Chapter

Around and Beyond the Square of Opposition

Part of the series Studies in Universal Logic pp 241-260

Fuzzy Syllogisms, Numerical Square, Triangle of Contraries, Inter-bivalence

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Abstract

A new Predicate Calculus, called “Distinctive” D, is presented in the form of the Hexagon of Opposition, in which the Particular Yba (= only some b are a) is the contradictory of the Universal Uba (= all or no b are a). Y is preferred, as a primitive, to I or O, because it is more “natural” than the others. Typical inferences of the systems are the obversions: Yba=Yba′ (= only some b are not a), Uba=Uba′ (= all or no b are not a).

D-Systems include traditional Syllogisms. Polygonal and Numerical developments are being developed, including intermediate quantifiers (“the majority of”, …). Polygons are finally absorbed into the Numerical D-Square NDS. Isomorphisms are discovered between bivalent D-systems and some Non-Standard Logics by depriving the subject-class of the quantifier and transferring its (pre)numerical attribute to the truth value of the judgement. These ‘(poly-)inter-bivalent’ logics admit intermediate values between true and false. In that way we get Fuzzy Logic, and present the Fuzzy Square.

Keywords

Predicate logic Quantifiers Logical diagram Square of opposition Hexagon of opposition Numerical syllogisms Non-standard logic Fuzzy syllogisms Synonyms

Mathematics Subject Classification

03A05 03B20 03B52 03B65 03B60 03B53 05C99