Fifth Meeting of the Association for Symbolic Logic

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Fifth Meeting of the Association for Symbolic Logic Nelson Goodman The Journal of Symbolic Logic, Vol. 4, No. 4. (Dec., 1939), pp. 176-177. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28193912%294%3A4%3C176%3AFMOTAF%3E2.0.CO%3B2-3 The Journal of Symbolic Logic is currently published by Association for Symbolic Logic.

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TEE JOVRN~L OF SNEOUC LOOIC Volume 4, Number 4. Deosmber 1930

FIFTH MEETMG OF THE ASSOCIATION FOR SYMBOLIC LOGIC THE ASSOCIATION FOR SYMBOLIC LOGICheld a meeting a t Harvard University, Cambridge, Massachusetts, on September 8, 1939, in conjunction with the Fifth International Congress for the Unity of Science. President H. B. Curry presided. Papers were read by Prof. Alonzo Church, Dr. Alfred Tarski (introduced by Dr. W. V. Quine), Prof. Barkley Rosser, Prof. S. C. Kleene, and Dr. Julius Kraft; and a paper by Prof. Karl Dfirr was presented by title. Brief abstracts of these papers appear below; longer abstracts of some of them were distributed as preprints a t the meeting and will be published in The journal of unilied science (Erkenntnis). Professors Rudolf Carnap and A. A. Bennett and Drs. W. V. Quine and J. C. C. McKinsey participated in the discussion following the reading of the papers. Many members of the Association offered papers in other sessions of the Congress during the week of September 3-9. NEL~ON GOODMAN ALONZO C a m c ~ . Schrdder's anticipation of the simple theory of types. Schroder's Algebra der Logik (1890) contains an explicit anticipation of the simple theory of types. Schroder's theory of types differs from the contemporary one only in that he does not distinguish the relations E and c, and accordingly does not distinguish a unit class from its single member (his theory is not in consequence inadequate, because these distinctions are actually superfluous in the presence of distinctions of type among the relations 2).Frege published in 1895 a violent but able criticism of this and related features of Schr6der's algebra; parts of this may be urged as significant also against the contemporary form of the theory of types. ALFREDTARSKI. New investigations on the completeness of deductive theories. 1. The concept of completeness and its importance for the methodology of the deductive sciences. 2. Positive results concerning completeness which are to be found in the literature: the complete deductive systems investigated by Post, Langford, Presburger, and others. The elementary logical structure and the limited mathematical content of these systems. 3. The results of the speaker: the completeness of elementary algebra and geometry; the logical structure and mathematical content of the systems of algebra and geometry investigated. (The systems in question are formalized entirely within the restricted functional calculus. The system of algebra contains the theory of addition and multiplication of real and complex numbers, and it is possible to formalize within its boundaries comprehensive parts of classical algebra. The whole of classical elementary Euclidean geometry and in addition some branches of higher geometry, e.g., the theory of conic sections, can be carried over almost intact into the system of geometry considered.) 4. The "effective" character of all positive proofs of completeness so far given-not only the problem of completeness but also the decision problem is solved in the positive sense for all the deductive systems mentioned above. 5. The far reaching negative result of Godel concerning completeness. The restricted scope remaining for future investigations in this field. BARKLEY ROSSER. The introduction of quantification into a three-valued logic. The heuristic principle used in introducing quantification into a three-va!::ed logic is to think of (z)f(z) as a logical product of the values of f(z) as z runs over all possible values. Defining (Ez)f(z) in the usual manner, as ~ ( 2 ) - f ( z ) , is consistent with the interpretation of (Ez)j(z) as the logical sum of the values of f(z). With these interpretations one can define the completeness of the restricted functional calculus for a three-valued logic, and choose an appropriate set of axioms and rules. To these axioms are added axioms for classes and descriptions analogous to the axioms for two-valued logic.

FIFTH M E E ~ GOF SOC CIA TI ON FOR SYIKBOLIC LOGIC

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Certain unexpected features of the resulting structure are indicated. Despite the presence of an analogue of the axiom of extensionality, a class is not uniquely determined by ite membership alone. Also there is no unique generalization of implication. Two dietinct types of implication are needed, and theee bear startling analogies to material and strict implication. S. C. KLEENE. On the term 'analytic' in bgical syntaz. This paper gives an alternative definition of 'analytic' for Carnap's Language 11. This definition is claimed to be equivalent to Carnap'e in his syntax language, and to make clearer the essential feature. To illustrate this feature, consider a sentence of the form SlvSl without free variables or descriptive symbols. The sentence ie true if and only if St ie true or S2 ie true. Thus the validity condition corresponding to 'v' can be stated using 'or' in the syntax language, without explaining that 'v' means 'or.' The dehition given here in detail is the one which is sketched in the author's review of Carnap, IV 82. JULIUSKRAFT. Logical or metaphysical interpretation of logistic? The two characteristic extremes which have arisen in connection with the methodological interpretation of logietic rest on a metaphysical proposition. The basis of the usual interpretations of logistic can be expressed in the form of two principles: (I) "If logistic cannot be interpreted ontologically, i t must be interpreted in a certain conventional manner," and (2) "If logistic cannot be interpreted empirically, it must be interpreted in a certain conventional manner." (1) and (2) can be comprehended under (3): "If logistic ie not capable of a certain material (inhaltlich) interpretation, then it can only be interpreted in a certain conventional manner." (3) ie the common metaphysical presupposition of the usual methodological interpretations of logistic and is itself an application of the metaphysical proposition (4) that there is only material knowledge. A formal or logical interpretation of logistic cannot be avoided, because (4) leads to contradictions. Such an interpretation reveals the continuity of logical abstractions from Aristotle to Russell and contributes to elucidation of the contemporary philosophical situation as a whole.

KARLD ~ ~ R RDie . mathematische Logik des Arnold Geutincz. Die ersten fiinf Kapitel einer Schrift des Arnold Geulincx, die 1663 erechienen iet, sind als eine mathematische Logik zu bezeichnen. Dm Schema, nach dem die Beweise in diesem Schriftstiick gefiihrt werden, iet identisch mit dem Schema, welches den Beweisen in den Elementen Euklide zu Grunde liegt. Es wird bewiesen, dai3 sich unendlich viele Aussagen, und insbesondere unendlich viele wahre Aussagen konstruieren lassen. Es werden Siitze aufgestellt und bewiesen, die ins Gebiet des Aussagenkalkiils fallen. Zu diesen Siitzen gehoren: 1) die Regeln von De Morgan; 2) der Satz, welcher beeagt, dai3 sich aus der konjunktiven Verkniipfung zweier Aussagen die disjunktive Verknfipfung dieser Aussagen ableiten Eat.