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Preliminary version, All comments are welcome! FREGE S PARADISE AND THE PARADOXES Sten Lindström Umeå university, Sweden Abstract The main objective of this paper

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Preliminary version, All comments are welcome! FREGE S PARADISE AND THE PARADOXES Sten Lindström Umeå university, Sweden Abstract The main objective of this paper is to examine how theories of truth and reference that are in a broad sense Fregean in character are threatened by antinomies; in particular by the Epimenides paradox and versions of the socalled Russell-Myhill antinomy, an intensional analogue of Russell s more well-known paradox for extensions. Frege s ontology of propositions and senses has recently received renewed interest in connection with minimalist theories that take propositions (thoughts) and senses (concepts) as the primary bearers of truth and reference. In this paper, I will present a rigorous version of Frege s theory of sense and denotation and show that it leads to antinomies. I am also going to discuss ways of modifying Frege s semantical and ontological framework in order to avoid the paradoxes. In this connection, I explore the possibility of giving up the Fregean assumption of a universal domain of absolutely all objects, containing in addition to extensional objects also abstract intensional ones like propositions and singular concepts. I outline a cumulative hierarchy of Fregean propositions and senses, in analogy with Gödel s hierarchy of constructible sets. In this hierarchy, there is no domain of all objects. Instead, every domain of objects is extendible with new objects that, on pain of contradiction, cannot belong to the given domain. According to this approach, there is no domain containing absolutely all propositions or absolutely all senses. 1. Introduction Hilbert spoke of Cantor s universe of sets as a paradise for mathematicians: No one shall drive us out of the paradise that Cantor has created for us 1 a paradise that seemed to have room for all the entities that a mathematician would ever need. However, this paradise was as we all know threatened by paradoxes. Similarly, what one might call Frege s paradise Frege s intensional ontology seems large enough to accommodate many, if not all, of the abstract entities that a philosophical logician may use when interpreting our language and thought: propositions (Gedanken), senses (Sinne), functions, relations, classes, and the two truth-values, the True and the False. Frege s paradise, however, is also threatened by paradoxes. First, of course, there is Russell s paradox, that proved Frege s theory of extensions (or classes) to be inconsistent. There are also the semantic paradoxes that threaten his intensional ontology and his theory of sense (Sinn) and denotation (Bedeutung). 2 The purpose of this paper is logical rather than historical. The main objective is to investigate how the semantical paradoxes threaten theories of propositions and their constituents that 1 Hilbert (1967). 2 I follow Church (1951), Kaplan (1964), and Anderson (1980, 1987) in translating Frege s term Bedeutung by denotation rather than reference. My reason is that I want to emphasize the technical character of Frege s notion of Bedeutung. 2 are in a broad sense Fregean in character, rather than to discuss whether, or to what extent, these theories were actually held by Frege. I will, of course, also discuss ways of modifying the Fregean framework in order to avoid the paradoxes. 2. Fregean minimalism Fregean theories about propositions and senses have recently received renewed significance, in connection with deflationary approaches to the concept of truth that take abstract propositions as the primary truth bearers. The most influential theory of this kind is Paul Horwich s minimalist theory of truth (or minimalism as it is also called). 3 Horwich s minimalism has three ingredients: (i) An account of the concept of truth: Horwich claims that the word true picks out an indefinable property of propositions the content of which is exhausted by (or is implicitly defined by ) a certain theory which he calls the minimal theory of truth, or MT for short. Roughly speaking, the axioms of MT are all the propositions that are expressed by (nonparadoxical) instances of the schema: (E) The proposition that p is true iff p. (ii) An account of the utility of the truth predicate: If the truth predicate only occurred in what we may call primary contexts: (a) The proposition that snow is white is true (or its sentential counterparts: Snow is white is true), then, it could be eliminated by means of the schema (E) and would thus be redundant (at least in extensional contexts). However, we also want to use the truth predicate to say things like: (b) (c) (d) (e) The continuum hypothesis is true. There are true propositions that are not supported by the available evidence. Every sentence is such that either it or its negation is true. Most statements that Clinton made in his deposition were true. In the latter sentences, however, the truth predicate cannot be eliminated by means of the (E) schema. According to Horwich, the sole purpose of having a truth predicate at all is to be able to express claims of this latter kind. (iii) An account of the nature of truth: The property truth does not have any underlying nature and the explanatory basic facts about truth are instances of the (E) schema. Horwich s minimal theory of truth is noncommittal with respect to the nature of propositions: 3 Cf. Horwich (1998). See also Lindström (2001) for a discussion of the logical aspects of Horwich s minimalism about truth. 