An intrinsic fuzzy set on the universe of discourse of predicate formulas

The validity degree vectors of formulae in two-valued predicate logic

“…But, because of the complexity of interpretations for first-order language, it's quite hard to complete this work in predicate logic, and so far there are only some sporadic results such as refs. [18,[30][31][32][33][34][35][36][37][38][39][40]. Among them, the work in ref.…”

“…Distinct to the former two, ref. [36] first established an axiomatic theory of truth degree of formulae from the point view of syntactic instead of semantics, but it's a pity that only the truth degrees of all the closed formulae in predicate logic was studied. Compared with the former three, the model proposed in ref.…”

Theory of (n) truth degrees of formulas in modal logic and a consistency theorem

“…. , n} and the binary relation R is determined by (32), but the valuation V : Φ → ℘(W ) can change arbitrarily. Let M n,R denote the set of all this kind of models.…”

“…[32] was only a preliminary test far from satisfactory because it defined truth degrees for predicate formulas simply by taking the average of the values obtained from two extreme cases. The present paper thinks of a basic idea for establishing a reasonable truth degrees theory in predicate logic, that is, to find out an appropriate part of predicate formulas with simple forms, and first define truth degrees for these predicate formulas and then extend the concept to general predicate formulas.…”

“…As in ref. [32], the present paper considers only finite models. Even so, the calculation in this paper is far more complex than the corresponding calculation in classical propositional logic, because the valuations of a formula in the system K can be any subset of the power set of a finite model, while that of the latter only need to be chosen from the two-element-set of {True, False}.…”