1 CONCEPTS AND OBJECTS Pavel Materna 0. Introduction Once upon a time, Pavel Tichý started a dialogue with me. He asked: “Do we define expressions, concepts, or objects?”. At that time, a bud of my present theory of concepts was already in my head; I construed definitions as linguistic means for associating concepts with some (“new”) simple expressions. Therefore I answered: “Well, for me, definitions concern expressions, so that we define expressions.” Tichý laughed. “Imagine”, he said, “that you should define the expression ‘prime number’. What would you say?” “O.K. Prime numbers are natural numbers divisible exactly by two numbers.” “Surely not; if what you define is an expression your definition would be: ‘the expression whose first letter is p, the second letter r, etc.” I explained my motivation: an essential feature of (explicit) definitions consists in assigning the definiendum with a meaning (i.e., with a concept). Tichý did not assent. He said: “Imagine such a function: its domain is the set of natural numbers, and it returns 1,2 or 3 depending on whether the given argument is divisible by at most two numbers, or by more than two and less than six numbers, or by more than six numbers, respectively. Have I defined this function?” I admitted this. Tichý: “But I have not introduced any new term for denoting this function: there is no definiendum here.” This dialogue was characteristic of Tichý’s aversion to the dominating trends connected with the s.c. ‘linguistic turn’.1 He even invented a new term: using an etymological paraphrase of psychologism (as an old and already defeated enemy of logic) he spoke about glossologism (as a rather new modish danger). His philosophy of logic and language was extremely hostile to Quinean behavioristic pseudo-semantics and late Wittgenstein’s linguistic games.2 I share Tichý’s views, resulting in his transparent intensional logic (TIL), and I already have made some attempts at applying his conception.3 The present study belongs to such attempts; its starting intuitions can be formulated as follows: (I) The (contingent, conventional) connection of an expression with an object (‘aboutness’) is realized due to the fact that expressions (of a language) encode (Gödel:”fix”)4 abstract entities which may be 1 2 called (their) meanings or concepts (of the respective objects).5 Whereas these objects are ‘flat’ in that they are structureless (at least in the sense that they do not mirror the structure of the respective expression), the meanings (concepts) are structured and their structure is mirrored by the respective expression. Moreover, two distinct languages (on approximately the same level of development) do not differ by encoding distinct meanings (concepts) but by the rules that connect expressions with these meanings. Thus meanings (concepts) are a kind of ‘ways’ (abstract ‘procedures’) which contain particular ‘intellectual steps’ 6 leading (in the best case) to the identification of the respective object. Further, concepts can identify other concepts; the latter play the role of ‘quasi-objects’. Finally, objects (and quasi-objects) which are identified by concepts are always identified a priori: objects identified by empirical concepts are intensions; they are never the contingent values of these intensions in the actual world. A paradigmatic example: the concept PLANET (encoded, e.g., by the English expression ‘planet’) identifies a property of individuals; it does not identify the set {Mercury, Venus,...}. (Maybe that this position could be called ‘neo-Fregean position’, according to Forbes in [Forbes 1987], p.5-6.) Many points in this ‘confession’ could be criticized from the very beginning, of course. The main objections would come from Quinean behaviorists7and ‘late Wittgensteinians8 ;the don’t ask for meaning, ask for use-slogan would refuse any assumption that there would be some abstract entity like meaning. Indeed, no nominalist can accept such an assumption. (On the other hand, if an explication of ‘concept’ (‘meaning’) were successful, the well-known Quine’s argument about circularity9 connected with the attempts at defining the triple analyticity-synonymy-meaning would break down.10 Now there are two ways of defending the above intuitions. One of them is ‘direct’: arguments supporting these intuitions could be formulated and allies in the history found. This way is not very effective: where the dissent concerns such basic intuitions as I have introduced in (I) no direct arguments use to be helpful (even if they point out some essential troubles connected with refusing these intuitions). A far more fruitful way consists in applying the following strategy: We start with our intuitions and show that they help us to solve many fundamental semantic problems11 and that they do not principally block solving other semantic problems. Indeed, choosing this strategy we do not intend not to use arguments. So on the one hand we do not pretend that we have proved, e.g., realism (an