Twenty-Fifth Annual Meeting of the Association for Symbolic Logic

Abstract

S OF PAPERS 38 of the m-adic notations for x and y is the m-adic notation for z. Replacing addition and multiplication by the single relation of m-adic concatenation in the above description of CA, we have (Smullyan, ibid.) a description of the m-rudimentary relations. It is shown that these are merely alternative descriptions of the same class of relations; i.e., for each m, the class of m-rudimentary relations is precisely CA. The ternary relation of exponentiation is in CA. In fact, the quaternary relation SP(n, x, y) = z is in CA where SP is the doubly but not primitive recursive function defined by SP(O,x,y) = x + y; SP(w + 1, x, O) = x; and SP(w + 1, x,y + 1) = SP(n,x, SP. (n + 1, x, y)). However, CA is a small subclass of the primitive recursive relations in the sense that there is a fixed positive integer K such that the characteristic function of every relation in CA is definable from the successor function by at most K uses of primitive recursion. Progress in relating CA to other small classes of relations defined in the literature is also described. (Received November 28, 1961.) R. B. ANGELL. A logical notation with two primitive signs. This paper presents two systems of logical notation, each using just two typographic signs or shapes, namely, the leftand right-hand parentheses. The first system is easily shown adequate for Quine's Mathematical Logic (1940, 1958). It contains: I. Primitive signs:) (. II. Symbols (or well-formed signs): 1. '()' is a symbol. 2. If S and S' are symbols, rSS'1 is a symbol. 3. If S is a symbol, r(S)p is a symbol. A. Atomic symbols: 1. '(()' is an atomic symbol. 2. If r(S)1 is an atomic symbol, F(SO()1 is an atomic symbol. (Variables are atomic symbols containing 2n (n >1) symbols '('.) B. Well-formed formulae: 1. If a and fi are variables, r((())(zfi))1 is a wff (in this case alone, an atomic formula). 2. If 0 is a wff, and a is a variable, r((a)I))l is a wff. 3. If 1D and T are wffs, then F((D) (T))1 is a wff. It is shown that Quine's variables, atomic formulae, quantified expressions, and stroke applications can be correlated unambiguously with expressions resulting from formation rules 3A, 3BI, 3B2 and 3B3 respectively. The class of wffs defined above are syntactically equivalent to Quine's class of logical formulae (cf. Quine, p. 124) and are thus adequate for all his definitions and statements. The second system is syntactically simpler, eliminating a special notation for quantifiers in favor of a single class constant. Formation rules are the same except that X, 1-3 are replaced by B'1. If a is a variable and f is either a variable or '((0()', then r((())-, is a wff. B'2. If T is a wff, F(o)l is a wff. B'3. If 4 and T are wffs, then r(QD*)1 is a wff. Rules B'2 and B'3 are associated with denial and conjunction respectively. By the following definitions membership is introduced, as well as a class constant 'E' which eliminates the need of quantifier notation: Dl. ot e fi1 for F((0)(,B)) . D2. FE0l for rx e En or r((()(,(( ())1l. An interpretation is provided such that every standard quantified statement is equivalent on the usual interpretation to a statement in this notation, and every statement in this notation is either equivalent to a standard quantified statement in its usual interpretation or else an unobjectionable addition. (The interpretation introduced requires a new interpretation of expressions usually called "propositional functions"). The paper does not attempt to provide a set of axioms for this system, as it establishes only syntactical equivalence of symbolisms. (Received April 23, x962.) G. KREISEL and W. W. TAIT. Induction and recursion. Let R be primitive recursive arithmetic (PRA) with the rule A (0), t (a') -< a', A (t(a)) ->A (a)

Cite this paper

@article{Nelson1960TwentyFifthAM, title={Twenty-Fifth Annual Meeting of the Association for Symbolic Logic}, author={David Nelson}, journal={J. Symb. Log.}, year={1960}, volume={28}, pages={297-308} }