Logic,topology, and integration

The central notion presented is that of closeness of (or difference between) two theories. In the first part, we give intuitive arguments in favor of considering topologies on the set of theories, continuous logics, and the average difference between two logics (i.e., the integral of their difference). We argue for the importance of the difference between theories in a wide range of applications and problems. In the second part, we give some basic definitions and results for one such type of topology. In particular, separation properties and compactness are discussed and examples given. The techniques employed for constructing the topology are also used for defining a -algebra of measurable sets on the set of theories, leading to the usual definition of the Lebesgue integral and a precise definition of the average difference of two logics.