“Everywhere” and “Here”.

The paper studies propositional logics in a bimodal language, in which the first modality is interpreted as the local truth, and the second as tl1e universal truth. The logic S4UC is introduced, which is finitely axiomatizable, has the f.m. p. and is determined by every connected separable metric space. This paper studies some modal logics with universal modality. A systematic research of this area of modal logic was started by papers of Bulgarian logicians in the 80s (cf. [1], [2]). Soon it became clear that the universal modality essentially increases the expressive power of a modal language. We consider the universal modality in the context of topological (neighbourhood) semantics. It is well-known [3) that in neighbourhood semantics a necessity operator D is understood as "locally true": a formula A is true at a point x iff A is true within some neighbourhood of x. In other words, a witness at x can say: "A is necessary true here" if all witnesses rather close to him can confirm that A is true. Having this interpretation in mind, it is natural to enrich the language also with the universal modality V interpreted as "everywhere". Then we get a language telling us about properties of topological spaces, which is more expressive than the language with D alone. This idea was suggested in [11); some results of the present paper were also stated there. Let us pass to formalities now. We consider propositional formulas built from a countable set P L = {p, q, . } of proposition letters, classical connectives (:>, 1.) and monadic modal operators D, V . We will use also derived connectives, classical (t\, V, -,, T, :=) and modal (<>, 3). We recall that <>A= -,0-,A, 3A = -,'t/-,A. A (normal ) bimodal logic (or merely, a logic ) is a set of formulas closed under the rules of Substitution, Modus Ponens, Dintroduction (A/DA), V -introduction (A/VA), containing all classical propositional tautologies and the formulas 1 The work on this paper was supported by the Russian Foundation for Basic Research (project No.96-0l-01378) Journal of Applied Non-Classical Logics. Volume 9no 2-3/1999, pages 369 to 379 D ow nl oa de d by [ Si m on F ra se r U ni ve rs ity ] at 2 1: 12 1 6 N ov em be r 20 14 370 Journal of Applied Non-Classical Logics. Volume 9no 2-3/1999 D(p :::> q) :::> (Dp :::> Dq), 'V(p :::> q) :::> ('Vp :J 'Vq). K2 denotes the minimal bimodal logic. The smallest logic containing a given logic L and a set of formulas r is denoted by L + f; for a formula A, L +A is an abbreviation for L + {A}. We say that the logic K 2 + r is axiomatized by the set r. In this paper the following logics will be often used: S4 * S5 = K2 + {Dp :J p, Dp :J DDp, 'Vp :J p, 'Vp :J Wp, 3'Vp :J p}, S4U = S4 * S5 + 'Vp :J Dp. As Kripke semantics for modal logics is well-known, we recall corresponding notions and terminology very briefly. A {bimodal) Kripke frame is a triple F = (W, R1 , R2), such that W # 0 ; R1, R2 ~ W x W. Elements of W are called worlds (or points ) . A Kripke model over the frame F is a pair (F, <p), in which <p: P L-+ 2w is any function (valuation ) . A topological frame (in this paper) is just a topological space. We denote the interior and the closure operation by i and c respectively. A topological model over a topological frame X is a pair (X, <p ), where <p : P L -+ 2x is an arbitrary function (valuation ) . The truth of a formula A at a point x in a (Kripke or topological) model M is denoted by M, x FA (or briefly, by x FA). Recall the inductive definition of M, x F A for the topological case: M,x FA¢> x E <p(A)( for A E PL);