Transposition of variables is hard to describe

Abstract

The function $p_{xy}$ that interchanges two logical variables $x,y$ in formulas is hard to describe in the following sense. Let $F$ denote the Lindenbaum-Tarski formula-algebra of a finite-variable first order logic, endowed with $p_{xy}$ as a unary function. Each equational axiom system for the equational theory of $F$ has to contain, for each finite $n$, an equation that contains together with $p_{xy}$ at least $n$ algebraic variables, and each of the operations $\exists, =, \lor$. This solves a problem raised by Johnson [J. Symb. Logic] more than 50 years ago: the class of representable polyadic equality algebras of a finite dimension $n\ge 3$ cannot be axiomatized by adding finitely many equations to the equational theory of representable cylindric algebras of dimension $n$. Consequences for proof systems of finite-variable logic and for defining equations of polyadic equality algebras are given.