# Order of precedence of “()”, “implies”, “forall” and “and”

I would like to know the order of precedence of $()$, $\implies$, $\forall$ and $\wedge$. For instance, how many "$()$" possibly could we remove for the following formula:

$(\forall a, ((b_0 \wedge b_1) \implies c)) \wedge (\forall d, (e \implies f)) \wedge ((\forall g, h) \implies i)$

where $a, b_0, b_1, c, d \ldots$ may represent either variable or formulae which contain other variables...

Thank you very much

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The expression is not well-formed. First comma has no reasonable interpretation, the quantifiers towards the end make no sense. Maybe you could rework the formula a bit? –  André Nicolas Aug 3 '11 at 15:25

I don't think there is a well-defined answer in the sense that there is not one universal accepted practice. I would use as many as I thought I needed in order for there to be no confusion - that's the important part.

I suppose I would write it as

$$(\forall a, \quad b_0 \wedge b_1 \implies c) \wedge (\forall d,\quad e \implies f) \wedge (\forall g, \quad h \implies i)$$

But this might be cheating, because I'm using spaces after the commas to indicate separation. Perhaps that is not uncommon?

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You could use a period instead of a comma, but that loosens the binding of the quantifier, necessitating the brackets you have. –  Zhen Lin Aug 3 '11 at 17:30

The original formula, as written, is not well-formed. If we are to interpret it as @mixedmath did, I would write $$\forall a\,((\,b_0 \wedge b_1) \implies c) \wedge \forall d\,(\,e \implies f) \wedge \forall g\,(\, h \implies i).$$

It can be visually useful to put parentheses around quantifiers, like this: $(\forall x)$. These parentheses are so tightly bound to the $\forall x$ that they do not cause visual parsing problems.

If in doubt, use parentheses. The problem with elaborate order of precedence rules is that one cannot reasonably expect everyone to be aware of them.

The use of commas as separators, except in expressions like $(\forall x,y)$, and of course $f(x,y)$, is quite non-standard.

Usually, there are implicit conventions in our field, and we simply imitate.

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