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...

If F : Set → Set is a functor which is bounded and preserves weak generalized pullbacks then a class of F-coalgebras is a covariety, i.e., closed under H (homomorphic images), S (sub-coalgebras) and ∑ (sums), if and only if it can be de0ned by a set of “coequations”. Similarly, quasi-covarieties, i.e., classes closed under H and ∑ , can be characterized by implications of coequations. These res...

Matthew Spinks [35] introduces implicative BCSK-algebras, expanding implicative BCK-algebras with an additional binary operation. Subdirectly irreducible implicative BCSK-algebras can be viewed as flat posets with two operations coinciding only in the 1and 2-element cases, each, in the latter case, giving the two-valued implication truth-function. We introduce the resulting logic (for the gener...