This paper describes an algorithm of proof search for implicational formula in intuitionistic logic. It is based on a natural deduction style formulation of long normal form of simply typed -calculus. The algorithm returns a deduction tree, which becomes a proof if the formula is provable. When the formula is unprovable, a counter-model is constructed by identifying the repeated nodes in the de...

We introduce a functional calculus with simple syntax and operational semantics in which the calculi introduced so far in the Curry–Howard correspondence for Classical Logic can be faithfully encoded. Our calculus enjoys confluence without any restriction. Its type system enforces strong normalization of expressions and it is a sound and complete system for full implicational Classical Logic. W...

– from a linguistic perspective – from a formal perspective • Locality of grammatical relations in HPSG 3 HPSG grammars from a linguistic perspective From a linguistic perspective, an HPSG grammar consists of • a lexicon licensing basic words • lexical rules licensing derived words • immediate dominance (id) schemata licensing constituent structure • linear precedence (lp) statements constraini...

In this contribution we present uniform goal-directed rules for the implicational fragments of the three main formalizations of fuzzy logic; namely, Łukasiewicz logic Ł, Gödel logic G, and Product logic Π. We begin with a historical overview of the goal-directed methodology, focussing in particular on the pioneering work of Dov Gabbay, then proceed by recalling the fundamental systems of fuzzy ...

It is well known that the model categories of universal Horn theories are locally presentable, hence essentially algebraic [2]. In the special case of quasivarieties a direct translation of the implicational syntax into the essentially equational one is known [1]. Here we present a similar translation for the general case, showing at the same time that many relationally presented Horn classes a...

It is widely known that closure operators on finite sets can be represented by sets of implications (also known as inclusion dependencies) as well as by formal contexts. In this paper we survey known results and present new findings concerning time and space requirements of diverse tasks for managing closure operators, given in contextual, implicational, or black-box representation. These tasks...

We present a system of natural deduction and associated term calculus for intuitionistic non-commutative linear logic (INCLL) as a conservative extension of intuitionistic linear logic. We prove subject reduction and the existence of canonical forms in the implicational fragment. We also illustrate by means of an example how the proof term calculus can be employed in the setting of logical fram...