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Algebraic logic perspective on Prucnal’s substitution. (English) Zbl 1403.03046
Summary: A term \(td (p,q,r)\) is called a ternary deductive (TD) term for a variety of algebras \(\mathcal{V}\) if the identity \(\operatorname{td}(p,p,r)\approx r\) holds in \(\mathcal{V}\) and \((\mathsf{c},\mathsf{d})\in\theta(\mathsf{a},\mathsf{b})\) yields \(\operatorname{td}(\mathsf{a},\mathsf{b},\mathsf{c})\approx td (\mathsf{a},\mathsf{b},\mathsf{d})\) for any \(\mathscr{A}\in\mathcal{V}\) and any principal congruence \(\theta\) on \(\mathscr{A}\). A connective \(f(p_{1},\ldots,p_{n})\) is called \(\operatorname{td}\)-distributive if \(\operatorname{td}(p,q,f(r_{1},\ldots,r_{n}))\approx f(td(p,q,r_{1}),\ldots,td (p,q,r_{n}))\). If \(\mathsf{L}\) is a propositional logic and \(\mathcal{V}\) is a corresponding variety (algebraic semantic) that has a TD term \(td\), then any admissible in \(\mathsf{L}\) rule, the premises of which contain only \(\operatorname{td}\)-distributive operations, is derivable, and the substitution \(r\mapsto td(p,q,r)\) is a projective \(\mathsf{L}\)-unifier for any formula containing only \(\operatorname{td}\)-distributive connectives. The above substitution is a generalization of the substitution introduced by T. Prucnal [Stud. Log. 30, 45–52 (1972; Zbl 0268.02013)] to prove structural completeness of the implication fragment of intuitionistic propositional logic.
03B55 Intermediate logics
03B60 Other nonclassical logic
03G27 Abstract algebraic logic
03G25 Other algebras related to logic
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