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.
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.