Topic Archive: measure theory
Long before Kripke semantics became standard in modal logic, Tarski showed us that the basic propositional modal language can be interpreted in topological spaces. In Tarski’s semantics for the modal logic $S4$, each propositional variable is evaluated to an arbitrary subset of a fixed topological space. I develop a related, measure theoretic semantics, in which modal formulas are interpreted in the Lebesgue measure algebra, or algebra of Borel subsets of the real interval $[0,1]$, modulo sets of measure zero. This semantics was introduced by Dana Scott in the last several years. I discuss some of my own completeness results, and ways of extending the semantics to more complex modal languages.