Foundations of fuzzy sets: A nonstandard approach

Boolean Methods and Pták's Sum

“…At the same time we import some "strange" relativity phenomena of mathematical concepts, which are usually associated with Boolean-valued models. This also forces us to give in the next section the appropriate interpretations for Β ® L, see [3].…”

“…This interpretation which is dual to the stochastic and depends on the general duality of Extension-Intention (see [3]), also gives a non-Cantorian interpretation of the logic L. Finally we should stress again that L[B]*, can be seen in two different ways: Externally, that is as a structure within the classical ZFC universe of sets and internally as a nonclassical structure with its own "truth" which is usually called "validity" and it is based on the truth function || • ||. Failing to differentiate these two ways can lead to contradictions.…”

“…Using the isomorphism of Theorem 3.1, and the interpretations we usually give to the Boolean powers (see [3], [4], [5], [6]), we shall try to give a different interpretation to Pták's sum which takes into account the relativity of the concepts.…”

“…Using the terminology of [3], [5], we way regard a partition of unity of B, as a random experiment or trial. Let T(B) := {Τ C Β : Τ = (*<)<€/, U A t¡ = 0 S i φ j L· \J U = Ib} «6/ be the collection of all random experiments in B.…”

“…Following [3], and considering the model L[B]*, we may regard L[B]* as a Β -fuzzy logic, which includes all the Β -probability density functions for each X G «S(B, L). This interpretation which is dual to the stochastic and depends on the general duality of Extension-Intention (see [3]), also gives a non-Cantorian interpretation of the logic L. Finally we should stress again that L[B]*, can be seen in two different ways: Externally, that is as a structure within the classical ZFC universe of sets and internally as a nonclassical structure with its own "truth" which is usually called "validity" and it is based on the truth function || • ||.…”