How to grammatically formalise mathematical statements?

Question

I am currently trying to find the best way to express some first-order logic statements. I have several such statements, and I am unsure which gramatical rules I should follow to express them.

For example, I have a statement of the following form:

Let $F\subseteq\{f\mid f:A\to B\}$ be some subset of all functions from $A$ to $B$ and fix $b'\in B$. Then, there exists a function $f\in F$ such that for every element $a\in A$, $f(a)=b'$.

Following some authors in my field whose style I quite like, I am currently expressing such statement formally like this: \begin{gather} (\exists f\in F)(\forall a\in A)(f(a)=b') \end{gather}

I have been suggested to write this: \begin{gather} \exists f\in F:\forall a\in A,f(a)=b' \end{gather} Or this: \begin{gather} \exists f\in F,\forall a\in A:f(a)=b' \end{gather}

I am wondering whether

1. some of the above expressions are correct / incorrect to express the statement in italics;

2. there exist other valid (and commonplace) expressions to express the statement in italics;

3. there exists one preferred formal way to express the statement in italics.

Also, some references that could help me clarify my doubts would also be appreciated.

Thank you all!

Answer 1

1. Is $$∀x\color\red,P(x)→Q$$ intended to be read as

   • “for each $x\color\red,\,P(x)$ being true implies that $Q$ is true” $$∀x\:[P(x)→Q]\tag1$$

   or, non-equivalently, as

   • “if for each $x\,P(x)$ is true, then $Q$ is true” ? $$\big[∀x\,P(x)\big]→Q\tag2$$

   On the other hand, $$∀x\,P(x)→Q$$ (without any comma) conventionally just means sentence $(2)$ above.

2. The colon in the complete statement $$∀x:x^2\ge0$$ clearly does not mean ‘such that’.

3. As seen, in logic formalisations, commas/colons/periods sometimes are inserted to supply breathing space (but may break the flow of reading and be logically non-meaningful), and sometimes are an inheritance from the corresponding verbal sentences. In these cases, the punctuation is superfluous.

   On the other hand, many authors insert a punctuation mark after a quantifier, replacing parentheses, specifically to delimit that its scope extends as far right as possible, that is, to signify reading $(1)$ above. This punctuation usage conflicts with the two cases in the previous paragraph.

   Hence, in the absence of a prefacing usage note, it is generally ambiguous or misleading to punctuate quantifiers (other than with parentheses), except in a shorthand like $(\forall x{,}y{\in}\mathbb R\;P(x,y)).$

Summary: $$∀x,P(x)→Q\\∀x\,{:}\,P(x)→Q\\∀x.P(x)→Q$$ may be understood either as $$∀x\:[P(x)→Q],\tag1$$ or as the non-equivalent \begin{align}∀x\,P(x)→Q\tag2\\\big[∀x\,P(x)\big]→Q.\tag2\end{align}

4. For human reading, I prefer writing $$∃f{\in}F\; ∀a{\in}A\; f(a)=b'\:→\:∀y\;y=b'$$ or the more explicit $$\big(∃f{\in}F\; ∀a{\in}A\; f(a)=b'\big)\:→\:\big(∀y\;y=b'\big).$$
5. None of your three suggestions \begin{gather} (\exists f\in F)(\forall a\in A)(f(a)=b')\\ \exists f\in F:\forall a\in A,f(a)=b'\\ \exists f\in F,\forall a\in A:f(a)=b' \end{gather} are ambiguous, but why use both comma and colon, especially in the third? The first example is fine.

Answer 2

In first order logic all the variables range on a unique set $X$, so no first order statement contains $\exists\ f \in F\ \forall a \in A$, no matter punctuation, parenthesis or italics. Moreover, if $f$ is a symbol for a function in one variable of the language, then any interpretation of $f$ will be a function $f \colon X \to X$.