Quantified Boolean algebra written in Elementary Algebra

Question

Propositional calculus can be "encoded" into elementary algebra using the formulas

$\neg x = 1-x$

$a \wedge b = ab$

Is there a way to encode the quantifiers $\forall$ and $\exists$ into elementary algebra in a similar way?

Answer 1

If you code truth values as numbers in the standard way with $0$ representing falsehood and $1$ representing truth, then $\forall$ is $\inf$ and $\exists$ is $\sup$. More precisely, if we write $\phi \mapsto \phi^*$ for the translation of logical formulas into arithmetic expressions, then we have:

$$ \begin{align} (\forall x\phi)^* &\equiv \inf\{x \mid \phi^* = 1\}\\ (\exists x\phi)^* &\equiv \sup\{x \mid \phi^* = 1\} \end{align} $$