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I study maths, and I have found that a useful way of thinking about two propositions A and B being equivalent is to regard them as being two different ways of saying the same thing, or equivalently, interchangeable.

However, what I find puzzling from a philosophical perspective is why it may seem that one of the equivalent statements is much more apparent than the other. For example, it has been proven that Euclid's parallel postulate is equivalent to all of the following:

  1. two lines that are parallel to the same line are also parallel to each other
  2. there exists a triangle with internal angles summing to 180°
  3. every triangle in the plane has internal angles summing to 180°
  4. the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the legs of the triangle (Pythagoras's theorem)

I have listed them in the (subjective) order "most obvious" to "least obvious". In particular, the idea that (2) and (3) are equivalent is really quite puzzling. Intuitively, (2) seems to be a much less stringent condition than (3) - it is much more apparent, even though both provably say exactly the same thing!

My initial thought was that the fact that we perceive a difference indicates only that we understand some topics better than others i.e. that all equivalent statements ought to have the same apparency, but that is not true for us because we do not have enough knowledge. But I am not convinced that this fully explains (2)-(3).

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    Equivalent means "same truth value" in propositional logic and means inter-derivability (in a certain context) in a more broad context. But this does not explain all intuitive nuances of "equivalent" and thus we have Intensional Logic.
    – Mauro ALLEGRANZA
    Commented Jul 7, 2021 at 19:29
  • Because equivalence is a relative notion, it is relative to the background assumptions under which the propositions are considered, and which may involve conceptions not even present in the propositions themselves. In Euclid's case they are the remaining axioms of geometry. Equivalencies are not apparent to us for the same reason that logical consequences of a group of assumptions are not, the longer an inference chain that establishes a consequence the less we are able to "see" it.
    – Conifold
    Commented Jul 7, 2021 at 22:37
  • @Conifold In this case, that may be the issue. Unfortunately, I don't know which of Euclid's other axioms each proposition requires in order to be equivalent. However, I believe the problem can persist even for other sets of equivalent propositions founded on exactly the same axioms. Are you certain that it doesn't?
    – legionwhale
    Commented Jul 8, 2021 at 18:50
  • @MauroALLEGRANZA I read only as much as I could understand, but I disagreed strongly from the opening. It is correct to say that the morning(/evening) star is Venus. However, I do not think it is correct to say that Venus is the morning star. The morning star is specifically referring to "Venus is in the morning sky", not "Venus". Thus, I do not recognise this as an equivalence or equality, and therefore I do not have any problems such as "The morning star is the evening star". C.f. all dogs are tetrapods but obviously, not all tetrapods are dogs. Is this any different? I don't believe it is.
    – legionwhale
    Commented Jul 8, 2021 at 18:58
  • It is the complexity of inference chains that matters for "apparency", not whether background assumptions are the same. Different equivalencies may have proofs of different length and complexity even if they use the same axioms. See What is the difference between depth and surface information? for one way to measure "apparency".
    – Conifold
    Commented Jul 8, 2021 at 23:07

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Your question is related to the question: how can a deductively valid argument tell you something that you don't already know?

Human beings are not logically omniscient, i.e. we do not know the logical consequence of everything we know. One of the purposes of a good proof is to take a potentially non-obvious logical consequence relation and express it as a sequence of simple and obvious steps. In the case of logical equivalence, there is a two-way logical consequence relation. If A entails B and also B entails A, then A and B are logically equivalent. But as with any logical consequence relation, the fact that A and B entail each other does not mean that it is obvious that they do.

To say of two propositions that they "say the same thing" suggests something stronger, perhaps something like they convey the same information or the same meaning to you. This is a more finely-grained distinction than logical equivalence. Meanings relate to intensions as well as extensions. Also, propositions that are logically equivalent are not always interchangeable. It is possible even for propositions that share the same intension to fail to preserve truth when substituted. This is sometimes referred to as hyperintensionality.

So, rather than thinking of logically equivalent propositions as "saying the same thing" or as being "interchangeable", it is better to think of them in terms of a pair of consequence relations.

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  • I think I myself was slightly confused by my choice of Euclid's postulates, since they are axioms rather than results. But I disagree that this is about logical omniscience. I agree that a result proved from axioms may be less apparent than the axioms -- I presume that apparency depends on the length of the inference chain. However, with two equivalent statements (say A and B), we have an inference chain from both sides. Therefore it is not clear why we should regard A as less apparent than B. Because we proved A first? Aliens who proved B first (finding it more apparent) would disagree.
    – legionwhale
    Commented Jul 28, 2021 at 23:56
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    I suppose you could fall back on saying that one of a pair of logically equivalent propositions just seems more reasonable in some intuitive way because it makes a better fit with our mental models. I remember a mathematician joking that in set theory, the well-ordering theorem is obviously false, the trichotomy is obviously true, and he couldn't decide whether the axiom of choice is true or not.
    – Bumble
    Commented Jul 29, 2021 at 0:58
  • Yes, I suppose so. When I asked this, I was wondering why it might be that certain propositions fit better with our mental models, especially in a pure field like maths, but I guess that none of us could really answer definitively. I like the joke. The axiom of choice seems pretty shaky to me, but I should probably study set theory before I make a judgement.
    – legionwhale
    Commented Jul 29, 2021 at 11:43

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