Student Work

4-2003

Thesis

Degree Name

Master of Arts in Mathematics (MA-MATH)

Mathematics

Scott Downing

Abstract

This thesis is concerned with mathematical logic, in particular it is an investigation of a branch of mathematical logic called modal logic. This branch of mathematical logic extends the propositional calculus by adding two unary operators □ and 0 to the standard set of logical operators. This extension of classical logic has many interpretations; traditionally it is said to be the logic of necessity, denoted by the box operator, and possibility, denoted by the diamond operator. The notion of necessity within modal logic is ubiquitous and lends itself to a vast sea of metaphysics. For example, if X is necessarily true, denoted O X , then it is said to be true in all possible worlds. This way of understanding modalities gave imputes for a semantics that provided fodder for the first completeness proofs in modal logic.

Modalities in logic have its roots in philosophy and dates back as far as Aristotle’s M etaphysics, but was brought into the limelight with the work of the philosopher mathematician Saul Kripke who in 1959, as a high school student, published the first completeness proof for a class of modal logics [Kripke]. His method used the so-called semantic-tableaux which was introduced by B eth’s The foundations of mathematics to obtain quick completeness proof for the propositional and predicate calculus. In this thesis, we are also interested in completeness for modal logics, but will use a more modern method known in the.literature as canonical model constructions . Moreover, we wish to provide a semantics for modal logics that is not the traditional possible world semantics. Our models will be topological in nature. Our goal is to provide a completeness proof for a particular modal logic called S4 which interprets the modal operators as the interior and closure operators on topological spaces. We will also prove that the logic S4 is complete with respect to the class of transitive and reflexive trees. This gives us two new completeness proof for the modal logic S4.