The use of "$therefore$" and "$because$"

Mathematics Educators Asked on September 6, 2021

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 $ARightarrow B$ and $A$ we could deduce $B$, so the three parts are: $ARightarrow B$, $A$ and $B$.

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

2 Answers

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

Answered by Peter Saveliev on September 6, 2021

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.)

Answered by Pedro on September 6, 2021

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