Natural Deduction in a Paracomplete Setting

In this paper we present the automated proof search technique in natural deduction paracomplete logic. Here, for some statements we do not have evidence to conclude if they are true or false, as it happens in the classical framework. As a consequence, for example, formulae of the type p_:p, are not valid. In this paper we formulate the natural deduction system for paracompletelogic PComp, explain its main concepts, define proof searching techniques and the searching algorithm providing examples proofs

Pandora: A reasoning toolbox using natural deduction style.

Abstract Pandora is a tool for supporting the learning of first order natural deduction. It includes a help window, an interactive context sensitive tutorial known as the ''e-tutor'' and facilities to save, reload and export to . Every attempt to apply a natural deduction rule is met with either success or a helpful error message, providing the student with instant feedback. Detailed electronic logs of student usage are recorded for evaluation purposes. This paper describes the basic functionality, the e-tutor, our experiences of using the tool in teaching and our future plans

On multiple conclusion deductions in classical logic

Kneale observed that Gentzen’s calculus of natural deductions NK for classical logic is not symmetric and has unnecessarily complicated hypothetical inference rules. Kneale proposed inference rules with multiple conclusions as a basis for a symmetric natural deduction calculus for classical logic. However, Kneale’s informally presented calculus is not complete. In this paper, we define a calculus of multiple conclusion natural deductions (MCD) for classical propositional logic based on Kneale’s multiple conclusion inference rules. For MCD we present elementary proof search that produces proofs in normal form. MCD proof search is motivated and explained as being a notational variant of Smullyan’s analytic tableaux method in its initial part and a notational variant of refutation proofs based on Robinson’s resolution in its final part. We consider MCD to have semantic motivation of both its inference rules and its proof search. This is unusual for the natural deduction calculi as they are syntactically motivated. Syntactic motivation is adequate for intuitionistic logic but not a natural fit for truth-functional classical propositional logic

Normal natural deduction proofs (in classical logic)

Abstract: "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: How do we bridge the gap between claims and assumptions in heuristically motivated ways? This informal question motivates the formulation of intercalation calculi. Ic-calculi are the technical underpinnings for (1) and (2), and our paper focuses on their detailed presentation and meta-mathematical investigation in the case of classical predicate logic. As a central theme emerges the connection between restricted forms of nd-proofs and (strategies for) proof search: normal forms are not obtained by removing local 'detours', but rather by constructing proofs that directly reflect proof-strategic considerations. That theme warrants further investigation.