Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional log...

In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension of this reducibility for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice with the disjoint union of multi-valued functions as greatest ...

While the purpose of conventional proof calculi is to axiomatise the set of valid sentences of a logic, refutation systems axiomatise the invalid sentences. Such systems are relevant not only for proof-theoretic reasons but also for realising deductive systems for nonmonotonic logics. We introduce Gentzen-type refutation systems for two basic three-valued logics and we discuss an application of...

We study different completeness definitions for two categories of lattice-valued Cauchy spaces and the relations between these definitions. We also show the equivalence of a so-called completion axiom and the existence of a completion.

In recent years, model theory has had remarkable success in solving important problems. Its importance lies in the observation that mathematical objects can be cast as models for a language. Ultraproduct is a method of constructing a new model from a family of models, In this paper, we deal with a new form of ultraproduct model for first-order lattice-valued logic LF(X) whose truth-value field ...

The setup of a mathematical propositional logic is given in algebraic terms, describing exactly when two choices of truth value algebras give the same logic. The propositional logic obtained when the algebra of truth values is the real numbers in the unit interval equipped with minimum, maximum and :x = 1 x for conjunction, disjunction and negation, respectively, is the standard propositional f...

in this paper, for a complete lattice l, we introduce interval-valued l-fuzzy ideal (prime ideal) of a near-ring which is an extended notion of fuzzy ideal (prime ideal) of a near-ring. some characterization and properties are discussed.

One of key issues for α-n(t) ary resolution automated reasoning based on lattice-valued logic with truthvalue in a lattice implication algebra is to investigate the α-n(t) ary resolution of some generalized literals. In this article, the determination of α-resolution of any 3-ary generalized literals which include the implication operators not more than 2 in LP(X). It not only lay the foundatio...

In this paper, we investigate the Lukasiewicz’s 4-valued modal logic based on the Aristotele’s modal syllogistic. We present a new interpretation of the set of algebraic truth values by introducing the truth and knowledge orderings similar to those in Belnap’s 4-valued bilattice but by replacing the original Belnap’s negation with the lattice pseudo-complement instead. Based on it, we develop a...

As an extension of interval-valued pseudo t-norms, pseudo-overlap functions (IPOFs) play a vital role in solving multi-attribute decision making problems. However, their corresponding algebraic structure has not been studied yet. On the other hand, with development non-commutative (non-associative) fuzzy logic, study residuated lattice theory is gradually deepening. Due to conditions operators ...