Birkhoff Garrett and MacLane Saunders. Algebra of classes. A survey of modern algebra , revised edition, by Birkhoff Garrett and MacLane Saunders, The Macmillan Company, New York 1953, pp. 335–355.
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In developing the semantics of a particular object language, Tarski's method is to provide, in a sufficiently strong syntax language, a definition of truth (of sentences of the object language). As an alternative to this, however, Tarski's 285/6 already contains the proposal of an axiomatic theory of truth, in which a notation for truth is introduced as a primitive notation in the meta-language, together with appropriate axioms containing it. The present paper is based on the idea of an axiomatic theory of the relation of (singular) denotation, in a similar sense. The object language T is a formulation of simple type theory, like that of Tarski in 28516, pp. 365-366. The variables are x\ (where n = 1, 2, 3, . . . and k = 1, 2, 3, . . . ) , the superscript n indicating the type. To Tarski's primitives an abstraction operator * is added, together with appropriate axioms of abstraction. The meta-language contains the whole of the object language T. It contains also the syntax of T, formulated in the manner of Tarski (28516, pp. 284-303); in particular there are variables a, b, c, . . . whose values are the expressions (well-formed or not) of T; and a notation PredCon"(a) is defined which may be read as meaning that a is an abstract, x%*B. There is further a notation Den for the relation of denotation, and the four following axiom schemata: (a Den xf) (a Den x~ff) D xf = tfjj*; a Den xf D ~(a Den xjj:), where m =£ n; a Den xf Z) PredCon(a); A Den x^B, where A is a structural description of the abstract x^B (the complete statement must of course include effective instructions for recognizing A as structural description of xfaB). I t is not maintained that all the obvious elementary properties of the relation of denotation are consequences of these axioms of denotation, but only that the axioms are sufficient for the immediate objective of the paper — which is to define a notation Tr(a), "a is a true sentence," and to show that Tr(^4) = S is provable whenever 5 is a sentence and A is a structural description of S. This objective is certainly attainable, but because of a technical error it is not reached in the present paper. Namely the definition (I) on page 313 introduces in fact an infinite list of notations Tr(c), one for each type n, rather than a single notation Tr(c) as claimed. This invalidates the definition (II) on page 314. And a like objection holds against the alternative definition