I present a new syntactical method for proving the Interpolation Theorem for the implicational fragment of intuitionistic logic and its substructural subsystems. This method, like Prawitz’s, works on natural deductions rather than sequent derivations, and, unlike existing methods, always finds a ‘strongest’ interpolant under a certain restricted but reasonable notion of what counts as an ‘inter...

The canonical base of a formal context plays a distinguished role in Formal Concept Analysis, as it is the only minimal implicational base known so far that can be described explicitly. Consequently, several algorithms for the computation of this base have been proposed. However, all those algorithms work sequentially by computing only one pseudointent at a time – a fact that heavily impairs th...

It is well known that for each l-term there is a corresponding combinatory term formed using the combinators K and S instead of the I-operator. Similarly for every combinatory term there is a i.-term. For weaker sets of combinators such as B, C and K or B, B’, I and W we show how such a correspondence or “translation” can be formulated and we determine in the case of several such sets of combin...

It is well known that knowledge-based systems would be more robust and smarter if they can deal with the inconsistent, incomplete or imprecise knowledge, which has been referred to as common sense knowledge. In this paper, we discuss fuzzy implications in the sense of common sense reasoning. F i rs t ly , we analyse the ra t i ona l i t y of some exist ing fuzzy implications based on the discus...

In this article we present a class of formulas φn, n ∈ Nat, that need at least 2 assumption occurrences to be proved in a normal proof in Natural Deduction for purely implicational minimal propositional logic. In purely implicational classical propositional logic, with Peirce’s rule, each φn is proved with only one assumption occurrence in Natural Deduction in a normal proof. Besides that, the ...

It is well-known that the size of propositional classical proofs can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective non-normal proofs. The aim of this work is to study how to reduce the weight of propositional deductions. We present the formalism of proof-graphs for purely implicational logic, which are graphs of a specific ...

In 1979 Richard Statman proved, using proof-theory, that the purely implicational fragment of Intuitionistic Logic (M→) is PSPACE-complete. He showed a polynomialy bounded translation from full Intuitionistic Propositional Logic into its implicational fragment. By the PSPACEcompleteness of S4, proved by Ladner, and the Gödel translation from S4 into Intuitionistic Logic, the PSPACE-completeness...