I apologize in advance for my lack of knowledge about the terminology of formal logic. I am only interested in informal logic to the extent that a practicing mathematician needs it to proceed. Despite years of experience in mathematics, I am finding myself confused about what a contradiction means. According to this site,
A contradiction is a conjunction of the form "A and not-A"... So, a contradiction is a compound claim, where you’re simultaneously asserting that a proposition is both true and false.
I doubt that this is mathematical definition though, as Wikipedia's article on contradiction defines that
a proposition is a contradiction if false can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false
- Main question: I'm confused as to the difference between a contradiction and a false statement. If I say that $x\in S\wedge x\not\in S$ then is this a contradiction or a false statement? There seems to be two ideas at play, one being a statement that is simply false like "The sky is red" versus something like $P\wedge \neg P$ where the $P$ can be any statement with a true/false value like a proposition or quantified predicate but regardless of whether $P$ is $0$ or $1,$ the value of $P\wedge\neg P$ is $0 $ (false). In the former case, there is no varying in the underlying components whereas in the latter we compute a truth table to find that we always get $0.$ I am running into the issue of distinguishing between the two because this article on proof by contradiction uses the $\bot$ symbol and I don't know whether it is refering to a false statement or a logical contradiction, where by a false statement I mean something like "The sky is red" and by a contradiction I mean a statement like $P\wedge\neg P$ whose truth table has all $0$'s in the final column (I don't know if these are the right definitions for the terms).
- Side question: Are all contradictions, that is those statements that evaluate to a truth table of all $0$'s in the final column, logically equivalent to a statement of the form $P\wedge \neg P$? A counterexample or proof would be appreciated.