These notes cover the content of a basic course in propositional algebraic logic given by the author at the italian School of Logic held in Cesena, September 18-23, 2000. They are addressed to people having few background in Symbolic Logic and they are mostly intended to develop algebraic methods for establishing basic metamathematical results (like representation theory and completeness, finit...

The logic MTL, Monoidal t-norm based logic was presented by F. Estva and L. Godo and discuss several properties of MTL-algebra in [6] The Boolean logic (BL) was introduced by P. Hajek and discuss the properties in [1].The fuzzy reasoning and implication operators was discussed by Ying in ([2],[3]). A proper logical system for fuzzy propositional calculus and a innovative arithmetical configurat...

The question whether material implication is a proper translation of “if ... then ...” in conditional sentences has been known and discussed as “material implication paradox”. The source of this paradox comes from the interpretation of material implication that any falsehood, or contradiction, entails every proposition. For example, consider truth conditions of the conditional sentence (1) and ...

Where ~t is a sequence of terms, ~x is a sequence of variables, and ~a is a sequence of assumptions. In the assumptions we write implication in the opposite direction than we normally do (i.e., “p(~t ) ⊂ g” rather than “g ⊃ p(~t )”). This is a standard convention in logic programming (e.g., Prolog) because it makes it easier for the programmer to understand the program; the meaning of implicati...

In 1977, R. B. Angell presented a logic for analytic containment, a notion of “relevant” implication stronger than Anderson and Belnap’s entailment. In this paper I provide for the first time the logic of first degree analytic containment, as presented in [2] and [3], with a semantical characterization—leaving higher degree systems for future investigations. The semantical framework I introduce...

Where ~t is a sequence of terms, ~x is a sequence of variables, and ~a is a sequence of assumptions. In the assumptions we write implication in the opposite direction than we normally do (i.e., “p(~t ) ⊂ g” rather than “g ⊃ p(~t )”). This is a standard convention in logic programming (e.g., Prolog) because it makes it easier for the programmer to understand the program; the meaning of implicati...

We study inference systems of weak functional dependencies in relational and complex-value databases. Functional dependencies form a very common class of database constraints. Designers and administrators proficiently utilise them in everyday database practice. Functional dependencies correspond to the linear-time decidable fragment of Horn clauses in propositional logic. Weak functional depend...

is obvious, and then what always hold in the Mamdani's case is the equivalence (p 2 q ! r) = (p ! r) ^ (q ! r): (14) Consequently, if I is a Mamdani–Larsen's implication (8) never holds, but (14) is always true. IV. CONCLUSION After remembering the four main types of implications used in Fuzzy Logic, the equality [p ^ q ! r] = [(p ! r) _ (q ! r)] (*) is analyzed bot in complemented lattices and...

In this paper, we propose a new static analysis method which is applicable for a classical linear logic programming language. Andreoli et al. proposed a static analysis method for the classical linear logic programming language LO, but their method did not cover multiplicative connectives which are important for a resource-sensitive feature of linear logic. Our method, in contrast, covers multi...

the logical treatise al-muhja has been attributed to avicenna in some bibliographies of logical treatises. since a proper understanding of the development of avicennian logic rests on the identification and consideration of the entire corpus of his logical works, the task of the present study is to investigate the validity of ascription of this treatise to avicenna. in carrying out an analysis ...