What is the definition of an embedding of one logic in another?

Question

An oft-cited result about intuitionistic logic is that

Proposition 1. Classical logic can be embedded into intuitionistic logic.

This is the Godel-Gentzen double-negation translation: given a formula $\phi$, we insert $\lnot \lnot$ throughout the formula to obtain a new formula $\texttt{encode}(\phi)$, such that $$ \phi \text{ is provable in classical logic} \iff \texttt{encode}(\phi) \text{ is provable in intuitionistic logic}. \tag{1} $$

People cite this as a result that intuitionistic logic is more expressive or more general. So there is an unspoken assumption that:

Proposition 2. Intuitionistic logic does not embed into classical logic.

My question is, what is the standard definition of embedding used such that Propositions 1 and 2 are both true?

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Why this is not obvious:

The condition (1) cannot be the definition of embedding used: it is easy to provide such an encoding $\texttt{encode}$ from any logic to any other non-trivial logic. Just send any provable statement to a simple provable sentence (like $1 = 1$), and any unprovable statement to simple unprovable sentence (like $0 = 1$).

There are less trivial encodings, too. Suppose we require the encoding to be computable, or that it preserves some structure (each step of a proof of $\phi$ can be translated into a step of a proof of $\texttt{encode}(\phi)$). Then I think it may be true that such a translation exists from intuitionistic logic (or any other logic) into classical logic: we just have to introduce sufficiently many axioms as part of the encoded formula to describe how the source logic works. So for a formula in intuitionistic logic $\psi$, we define $\texttt{encode}(\psi)$ in classical logic as the statement that the conjunction of all those axioms implies "$\psi$ is provable", where "is provable" is also encoded in the target language. (Some consistency assumptions may be necessary on the set of axioms to make sure it faithfully describes which statements of intuitionistic logic are provable.)

Answer 1

Only a partial answer:

There isn't a single definition of "logic," but if we think of a logic as just a set of "sentences" together with a "deduction relation" (relating sets of sentences to individual sentences: a la "$\Gamma\vdash_\mathcal{L}\varphi$"), then a natural notion of embedding of $\mathcal{L}$ into $\mathcal{L}'$ is an injection $f$ from $Sent_\mathcal{L}$ to $Sent_{\mathcal{L}'}$ such that for all $\Gamma\subseteq Sent_\mathcal{L}$ and $\varphi\in Sent_\mathcal{L}$ we have $$\Gamma\vdash_\mathcal{L}\varphi\iff \{f(\gamma): \gamma\in\Gamma\}\vdash_{\mathcal{L}'}f(\varphi).$$

This does not allow for the kind of trivial embedding you've described, but the double negation translation is an embedding in this sense. As an example of a nontrivial result about these embeddings, note that it follows that no non-compact logic can embed into a compact one - which makes sense.

However, there's still not much structure here, and while this is a better definition of embedding it's not clear whether it's "the right one." We might, for example, want to demand more structure.$^1$ This reflects the interesting debate I essentially ignored at the start of this answer, namely what exactly a logic is. This discussion on the FOM mailing list has a number of useful sources, with a positive result by Jerabek and a negative result by Epstein in particular depending on how exactly the question is phrased.

$^1$Here's an example of some additional structure we might want. Intuitionistic logic has "triple negation elimination:" $$\neg\neg\neg \varphi\dashv\vdash_{int} \neg \varphi.$$ So we might want that to be reflected on the classical side: that there should be some "nice" operator $o$ on classical sentences such that

• $ooo\varphi\dashv\vdash o\varphi$ and

• $f(\neg\varphi)=of(\varphi)$ where $f$ is our translation.

This is a tall order (and in fact I think doesn't exist in general). More generally, we could demand that "expressible operations" go to "expressible operations." This is a huge restriction and reflects the fact that we generally think of a logic as more than just a set of sentences with a deduction relation - in particular, that we usually think of that set of sentences itself having some algebraic structure (the syntax).