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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?
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 ] }
