What is the justification in using logical equivalence to reformulate propositions?

Question

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 falseIFF the second formulation is false”.

My 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!

Answer 1

Perhaps surprisingly, I'm going to argue that you are essentially correct: when we assert that an argument is valid, we're doing so in the context of some particular logical system. People who disagree about what logical system is the "right" base to use will disagree over what a valid argument is. And contra Jose Carlos Santos' answer, such disagreements do exist - for example, consider intuitionism, which rejects the general equivalence between an implication and its contrapositive. There are also serious mathematicians who reject the idea of a "one true logic" in the first place.

The completely unambiguous way to phrase a validity claim, then, is:

In such-and-such logical system, the following argument is valid.

This is a completely concrete claim, and one which we can all verify regardless of our foundational stances.

(Granted, this does of course assume that some very basic level of common understanding exists: we have to all understand what the rules of the logical system are, and how arguments are formed. Rejecting that basic commonality makes everything impossible (see e.g. here), and this level of skepticism is a whole different thing from the higher-level logical debates relevant to this question. While much interesting ink has been spilled on how much skepticism we can adopt while preserving the ability to communicate (see e.g. Wittgenstein or Kripke), that's a fundamentally different issue from what you raise here.)

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At this point it's important to mention that the study of alternate logical systems is valuable even if we genuinely accept classical logic. For example, intuitionistic logic has found applications in the proof theory of classical theories, in computability theory, and in topos theory.

This is to my mind a fascinating situation, and the primary reason why I consider myself a logical pluralist (although I almost always work inside classical logic).

Answer 2

Your assumption that “these sorts of ‘equivalencies’ solely depend on the definitions/structures that mathematicians used to initially create propositional logic” is wrong. This kind of proof was already used in Philosophy at least two generations before Aristotle formalized Logic. It simply says that to prove that “If I have $P$, then I also have $Q$.” is exactly the same thing as to prove that “If I don't have $Q$ then I also don't have $P$.” This is a basic rule of thought, and no mathematician doubts it.

Answer 3

As for your answer, I refer you to the response of Jose Carlos Santos and Noah Schweber. I believe Carlos was simply pointing out that mathematicians who utilize the contrapositive (or any equivalency unique to the particular system they're using) don't do so without justification, and the justification is provided by the logical system itself. I would like to add something to feed your line of inquiry:

If you are intrigued by the notion that such "rules of thought" which seem so concrete and objective could actually be quite... subjective or lacking justification... then I encourage you deepen your learning by exploring how all formal systems of reasoning, including mathematics, are developed.

In short, they all rest on a set of axioms whose truthfulness and consistency cannot be proven, but is simply assumed. This is true of logical systems, mathematics, and even science in general. Unfortunately, this is an unavoidable tragedy because attempting to prove the axioms of a system will send you on an infinite chain of reasoning or even on the doorstep of contradiction. We make peace with this dilemma by making our axioms as few as possible and choosing axioms whose truthfulness appears to be self-evident.

With this mind, be aware that the practical utility of any formal system of reasoning is dependent on the extent to which the axioms of the system apply to the situation you're dealing with. In fact, this is how many "new" systems of reasoning are developed. People discover a domain in which one or more axioms of system A no longer apply, and then... viola! System A must be reconfigured or extended in some way to make sense of the new domain.