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I seek an equational axiomatization of the quantifiers of predicate logic (that permits empty domains).

Start with the equational axioms for Boolean algebra. Add the following axioms, which come in dual pairs:

Renaming (for $y$ not free in $a$): \begin{align} \forall x a &= \forall y a[x / y] \\ \exists x a &= \exists y a[x / y] \end{align}

Exchange: \begin{align} \forall x \forall y a &= \forall y \forall x a \\ \exists x \exists y a &= \exists y \exists x a \end{align}

Distributivity: \begin{align} \forall x (a \land b) &= \forall x a \land \forall x b \\ \exists x (a \lor b) &= \exists x a \lor \exists x b \end{align}

Empty distributivity: \begin{align} \forall x \top &= \top \\ \exists x \bot &= \bot \end{align}

However, there are some tautologies I can't derive, such as

\begin{align} \exists x a &\leq a & \text{$x$ not free in $a$} \\ a &\leq \forall x a & \text{$x$ not free in $a$} \end{align}

\begin{align} \exists x (a \land b) &\leq \exists x a \\ \forall x a &\leq \forall x (a \lor b) \end{align}

\begin{align} \forall x a \land \exists x b &\leq \exists x (a \land b) \\ \forall x (a \lor b) &\leq \exists x a \lor \forall x b \end{align}

where $a \leq b$ means $a \land b = a$ or $a \lor b = b$. What is the simplest way to complete or modify this equational axiomatization so that all tautologies (in the language generated by $\{\land, \lor, \top, \bot, \forall, \exists\}$) are derivable?

Some of the rules of passage fail on empty domains. For example, $Qx(\beta \land \alpha) \leftrightarrow \beta \land Qx \alpha$ (for $x$ not free in $\beta$) fails on the empty domain when $Q = \forall$ and $\beta = \bot$.

user76284
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  • What restrictions are you placing on the vocabulary for your logic? If you look at FOL with constant symbols only (and $=$, which is part of the logic), you get cylindric algebra. If you want the ability to express functions and relations as well, I don't know what the answer is ... I tried looking around for multi-sorted generalizations of cylindric algebra (like maybe polyadic algebra), but I haven't found a reference yet that I understand. – Greg Nisbet Jan 22 '23 at 18:34
  • @GregNisbet Preferably the latter. – user76284 Jan 22 '23 at 19:48

1 Answers1

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I think that Valby's paper The Universal Theory of First Order Algebras and Various Reducts answers your question.

As Greg Nisbet mentions in the comments, this work is closely related to the theory of cylindric algebras, which goes back to Tarski. But Valby works in a multi-sorted setting and allows empty structures, and I find his paper much easier to understand than the literature on cylindric algebras.

Alex Kruckman
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  • Thank you for your answer. I wonder if there's an equational axiomatization that's closer in spirit to that presented in the question. (I may have to read the paper more closely.) – user76284 Jan 30 '23 at 04:02
  • @user76284 The equational axioms for Boolean algebras are Valby's axioms (1) and (6). Your Distributivity and Empty Distributivity for $\exists$ are Valby's axiom (7). Their duals for $\forall$ are derivable in Valby's system after defining $\forall$ as $\lnot \exists \lnot$. Valby works with the convention that every variable context has the form $x_1,\dots,x_n$. The role of your Renaming is to generalize to allow arbitrary variable contexts. – Alex Kruckman Jan 30 '23 at 15:23
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    So the main differences between your system and Valby's is that (a) you take Exchange as primitive, while it's derivable in Valby's system, and (b) Valby's system includes all the other axioms that you need to completely axiomatize the behavior of quantifiers. I wouldn't say they're really very different in spirit. Also, note that Valby's axiom (0) is the only one that isn't equational, and he proves that if you drop it, the remaining axioms axiomatize the equational theory. – Alex Kruckman Jan 30 '23 at 15:24