I am aiming to understand the style of argument that mathematical logic is a form of, so in some ways my question might seem more philosophical than mathematical but please bear with me.

So, essentially, we assume a statement to be true. We use certain logical deductions allowed in the mathematical framework we are working under and arrive at the "not" of that statement- so, in other words, if the statement were true, it would also have to be false as per our logical deductions. This is what we call a contradiction. From this, we conclude that our assumed statement must have been false.

Now, I understand that if we have statement P, then P and not P is always false (a tautology). But what I am looking for are natural, intuitive reasons as to why we described certain things to be tautologies.

Consider this- Water being wet and not wet at the same time does not make sense. We have never come across something in nature that has a certain property and doesn't have that same property at the same time. So, I can see a justification for why P and not P would be false.

For proof by contradiction to "make sense", are we not saying that contradictions can only arise from false statements under sound logical deductions? How do we know this to be true? In real life, I cannot come up with an example where a true statement is creating a contradiction under logical deductions, so I can believe that only false statements do that. But is that not an inherent assumption that I am making based on what I see in nature? What if there are false statements that produce contradictions but we just haven't thought of them yet?

That, at the core, is my basic question. Please note that I am currently taking an introductory course in proof writing and I do not have any experience in formal logic.

  • $\begingroup$ If you look at the list of "Related" questions, you will see several where your questions may already have been raised and answered. Try it! $\endgroup$ – Gerry Myerson Oct 23 '18 at 3:06
  • $\begingroup$ I did try looking through them, but the most satisfactory answer I got was "mathematics is consistent and every time we have adopted the method of proof by contradiction, we have never been wrong.. yet". So, does this not mean that it is an assumption that has worked in countless many cases such that mathematicians accept it to be a fundamental principle of sorts, but at the core of it, it is just an assumption? $\endgroup$ – childishsadbino Oct 23 '18 at 6:04
  • $\begingroup$ At the core of it, everything is just an assumption. $\endgroup$ – Gerry Myerson Oct 23 '18 at 9:14

The crux of the proof by contradiction isn't that a given statement is false; it is that R and $\lnot$R cannot both occur at the same time. So if we want to prove P $\rightarrow$ Q and we show that $P \lnot Q \rightarrow (R \land\lnot R)$ is true, then it must be that $P \land \lnot Q$ is false. This last statement is logically equivalent to P $\rightarrow$ Q.

  • $\begingroup$ I realize that, but the move from "R and not R" being false implying P and not Q is false is what I am calling into question. If you define implications in the way we do in math, this makes total sense. My question is, WHY did we say that if something is false, then under logical deductions, the statement producing the falsity must be false? Intuitively, like I said in my question, I can't think of an example where something true produces a contradiction under logical arguments. But what if there are cases like that? $\endgroup$ – childishsadbino Oct 23 '18 at 6:08
  • $\begingroup$ @childishsadbino We are trying to see that if P occurs, then necessarily Q occurs. in other words, is Q is ALWAYS a consequence of P? The implication P$\rightrrow$Q is false if and only if P is true and Q is false. Any other case gives a true implication. For example, "If 2+2=100, 2 is a chair." is a true statement, whereas "If 2+2=4, 2 is a chair." is a false statement. Contradictions cannot arise. No statement can be both true and false. Depending on the assumptions, it is one or the other. If a contradiction arises, then the assumptions were wrong. $\endgroup$ – Joel Pereira Oct 23 '18 at 17:19

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