Here's a familiar conversation:


**Me:** Do you think Conjecture A and Conjecture B are equivalent?


**Friend:** Yes, because I think they're both true.


**Me:** [eye roll] You know what I mean...


Does there exist a rigorous notion of what I mean? Perhaps something about the existence of a proof of $p \Leftrightarrow q$ that doesn't "come close" to revealing the truth value of either $p$ or $q$? Unfortunately, the idea of "coming close" sounds rather subjective and reminiscent of [Timothy Chow's answer here](https://mathoverflow.net/a/46979/29873). Perhaps there's another approach?


**Edit:** Judging by the immediate response, it seems that my question requires additional clarification. I am wondering whether there is a rigorous notion in logic for pairs of statements that are logically equivalent in a nontrivial sense. Observe that the fundamental theorem of algebra is equivalent to $0=0$, but in a trivial sense since they are both true. On the other hand, one might consider $p\Leftrightarrow q$ to be "nontrivially true" if there exists a proof of $p\Leftrightarrow q$ despite $p$ not being provable. Is there a more general theory that applies in the case where $p$ is provable?



What does it mean to suspect that two conjectures are logically equivalent?
First of all, in practice when we say "Conjecture A is equivalent to Conjecture B," what we mean is "We *have a proof* that Conjecture A is true iff Conjecture B is true." We can have such a proof without having a proof of either conjecture, so this is a meaningful situation. Of course, it will (hopefully) *later become* trivial, when we prove or disprove the conjectures (and so reduce it to "they're both true" or "they're both false").


But your question has more to it than that: suppose we want to say that two *theorems we already know to be true* are equivalent. How can we do that? (Note that this is something we in fact do all the time - e.g. when we say "the compactness of the real numbers is equivalent to their satisfying the extreme value theorem.")


The simplest approach to this is by considering extremely weak axiom systems, which aren't strong enough to prove either result but *can* prove the equivalence. That is, we work over some very weak "base theory."


Historically, of course, the most well-known example is the study of equivalences/implications between versions of the axiom of choice over the theory ZF; as a fun fact, [there's a famous story](https://mathoverflow.net/a/13903/8133) that when Tarski tried to publish a certain equivalence over ZF, one editor rejected it on the grounds that the equivalence between two *true* statements isn't interesting and the other rejected it on the grounds that the equivalence between two *false* statements isn't interesting. *(I believe there were also hints of interest in equivalences between true principles in the study of [absolute geometry](https://en.wikipedia.org/wiki/Absolute_geometry), but I'm not certain - it's been a while since I looked at the history of non-Euclidean geometry.)* However, ZF is "too strong" for most statements of mathematical interest, so we want to go deeper into things.


This is one of the motivations behind [**reverse mathematics**](https://en.wikipedia.org/wiki/Reverse_mathematics): we look at equivalences/implications/nonimplications over a very weak theory, RCA$\_0$, which intuitively corresponds to "computable" mathematics with "finitistic" induction only.


For example, here are some statements which are all equivalent to each other in the sense of reverse mathematics:


* Every commutative ring which is not a field or the zero ring has a nontrivial proper ideal.
* $[0,1]$ is compact.
* Every infinite binary tree has an infinite path.


*(There is actually a serious issue here which doesn't really crop up when proving equivalences over ZF, namely that we have to figure out how to express the statements we care about in the language of our base theory; this is an issue with weak theories like RCA$\_0$ whose language is quite limited. I'm ignoring this issue completely here.)*


And we sometimes want to go weaker still. Equivalences over theories much weaker than RCA$\_0$ have been studied, albeit not (in my understanding) as extensively.


