In the four papers available on our web site (of which this is the first), we propose to develop an inductive logic. By “inductive logic” we mean a set of principles that distinguish between successful and unsuccessful strategies for scientific inquiry. Our logic will have a technical character, since it is built from the concepts and terminology of (elementary) model theory. The reader may therefore wish to know something about the kind of results on offer before investing time in definitions (...) and notation. Providing such an informal overview is the purpose of the present essay. We begin with discussion of the central concept under investigation, namely, theory acceptance. (shrink)

The Interpolation Theorem, first formulated and proved by W. Craig fifty years ago for predicate logic, has been extended to many other logical frameworks and is being applied in several areas of computer science. We give a short overview, and focus on the theory of software systems and modules. An algebra of theories TA is presented, with a nonstandard interpretation of the existential quantifier . In TA, the interpolation property of the underlying logic corresponds with the quantifier combination property . (...) It is shown how the Modularization Theorem, the Factorization Lemma and the Normal Form Theorem for module expressions can be proved in TA. (shrink)

The topics of structural proof theory and logic programming have influenced each other for more than three decades. Proof theory has contributed the notion of sequent calculus, linear logic, and higher-order quantification. Logic programming has introduced new normal forms of proofs and forced the examination of logic-based approaches to the treatment of bindings. As a result, proof theory has responded by developing an approach to proof search based on focused proof systems in which introduction rules are organized into two alternating (...) phases of rule application. Since the logic programming community can generate many examples and many design goals, the close connections with proof theory have helped to keep proof theory relevant to the general topic of computational logic. (shrink)

In this paper we define and examine frame constructions for the family of manyvalued modal logics introduced by M. Fitting in the '90s. Every language of this family is built on an underlying space of truth values, a Heyting algebra H. We generalize Fitting's original work by considering complete Heyting algebras as truth spaces and proceed to define a suitable notion of H-indexed families of generated subframes, disjoint unions and bounded morphisms. Then, we provide an algebraic generalization of the canonical (...) extension of a frame and model, and prove a preservation result inspired from Fitting's canonical model argument in [FIT 92a]. The analog of a complex algebra and of a principal ultrafilter is defined and the embedding of a frame into its canonical extension is presented. (shrink)

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

Students often study logic on the assumption that it provides a normative guide to reasoning in English. In particular, they are taught to associate connectives like “and” with counterparts in Sentential Logic. English conditionals go over to formulas with → as principal connective. The well-known difficulties that arise from such translation are not emphasized. The result is the conviction that ordinary reasoning is faulty when discordant with the usual representation in standard logic. Psychologists are particularly susceptible to this attitude.