In this paper, we consider a set of quite interesting three- and four-valued logics prove the normalisation theorem for their natural deduction formulations. Among in question are Logic Paradox, First Degree Entailment, Strong Kleene logic, some implicative extensions, including RM3 RM3⊃. Also, present detailed version Prawitz’s proof Nelson’s logic N4 its extension by intuitionist negation.

In this paper we present a short history of logics: from particular cases of 2-symbol or numerical valued logic to the general case of n-symbol or numerical valued logic. We show generalizations of 2-valued Boolean logic to fuzzy logic, also from the Kleene’s and Lukasiewicz’ 3-symbol valued logics or Belnap’s 4-symbol valued logic to the most general n-symbol or numerical valued refined neutro...

The purpose of this paper is to make some practically relevant results in automated theorem proving available to many-valued logics with suitable modifications. We are working with a notion of many-valued first-order clauses which any finitely-valued logic formula can be translated into and that has been used several times in the literature, but in an ad hoc way. We give a many-valued version o...

Łukasiewicz three-valued logic Ł3 is often understood as the set of 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using D...

The Polish logician Roman Suszko has extensively pleaded in the 1970s for a restatement of the notion of many-valuedness. According to him, as he would often repeat, “there are but two logical values, true and false.” As a matter of fact, a result by Wójcicki-Lindenbaum shows that any tarskian logic has a many-valued semantics, and results by Suszko-da Costa-Scott show that any many-valued sema...

The Polish logician Roman Suszko has extensively pleaded in the 1970s for a restatement of the notion of many-valuedness. According to him, as he would often repeat, “there are but two logical values, true and false.” As a matter of fact, a result by Wójcicki-Lindenbaum shows that any tarskian logic has a many-valued semantics, and results by Suszko-da Costa-Scott show that any many-valued sema...

In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix)....

Inconsistency is a pervasive problem in software engineering, where different aspects of a system are described in separate models. Resolving all the inconsistencies in a large set of models is often infeasible, in which case automated reasoning tools based on classical logic have limited application. In this paper we describe an automated tool for paraconsistent reasoning, using multi-valued l...

This paper focuses on approximate reasoning based on the use of similarity spaces. Similarity spaces and the approximated relations induced by them are a generalization of the rough set-based approximations of Pawlak [17, 18]. Similarity spaces are used to define neighborhoods around individuals and these in turn are used to define approximate sets and relations. In any of the approaches, one w...