Meeting of the Association for Symbolic Logic, New York 1969

Abstract

S OF PAPERS 349 language) discourse, it then becomes possible to collapse the time-honored distinction between (I) and (11), at least in so far as many subsets of (I1-A) are concerned. That is to say, various examples of (II-A) may be interpreted as instances of (1). Specifically, (11-A-1) appeals to ignorance become reducible to the modal logical fallacies: (II-A-I-i) D P/.. W'P, and (II-A-I-ii)-DP/.. -OP. Also, (II-A-2) complex questions may be explained in terms of one of DeMorgan's theorems: [(P& Q)v (P& Q)] & -(P& Q)/.. P& -Q. Similarly, (1I-A-3) post hocC etgo propter hoc arguments may be taken to be affirmations of the consequent: (PQ) & Q./.. P. Regarding other fallacies, only conjectures can be offered at this time. Finally, the formalistreductionist thesis of this paper is presented simply as a sufficient means by which to account for the assumed invalidity in (I-A-i, 2, 3); not as a ilecessary condition thereof. CHARLES WISEMAN. The theory of Inotal groups. There is no formalized theory that (a) relates the concepts of negation and contrariety, and (b) permits the application of these and related concepts to terms, formulas or constants in a language, and to senses and denotations assigned to such expressions. The theory of Inotal groups is a syntactically formulated axiomatic theory that accomplishes both (a) and (b). Let L be a language that contains the standard logical constants -, A, V, --, +y, A, V, for the concepts of negation, conjunction, disjunction, material implication, material equivalence, universal quantification and existential quantification (resp.). Assume that there is a formal system with a standard set of axioms and inference rules to govern these constants. Add to L three 1-place sentential connective logical constants I (affirmation), (D (duality), and 0 (contrariety). Add to the formal system for L the following seven modal group axioms: (Al) FI ?+F F"x F, (A2) F? F?, (A3) F F?0, (A4) Fx?+FxI, (A5) FXY + FYX, (A6) F(Xy)zFx(yz) (A7) Fe-* F-, where x, y, z = 1, 0, 0, Fx, FXY, FVyZ abbreviate xF, yxF, zyxF. Note that (A7) expresses the fact that contrariety entails negation. E.g. if a sentence is contravalid, i.e., its negation is valid, then it is not valid; if John is unlucky, then John is not lucky. Given these axioms, the most important modal group concepts for formulas may be defined. E.g., MF is a modal group generated by F iff MF = (FI, F0, F0, F-); IF isanl incotisistencygrouipgenerated by Fiff IF = (Fl A FI),Fl v F6, FA F?, F v F?). The components of (FI, F-), (F?, F0) are contradictories, (Fl, F0) are contraries, (F?&, F-) are subcontraries. F1 A F0, Fl v F0, Fl v Fis the inconsistency, decitability, tautology formula for F (resp.). Modal group concepts may be defined, in general, for formulas, terms and complex term or complex formula forming constants, if such occur in L. Examples of modal and inconsistency groups may be found in sentential, predicate, alethic and epistemic logic as well as in theories of preference, action, predication, determinacy, sets, logical syntax, and logical semantics. M. J. CRESSWELL. The completeness of SI an(I SOIme related systems. Lewis' modal system SI dates from 1932 but has resisted algebraic and semantical study. We This content downloaded from 157.55.39.255 on Mon, 01 Aug 2016 06:00:42 UTC All use subject to http://about.jstor.org/terms 350 ABSTRACTS OF PAPERS define SI-algebras and obtain a semantics in the style of Kripke. (But we use the terminology of [1].) An SI-algebra is an ordered quadruple where is a Boolean algebra, K is closed with respect to the monadic operation *, and the following conditions are satisfied (for a, b e K): 1. a ' * a. 2. (*a + *b + *0) = (*(a + b) + *0). 3. *0 (* a + *b) provided a x b = 0. 4. If * a C * 0 then a = 0. Where is regarded as a matrix with the set D of designated elements defined as {a E K: * 0 C a}, SI-algebras can be proved characteristic for SI and by an adaptation of McKinsey's method SI can be shown to have the finite model property. Semantic models for SI can be defined using 'possible worlds' but they appear to have no intuitive significance. Relaxing or dropping some of the conditions (particularly I or 3) would seem to provide semantics for various other systems in the neighbourhood of SI. Strengthening 3 to * 0 ' * a reduces 2 to * a + * b = * (a + b) and * 0 = 0 also makes 4 redundant.