TITLE: What exactly is a contradiction and how does it differ from falsity?
QUESTION [13 upvotes]: 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

Two questions:

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.

REPLY [12 votes]: The following is less concrete than lemontree's answer and amWhy's comments, which I think are more on-point. However, I do think the following is worth saying, so I'm putting it here.
The snappy version, as you suspect, is:

A contradiction is never true in any situation. A statement is called false if it fails in the particular situation (or class of situations) we care about - but a false statement may nonetheless hold in a different situation (whereas a contradiction cannot).

Below I'll describe the two main ways of making this precise.

Semantic version
The "semantic" view of logic is that a logical system $\mathcal{L}$ is used to describe objects (or structures): basically, such an $\mathcal{L}$ consists of a class of sentences $Sent_\mathcal{L}$, a class of applicable structures $Struc_\mathcal{L}$, and a relation $\models_\mathcal{L}$ between applicable structures and sentences with $$\mathfrak{A}\models_\mathcal{L}\varphi$$ being interpreted as "the sentence $\varphi$ is true in the structure $\mathfrak{A}$."
A contradiction in the sense of $\mathcal{L}$, then, is a sentence which is not true in any structure: a $\psi$ such that for every $\mathfrak{A}$ we have $\mathfrak{A}\not\models_\mathcal{L}\psi$. By contrast, when we decide to focus on a particular structure $\mathfrak{S}$, we say that $\varphi$ is false iff $\mathfrak{S}\not\models_\mathcal{L}\varphi$.

Syntactic version
We can also refrain from talking about structures entirely. The "syntactic" view of logic is that a logical system is used to manipulate sentences (without necessarily assigning them particular meanings). Basically, such an $\mathcal{L}$ consists of a class of sentences $Sent_\mathcal{L}$ and a relation $\vdash_\mathcal{L}$ between sets of sentences and individual sentences with $$\Gamma\vdash_\mathcal{L}\varphi$$ being interpreted as "the sentence $\varphi$ is deducible from the set of sentences $\Gamma$."
A contradiction in this framework is then a sentence from which we can deduce anything: $\varphi$ is a contradiction in the sense of $\mathcal{L}$ iff for all $\psi$ we have $\{\varphi\}\vdash_\mathcal{L}\psi$. By contrast, when we say that a sentence $\varphi$ is false, what we mean is that we have in mind some particular "background set of sentences" $\Gamma$ and $\Gamma\cup\{\varphi\}$ would let us deduce anything (think of this $\Gamma$ as our set of axioms).

Connecting the two
It's worth noting that every semantic logic induces a syntactic logic: given a semantic logic $\mathcal{L}=(Sent_\mathcal{L}, Struc_\mathcal{L},\models_\mathcal{L})$ we get a syntactic logic $\mathcal{L}'=(Sent_{\mathcal{L}'}, \vdash_{\mathcal{L}'})$ defined as follows:

$Sent_{\mathcal{L}'}=Sent_\mathcal{L}$, that is, we use the same sentences for both logics.

We set $\Gamma\vdash_{\mathcal{L}'}\varphi$ iff whenever $\mathfrak{A}\in Struc_\mathcal{L}$ with $\mathfrak{A}\models_\mathcal{L}\psi$ for each $\psi\in\Gamma$, we have $\mathfrak{A}\models_\mathcal{L}\varphi$.


Note that this makes the two notions of "contradiction" line up: if $\varphi$ fails in every structure, then vacuously we have $\{\varphi\}\vdash_{\mathcal{L}'}\psi$ for every $\psi$.
There is also a way to go "syntax-to-semantics" which again makes the two notions of "contradiction" line up, but it's a bit less natural (basically we interpret "structure" as "set of sentences which doesn't deduce everything and is maximal with that property").

A caveat
Actually, the above isn't entirely accurate: there are logical systems where sentences of the form "$P\wedge\neg P$" do not let you deduce everything (these are called "paraconsistent logics;" another relevant (hehe) term is "relevance logics"). This leads to a more nuanced notion of "contradiction" and its relatives. But that's a more advanced topic which I wouldn't approach before first understanding the classical picture.