Boolean-Valued Sets as Vague Sets

Abstract

The standard fuzzy logic and fuzzy set theory are degree-functional and thus susceptible to the problem of penumbral connections. This chapter attempts to radically revise them in order to remove this feature. Whereas the original theory assigns any real number in [0,1] to a proposition or set membership, the revised theory, called Boolean-valued set theory, assigns a value in a Boolean lattice structure B = ⟨D,∧,∨,¬,0,1⟩; consequently, “p or not-p” will receive value 1 and “p and not-p” will receive value 0 for any proposition p in Boolean-valued set theory. The resulting view of vague sets is essentially that depicted in Boolean-valued models of ZFC set theory introduced by Scott and Solovay. On this view, propositions and set membership have many values between 0 and 1, but the values are Boolean-structured and only partially ordered, so we cannot turn them into meaningful numerical degrees in [0,1]. The identity relation between two sets is also given various values, depending on the values of the membership relations involving the two sets; so this theory also endorses vague identity among sets.