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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.
##### MSC:
 03B55 Intermediate logics 03B60 Other nonclassical logic 03G27 Abstract algebraic logic 03G25 Other algebras related to logic
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