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Given:

∀y∃x Ryx
∃x∀y Ryx

How do we know if these formulas are true? Are they true as long as some instances are true or must all be true? In other words, how can we identify the main quantifier?

Which quantifier - the existential or universal - takes precedence in this case and how can we know for other formulas? Is there an order of operations like in arithmetic or does precedence move in a direction?

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  • No "precedence" at all; you have to start from the outermost one. Commented Oct 12, 2020 at 5:58
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    This is not an argument, it is a closed formula of predicate calculus. T evaluate its truth value you start by picking a value of y and seeing if you can find a value of x for which Ryx is true. If you fail to find such x for even one y the formula is false.
    – Conifold
    Commented Oct 12, 2020 at 6:31
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    – J D
    Commented Oct 12, 2020 at 13:55

1 Answer 1

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Firstly, what you have written there is a proposition, not an argument. An argument has premises and a conclusion. What you are asking is how do we interpret the quantifier scope in propositions that have more than one quantifier. The short answer is to read the proposition left-to-right and take the leftmost quantifier as having wider scope. So,

(∀y)(∃x)Ryx should be read as: for any y...there is some x such that...Ryx. 

On the other hand,

(∃x)(∀y)Ryx should be read as: there is some x such that...for any y...Ryx. 

The two mean different things: if we restrict the quantifier y to range over rats, and x over tails, then the first can be interpreted to mean every rat has a tail, while the second would mean there is some one tail that every rat has.

If that is not sufficiently clear, think of each successive quantifier as being parenthesised. So

(∀y)(∃x)Ryx is equivalent to (∀y)[ (∃x)Ryx ] 

or with three quantifiers,

(∃z)(∀y)(∃x)Qzyx is equivalent to (∃z){ (∀y)[ (∃x)Qzyx ] }
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  • Hi thanks for your answer! So, in order to find out the main quantifier for a proposition, is it always the case that we read it from left to right? Commented Oct 12, 2020 at 5:56
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    Yes, that's right.
    – Bumble
    Commented Oct 12, 2020 at 7:22

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