Łukasiewicz logic and fuzzy set theory

Abstract

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 Ł α, of Łukasiewicz. In fact, the approach provides a dialogue interpretation of Ł ∞ 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 Ł ∞ (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 a simple theorem is stated and proved using Ł ∞ 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.