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

Question

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?

Answer 1

"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 $\vdash$, $\models$. 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.

Answer 2

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.)

Answer 3

$\therefore$ 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.

Answer 4

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.

Answer 5

1. $$A \implies B \implies C \tag{1}$$ 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 (convince yourself using the truth assignments $(0,0,0)$ and $(1,0,0)$): $$A \implies (B \implies C).\tag{1a}$$ $$(A \implies B) \implies C\tag{1b}$$ $$(A \implies B)\: \text{ and } \:(B \implies C)\\\text{(A being true implies that C is true)}\tag{1c}$$ The third is the most common interpretation, and can be read as “in case $A$ is true, then $C$ too is true”.

2. • If the difference between lines $(2)$ and $(1\text c)$ seems trivial or just a matter of contextually inferring the stronger meaning as necessary, then observe that $$x>-4;\:\:\therefore\, x>4$$ tells us that $x\not\in(-\infty,4],$ whereas $$x>-4\implies x>4$$ tells us that $x\not\in(-4,4].$

   • Another example: \begin{align}\lvert2x\rvert=x-1&\implies x=-1 \;\text{ or }\; \frac13\end{align} is true while $$\lvert2x\rvert=x-1; \;\text{therefore, }\; x=-1 \;\text{ or }\; \frac13$$ is false (explanation here and here).

   • And two illustrations that overloading the $\text‘{\implies}\text’$ symbol imposes a burden of disambiguation on the reader, breaking their flow of reading and skimming.

3. All in all, I think it is bad practice to treat $\text‘{\implies}\text’$ and $\text‘\therefore\text’$ as interchangeable.

   Here are several articles expounding their distinction:

   • implication versus conclusion
   • “implies” versus “therefore”
   • guide to writing mathematics (p. 17)
   • abuse of the implication symbol