Some Logics Related to von Wright's Logic of Place

Abstract

In this paper we study some logics related to the logic of place introduced by von Wright and studied by Segerberg. For every $n \geq 1$ we study the logic of the class of frames whose accessibility relation R satisfies the following condition: if $x \neq y$ then there is $j \leq n$ such that $xR^j y$. For a fixed $n \geq 1$ the logic is the one axiomatized by K $ + [n] \phi \rightarrow [n+1]\phi+ \phi \rightarrow [n]\langle n\rangle\phi$, which we call Kn.4B, where $[n]\phi \hbox{ is } \phi \wedge \square\phi \wedge \dots\wedge \square^n\phi $. We prove that these logics are canonical and hence complete, and that they have the finite model property, being thus decidable. We also characterize their classes of frames. In the way of studying them we also study the logics ${\bf K} + [n]\phi \rightarrow [n+1]\phi $, called Kn.4, and ${\bf K} + \phi \rightarrow [n]\langle n\rangle\phi $, called Kn.B. A translation between these logics and S5 is also presented, and the relation among them all is established.