Hopefully I can sufficiently outline my question in the following paragraph. I just finished reading Daniel Solow’s book “How to Read and Do Proofs”. It was a great book for a beginner like me, and I learned a great deal. However, there are certain portions of the book that I purposefully tried not to “overthink” in order to take away the major points. Now that I have finished… :)

One such section is related to reformulating propositions into logically equivalent structures that may be easier to prove than the original proposition. The **contrapositive method** is one such example of this.

*As a quick showcase:*

Proposition: The function $f(x) = x^3$ is injective.

By definition of injective, this means, “If $u \neq v$, then $u^3 \neq v^3$”. This is not a straightforward proof to me…but by using the contrapositive form, “If $u^3=v^3$, then $u=v$” I can quickly find the proof.

Now, I am quite aware of what logical equivalence means with respect to truth tables. Using the contrapositive as an example, if I have a premise $P$ and a premise $Q$, then the implication $P \Rightarrow Q$ has the same truth table as $\neg Q \Rightarrow \neg P$.

My interpretation of this is that, “*these two propositions are true or false under the exact same circumstances*” …which I guess is why we can treat them as equivalent. Or on a similar note, “*The first formulation is true IFF the second formulation is true*

**and**

*the first formulation is false*”.

**IFF**the second formulation is falseMy confusion/curiosity is as follows: these sorts of “equivalencies” solely depend on the definitions/structures that mathematicians used to initially create propositional logic. As such, it seems to me that **“logical equivalency” is valid only because there is a general consensus amongst mathematicians who concur that it IS valid**. Is this a correct statement? Or am I missing something.

Are there mathematicians who do not believe in these sorts of reformulations as being “equivalent”?

Thank you!