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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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  • $\begingroup$ Yeah, I read the $\implies$ symbols in this context as "Which implies that,..." I read the "$\therefore$" as "Therefore, ...". In normal language, we'll prefer "Which implies that,..." more in the middle of an argument, and "Therefore,..." at the end or at a natural "punctuation" point in the argument. $\endgroup$ Nov 16 '12 at 20:16
  • $\begingroup$ Zilch. ${}{}{}{}$ $\endgroup$ Nov 16 '12 at 20:20
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    $\begingroup$ In particular, "$\implies$" is most often used to indicate that the conclusion follows from relatively recent statements. If you needed to prove something first for $n$ odd then for $n$ even, if each case was a relatively long proof, I don't think you'd want to write, after the two cases were proven: "$\implies$ X(n) is true for all integers $n$." That's a case where "$\therefore$" makes more sense, even when it is not the "end" of your proof. $\endgroup$ Nov 16 '12 at 20:22
  • $\begingroup$ Once difference I notice is that I haven't used $\therefore$ since secondary school (high school), while I use $\implies$ and $\iff$ these days. $\endgroup$ Nov 16 '12 at 20:30
  • $\begingroup$ I have a related question. I once used $\implies$ in a proof for a class to say one thing implies another. My professor said that it is not okay to use it in that case. He said it would be okay if you are proving an if and only if argument, to indicate which direction you're proving $\Rightarrow$: and $\Leftarrow$: Do any others of you agree with this? $\endgroup$
    – GeoffDS
    Nov 17 '12 at 0:43
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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.

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  • $\begingroup$ Yes, "imply each other" was kind of poorly worded. I meant that $\implies$ is appropiate when you have a series of logical implications from one statement to another. $\endgroup$
    – Javier
    Nov 16 '12 at 20:11
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    $\begingroup$ Yes, I pretty much gathered that you knew what you meant. It's just that many people experience significant confusion about what is meant by an implication, so for the sake of clarity and precision, I wanted to point that out. $\endgroup$
    – amWhy
    Nov 16 '12 at 20:18
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    $\begingroup$ @Javier Perhaps this example will help: an implication, in and of itself, asserts no conclusion, save for the implication. "If the grass is purple, then elephants can fly." It's a "valid" statement (since anything follows from a false assertion) even though if seems absurd. And certainly, elephants being able to fly has nothing to do with the color of grass, nor is "elephants can fly" because "grass is purple". In contrast, $\;\therefore\;$ typically carries with it the idea that the conclusion follows because of the premises, (never in spite of the premises). $\endgroup$
    – amWhy
    Nov 17 '12 at 17:55
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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.)

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  • $\begingroup$ I'll add here that A. N. Prior's textbook Formal Logic has parts of it which read like the following: "Rule: Detachment ($\alpha$, D$\alpha$D$\beta$$\gamma$ $\rightarrow$ $\gamma$) and (In all cases the sole rule beside substitution is E-detachment: $\alpha$, E$\alpha$$\beta$ $\rightarrow$ $\beta$. And in my opinion Prior's symbolism comes as clearer here than writing {E$\alpha$$\beta$, $\alpha$} $\vdash$ $\beta$, since the "$\rightarrow$" symbol suggests that one transitions from the left-hand side to the right hand side. So, as this answer says let context disambiguate. $\endgroup$ Nov 10 '14 at 19:37
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$\therefore$ and $\implies$ are quite different!

"Hence" and "therefore" and "as a consequence" are all synonyms. The usage is "A, therefore B", meaning "A is true, and it follows that B is true." Note that the truth of A is being asserted. Latex \therefore ($\therefore$) gives the dot triangle that has long been used to mean "therefore".

"Because" is the same in reverse. "B because A" means that B is true because A is true. This contains the assertion that A is true. Latex \because ($\because$) gives the inverted dot triangle that has long been used to mean "because".

