Fri Sep 25, 2009
60 Evans Hall, 4:10–6 PM
Dana S. Scott (Carnegie Mellon University)
Mixing Modality and Probability
Orlov first  and Gödel later  pointed out the connection between the Lewis System S4 and Intuitionistic Logic. McKinsey and Tarski gave an algebraic formulation and proved completeness theorems for propositional systems using as models topological spaces with the interior operator corresponding to the necessitation modality. Earlier, Tarski and Stone had each shown that the lattice of open subsets of a topological space models intuitionistic propositional logic. Expanding on a suggestion of Mostowski about interpreting quantifiers, Rasiowa and Sikorski used the topological models to model many first-order logics. After the advent of Solovey’’s recasting of Cohen’’s independence proofs as using Boolean-valued models, topological models for modal higher- order logic were studied by Gallin and others. For Boolean-valued logic, the complete Boolean algebra M = Meas([0,1])/Null of measurable subsets of the unit interval modulo sets of measure zero gives every proposition a probability. Perhaps not as well known is the observation that the measure algebra also carries a nontrivial S4 modality defined with the aid of the sublattice Open([0,1])/Null of open sets modulo null sets. This sublattice is closed under arbitrary joins and finite meets in the measure algebra, but it is not the whole of the measure algebra. By working by analogy to the construction of Boolean-valued models for ZF, we can construct over M a model for a modal ZF (MZF) where membership and equality predicates have interesting and natural modal properties. In such a universe the real numbers correspond to random variables, and — following a suggestion of Alex Simpson (Edinburgh) — there is also an well motivated modeling of random reals. A modal set theory, however, requires a reexamination of comprehension principles, and much work remains to be done to organize methods of proof to take account of the new distinctions encountered. It is also possible to recast well known theorems of Ergodic Theory as principles about this modal universe, and the question to be considered is whether the new perspective can lead to some new results.