Intereting Posts

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How to use the Mean Value Theorem to prove the following statement:

Is there some understandable rationale why $\forall x\, \exists y\, P(x,y) \not \equiv \exists y\, \forall x\, P(x,y)$?

I’m looking for a sentence I can explain to students, but I am failing every time I try to come up with one.

Let $P(x,y)$ mean that $x$ is greater than $y$.

- Is a proposition about something which doesn't exist true or false?
- Two styles of semantics for a first-order language: what's to choose?
- Quantifiers, predicates, logical equivalence
- How does “If $P$ then $Q$” have the same meaning as “$Q$ only if $P$ ”?
- Is the “domain of discourse” in axiomatic set theory also a “set”?
- Why is quantified propositional logic not part of first-order logic?

- $\forall x\, \exists y\, P(x,y)$ means that for all $x$, there is a number $y$, such that $x$ is greater than $y$.
- $\exists y\, \forall x\, P(x,y)$ means that there is some $y$, that every number $x$ is greater than.

These don’t seem to mean different things to me. Is this perhaps an example where they do mean the same thing or am I just translating to English incorrectly?

- Difference between “for any” and “for all”?
- Logic: Using Quantifiers To Express “At Least 2?”
- Is a proposition about something which doesn't exist true or false?
- Prove the statement $\forall x\in\mathbb N (x > 1\to\exists k\in\mathbb N\exists m \in\mathbb N (m \equiv 1 (\text{mod }2) \wedge x = 2^km))$.
- How to prove this sequent using natural deduction?
- Quantifiers, predicates, logical equivalence
- Why can't we use implication for the existential quantifier?
- What is the justsification for this restriction on UG?
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- Classes, sets and Russell's paradox

Take $P(x, y)$ to mean $y$ is a parent of $x$.

Then $\forall x \exists y P(x, y)$ means everybody has a parent, while $\exists y \forall x P(x, y)$ means there is someone who is the parent of every son and daughter.

Let $P(O,C)$ mean car $C$ is owned by owner $O$.

Then $\forall C \ \exists O \ P(O,C)$ means every car has an owner.

However, $\exists O \ \forall C \ P(O,C)$ means that some owner owns all cars.

Clearly these mean different things.

Take $P(x,y)$ to be “$x$ is friends with $y$”.

One statement says ‘everyone has a friend’, the other says ‘someone is friends with everyone’.

Oops! My first answer was to the question I thought you were asking. Here is the answer to the question you actually asked: if $P(x, y)$ means $x > y$, then $\forall x \exists y P(x, y)$ is true iff there is no greatest number and $\exists y \forall x P(x, y)$ is true if there is a smallest number. Your translations into natural language are OK, but these statements are not the same: they are both true for the natural numbers, but the first is true and the second is false for the integers. So the two statements certainly don’t mean the same thing.

Order tells you the valid dependencies. For example, if we say $$\forall x \exists y P(x,y),$$

what we’re really saying is, for any $x$, we can find a $y$ dependent on that $x$ such that $P(x,y)$.

It may help to Skolemize. The formula becomes: there exists a functions $y(*)$ such that for all $x$ we have $P(x,y(x))$.

I stumbled on this page asking the same question, but I think I understand now why $\forall x\exists yP(x,y) \not\equiv \exists y\forall xP(x,y)$. It seems that the quantifiers for *x* and *y* are essentially nested in each other, such that the second quantifier is **dependent** on the first.

Let’s use the parent example where $P(x,y)$ is a predicate function meaning that *y* is the parent of *x*, which can also be stated that *x* is the child of *y*. Then, $\forall x\exists yP(x,y)$ can be interpreted as “For all children, there exists a parent.” Switching the order of the quantifiers gives $\exists y\forall xP(x,y)$, which can be interpreted as “For some parent, there exists all their children.” It seems that the second interpretation has the domain of children limited to the children of the parent, whereas in the first it was all children.

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