Algebraic Logic Perspective on Prucnal’s Substitution

Abstract

A term $\mathit{td}(p,q,r)$ is called a ternary deductive (TD) term for a variety of algebras $\mathcal{V}$ if the identity $\mathit{td}(p,p,r)\approx r$ holds in $\mathcal{V}$ and $(\mathsf{c},\mathsf{d})\in \theta (\mathsf{a},\mathsf{b})$ yields $\mathit{td}(\mathsf{a},\mathsf{b},\mathsf{c})\approx \mathit{td}(\mathsf{a},\mathsf{b},\mathsf{d})$ for any $\mathcal{A}\in \mathcal{V}$ and any principal congruence $\theta$ on $\mathcal{A}$. A connective $f(p_1,\dots ,p_n)$ is called $\mathit{td}$-distributive if $\mathit{td}(p,q,f(r_1,\dots ,r_n\left)\right)\approx$ $f\left(\mathit{td}\right(p,q,r_1),\dots ,\mathit{td}(p,q,r_n\left)\right)$. If $\mathsf{L}$ is a propositional logic and $\mathcal{V}$ is a corresponding variety (algebraic semantic) that has a TD term $\mathit{td}$, then any admissible in $\mathsf{L}$ rule, the premises of which contain only $\mathit{td}$-distributive operations, is derivable, and the substitution $r\mapsto \mathit{td}(p,q,r)$ is a projective $\mathsf{L}$-unifier for any formula containing only $\mathit{td}$-distributive connectives. The above substitution is a generalization of the substitution introduced by T. Prucnal to prove structural completeness of the implication fragment of intuitionistic propositional logic.