Some Post-complete extensions of ${\rm S}2$ and ${\rm S}3$.

Abstract

We shall take Λf, v, and Ί as primitive connectives.Let £ be the set of all wffs with these connectives.If α, βe «£, we shall write a -3 β for ΊMΊ(iαvβ), and α = β for i[τ(α -3 β) v l(β -3α)].We let f and t denote the wffs p Λ Ίp and Ί pvp, respectively.If ae £, we denote by -C[α] the smallest subset of -C containing a and closed under the connectives M, v, and Ί.A modal logic L is a proper subset of -C which is closed under the rules of uniform substitution and modus ponens, and contains all tautologies.If L L and L 2 are modal logics, then L λ is anhas no proper extensions.Let p(L) be the number of Post-complete extensions of a modal logic L. Several papers have considered the problem of evaluating p(L), for various modal logics L [1, 2, 3].It has long been known that p(S2) ^ tt o Segerberg claims in [3] to prove that p(S3) = 2 X °: his proof is incorrect, but it may easily be modified to show that p(S2) = 2*° and that p(S3)^*V Whether or not p(S3) = tf 0 remains an open question, to which this author believes the answer is probably affirmative.Most of the work on Post-complete systems uses the classical results of Lindenbaum and Tarski [4], and is therefore highly non-constructive.In fact, the only explicitly described Post-complete extensions of S3 in the literature known to the author are the systems S9 of [5] and F and Tr of [3], This paper applies a variant of a theorem of Belnap and McCall [6] to construct some Post-complete extensions of the Lewis systems S2 and S3.Let 3W•= (B, D, *) be any matrix for a modal logic, where ΰisa Boolean algebra, D a set of distinguished elements, and * interprets the possibility operator.Each element αe-C[f] determines an element V m (a) of B, when interpreted in Wl in the usual way.Definition The matrix 3W is a functionally complete matrix (FCM) if: (i) for any xe B, there is an a e X[f ] such that V m (a) = x.and (ii) for every xe B, either xe D or -xe D.

Locations

• Notre Dame Journal of Formal Logic - View - PDF