Modal logic with functorial variables and a contingent constant.
C. A. Meredith, Arthur N. Prior · Notre Dame Journal of Formal Logic · 1965
The World as a Propositional Constant 1.The present section is by Prior; the two which follow it, by Meredith, Meredith's sections were originally produced in 1953 and circulated among colleagues; subsequent references to them in the literature, e.g. in [3], [4], [5], [6] and [7], may be clarified if these two notes are now made more widely available.They were provoked by Lukasiewicz's development of the modal system which he presented in [2], The importance of this system as a limiting case has been made clear by Smiίey in [9]; a number of commentators have noted its intuitive peculiarities.Meredith was interested in it as a first attempt to incorporate functorial variables in a modal system, and sought in the system (C, Γ, 0, δ, p) below to incorporate the same feature in a more normal type of modal logic, namely Lewis's S5.This system is equivalent to S5 supplemented by the qualified law of extensionality CTΈpqCδpδq (Meredith takes over Lukasiewicz's symbol Γ for necessity and Δ for possibility).2. The system (C,Γ,0,n,δ,p) introduces the more original feature of a constant n to represent "the world" in the Wittgensteinian sense of "everything that is the case."Its most distinctive feature is the law CpTCnp, "What is true is necessarily implied by the totality of what is the case"necessarily because this totality is equivalent to a conjunction of which all true propositions are conjuncts, and we have TCKpqp.In a sense, of course, unless all truths are necessary, the totality of what is the case might not have contained (and so implied) the given truth p; but in the symbol n, "the totality of what is the case" is not given by this description of it but given simply as the actual totality of what is the case.Meredith's proof, in his second item, of the independence of this fundamental law CpTCnp, is instructive.To distinguish a contingent truth from a necessary one we need two possible worlds, a contingent truth holding in one of them only and a necessary truth in both.Two such worlds generate four truth-values, "truth in both", "true in the actual world but not in the other", "true in the other world but not in the actual one", "true in