A Construction Method for Modal Logics of Space
Spencer Gerhardt
Abstract:
Given the important role of spatial intuitions in cognition, and the
apparent unreliability of these intuitions, there is something natural
about looking at spatial structures from an axiomatic
standpoint. Indeed, it is not surprising that the first and best known
application of the axiomatic method was to provide a development of
geometry.
In the past century, with the development of formal logic and
subsequent discovery that elementary number theory is not
axiomatizable, it became both possible and independently interesting
to examine spatial structures within given logical
formalizations. While the central framework for examining spatial
structures axiomatically has been first order logic, from time to time
other logics with spatial interpretations have been considered as
well.
Recently, one popular area of investigation has been looking at modal
logics with spatial interpretations. This subject can be traced back
to McKinsey and Tarksi's [14] work on boolean algebras with closure
operators in the 1940s. However, within the past ten years a more
general program of providing a modal analysis of space has emerged.
By and large, the techniques used in investigating modal logics of
space have been model theoretic in nature, involving the transfer of
geometric or topological structure from the desired mathematical
object to some Kripke frame. While this works well in cases where the
relevant modal logic has nice Kripke frame characterizations, in other
cases this way of proceeding can become quite difficult.
In this thesis we will examine a more syntactic approach to
establishing completeness results in modal logics of space. The
technique we will use has the virtue of constructing the desired
mathematical structures directly, rather than working indirectly
through Kripke frames. This allows for a good deal of control over
what models of the relevant logic look like, and avoids sometimes
unpleasant detours through Kripke semantics.
In the second chapter, we will use our construction method to give new
proofs of the completeness of S4 with respect to Q, and S4 S4 with
respect to Q × Q,1,2 . We will also provide a much simpler
axiomatization of Q, in the combined language P + F + P and an
axiomatization of Q,< in the Since/Until language. In the third
chapter, we will discuss the advantages and disadvantages of the
construction method in comparison to the standard model theoretic
approach.
Keywords: Spatial Logic, Modal Logic