Normality, Non-contamination and Logical Depth in Classical Natural Deduction

Natural Deduction for Non-Classical Logics

We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large famili...

Classical Natural Deduction

In the tradition which considers formal logic as an organon of thought a central role has been played by the “method of analysis”, which amounts to what today, in computer science circles, is called a “bottom-up” or “goaloriented” procedure. Though the method was largely used in the mathematical practice of the ancient Greeks, its fullest description can be found in Pappus (3rd century A.D.), w...

Non-idempotent types for classical calculi in natural deduction style

In the first part of this paper, we define two resource aware typing systems for the λμ-calculus based on non-idempotent intersection and union types. The nonidempotent approach provides very simple combinatorial arguments –based on decreasing measures of type derivations– to characterize head and strongly normalizing terms. Moreover, typability provides upper bounds for the lengths of the head...

Normal Natural Deduction Proofs (in Non-classical Logics)

Normal Natural Deduction Proofs (in classical logic)

Natural deduction (for short: nd-) calculi have not been used systematically as a basis for automated theorem proving in classical logic. To remove objective obstacles to their use we describe (1) a method that allows to give semantic proofs of normal form theorems for nd-calculi and (2) a framework that allows to search directly for normal nd-proofs. Thus, one can try to answer the question: H...