Consistent, independent, and distinct propositions.
Anjan Shukla · Notre Dame Journal of Formal Logic · 1972
THE PROBLEM STATED.This is a sequel to [10] and acquaintance with it is presupposed.It was shown there that the existence postulate can be proved in certain non-regular systems.It followed, loosely speaking, that in those systems we had two consistent and independent and four distinct propositions.We now propose to construct a system in which there are denumerably many consistent, independent, and distinct propositions.Note that this is a propositional system.We, therefore, do not yet enter into the controversies surrounding quantified modal logic.At a future date we intend to add quantifiers to our system and we shall see what happens then.But we do claim that Lewis would have found our propositional system highly satisfactory.We now give precise definitions of some of the terms that will be used.The notation employed is that of [8]. DEFINITIONS.We first propose to define a proposition.We wish to say, roughly, that a wff B is a proposition if and only if every substitutioninstance (SI) of B is equivalent to B. But it is necessary to proceed with caution.In the definitions that follow P is a propositional calculus which has, among its rules, the rule of substitution on variables.For the first two definitions P can be thought of as a propositional calculus in a very wide sense; the next two assume that it has, among its connectives ~ and Λ; the remaining ones have, in addition, the connective O.These three connectives may be primitive or defined.Small letters stand for variables and capital letters denote formulas.Definition 1.Let λ be a connective (primitive or defined) of P such that tppλp; and further, the following rule (primitive or derived) is available in P: "If B results from A by substitution of N(M) for M(N) at one or more places in A (not necessarily for all occurrences of M(N) in A), if hpMλN and fpA, then tψB [1, p. 101; with a slight variation]/' Then λ is said to be an ^-connective of P.Comments.In the systems T, S4, and S5, both substitutivity of strict equivalents (SSE) and substitutivity of material equivalents (SME) are