Articles of quantifiers

Difference Between “$\forall x \exists y$” and “$\exists y \forall x$”

Possible Duplicate: Confused between Nested Quantifiers I asked the question about two sentences. interpreting mixed quantifier But, I don’t know the meaning difference between $$∀x∃y(\text{Cube}(x) → (\text{Tet}(y) ∧ \text{LeftOf}(x, y))),$$ and $$∃y∀x(\text{Cube}(x) → (\text{Tet}(y) ∧ \text{LeftOf}(x, y))),$$ “Every cube is to the left of a tetrahedron” “There is a tetrahedron that is to the right […]

Lack of implication and logical quantifiers

After some botched attempts at formulating my question correctly, I stopped trying to simplify it and just ask straight out. Here’s the situation. We have a statement that we’re trying to prove: For all $x \in \mathcal X$, $A(x) \Rightarrow B(x)$. We attempt to do this by contradiction. Define $\mathcal X’ := \{x \in \mathcal […]

Why is “for all $x\in\varnothing$, $P(x)$” true, but “there exists $x\in\varnothing$ such that $P(x)$” false?

This question already has an answer here: Why is predicate “all” as in all(SET) true if the SET is empty? 6 answers

Quantifier Notation

What’s the difference between $\forall \space x \space \exists \space y$ and $\exists \space y \space \forall \space x$ ? I don’t believe they mean the same thing even though the quantifiers are attached to the same variable, but I’m having a hard time understanding the difference. Any examples to make the distinction clear would […]

What makes Tarski Grothendieck set theory non-empty?

I’m fighting with Grothendieck set theory for some time now. This is the framework for the automated proof checking system of Mizar and hence there is a formalized version of the axioms here too, and here in the “Content” link provided as a pdf. What I want to understand is how the theory contains sets […]

Symbols for Quantifiers Other Than $\forall$ and $\exists$

The symbols $\forall$ and $\exists$ denote “for all” and “there exists” quantifiers. In some papers, I saw the (not so common) quantifiers $Я$ and $\exists^+$, denoting “for a randomly chosen element of” and “for most elements in”, respectively. Are there other symbols for quantifiers? I’m specially interested in quantifiers for: for all but finitely many […]

Intuitive Reason that Quantifier Order Matters

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. Example Let $P(x,y)$ mean that $x$ is greater than $y$. $\forall x\, \exists y\, […]

Difference between “for any” and “for all”?

In a textbook (on economics, not “pure” mathematics), one definition requires that some condition holds for any $x,\ x’ \in X$, and right afterwards another one requires that some other condition holds for all $x,\ x’ \in X$. My question: is there a difference between the two (for any, for all)? Though searching for previous […]

How to prove this sequent using natural deduction?

How do I prove $$S\rightarrow \exists xP(x) \vdash \exists x(S\rightarrow P(x))$$ using natural deduction? Just an alignment of which axioms or rules that one could use would be much appreciated.

Restrictions on universal specification (in first-order logic)?

I’m currently working my way through the details of first-order logic (using Suppes’ Introduction to Logic), and I have a question about universal specification (US) (aka universal instantiation or elimination). Specifically, my question has to do with the choosing of a name for the ‘ambiguous object’ that is used to replace the dropped universally quantified […]