I’ve seen both symbols used to mean “therefore” or logical implication. It seems like $\therefore$ is more frequently used when reaching the conclusion of an argument, while $\implies$ is for intermediate claims that imply each other. Is there any agreed upon way of using these symbols, or are they more or less interchangeable?
“It seems like $\therefore$ is more frequently used when reaching the conclusion of an argument, while $\implies$ (alternatively $\rightarrow$) is for intermediate claims that imply each other.”
Your supposition is largely correct; my only concern is your description of $\implies$ being used to denote intermediate claims (in a proof or an argument, for example) that imply each other. The $\implies$ denotation, as in $p \implies q$, merely conveys that the preceding claim ($p$, if true) implies the subsequent claim $q$; i.e., it does not denote a bi-direction implication $\iff$ which reads “if and only if”.
‘$\implies$’ or ‘$\rightarrow$’ is often used in a “modus ponens” style (short in scope) argument: If $p\implies q$, and if it’s the case that $p$, then it follows that $q$.
Typically, as you note, $\therefore$ helps to signify the conclusion of an argument: given what we know (or are assuming as given) to be true and given the intermediate implications which follow, we conclude that…
So, put briefly, $\implies$ (“which implies that”) is typically shorter in scope, usually intended to link, by implication, the preceding statement and what follows from it, whereas ‘$\therefore$’ has typically, though not always, greater scope, so to speak, linking the initial assumptions/givens, the intermediate implications, with “that which was to be shown” in, say, a proof or argument.
Added:
I found the following Wikipedia entry on the meaning/use of the symbol’$\therefore$’, from which I’ll quote:
To denote logical implication or entailment, various signs are used in mathematical logic: $\rightarrow, \;\implies, \;\supset$ and ⊢, ⊨. These symbols are then part of a mathematical formula, and are not considered to be punctuation. In contrast, the therefore sign $[\;\therefore\;]$ is traditionally used as a punctuation mark, and does not form part of a formula.
It also refers to the “complementary” of the “therefore” symbol$\;\therefore\;$, namely the symbol $\;\because\;$, which denotes “because.”
Example:
$\because$ All men are mortal.
$\because$ Socrates is a man.
$\therefore$ Socrates is mortal.
There are four logic symbols to get clear about:
$$\to,\quad \vdash,\quad \vDash,\quad \therefore$$
As for ‘$\Rightarrow$’, this — like the use of ‘implies’ — seems to be used informally (especially by non-logicians), in different contexts for any of the first three. So I’m afraid you just have to be careful to let context disambiguate. (And NB in the second and third uses where ‘$\Rightarrow$’ is more appropriately read as ‘implies’ there’s no scope difference with ‘$\therefore$’. In either case, we can have many wffs before the implication/inference marker.)
Contrast these:
I deny that I was planning to rob this bank. If I had been planning to rob this bank, I would be wearing a ski mask.
I was planning to rob this bank. Therefore I am wearing a ski mask.