In this paper, we show how an appropriate possible world semantics exhibits the fundamental diierences between various forms of logical reasoning such as deduction, abduction and induction. The ensuing model theoretic formal-isation provides a solid foundation for exact further discourse on the issue, such as deciding whether a particular instance of reasoning should be classiied as abduction o...

We introduce a method to associate calculi of proof terms and rewrite rules with cut elimination procedures for logical deduction systems (i.e., Gentzen-style sequent calculi) in the case of intuitionistic logic. We illustrate this method using two different versions of the cut rule for a variant of the intuitionistic fragment of Kleene's logical deduction system G3. Our systems are in fact cal...

Deduction is the ability to draw necessary conclusions from previous knowledge. Here we propose a novel approach to understanding the neural basis of deduction, which exploits fine-grained inter-participant variability in such tasks. Participants solved deductive problems and were grouped by the behavioral strategies employed, i.e., whether they were sensitive to the logical form of syllogistic...

This paper is a study of similarities and differences between strong and weak quantum consequence operations determined by a given class of ortholattices. We prove that the only strong orthologics which admits the deduction theorem (the only strong orthologics with algebraic semantics, the only equivalential strong orthologics, respectively) is the classical logic. In the papers concerning quan...

Objects like rewrite rules and strategies are of very general use and interest in computer science. They are especially useful when modeling inference based systems like logical frameworks, theorem provers, constraint solvers, expert systems, transition systems. In order to formalize the concept of rule application, we have designed a calculus of explicit rule application called the rewritingor...

One of the main ideas of calculi of natural deduction, as introduced by Jaskowski and Gentzen, is that assumptions may be discharged in the course of a derivation. As regards sentential logic, this conception will be extended in so far as not only formulas but also rules may serve as assumptions which can be discharged. The resulting calculi and derivations with rules of any finite level are in...

Spatial logics are used to reason locally about disjoint data structures. They consist of standard first-order logic constructs, spatial (structural) connectives and their corresponding adjuncts. Lozes has shown that the adjuncts add no expressive power to a spatial logic for analysing tree structures, a surprising and important result. He also showed that a related logic does not have this adj...

We propose a combinatory reformulation of the product free version of the categorial calculus LL, i.e. the associative Lambek calculus that admits empty premises. We prove equivalence of the combinatory with the standard Natural Deduction presentation of LL. The result offers a new perspective on the relation between the type logical and the combinatory branch of the Categorial Grammar research...

A logical framework is presented for representing and reasoning about nondeterministic programs that may not terminate. We propose a logic PDL(; ;, ||, d(∗)) which is an extension of dynamic logic such that the program constructors related to demonic operations are introduced in its language. A complete and sound Hilbert-style proof system is given and it is shown that PDL(; ;, ||, d(∗)) is dec...

We give a direct, purely arithmetical and elementary proof of the strong normalization of the cut-elimination procedure for full (i.e. in presence of all the usual connectives) classical natural deduction.