# Origin and usage of $\therefore$ and $\because$

I've recently read a book which used the sign $\therefore$ (for "therefore"). It was more or less clear from the context what was meant, but I looked it up among the AMS LaTeX symbols just to be sure. They also listed the similar $\because$ ("because") sign.

I might have seen this notation before, but very rarely if at all. Is this because I've studied in Germany and these symbols are used more often in other countries? Or are they very old and out of fashion? (The book I mentioned was rather new, though.) Or did I just read the wrong books?

Any information tidbits about where $\because$ and $\therefore$ come from and about their current or historical usage?

And, FWIW, I'd also be interested in whether there's some graphical or typographical clue hidden in these symbols (like the "vell" in $\lor$). How do you memorize which is which?

• The relevant Wikipedia article may be of interest to you: en.wikipedia.org/wiki/Therefore_sign. – Gahawar Sep 6 '14 at 14:07
• Ah, thanks. I had searched Google, but it didn't occur to me to search Wikipedia directly. ("Therefore" and "because" aren't the ideal search terms...) – Frunobulax Sep 6 '14 at 14:12
• In the UK it is old and out of fashion. – almagest Sep 6 '14 at 14:19
• I'd expect to find this symbol in any introductory logic book/course and also in discrete mathematics courses. – Git Gud Sep 6 '14 at 14:29
• I don't think it's widespread in France either, except maybe(?) in formal logic – StayHomeSaveLives Sep 8 '14 at 16:40

The symbol $\therefore$ is used in the book An introduction to formal logic, by Peter Smith, a very nice book indeed. It is a symbol of the formal language of logic, used as inference marker, to signal that an inference is being drawn from the premises. I quote from p. 110 (PL is the language of propositional logic, later in the book extended to full first-order theory logic):
Rather oddly, counting $\therefore$ as part of PL is not standard; but we are going to be very mildly deviant. However, our policy fits with our general view of the two-stage strategy for testing inferences. We want to be able to translate vernacular arguments into a language in which we can still express arguments; and that surely means not only that the individual wffs of PL should be potentially contentful, but also that we should have a way of stringing wffs together which signals that an inference is being made
Then, on p. 119, there is a nice discussion about the difference between $\vDash$ (a metalinguistic symbol, which does not belong to PL, for talking about the relationship of some wffs in PL) and $\therefore$ (a symbol of PL)