On the extension of classical propositional logic by means of a triangular norm

✍ Scribed by G. de Cooman; E. E. Kerre; B. Cappelle; Ruan Da; F. Vanmassenhove

• Publisher: John Wiley and Sons
• Year: 1990
• Tongue: English
• Weight: 682 KB
• Volume: 5
• Category: Article
• ISSN: 0884-8173
• DOI: 10.1002/int.4550050307

✦ Synopsis

In this article, we introduce a generalized extension principle by substituting a more general triangular norm T for the min intersection operator in Zadeh's extension principle. We also introduce a family of propositional logics, sup-T extension logics, obtained by the extension of classical-logical functions. A few general properties of these sup-T extension logics are derived. It is also shown that classical binary logic and the Kleene ternary logic are special cases of these logics for any choice of T, obtained by a convenient restriction of the truth domain. The very practical decomposability property of classical logic is furthermore shown to hold for the sup-min extension logic, albeit in a somewhat more limited form.

I. THE TRUTH DOMAINS U AND FN(U)

The basic assumption of this article is that a semantic notion of truth exists, and that truth can take degrees. In general, therefore, it seems reasonable to represent truth by a set Tr, provided with a total order s (representing the relation "is not less true than"), with a smallest element (representing absolutely false) and a greatest element (representing absolutely true). The elements of the truth domain Tr are called truth values or degrees oftruth. Once a particular choice for the truth domain Tr has been made, we can define a proposition p as any affirmative sentence, to which a truth value can meaningfully be ascribed.

In classical logic, the truth domains is the set U = {true, false}, with the total order 5 , defined by false I true. A proposition in classical logic is then any affirmative sentence that is either (abolutely) true or (absolutely) false.