# How to grammatically formalise mathematical statements?

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!

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}

1. For human reading, I prefer writing $$∃f{\in}F\; ∀a{\in}A\; f(a)=b'\:→\:∀y\;y=b'$$ or the more explicit $$\big(∃f{\in}F\; ∀a{\in}A\; f(a)=b'\big)\:→\:\big(∀y\;y=b'\big).$$
2. 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.

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 $$f \colon X \to X$$.