FRAGMENTS OF MODAL AND INTUITIONISTIC PROPOSITIONAL CALCULI

RALPH HENRY MOON, Purdue University

Abstract

Kripke-style semantics are used to obtain deductive and semantic characterizations of various fragments of the extensions of the modal system, S4, and of the intuitionistic propositional calculus, H. -fragments and -fragments, that is, strict implicational and strict equivalential fragments, are isolated from the following extensions of S4: S5, each of the systems in the S4.3.n family, K3, Z6, S4.3 and each of the subsystems in K2. As well, the following systems are shown to share the same strict implicational and strict equivalential fragments: S4.3, Z6 and K3; Fin(K1.1) and Fin(K2.1); Fin(Z3) and Fin(Z5); Fin(S4.1) and Fin(S4.2.1); and, all of the subsystems of K2. {<(GAMMA), (alpha)> (epsilon) Fin(L) if and only if (GAMMA) is finite and (alpha) is deducible from (GAMMA) in L.} The ( ,N)-fragments of the following systems are obtained: S4, S4.2, S4.3, each member of the S4.3.n family, S4.4, S5, each member of the S4A1tn family, each member of the S5A1tn family, Fin(Z1), Fin(Z6), Fin(Z8), Z9, Fin(K1), K2, K3, K3.2 and K4. The {L,C}-fragments of the systems covered by the first six items in the preceding list are also isolated, as well as the {L,C}-fragments of the following: S4.04, S4.1.2, Z2, Z8, Z9, Fin(K1.1), K1.2, Fin(K3.1), K4, V1 and V2. Further results come from these by way of a proof that the systems which share the same strict implicational and strict equivalential fragments also share the same {L,C}-fragments. The implicational and equivalential fragments of H, Dummett's LC and its extensions are isolated. It is also shown that H and the system KC have the same implicational and equivalential fragments. A proof is then given that no axiomatic system having equivalential detachment ("from (alpha) and (alpha)(TBOND)(beta) to infer (beta)") as its sole rule of inference can axiomatize the equivalent fragment of H, LC or any of the extensions of LC, except classical propositional logic.

Degree

Ph.D.

Subject Area

Philosophy

COinS