3 As far as the minimal theory of truth is concerned, propositions could be composed of abstract Fregean senses, or of concrete objects and properties; they could be identical to a class of sentences in some specific language, or to the meanings of sentences, or to some new and irreducible type of entity that is correlated with the meanings of certain sentences. 4 Recently, however, Christopher Hill (2002) has developed a kind of minimalism along broadly Fregean lines. Hill s discussion of truth does not take as its starting point the truth and falsity of linguistic items. Instead, he is concerned with the semantic properties of thoughts, i.e., the (alleged) propositional objects of such psychological states as beliefs, desires and intentions. Hill s goal is to explain what it means to say that a thought is true. Secondarily, he wants to explain what it means to say of the constituents of thoughts that they refer to or are satisfied by things. The constituents of thoughts he refers to as concepts. Hill makes the following fundamental assumptions about thoughts: (i) (ii) (iii) Thoughts have logical structures. Thoughts have concepts as their constituents. Thoughts are individuated by their logical structures and their constituent structures. Thoughts are themselves a particular kind of concepts. Hence, thoughts can have other thoughts as constituents. I take these assumptions to imply the following principles: (C) (MD) The principle of compositionality for thoughts. Thoughts that are built up in the same way from the same concepts are identical (assumptions (i) together with (ii)) The principle of maximum distinction for thoughts: Two thoughts are identical only if they are built up in the same way from the same concepts. (Assumption (iii)). Hill proposes a theory of truth that he calls the simple substitutional theory of truth (or simple substitutionalism, for short). The theory consists of one single axiom: (S) x(true(x) (Sp)((x = the thought that p) Ÿ p)), where, the universal quantifier x ranges over the domain of absolutely all objects, and (Sp) is an unusual technical device, namely a substitutional propositional quantifier, having the class of absolutely all propositions as its substitution class. Hill s main idea is that the sentence (S), or perhaps more accurately the thought that it expresses, should serve as as an explicit definition of a minimalist concept of truth. There are, however, several problems with this idea. First, there is the difficulty of explaining the meaning of the substitutional quantifier (Sp) without making use of the concept of truth, the very concept that we want to define. Hill tries to solve this problem by a deductive approach, 4 Horwich (1998), p. 17. 4 using natural deduction style rules of inference, instead of truth-clauses, when explaining the meaning of the substitutional quantifiers. It is doubtful, however, whether it is possible to obtain a proper understanding of Hill s rules of inference, without being told that they preserve truth. Hence, it is not obvious that Hill has succeded in explaining his substitutional quantifiers in a way that does not presuppose the notion of truth. Perhaps, it is preferable to view sentences of the form (Sp)(...p...) as abbreviations of infinite disjunctions. That is, (S) is viewed as an abbreviation of: (S ) x[true(x) (x = the thought that snow is white) Ÿ snow is white) (x = the thought that grass is green) Ÿ grass is green) (x = the thought that = 5) Ÿ = 5)...)], where the infinite disjunction is supposed to contain one disjunct for every proposition. However, (S ) can be an adequate definition of truth, only if the language that we are employing is universal in the sense of containing for every thought (proposition) a sentence that expresses that thought. Given a language L that is universal in this sense, we can define truth in an extension L of L in the following way: (S ) x[true(x) j Œ L (x = the thought that j) Ÿ j)]. From (S ), we can infer every instance in L of the schema: (E) True(the thought that j) j. However, the kind of minimal theory of truth considered by Hill, as well as the modified version considered here, is threatened by paradoxes. Here, I am only going to consider a version of the Epimenides paradox. We make the intuitively coherent assumption that Epimenides entertains one and only one thought, namely the following one: (L) Ÿ j Œ L (E( j ) Æ ÿj), where Ÿ is the sign for infinite conjunction, E abbreviates Epimenides entertains, and j abbreviates the thought that j. Intuitively, this is a Liar sentence saying that no proposition that Epimenides entertains is true. Due to the postulated universality of the language L, there is a sentence l of L that expresses the thought L. That is: (1) l = Ÿ j Œ L (E( j ) Æ ÿj) , We assume the following principle for sentences of L : (2) j = y Æ (j y). From (1) and (2) we get: (3) l Ÿ j Œ L (E( j ) Æ ÿj) 5 But, we also have: (4) Ÿ j Œ L (E( j ) ( j = l )), i.e., L is the one and only thought that Epimenides entertains. Now, we can reason as follows: (5) l assumption (6) Ÿ j Œ L (E( j ) Æ ÿj). from (3) and (5). (7) E( l ) from (4) (8) ÿl from (6) and (7) (9) ÿl (5) - (8) by ÿ-introduction (10) j Œ L (E( j ) Ÿ j) from (3) and (9) (11) l from (4) and (10) (12) ^ One reasonable conclusion from this argument seems to be that there cannot be a universal language in which all thoughts are expressible. In other words, for any language L, there are thoughts that are not expressible in L. Hill s original and interesting account of truth and reference is a version of what I would like to call Fregean minimalism, a deflationary approach that takes Fregean propositions ( thoughts ) and their constituents ( senses ) as the primary bearers of truth and reference. Fregean minimalism is an attractive approach to truth. It is threatened, however, by the kind of logical difficulties ( paradoxes ) that I am going to discuss in this paper. Unless we can come to terms with these difficulties, there is little hope of developing a satisfactory theory of truth and reference along Fregean lines. 3. The Fregean ontology and the theory of sense and denotation Let us briefly recapitulate Frege s theory of sense and denotation and its accompanying ontology The distinction between objects and functions. Frege makes a fundamental ontological distinction between objects and functions. We may think of objects and functions as constituting two separate mutually exclusive domains. The basic notion when distinguishing functions from objects is functional application. A function f is the kind of entity that can be applied to one or several entities a 1,..., a n (within its domain or range of definition) to yield an object f(a 1,..., a n ), called the value of f for the arguments a 1,..., a n. An object, on the other hand, cannot be applied to anything. Frege indicates this distinction by saying that functions are incomplete (or, unsaturated), while objects are complete (or, saturated). Functions are divided into levels, roughly, as follows. First-level functions only take objects as arguments. 6 Second-level functions take first-level functions as arguments, and so on. 5 The result of applying a function to something is for Frege always an object. Hence, there is in Frege s ontology no room for functions that yield functions as values. In addition to functions, we may expect Frege s ontology also to contain properties and relations. Intuitively, we speak of items as having properties and standing in relations to each other. Frege, however, identifies properties and relations with functions of a special kind that he calls concepts. Among the objects, there are the two truth-values the True (t) and the False (f). These are abstract logical objects. A Fregean concept F is a function, which for any argument (or sequence of arguments) within its domain, yields one of the truth-values, t or f, as value. So, instead of saying that an item a has the property F, Frege says that F(a) = t. And instead of saying that, a 1,..., a n (in that order) stand in the relation F to each other, Frege says that F(a 1,..., a n ) = t. Intuitively, F(a 1,..., a n ) = f means that F is defined for the arguments a 1,..., a n, although a 1,..., a n do not stand in the relation F to each other. In the following, we shall use the term attribute with the same meaning as Frege s concept. 6 Thus, an n-ary attribute is an n-ary function taking a sequence of n items as arguments and yielding one of the truth-values t or f as values. Usually, we are only going to considering first-level attributes, i.e., attributes taking objects as arguments and yielding truth-values as values. A property is a unary attribute and, for n 2, an n-ary relation is an n-ary attribute. If f is an n-ary (first-level) attribute, and a 1..., a n are objects, then f(a 1,..., a n ) is either t (the True) or f (the False) Sense and denotation. Any well-formed expression E of a logically proper language has a denotation (Bedeutung), den( E ), and a sense (Sinn), sense( E ). The sense of a well-formed expression E is said to be the mode of presentation of its denotation. We shall say that an expression denotes its denotation and expresses its sense. den( E ) is the object designated by (or presented by) sense( E ). Singular terms ( names ) and general terms ( predicates ) have, respectively, objects and attributes (i.e., Frege s concepts ) as their denotations. A sentence has