unattainable goal, of course), but on the other hand we will try to adduce arguments supporting, e.g., realism. The central theme of the present study is the problem of explicating the notion of concept. Yet no survey of all or even of most historically realized theories of concept can be expected: 12 this is an original conception (based on TIL), so that offering a neutral (= non-critical) outline of theories of concept is not our intention. The arrangement of our study has been inspired by the following scheme (partially derived from (I)): expression represents 2 3 ↓ (S) denotes construction → concept generates identifies (another concept or ↓ →an object: a) no object b) a “non-empirical object” (extension) a 1st order object c) an “empirical object” (intension) d) a higher order object (a function with constructions/concepts as arguments or values) So the contents can be characterized as follows: Ch.1 justifies the global structure of (S) and concerns Frege’s and Church’s semantics including Tichý’s fundamental criticism. Ch.2 defines what is called ‘first order objects’ here. The definitions are based on a simple hierarchy of types and a non-Kripkean possible-world semantics. Ch.3 is the key chapter. The notion of construction, first introduced in TIL, makes it possible to solve many problems which arose due to a basic misunderstanding, viz. to mixing up objects with the way they have been constructed.13 This notion is the most characteristic one of TIL, and the latter cannot be understood without understanding the notion of construction.14 The inductive definition of constructions is reproduced here with slight modifications.15 Constructions not only serve to constructing objects. They can be also construed as a kind of objects (this time no more first order objects). This role of constructions cannot be logically dealt with within the simple hierarchy of types. Ch.4 reproduces the definition of a version of ramified hierarchy of types, as it has been introduced in [Tichý 1988] Ch.5 defines concepts. A brief and very incomplete history of this notion is adduced with a non-standard emphasis on Bolzano’s theory of concepts. An important conclusion follows from those “historical” comments: set-theoretical construals of concepts are a blind alley in the theory. A competitive construal of concepts as structural entities16 can be based on the notion of construction. Then a comparison with some traditional suggestions can be made and some new perspectives demonstrated. Besides, the scheme (S) can be justified and explained more exactly. Ch.6 introduces a most important notion of conceptual systems: this will be needed when definitions will be treated in Ch.8. Not less important is Ch.7, where the connection between abstract concepts and conceptual systems on the one hand and languages on the other hand is investigated. This chapter shows that semantics of natural languages can be based on the notion of concept and that the futility of the sceptical objections to the theories of meaning can then be convincingly demonstrated. 3 4 The last sections of Ch.7 are a kind of preparation for Ch.8, where a possibility of building up a theory of definitions based on the present theory of concept is sketched, the expressive power of a language is defined and some comments about ‘implicit definition’, explication and recursive definition are given. Ch.9 briefly sums up the main points of the study. To ensure smoothness of the text, references are located in remarks, which are concentrated in a pseudo- chapter following Ch.9. The main idea of the book has been formulated in some articles3 and - in Czech - in a book17. Unfortunately, I had no opportunity to discuss this idea with Pavel Tichý; he died before his planned return to Czech Republic, and he only knew some global features of my conception. So I am fully responsible for any possible misinterpretation of TIL. On the other hand, I admit that among my views there can be some which Tichý would not approve. All the same, without him no new theory of concept would be created - at least not by me. 1. Church’s scheme We will probably agree that if the word concept were substituted for by the word meaning or sense in the scheme (S), our intuitions would not change (letting aside the up to now enigmatic word construction). One of the aims of the present chapter is to show that the possibility of such a substitution has already been explicitly stated and motivated by Alonzo Church1 and that this Church’s departure from Frege’s construal of concepts2 is a progressive step (which has been ignored by most later logicians and philosophers).3 To appreciate Church’s idea one has to study history from the point where the contemporary logical analysis of natural language (LANL) (or perhaps: logical semantics of natural language) has begun; I mean Frege.2 The story of the “Frege’s problem” is well-known.4 If it should be briefly recapitulated, then some new views should accompany this recapitulation.