Algebraic semantics for the (↔, ¬¬)‐fragment of IPC

Abstract

MSC (2010) 03B20We show that the variety of equivalential algebras with regularization gives the algebraic semantics for the (↔, ¬¬)-fragment of intuitionistic propositional logic. We also prove that this fragment is hereditarily structurally complete.

On Rules

“…It has long been known that the implicational fragment of IPC is hereditarily structurally complete [37] (the first occurrence of Prucnal's trick). The same holds for the implication-conjunction and some other fragments of IPC [32,47]. In [32] Mints showed that any admissible underivable rule of IPC must contain both implication and disjunction.…”

Algebraic semantics for the ‐fragment of and its properties

“…Hereditary structural completeness is established for many fragments of , including [, Corollary 4.7] and [, Corollary 5.2]. On the other hand, it does not hold for all fragments whose signature includes both implication and disjunction , as well as for some other fragments, as, e.g., for (cf.…”

Algebraic Logic Perspective on Prucnal’s Substitution

“…Hence, H is hereditarily ¹^; ! ; 1; $; ::º-structurally complete or, for instance, ¹$; ::º-structurally complete (see Słomczyńska [34]).…”