On a nonthesis of classical modal logic.
Robert W. Murungi · Notre Dame Journal of Formal Logic · 1974
Classical modal systems subjoined to the full classical statement calculus accept as a thesis the conditional (1) CLpp.Such systems, however, reject as a thesis(2) CpLp.The common argument given in the literature for rejecting (2) is that, in any modal system in which (1) is a thesis, accepting (2) would yield(3) ELpp and thus reduce such a modal system into the so-called "The Trivial System"-that is, into an assertoric one with the redundant notation L.Let us look more closely into the argument for accepting (1) and rejecting (2).To block derivation of (3) from ( 1) and (2) one must reject either (1) or (2) though one needs not reject both.But it does not follow from this that one must accept or even reject (1) or that one must accept or reject (2).Nor does it follow that one must accept at least one of (1) or (2).Therefore, the argument for blocking (3) cannot be the argument for accepting (1) and rejecting (2).Now suppose one accepts (2) rather than (1).Under this supposition, I wish to establish a contention of Prior's in [2], p. 199, that the resultant statement modal system is formally consistent-that is, consistent without reference to any intended interpretation of modal functors.Let M'(T') be a modal system which is like Feys-von-Wright M(T) except that (2) in M'(T') replaces (1), the Axiom of Necessity, in M(T) and that the Rule of Necessitation, (Rn) a/La which is a primitive rule of inference in M(T) becomes a derived rule in M'(T'), as Cresswell has shown in [1], p. 31.I prove that for no a of M'(T') are both a and Na theses-that is, M f (T f ) is (simply) consistent.To