Difference between $\implies$ and $\;\therefore\;\;$?

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?

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“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:

$$\to,\quad \vdash,\quad \vDash,\quad \therefore$$

  1. ‘$\to$’ (or ‘$\supset$’) is a symbol belonging to various formal languages (e.g. the language of propositional logic or the language of the first-order predicate calculus) to express [usually!] the truth-functional conditional. $A \to B$ is a single conditional proposition, which of course asserts neither $A$ nor $B$.
  2. ‘$\vdash$’ is an expression added to logician’s English (or Spanish or whatever) — it belongs to the metalanguage in which we talk about consequence relations between formal sentences. And e.g. $A, A \to B \vdash B$ says in augmented English that in some relevant deductive system, there is a proof from the premisses $A$ and $A \to B$ to the conclusion $B$. (If we are being really pernickety we would write ‘$A$’, ‘$A \to B$’ $\vdash$ ‘$B$’ but it is always understood that $\vdash$ comes with invisible quotes.)
  3. ‘$\vDash$’ is another expression added to logician’s English (or Spanish or whatever) — it again belongs to the metalanguage in which we talk about consequence relations between formal sentences. And e.g. $A, A \to B \vDash B$ says that in the relevant semantics, there is no valuation which makes the premisses $A$ and $A \to B$ true and the conclusion $B$ false.
  4. $\therefore$ is added as punctuation to some formal languages as an inference marker. Then $A, A \to B \therefore B$ is an object language expression; and (unlike the metalinguistic $A, A \to B \vdash B$), this consists in three separate assertions $A$, $A \to B$ and $B$, with a marker that is appropriately used when the third is a consequence of the first two. (But NB an inference marker should not be thought of as asserting that an inference is being made.)

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.