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I have a question on proof by contradiction. The Wiki page steps are to assume $\lnot P$ and derive a contradiction, formulated $\lnot p \implies \bot$. The law of non-contradiction is $\lnot(q \land \lnot q)$. Wiki explains $\bot$ is the logical contradiction or a false statement. It's my understanding the logical contradiction is ($p \land \lnot p$).

I have some confusion about the term contradiction and the $\bot$ symbol. Wiki writes a contradiction is logical incompatibility or incongruity between two or more propositions. Then Wiki states that In modern formal logic, the term is mainly used instead for a single proposition, often denoted by the falsum symbol. $\bot$. Wiki also has an article to assume $\lnot P$ and derive $\lnot Q \land Q$.

When following the steps, should one derive a contradiction or simply a false statement?

Did perhaps the Wiki authors really intend $\lnot p \implies (q \land \lnot q)$?

bof
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Nick
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  • I'm not sure what exactly you're asking here. It seems that what you're saying is that a contradiction is the falsum symbol and then talking about proving p implies q by contradiction? –  Nov 19 '20 at 00:46
  • I'm asking about the steps to proof by contradiction and the discrepancies in the Wiki articles. I do not mention $p \implies q$, where did you see that? – Nick Nov 19 '20 at 00:59
  • Except your first wiki article isn't the main proof by contradiction page it's just on contradiction –  Nov 19 '20 at 02:47
  • I think you're overcomplicating it. The point of proof by contradiction is to assume $\neg p$ and show that it implies $p$, which is a contradiction, because if $\neg p \implies p$ were true, that would further imply that $\neg p \wedge p$ is true, which definitively ⊥ (false). – Graviton Nov 19 '20 at 02:49

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