フレーゲにおける論理哲学の形成 : 意味論の視点から

Abstract

The purpose of the present essay is to give an outline of Frege's logicosemantic theory as seen in his three main works: Begriffsschrift (BS), Die Grundlagen der Arithmetik (GLA) and Grundgesetze der Arithmetik (GGA). After I mention Frege's main contributions to modern logic in BS, I argue especially (1) two necessary conditions for the identity of conceptual contents, (i) the possible consequences from judgements, (ii) the principle of the indiscernibility of identicals, and (2) the paradox of content-identity. By resorting to the mention and use of a symbol, Frege tries to explain the difference in cognitive significance between “a=a” and “a=b” only epistemologically in BS., namely, that a separate name for the same content corresponds to each of the different modes of determining the content. In GLA Frege criticizes the psychologism and formalism of mathematics and firstly exposes the logistic approach of arithmetic. In such an inquiry Frege distinguishes a proper name, which stands for an object, from a concept word, which signifies a concept. A definite description must satisfy the uniquely existential condition and a concept word should satisfy the sharp boundary of its application. Frege introduces the orders among concepts and distinguishes properties from characteristics. Correspondingly the primitive types of Fregean possible states of affairs could be distinguished: (a) an object's falling under a concept, and (b) a concept's falling under another higher order concept. I argue in detail the so-called context-principle. The negative effect of it is to prevent a psychological approach to the meaning of a word. But the main problem is just to explicate the sense of a proposition in which a number word occurs. The context of a proposition in question is nothing but a recognition-statement or a numerical identity and so the point is to ask for the meaning of a numerical word via the explication of the sense of a numerical identity. Frege does not regard a contextual definition of a numerical identity with an equivalence-statement as satisfactory, because it is circular. He explicitly defines the number belonging to a concept by resorting to the extension of a concept and to the logical definition of “equinumeracy”, though the former is only assumed to be known. Thus the context-principle in GLA is closely connected with Frege's logistic approach to arithmetic. In his conceptual notation of GGA Frege divides all well-formed expressions into the two syntactic categories: (i) names, letters, marks and (ii) conceptual notation-propositions, the presentation of a judgement in the conceptual notation. Marks are expressions in which letters (free variables) occur. Names are classified into proper names and function-names, whose levels are distinguished. Correctly-formed expressions are formed from the seven primitive constant nction-funames by definition, subject to some formation rules which I formulate explicitly. The leading principle of Frege's semantics in GGA is that correctly-for-med names and so primitive names must always denote something. Frege presents six general denoting-conditions for the function-names and for proper names. He gives his unique so-called chosen object theory of description distinct from that of Russell's. Now the most fundamental presupposition of Frege's theory of denotation is the fact that a name of a truth-value denotes a truth-value.

Further every correctly-formed name in GGA must express a sense. The sense of the name of a truth-value is the thought that the conditions, under which the name denotes the True, are fulfilled, that is, the fulfillment of a truth-condition. The truth-conditions of primitive sentences are as follows: (T1) A primitive sentence 'φ(Γ) 'denotes the truth-value, the True, if and only if an object Γ falls under the concept φ(ξ). (T2) A primitive sentence 'Mβ(φ(β)) ' denotes the True if and only if a first level concept φ(ξ) falls under the concept of the second level Mβ(ҩ(β)). (T1) and (T2) show are markable similarity with Tarski's convention (T). Component names of a name of a truth-value contribute to the expression of the thought, and this contribution of the component is its sense, which is part of the thought. One could hear an echo of the context-principle concerning sense, that is, to ask for the sense of a component name of a truth-value name within the thought which the truth-value name expresses. If so, the denoting-conditions of names in GGA could be taken as their senses. One could also hear an echo of the context-principle concerning denotation, namely, what a component name denotes depends on what a specific truth-value name in which it occurs denotes. After Frege's clear distinction of sense and denotation in GGA one might still insist on the original context-principle in GLA concerning the dependency of denotation on sense, because the denotations of component names are asked for through the sense or thought of a truth-value name in which they occur.