Łukasiewicz logic and fuzzy set theory

✍ Scribed by R. Giles

• Publisher: Elsevier Science
• Year: 1976
• Weight: 1008 KB
• Volume: 8
• Category: Article
• ISSN: 0020-7373
• DOI: 10.1016/s0020-7373(76)80003-x

✦ Synopsis

A new form of logic is described, originally developed for the formalization of physical theories, the essential feature being a "fuzzification" of the concept of a proposition. A proposition is not regarded as being necessarily true or false; it is defined not via truth conditions but in terms of a definite commitment that is assumed by the speaker. In the case of an atomic proposition the commitment amounts to a bet on the outcome of some agreed test; for a compound proposition it leads to a dialogue between the speaker and an opponent. The resulting logic corresponds closely to the infinitevalued logic L=, of Lukasiewicz. In fact, theapproach provides a dialogue interpretation of Leo and leads to a convenient method for establishing logical identities.

Set theory is then developed, not by taking set as a primitive concept but by assuming each set A is determined by a property P characteristic of its members: A = {x:P(x)}. When this is expressed formally the result can be read in two ways according to whether the underlying logic is classical logic or I-.oo (with the above interpretation). If the propositions P(x) are classical we get ordinary sets; if they are propositions in the new "fuzzy" sense we get fuzzy sets (f-sets). The situation is illustrated by a number of definitions and theorems involving simple operations on f-sets. Lastly, the notion of a convex f-set is defined, and asimple theorem is stated and proved using Leo and the dialogue method of proof. All statements and proofs are expressed in terms of the new logic. In particular, use of the quantitative notion of "grade of membership" in a fuzzy set is entirely avoided.