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Is this true that it is a commonly agreed rule that $\forall x\in A:P(x) \wedge Q$ and $\forall x\in A:P(x) \Rightarrow Q$ should be interpreted correspondingly as $(\forall x\in A:P(x)) \wedge Q$ and $(\forall x\in A:P(x)) \Rightarrow Q$?

The question is about implied parentheses. Are the other interpretations $\forall x\in A:(P(x) \wedge Q)$ and $\forall x\in A:(P(x) \Rightarrow Q)$ common?

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    $\begingroup$ I would read this as $\forall x \in A : (P(x) \wedge Q)$. But one should add parentheses wherever there's ambiguity (e.g., here.) $\endgroup$
    – user98602
    Jun 10, 2014 at 20:05
  • $\begingroup$ I would have said that in the absence of any parentheses, quantifiers always extend as far as possible to the right, agreeing with Mike Miller's reading. $\endgroup$
    – MJD
    Jun 10, 2014 at 20:22
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    $\begingroup$ @MJD Exactly the opposite happens. The convention is that if there are no parentheses, then the scope of the quantifier is only the closest predicate. $\endgroup$
    – Git Gud
    Jun 10, 2014 at 20:26
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    $\begingroup$ The rule is that there is no rule that can be counted on. $\endgroup$ Jun 10, 2014 at 20:47
  • $\begingroup$ @GitGud Cite please? $\endgroup$
    – MJD
    Jun 10, 2014 at 20:53

1 Answer 1

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The reason that it is customary to "bind variables tightly" is to ensure we have clarity regarding the scope of a quantified variable. Absent any parentheses, the tightest of bounds applies.

To omit parentheses is sloppy, at best, but it happens (to the misfortune of those who have to mind-read with respect to the author's intention).

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