Annual Meeting of the Association for Symbolic Logic

S OF PAPERS 261 II. Interpretation S is satisfaction in Tarski's sense. The values of the variables are terms extensionally interpreted to denote indifferently and ambiguously any one of a number of individuals; in special cases they are empty or denote individuals. Where aSb, any arbitrary individual (ambiguously) denoted by 'a' satisfies any arbitrary individual (ambiguously) denoted by 'b' . Where 'a' is not empty, the entities satisfied are tokens of open sentences. The interpretation of terms closely corresponds to that of Lesniewski's Ontology; in fact, the primitive relation <• of Ontology (with other object-level properties and relations) is here introduced as a defined notion. However, in Ontology it appears that names are introduced as undefined constants, predicates (proposition-formative functors) by subsequent definition. Here conversely descriptions (usually ambiguous) corresponding to predicate constants are introduced by a concretion operator. One group of such operators, analogous to abstraction operators of set theory, introduce semantic descriptions of open-sentence tokens. Syntactic properties are unspecified; only demanded is that tokens differing semantically be physically distinct. The existence of such tokens for each satisfaction condition is asserted by reflexion axioms, which in effect argue from the use of open sentences to their mention. Reflexive paradoxes appear to be avoided by exclusion of token self-satisfaction. Other contradictions are avoided only by careful distinction between distributive and collective properties of concretions; collective properties are interpreted as material mode correlates of meta-level distributive properties. The system permits construction of the natural numbers in close analogy to set theory. I t seems at least worthy of exploration as a nominalist foundation of arithmetic. (Received November u, 1964; revised February 4, 1965.) PETER B. ANDREWS. A transfinite type theory with type variables. Consideration of certain problems involved in formalizing scientific theories which use mathematics leads to an interest in logistic systems in which mathematics can be formalized but in which an axiom of infinity, and other features which might be incompatible with the scientific theories, are avoided. A system Q of transfinite type theory which satisfies these requirements is here presented. Q is a natural generalization to transfinite types of a formulation of simple type theory with A-conversion by Henkin in which the notion of equality plays a central role. Q may be very roughly described as the result of grafting a system of firstorder logic (with equality) for type symbols onto a simplified version of L'Abb6's first system of transfinite type theory with A-conversion. The presence of variables ranging over type symbols in Q gives it special power. Unlike previous formulations of transfinite type theory, Q has no special axioms asserting the existence of entities of transfinite type or specifying the cardinality of the domain of individuals. Q is proved consistent by showing that it has a sound interpretation. The independence of some of the axioms of Q is established. Basic laws of logic are developed within Q. Russell's paradox manifests itself as a theorem about type symbols. A Theorem of Infinity is proved at the transfinite level. The semantics of finite type theory are formalized in Q, and it is shown that the truth definition provided satisfies Tarski's criterion for an adequate definition of truth. (Received October ig, ig64.) F. M. SIOSON. Equational bases for Newman and Boolean algebras. In an earlier paper entitled, "A characterization of Boolean algebras and rings". Journal of the London Mathematical Society, 16 (1941) 256-272, M. H. A. Newman introduced an algebraic system (N, + , •, 0, 1) which is a direct sum of a Boolean algebra and a non-associative Boolean ring. He also essentially showed that such (Newman) algebras are equationally definable (if a unary operation — is 262 ABSTRACTS OF PAPERS admitted) and satisfy the following equations: