# Words used in quantifier proofs

I'm creating a list of "gotcha words" that are often used in writing proofs (particularly quantifier proofs), but frequently in more than one possible way, and that beginners frequently misuse or misunderstand. But it seems likely that there are some I will forget to include, so I'm wondering whether anyone else has already created such a list and perhaps spent more time ensuring its completeness?

So far, here are the words I've got:

• "suppose" is used when proving a $\forall$ statement (suppose $x$ is an arbitrary real number) or using a known $\exists$ statement (we may suppose $x$ is a real number such that $x^2=2$).
• "assume" is used like "suppose", but more commonly restricted to statements rather than variables.
• "choose" is used when proving an $\exists$ statement (to show that there exists a $y$ with $xy=y+1$, choose $y = 1/(x-1)$) and using a known $\exists$ statement (we may choose an algebraic closure of the field $K$)
• "define" is used when proving an $\exists$ (define $y=1/(x-1)$)
• "arbitrary" is used when proving a $\forall$ (let $x$ be an arbitrary real number)
• "let" is used when proving a $\forall$ (let $x$ be an arbitrary real number), proving an $\exists$ (let $y=1/(x-1)$), or using an $\exists$ (let $L$ be an algebraic closure of $K$).

Edit: The current draft of my list is here.

• Not really an answer, but please, please let them know the difference between "thus" and $\implies$. I don't know the state of affair in your country, but in France the confusion is really getting in our way. Nov 6, 2015 at 22:17
• I second @Benoit's sentiment, and talked about the same here. Feb 6, 2021 at 16:08

(This is not exactly an answer to your question, but I hope it is useful.)

### Fix, Set

(These are how I understand these terms. I'm not sure if my usage is popularly used.)

Fix means "choose an arbitrary value for this constant."

Fix an $m\in\mathbb{N}$. Note that $\frac{\mathsf{d}}{\mathsf{d}x}m=\frac{\mathsf{d}}{\mathsf{d}x}(\underbrace{1+1+\ldots+1}_{m\text{ times}})=0$

Set means "choose this specific value for this variable."

Set $r=1$ for the case of a unit circle.

### Because (since), Therefore (thus)

Because ($\because$) or since means "given the premise."

Therefore ($\therefore$) or thus means "the logical consequence is."

$\because$ All gods are immortals. $\because$ Zeus is a god. $\therefore$ Zeus is an immortal.

Also,

To denote logical implication or entailment, various signs are used in mathematical logic: $\rightarrow$, $\implies$, $\supset$, $\vdash$, $\models$. These symbols are then part of a mathematical formula, and are not considered to be punctuation. In contrast, the therefore sign is traditionally used as a punctuation mark, and does not form part of a formula.

• I'm not sure if my answer addresses your question correctly. Tell me if it does not, and I will delete it.
– JRN
Nov 6, 2015 at 2:03
• Well, the literal question that I asked was "has anyone made such a list", so this doesn't really answer that question. But I'm also very happy to just receive suggestions for additional words that should be included in the list. In that vein, your comment is more directly along the lines of what I was thinking of than this answer is, but I wouldn't delete the answer; "therefore" and "because/since" are also important words to know. Nov 6, 2015 at 4:11
• Your use of "say" doesn't look like any step that I'm familiar with in a proof. What does it mean for something to be both arbitrary and specific? The only place I can imagine writing that is in an informal discussion of examples rather than in a proof; is that what you had in mind? Nov 6, 2015 at 5:34
• You are correct. I forgot that the discussion was about proofs, not examples. I'm removing that part now.
– JRN
Nov 6, 2015 at 6:04
• I'm making my answer community-wiki so that others can add their own examples of words.
– JRN
Nov 6, 2015 at 6:19

Have you looked at Charles Wells's "Handbook of Mathematical Discourse"? It has quite a bit on this (and more generally on the use of language in mathematics teaching.)