We consider propositional logic. Three-valued logics are old: the first one is Lukasiewicz three valued logic from 1920 [8]. Gödel in [5] from 1932 studied a hierarchy of finite-valued logics, containing Gödel three-valued logic. Our main interest pays to Kleene three-valued logic [6]. Other threevalued logics will not be considered here. Let us agree that the three truth values are 0, 1 2 , 1 ...

We study the positive fragment of a system of partial logic which has been developed by Ackermann in the course of his typefree foundation of mathematics. One of the prominent features of this system is its implication, which has been supplied with a deductive interpretation according to which A ! B expresses that B is derivable from A. The first section provides Ackermann’s original Hilbert-st...

We argue that the usual trace-based notions of implication and equivalence for linear temporal logics are too strong and should be complemented by the weaker notions of open implication and open equivalence. Although open implication is harder to compute, it can be used to advantage both in model checking and in synthesis. We study the difference between trace-based equivalence and open equival...

This paper deals with numerical functions able to functionally express operators defined as , and verifying either Modus Ponens or Modus Tollens, or both. The concrete goal of the paper is to search for continuous t-norms and strong-negation functions for which it is either (Modus Ponens), or (Modus Tollens), or both, for all in and a given . Functions are taken among those in the most usual fa...

This paper addresses the apparent mismatch between the normative and descriptive literatures in the cognitive science of conditional reasoning. Descriptive psychological theories still regard material implication as the normative theory of the conditional. However, over the last 20 years in the philosophy of language and logic the idea that material implication can account for everyday indicati...

The problem with the standard means by which Prolog programs are extended — assert — is that the construct is not semantically well-behaved. A more elegant alternative (adopted, for example, in λProlog) is implication with its intuitionistic meaning, but the assumptions so added to a logic program are of limited applicability. We propose a new construct rule, which combines the declarative sema...

BCK-algebras, introduced in the 1960s by Imai and Iséki [3], form a well-known class of algebras intensively studied in algebraic logic during the last decades. The reason of their introduction was practically twofold: they describe general properties of algebras of sets with the set subtraction as a binary operation, and secondly, they form a natural generalization of algebraic counterparts of...