"Implies" is completely different. "A implies B" means that IF A is true, then B is also true. It makes no statement about the truth of A. Latex \implies ($\implies$), \Rightarrow ($\Rightarrow$), and \Longrightarrow ($\Longrightarrow$) all give the double right arrow that is often used to mean "implies". Sometimes a single right arrow is used, which has the same meaning.

It is very common to use the \implies symbol instead of "therefore", but since "implies" and "therefore" have significantly different meanings, this is very bad writing.

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  • $\begingroup$ Agreed! That is indeed a big difference between 'implies' (denotes by $\Rightarrow$) and 'therefore' (denoted by $\therefore$) $\endgroup$
    – Bram28
    Aug 21 '20 at 0:19
  • $\begingroup$ @RyanG I would agree with you. I'll remove that comment. That should never be put in a post anyway, since other people can of course always add further answers. Thanks for your note! $\endgroup$
    – Bram28
    Sep 6 at 16:55
  • $\begingroup$ @Bram28 I wasn't actually quibbling with zaslav's first sentence (just deleted), which I in fact thought was a pertinent warning considering that the incorrect answer has numerous more upvotes and uptick than the correct ones. My above comment was just informational; apologies for any confusion caused. $\endgroup$
    – Ryan G
    Sep 6 at 17:14
  • $\begingroup$ @RyanG Ha ha, yes, I agree with that as well ... worst answer was upvoted the most. Oh well! $\endgroup$
    – Bram28
    Sep 6 at 18:10
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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.

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    $\begingroup$ I'm not sure I understand. In which case would you use each symbol? $\endgroup$
    – Javier
    Nov 17 '12 at 12:52
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    $\begingroup$ @Javier $\quad R$: I was planning to rob this bank. $S$: I am wearing a ski mask. Case $(1)\;$: $\lnot R.\;$ $R \implies S.\quad$ Case $(2)\;$: $R.\;$ $\;\therefore S$. (I may be wrong.) $\endgroup$
    – amWhy
    Nov 17 '12 at 17:31
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    $\begingroup$ You use the symbol that means "If . . . then . . ." in the first case and the symbol that means "Therefore" in the second case. $\endgroup$ Nov 17 '12 at 19:01
  • $\begingroup$ @JavierBadia : Your surmise is correct. $\endgroup$ Nov 18 '12 at 17:49
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$$A \implies B \implies C \tag{1}$$ has a different meaning and reading from $$A;\:\:\therefore B;\:\:\therefore C \tag{2} \\\text{($A$ is true; therefore $C$ is true)}.$$ Line $(2)$ asserts premise A, that $A \implies B$ and that $B \implies C,$ and concludes that $C$ is true.

Line $(1),$ on the other hand, neither specifies that sentences $A$ and $B$ are actually true, nor asserts that sentence $C$ is true. It in fact has three possible meanings, no pair of which is equivalent:

  • $$A \implies (B \implies C).\tag{1a}$$
  • $$(A \implies B) \implies C\tag{1b}$$
  • $$(A \implies B)\: \text{ and } \:(B \implies C)\\\text{(A being true implies that C is true)}\tag{1c}$$

The third is the most common interpretation, and can be read as “in case $A$ is true, then $C$ too is true”. If the difference between lines $(2)$ and $(1\text c)$ seems trivial and just a matter of contextually inferring the stronger meaning as necessary, then contrast this true claim $$\lvert2x\rvert=x-1\implies\pm2x=x-1\\\therefore\lvert2x\rvert=x-1\implies x=-1 \;\textbf{ or }\; \frac13$$ with this patently false conclusion and unsound argument $$(\lvert2x\rvert=x-1.) \;\text{Therefore, }\; x=-1 \;\textbf{ or }\; \frac13.$$ (The anomaly is explained here.)


All in all, I think it is bad practice to treat $\implies$ and $\,\therefore\,$ as interchangeable.

Here are two blog posts expounding further on, and a reddit post plainly summarising, their distinction:

  1. abuse of the implication symbol
  2. “implies” versus “therefore”
  3. https://www.reddit.com/r/math/comments/20mojw/therefore_there4_vs_implication_rarr/cg4qxg0
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