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In schools, many students learn the usage of "$\therefore$" and "$\because$" in proofs. Such three-dot notation are popular in many high-school books and exams, but are almost never used in university-level texts. (It seems that, at degree level, this notation only appears in some books about mathematical logic.)

Very often, it is somewhat awkward to use "$\therefore$" and "$\because$" for proofs, because modus ponens, the most commonly used principle of deduction, contains three parts, while "$\therefore$" and "$\because$" are just two symbols. Modus ponens states that from $A\Rightarrow B$ and $A$ we could deduce $B$, so the three parts are: $A\Rightarrow B$, $A$ and $B$.

We will of course write $B$ after "$\therefore$", but it is a good question where to put $A\Rightarrow B$ and $A$. We may either put both $A$ and $A\Rightarrow B$ after "$\because$", or put $A$ after "$\because$" and $A\Rightarrow B$ in brackets after "$\therefore B$".

In the end, the three-dot notation does not make the logic structure entirely clear. "$\therefore $" clearly indicates the conclusion, but the meaning of "$\because$" is not entirely clear - it could be either a theorem $A\Rightarrow B$ or a condition $A$. Sometimes, $A$ is too long (takes too many words) to be written out fully, which causes confusion.

Is there any better alternative to the three-dot notation? It is, after all, completely clear to just write everything in words.

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    $\begingroup$ The "therefore" symbol I used in school had 2 dots on the bottom and one dot on the top. I have never seen an upside-down "therefore" symbol before. What does it mean? $\endgroup$ – Adam Rubinson Jun 23 at 12:54
  • $\begingroup$ @AdamRubinson "$\because$" means "because". To type it, write "\because". $\endgroup$ – Ma Joad Jun 23 at 13:42
  • $\begingroup$ I see. Thank you $\endgroup$ – Adam Rubinson Jun 23 at 14:26
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    $\begingroup$ Advice: Use these symbols only if you have to write very fast. Otherwise, write out "therefore" and "since" as words. $\endgroup$ – Gerald Edgar Jun 23 at 22:47
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Is there any better alternative to the three-dot notation?

The usual general advice is to use words instead of symbols.

The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that you are explaining the subject to a friend on a long walk in the woods, with no paper available; fall back on symbolism only when it is really necessary.

(Paul Halmos, How to Write Mathematics, p. 40.)

This applies particularly to the three-dot notation.

Do not misuse the implication operator ⇒ or the symbol ∴. The former is employed only in symbolic sentences; the latter is not used in higher mathematics.

Bad: a is an integer ⇒ a is a rational number.
Good: If a is an integer, then a is a rational number.
Bad: ⇒ x = 3.
Bad: ∴ x = 3.
Good: hence x = 3.
Good: and therefore x = 3.

Bad Theorem. n odd ⇒ 8|n² − 1.
Bad proof.
n odd ⇒ ∃j ∈ Z, n = 2j + 1;
∴ n² − 1 = 4j(j + 1);
∀j ∈ Z, 2 | j(j + 1) ⇒ 8 | n² − 1

This is a clumsy attempt to achieve conciseness via an entirely symbolic exposition.Combining words and symbols and adding some short explanations will improve readability and style.

(Franco Vivaldi, Mathematical Writing, p. 4 and 132.)

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  • $\begingroup$ I believe in doing both when writing. $\endgroup$ – Peter Saveliev Jun 24 at 11:28
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    $\begingroup$ @PeterSaveliev I am unclear about what you mean by "doing both". Pedro does not seem to be suggesting an either / or approach, but instead seems to be saying "Use plain language until plain language becomes cumbersome." This seems to be advocating "both". $\endgroup$ – Xander Henderson Jun 24 at 18:46
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    $\begingroup$ I'm not a fan of those two three-dots notations, because (1) in handwriting it is all too easy to write them ambiguously or illegibly (2) they lend themselves to ambiguous/bad logic/grammar, because people often don't test their writing by re-reading (when it's not in genuine words). $\endgroup$ – paul garrett Jun 24 at 19:07
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    $\begingroup$ @PeterSaveliev It sounds like you are asking people to write everything twice: once in plain language, once in notation. This doesn't seem like a good idea to me, and I imagine that most journal or book editors would show you the door. $\endgroup$ – Xander Henderson Jun 24 at 19:10
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    $\begingroup$ @PeterSaveliev If you have an answer to the question, post an answer. It seems like you are suggesting a competing or additional answer. That would be excellent to have. $\endgroup$ – Chris Cunningham Jun 24 at 19:40
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The context isn't entirely clear so I'll assume this is about teaching. Then, I support Pedro's answer but also want to add that doing both verbal and symbolic versions may be a good idea. For example:

Theorem. A polynomial has a higher order than another if and only if its degree is higher.

In other words, for any two polynomials $P$ and $Q$, we have: $$P=o(Q) \ \Longleftrightarrow\ \deg P<\deg Q\, .$$

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    $\begingroup$ I don't think this scales well. I'm picturing a proof where this is done for every statement and it seems like it would be a nightmare to read. For summary or major conclusion points it could be fine. $\endgroup$ – Daniel R. Collins Jun 24 at 22:51
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    $\begingroup$ @Daniel R. Collins Yes, I meant this as a theorem. $\endgroup$ – Peter Saveliev Jun 24 at 23:29

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