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2016 Algebraic Logic Perspective on Prucnal’s Substitution
Alex Citkin
Notre Dame J. Formal Logic 57(4): 503-521 (2016). DOI: 10.1215/00294527-3659423

## 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 $\mathit{td}(p,p,r)\approxr$ 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 $\mathscr{A}\in\mathcal{V}$ and any principal congruence $\theta$ on $\mathscr{A}$. A connective $f(p_{1},\dots,p_{n})$ is called $\mathit{td}$-distributive if $\mathit{td}(p,q,f(r_{1},\dots,r_{n}))\approx$ $f(\mathit{td}(p,q,r_{1}),\dots,\mathit{td}(p,q,r_{n}))$. If $\mathsf{L}$ is a propositional logic and $\mathcal{V}$ is a corresponding variety (algebraic semantic) that has a TD term $\mathit{td}$, 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.

## Citation

Alex Citkin. "Algebraic Logic Perspective on Prucnal’s Substitution." Notre Dame J. Formal Logic 57 (4) 503 - 521, 2016. https://doi.org/10.1215/00294527-3659423

## Information

Received: 30 September 2012; Accepted: 30 January 2014; Published: 2016
First available in Project Euclid: 13 August 2016

zbMATH: 06663938
MathSciNet: MR3565535
Digital Object Identifier: 10.1215/00294527-3659423

Subjects:
Primary: 03B55, 03B60
Secondary: 03G25, 03G27

Rights: Copyright © 2016 University of Notre Dame

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Vol.57 • No. 4 • 2016