as its denotation one of the truth-values, t or f. With the possible exception of so-called oblique contexts, the denotation of a complex expression is functionally determined by the denotations of its parts. For instance, if Pt 1...t n is a sentence, where P is an n-ary predicate, and t 1,..., t n are singular terms, then 5 To be more exact, there is a type hierarchy of items of different types. First, there is the type i of all objects; Then, for any n 1 and types a 1,..., a n, there is a type [a 1,..., a n ] of all functions f taking n arguments a 1,..., a n from respective types a 1,..., a n and yielding an object as value. We can assign levels to types as follows. The type i has level 0. If the maximal level among the types a 1,..., a n is n, then the type [a 1,..., a n ] has level n+1. Items of a type of level n are called items of level n. Hence, objects are entities of level 0. Functions from objects to objects, are first-level functions, etc. 6 By adopting this terminology, I can follow Alonzo Church in using the term concept for those items that can serve as appropriate senses of linguistic expressions. 7 (i) den( P ) is an n-ary relation, den( t 1 ),..., den( t n ) are objects, and (ii) den( Pt 1...t n ) = den( P )(den( t 1 ),..., den( t n )); which is either t or f. We shall assume that the principle of compositionality also applies to senses, i.e., that the sense of a complex expression is a function of the senses of its constituents and the way in which it is built up from these constituents. 7 Following Church (1951), and deviating from Frege, we speak of the entities that can serve as appropriate senses of expressions as concepts. Thus, a concept is anything that is capable of being the Fregean sense of a linguistic expression. For any concept x, x is a concept of y iff x is capable of serving as the sense of an expression denoting y. We also say that x designates y if x is a concept of y. We follow Church in using the symbol D for the concept relation, which holds between a concept and the entity that it is a concept of. Every concept is a concept of at most one entity, i.e., If D(x, y) and D(x, z), then y = z. The denotation of an expression A is uniquely determined by its sense and by the concept relation in the following way: x = den(a) iff for some y, y = sense(a) and D(y, x). In other words, the denotation relation is the relative product of the relation between an expression and its sense and the concept relation. A meaningful expression A will lack a denotation, if sense(a) is an empty concept, i.e., if there is no y such that D(sense(A), y). There are various kinds of concepts corresponding to the different categories of entities in the Fregean ontology and to the different kinds of meaningful expressions in a logically wellconstructed language. A singular concept is a concept that can serve as an appropriate sense of a singular term in some (actual or merely possible) language, i.e., a singular concept is a concept of an object. Function concepts, property concepts, and relation concepts are appropriate senses of function terms, concept terms, and relation terms, respectively. A (Fregean) proposition is the appropriate sense of a (non-indexical) declarative sentence, i.e., it is a concept of a truth-value. A proposition P is true if it is a concept of the truth-value t; and it is false if it is a concept of the truth-value f. We also sometimes speak of Fregean propositions as thoughts (Gedanken). Thoughts, or propositions, are abstract objects that are true or false and can serve as the contents of propositional attitudes, like belief, desires, and intentions. 7 Although clearly part of a Fregean perspective on language, the principle of compositionality was never explicitly formulated by Frege. It appears that the first attribution of the principle to Frege is in Carnap (1947, p. 121), where Carnap formulates versions of the principle of compositionality for denotations as well as for senses. See also Church (1956), pp. 8, 9 and Kaplan (1964) for discussions of compositionality within the context of Fregean semantics. Whether, or in what form, Frege himself was committed to the principle is a difficult and much debated historical question (cf. Pelletier (2001), Janssen (2001)). 8 If we allow for meaningful sentences that lack truth-values, then the senses of such sentences must be propositions (thoughts) that are neither true nor false. Such propositions, that we might call empty propositions, would so to speak aspire to a truth-value without actually having one. Admitting empty propositions and empty concepts could, of course, be useful in the treatment of the Liar paradox and other semantic paradoxes. The question whether a Fregean theory of sense and denotation can allow for propositions and other senses being empty, is a controversial one. However, at least on some interpretations of the sense-denotation distinction, it makes good sense to speak of senses that do not determine any denotat

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