# The use of “$\therefore$” and “$\because$”

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.

• 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? – Adam Rubinson Jun 23 '20 at 12:54
• @AdamRubinson "$\because$" means "because". To type it, write "\because". – Ma Joad Jun 23 '20 at 13:42
• I see. Thank you – Adam Rubinson Jun 23 '20 at 14:26
• Advice: Use these symbols only if you have to write very fast. Otherwise, write out "therefore" and "since" as words. – Gerald Edgar Jun 23 '20 at 22:47

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.
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.)

• @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". – Xander Henderson Jun 24 '20 at 18:46
• 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). – paul garrett Jun 24 '20 at 19:07
• @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. – Xander Henderson Jun 24 '20 at 19:10
• @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. – Chris Cunningham Jun 24 '20 at 19:40
• I do like the notation for scratch work. Writing can be painful. Making it short and unambiguous is nice, and you can come back and rewrite if it ends up being something you want to share – Artimis Fowl Jul 11 '20 at 17:33

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: $$P=o(Q) \ \Longleftrightarrow\ \deg P<\deg Q\, .$$

• 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. – Daniel R. Collins Jun 24 '20 at 22:51
• @Daniel R. Collins Yes, I meant this as a theorem. – Peter Saveliev Jun 24 '20 at 23:29