence nobody can identify, know, any fact. This undoubtedly annihilates our basic pre-theoretical intuition that facts are knowable.

Procedural Explication of Fact and Possible World 4

Procedural conception of language. Before I expose procedural conception of fact which has not those undesirable consequences, I briefly sketch our intuitive (in fact procedural) conception of language. It is very natural to conceive language as a human tool enabling our survival in the external world, as a medium of knowledge of facts. Human intellect disposes of thoughts (in the non-psychological sense, cf. [1]) and these are, as it is common to conceive it, compound from concepts (cf. for example [4]). A thought “Xenakis is a composer”, for example, consists in the concept of Xenakis and the concept of being a composer, concepts which are combined a special unique way. We have already seen that the thought is somehow structured. It is common to think that thoughts stand between a subject and the external world, that there is such a filter of human intellectual apparatus. A subject handles with many thoughts and she wants to decide which of them are true and which are not, he wants to decide Yes or No as regards these thoughts. A human subject considers a collection of true thoughts as a true knowledge about the external world. It is natural to conceive thoughts as procedures which state the result of application of certain empirical test (for example, “being a composer”) to certain individual(s). Sentences of language are depicting thoughts, they record them. Sentential meanings should be then best modelled as thoughts, sub-sentential meanings as concepts; concept and thought should be explicated within some suitable logical apparatus. We have seen above that intensional logic is not the best apparatus at all for any structure is disowned by set-theoretical (or: functional) approach which underlies intensional logic. One should then return back to the procedural view which Tichý exposed in [6] when he considered language similar way as we suggest in this paragraph. However, for a long time Tichý thought that mutually equivalent procedures (he later called constructions) should be forgotten and replaced by the flat function-intension the procedures generate (btw. in these times he conceived intensions as finite, thus enumerable entities, [9], 187). It took more than fifteen years than Tichý himself realized that constructions should not be omitted, that they are what the natural language expressions are about, that constructions are indispensable, irreducible entities. Procedural conception of fact. Our procedural conception of fact and possible world finds its roots in the procedural conception of language. Language and fact will be explained unified way (this is the first of many other benefits). Tichý’s concept of propositional construction is the explication of Frege’s notion of thought (as Tichý himself largely argues in [7]) but, what is more, of Moore’s notion of proposition, even of the notion of proposition which is frequently used in recent philosophy, i.e. a notion of proposition which is somehow

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structured.3 Immediately, we can agree with Frege that “A fact is a thought that is true” ([1], 307), and with those who thought that “fact is a proposition (which is structured) that is true”. Briefly: fact is a propositional construction which may be true (in actual world)4 When facts are conceived as propositional constructions, there is no problem how to confirm our intuitions concerned with structuredness of facts. Constructions are structured entities, the way how the items (they consist of) are combined together is given by the definition of constructions. Propositional constructions can contain (concepts of, i.e. trivializations of) conjunction, disjunction, existential quantifier, relation, and so on, and this is why they can be called conjunctive, disjunctive, existential, relational, etc. Propositional constructions (possible facts) may contain concept of individual and concept of property what is in full conformity with our substantial conviction that facts consist in such things. Another benefit of our proposal is avoidance of the slingshot argument. Mathematical facts as well as empirical facts are propositional constructions. Despite some of them constructs the same proposition (in fact, any proposition is constructible by infinitely many equivalent constructions), they differ, they do not collapse into one Big Fact (one proposition). Tichý himself did not say anything about mathematical facts with respect the slingshot argument; only in §38 ([7]), which stands beyond his (nearly) latest treatment of facts, he wrote that intensional aboutness (that expressions are about intensional entities) should be omitted in favour of constructional aboutness what he documented by (unfortunately only) mathematical examples. On the other hand, Materna realized the danger of slingshot argument and for the case of mathematics he happily stated that he avoids the slingshot because for him mathematical facts are constructions; on the other hand, the constructional view of empirical facts seems to him fulsome (cf. [2], 22). Our conception does not share Materna’s a dualism, the dualism of explanation seems to us be an undesirable account.5 3

Well, the propositional construction does not accurately correspond to the Russellian structured thought (or singular propositions) because it is conceived as consisting of material entities, not of concepts of these entities. 4 Some conceptions of fact suggest that facts are events. We do not agree with that conception, Xenakis’ being a composer is not an event, despite it is reasonable to consider it as being a fact (due to Tichý’s theory of events, event consists, in the present case, only from conjunction of Xenakis’ being a composer with the proposition that it is, say 1/1/2006 12 p.m., i.e. events contain, in accordance with our intuitions concerning events, time specifications). 5 One may ask why “Xenakis is a composer” is the same fact as “Xenakis is a composer and ∀x∀y∀z∀n[(xn+yn=zn)→(n