Every world can see a reflexive world

Abstract

Let ℭ be the class of frames satisfying the condition

$$\forall x\exists y(Ry \wedge yRy)$$

(“every world can see a reflexive world”).

LetKMT be the system obtained by adding to the minimal normal modal systemK the axiom

$$M((Lp_1 \supset p_1 ) \wedge ... \wedge (Lp_n \supset p_n ))$$

for eachn ⩾ 1.

The main results proved are: (1)KMT is characterized by ℭ. (2)KMT has the finite model property. (3) There are frames forKMT which are not in ℭ, but allfinite frames forKMT are in ℭ. (4)KMT is decidable. (5)KMT is not finitely axiomatizable. (6) The class of all frames forKMT is not definable by any formula of first-order logic. (7) 〈W, R〉 is a frame forKMT iff for everyx εW, the worlds thatx can see form a sub-frame which is not finitely colourable.