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Intuitionistic and classical propositional logic, and even classical first-order logic with identity, have algebraic counterparts. Algebraizable logics, 1989, by Willem J. Blok and Don Pigozzi, is a classical reference.

Is more now known about whether stronger systems are algebraizable?

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    $\begingroup$ Sure, instead of saying $x^3+y^3=z^3\to xyz=0$, say $\max((xyz)^2(1-(x^3+y^3-z^3)^2))=0$. $\endgroup$
    – user44143
    Commented Jul 22, 2020 at 22:16
  • $\begingroup$ At least that was discovered after 1989. :) $\endgroup$
    – Frode Alfson Bjørdal
    Commented Jul 22, 2020 at 22:31
  • $\begingroup$ For this case, Euler had everything needed by 1770. $\endgroup$
    – user44143
    Commented Jul 22, 2020 at 23:12
  • $\begingroup$ Indeed. Anyway, I would like to know some about what such algebras look like if they exist. $\endgroup$
    – Frode Alfson Bjørdal
    Commented Jul 22, 2020 at 23:34

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