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On the predicate logics of continuous t-norm BL-algebras. (English) Zbl 1070.03013
Let \(\mathbf{C}\) be a class of BL-algebras whose lattice reduct is the real unit interval \([0,1]\) with the usual order.
Firstly, the paper deals with the task when the set Taut(\(\mathbf{C}\forall\)) of predicate formulas valid in all members of \(\mathbf{C}\) is recursively axiomatizable. The author shows that Taut(\(\mathbf{C}\forall\)) is recursively axiomatizable iff \(\mathbf{C}\) consists only of \([0,1]_G\) (i.e., the Gödel algebra on \([0,1]\)). If \(\mathbf{C}\) contains at least one algebra not isomorphic to \([0,1]_G\) then he recursively reduces Taut(\(\L\forall\)) to Taut(\(\mathbf{C}\forall\)), where \(\L\forall\) denotes Łukasiewicz predicate logic. As it is known that Taut(\(\L\forall\)) is \(\Pi_2\)-complete, he gets that Taut(\(\mathbf{C}\forall\)) is \(\Pi_2\)-hard. If \(\mathbf{C}\) contains only \([0,1]_G\) then Taut(\(\mathbf{C}\forall\)) is recursively axiomatizable since it is well known that Taut(\([0,1]_G\forall\)) is in \(\Sigma_1\).
Secondly, it is known that each t-norm is isomorphic to an ordinal sum of the three basic t-norms (Gödel, Łukasiewicz, product). The author proves that if \(\mathbf{C}\) contains a member whose monoidal operation is an ordinal sum containing either product t-norm or Łukasiewicz t-norm which is neither the first component nor the last component in the ordinal sum then Taut(\(\mathbf{C}\forall\)) is not arithmetical. In the case with the product component he recursively reduces Taut(\(\Pi\forall\)) (i.e., tautologies of the product predicate fuzzy logic) to Taut(\(\mathbf{C}\forall\)). Since Taut(\(\Pi\forall\)) is not arithmetical, the result follows. In the case with Łukasiewicz component he recursively reduces true arithmetic to Taut(\(\mathbf{C}\forall\)). Thus there remain only finitely many cases when the complexity of Taut(\(\mathbf{C}\forall\)) is not known.
Finally, the monadic fragment of Taut(\(\mathbf{C}\forall\)) is investigated. The author shows that if \(\mathbf{C}\) contains a member which is not isomorphic to any of product algebra on \([0,1]\) and MV-algebra on \([0,1]\), then the monadic fragment of Taut(\(\mathbf{C}\forall\)) is undecidable. For this purpose he reduces the theory of two equivalence relations to the monadic fragment of Taut(\(\mathbf{C}\forall\)). Since the theory of two equivalence relations is undecidable, the result follows. Moreover, the author shows that it is possible to use the same method to prove that the monadic fragment of Hájek’s basic predicate fuzzy logic is undecidable.

MSC:
03B52 Fuzzy logic; logic of vagueness
03D15 Complexity of computation (including implicit computational complexity)
03G25 Other algebras related to logic
03D35 Undecidability and degrees of sets of sentences
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