Connexive Implication, Modal Logic and Subjunctive Conditionals.

McCall’s article on “connexive implication” [4] credits Meredith with the suggestion that the matrices for →, ·, and ∼ in connexive implication can be replaced by the matrices for · and ∼ supplemented by a matrix for the unary necessity operator, 2. Since the matrices for McCall’s connexive implication are those of this writer’s subjunctive conditional [1], it follows that the latter, also, may be reduced to a modal logic. In this paper two axiom sets for a modal logic based on Meredith’s observa- tion are constructed. Both are shown consistent, and they are shown complete with respect to (1) this writer’s system, PA1, and (2) McCall’s system CC1, respectively. Certain differences between these two systems are pointed out, together with observations concerning Post-completeness and functional com- pleteness in the latter. Finally, a brief discussion is presented concerning some philosophical implications of finding connexive implication, or subjunctive con- ditionals, reducible to a modal logic. Professor Storrs McCall and I share an interest in logical systems which contain the non-classical theorems: 1. ∼(p → ∼p)—It is false that if p then not-p, 2. (p → q) → ∼(p → ∼q)—If (if p then q) then it is false that (if p then not-q); although our motives differ. I called my system PA1 (in [1]) a logic of subjunctive conditionals; he called his system, CC1 (in [4]) a system of “connexive implication” and allied himself with those who, according to Sextus Empiricus, “say that a conditional is sound when the contradictory of its consequent is incompatible with its This paper was supported by NSF Grants GS 630 and GS 1010.

Authors

• Richard Bradshaw Angell

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