What is the name of the symbol $=$ in English? Wikipedia says people use "equals sign" than "equality". Is it right? Is it to avoid ambiguity between the equality formula $(x = y)$ and the equality symbol $=$?
The main my question is the following:
can I call the symbol $=$ equality when it is obvious from the context?
I'm writing a text like:
The language of set theory uses the following symbols:
- truth: $\top$
- falsity: $\bot$
- negation: $\neg$
- conjunction: $\land$
- disjunction: $\lor$
- implication: $\Rightarrow$
- biconditional: $\Leftrightarrow$
- universal quantifier: $\forall$
- existential quantifier: $\exists$
- equality: $=$
- membership: $\in$
Does this sound strange? I'm not sure because I have seen some mathematicians use the word "equality" for the symbol but perhaps they are not native English speakers.
Reply to some comments:
To @xander-henderson:
For example, there are many famous propositions like Cauchy-Schwarz inequality. It is a proposition or a well-formed formula, not a symbol itself.
To @jmoravitz:
In Japanese, the proposition $x = y$ is only called "等式" or "方程式", and the symbol $=$ is only called "等号". The proposition $x \le y$ is called "不等式" and the symbol $\le$ is called "不等号". So I don't have any idea whether it is okay or not. Calling both "inequality" is like calling both "不等". So I'm afraid "equals sign" feels like too colloquial, but I'm not a native English speaker, it's unreliable. I have seen at least Japanese and Ukrainian mathematician uses "equality" as a symbol in some obvious contexts. But Wikipedia says "equals sign" is popular. So I asked.
Best Answer
In Logic, almost all logicians do use "equality" to refer to "=" as in "FOL with[out] equality", as short for "equality symbol". Mathematicians in general prefer to refer to "=" as "equals sign" or "equality sign" or "equality symbol" (with the explicit "symbol"). Yes, "equality" can also refer to a formula of the form "$s = t$". Somewhat amusingly, a formula of the form "$s ≠ t$" is instead called an "inequation" and not an "inequality".
To answer your question, no, it is not a grammatical mistake. This is why we say "FOL with[out] equality". If "equality" can only refer to equations rather than the symbol "=", then this phrase would be ungrammatical. But it is perfectly fine. For an example right here on Math SE, see this post where Noah Schweber repeatedly unambiguously uses "equality" to refer to "=".
And I would prefer not to think of this as 'just a matter of context', because one cannot just drop arbitrary words in general; ultimately it still comes down to whether it is acceptable in current mathematical vernacular. In English, it is as I said above. In other languages, it may be a totally different story.