# Why do equivalent propositions sometimes differ in apparency?

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).

• 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. 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. 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? 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. 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". Jul 8, 2021 at 23:07