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I apologize if this question is too vague, but it's sort of difficult to describe something well when your question about it is "How do I describe this better?" I'll try my best though.

I'm using Heyting and Boolean algebras to compare intuitionistic and classical logic. Particularly, I'm looking at how the existence of a Heyting algebra that is not also a Boolean algebra proves that it truly is not possible to prove the law of the excluded middle using the axioms of intuitionistic logic.

What I'm struggling with, is finding a term to effectively describe what this means. Some things that come to mind are "intuitionistic logic is 'valid'" or "intutionistic logic is 'separate' or 'distinct' from classical logic" or "intuitionistic and classical logic are 'not equivalent'".

I guess what I'm really asking here is, is there an actual vocabulary word that means that two different systems of logic are not actually just the same thing described in different ways? Or if there isn't, then what would you think is the best way to describe this phenomena?

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The phrase "intuitionistic logic is not equivalent to classical logic" would be universally correctly understood; other ways you could say this include "intuitionistic logic is strictly weaker than classical logic" and "intuitionistic logic is a proper sublogic of classical logic".

Let me say a bit about these last two. The first reflects the fact that classical logic deduces more things than intuitionistic logic. The latter is identical, but uses language which is suggestive of a set-theoretic interpretation of logics; and this is no accident. There are a few different definitions of what a "logic" is, but by and large they share the following observation: that a logic $\mathcal{L}$ is determined by its relation "$\Gamma\vdash_\mathcal{L}\varphi$" for $\Gamma$ a set of formulas and $\varphi$ a single formula. That is, we can identify $\mathcal{L}$ with the set (actually proper class but whatever) $$Ded_\mathcal{L}:=\{(\Gamma,\varphi): \Gamma\subseteq Form, \varphi\in Form, \Gamma\vdash_\mathcal{L}\varphi\}.$$ Under this identification, we really have $\mathcal{L}_{intuitionistic}\subsetneq\mathcal{L}_{classical}$, hence the phrase I used above.

(Note that the description of a logic given above is purely syntactic - it doesn't pay attention to what models of theories in the given logics consist of. So e.g. it can't distinguish between intuitionistic logic with the semantics of appropriate Kripke frames and intuitionistic logic with the semantics of Heyting algebras.)

Ultimately of course, if you're worried about ambiguity you should explicitly define your terms. Here, the definition of $Ded_\mathcal{L}$ above proves quite useful in concisely definition such things.

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  • $\begingroup$ I believe I've also heard people say that intuistionistic logic is "finer" because there are more logical equivalence classes. $\endgroup$
    – Mark S.
    Commented May 19, 2017 at 1:02

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