Infinite—valued Logical Calculi
Part of the Advances in Soft Computing book series (AINSC, volume 15)
In Chapter 4, many—valued logics were discussed. In that case, sets of truth values were finite and in Chapter 12 we have demonstrated that rough sets may be interpreted as Łukasiewicz algebras of 3—valued Łukasiewicz calculi. However, the Łukasiewicz semantics, introduced in [Łukasiewiczl8] and discussed also in [Łukasiewicz20, 30], and presented in Chapter 4, does allow for infinite sets of truth values. In this case, one may admit as a set of truth values any infinite set T ⊑ [0, 1] which is closed under functions
in the sense that c(x, y), n(x) ∈ T whenever x, y ∈ T; we recall that the function c is the meaning of implication C and the function n is the meaning of negation N in the Łukasiewicz semantics. One shows (cf. [McNaughton5l] quoted in [Rose—Rosser58]) that any such T needs to be dense in [0, 1]; it is manifest that 0,1 ∈ T for each admissible T. Thus, sets Q 0 = Q ∩ [0, 1] of rational numbers in [0,1] as well as the whole unit interval [0, 1] may be taken as sets of truth values leading to respective infinite valued logical calculi. The additional merit of these two sets of truth values is that both Q, R l are from algebraic point of view, algebraic fields so they may be regarded as vector spaces over themselves, and we will make use of this fact later on.
KeywordsFuzzy Logic Inference Rule Inductive Step Residuated Lattice Inductive Assumption
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