Michiel is visiting HCRC until June 1999, collaborating with Keith Stenning. His talk Vision, Language and Logic on March 11 for the Institute for Representation and Reasoning applied the logic presented here to the connection between vision and natural language.

Abstract

One idea which probability theory may have to offer to logic, is the generalisation of existential quantification embodied in its concept of conditional expectation. A fruitful way of translating this notion into logic appears to be to consider quantification with respect to a resource; classical existential quantifcation `there exists an x ...' applied to a formula with only x free, is then the limiting case where the resource is the 2-element Boolean algebra. Other algebra structures instead of the 2-element Boolean algebra may then capture the fact that one has various sorts of partial information at one's disposal; this is formalised by means of a quantifier conditional on the resource.

The talk will address the following questions. Which structures are suitable as resources? It will turn out that some amount of completeness is necessary. How does the choice of resource influence the logic of the conditional quantifier? The talk will study two cases: co-Heyting algebras and sigma-algebras. It turns out that in some cases the continuum hypothesis is necessary to make the idea work.