Questions tagged [quantifiers]

The quantifiers $\forall$ ("for all") and $\exists$ ("there exists") distinguish predicate calculus from propositional logic.

Quantifiers specify the quantity of objects that satisfy a given formula.

The quantifiers $\forall$ (for all) and $\exists$ (there exists) are the most common, but others such as $\exists!$ (there exists a unique) are also in usage.

Only use this tag if your question is about the usage of a quantifier in a formula. Be sure not to use this tag for any question with quantifiers.

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Why is this true? $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$

Why is this true? $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$
Mats
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"If everyone in front of you is bald, then you're bald." Does this logically mean that the first person is bald?

Suppose we have a line of people that starts with person #1 and goes for a (finite or infinite) number of people behind him/her, and this property holds for every person in the line: If everyone in front of you is bald, then you are…
Færd
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Why can't we use implication for the existential quantifier?

I'm not quite sure that I really understand WHY I need to use implication for universal quantification, and conjunction for existential quantification. Let $F$ be the domain of fruits and $$A(x) : \text{is an apple}$$ $$D(x) : \text{is…
Adam Thompson
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Is $\forall x\,\exists y\, Q(x, y)$ the same as $\exists y\,\forall x\,Q(x, y)$?

is $\forall x\,\exists y\, Q(x, y)$ the same as $\exists y\,\forall x\,Q(x, y)$? I read in the book that the order of quantifiers make a big difference so I was wondering if these two expressions are equivalent or not. Thanks.
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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…
Bernd
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In proofs, are "for each" and "for any" synonyms?

In proofs, are "for each" and "for any" synonyms? Or some context is usually required to determine this?
Loli
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Does the unique existential quantifier commute with the existential quantifier?

Given some function involving two variables, $\mathit p(x,y)$, is the formula $$\mathit \exists!x\exists yp(x,y)$$ equivalent to $$\mathit\exists y\exists!xp(x,y)$$ I have tried writing out the formal definition for the unique existential…
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What is dual to "There exists unique?"

I know that "for all" $(\forall)$ and "there exists" $(\exists)$ are dual, in the sense that $$\neg \forall \neg = \exists,\quad \neg \exists \neg = \forall$$ What is dual to "there exists unique"? In other words, how should we interpret $$\neg…
goblin GONE
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What's the difference between $∀x\,∃y\,L(x, y)$ and $∃y\,∀x\,L(x, y)$?

Everybody loves somebody. $∀x\,∃y\,L(x, y)$ There is somebody whom everybody loves. $∃y\,∀x\,L(x, y)$ What's the difference between these two sentences? If they are same, can I switch $\exists y$ and $\forall x$?
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Different standards for writing down logical quantifiers in a formal way

What are standard ways to write mathematical expressions involving quantifiers in a (semi)formal way ? In different posts of mine concerning similar question I have encountered for a generic expression of the type "for all $x\in I$ and $y\in J$…
temo
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categorical interpretation of quantification

Many constructions in intuitionistic and classical logic have relatively simple counterparts in category theory. For instance, conjunctions, disjunctions, and conditionals have analogues in products, coproducts, and exponential objects. More…
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Is the order of universal/existential quantifiers important?

If you have a formula with existential quantifiers, it is important in which order they appear. Just to make an easy example: $\forall$ man $\exists$ woman: the woman is the true love of the man which is obviously a different statement…
Martin Thoma
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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 $P(x,y)$…
danmcardle
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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…
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Who was the first to use $\bigwedge$ and $\bigvee$ as universal and existential quantifiers?

It's not surprised that somebody uses $\bigwedge$ and $\bigvee$ as universal and existential quantifiers since $$ \bigwedge_{x\in A}\,\varphi(x)\Leftrightarrow \varphi(a_0)\wedge\varphi(a_1)\wedge\varphi(a_2)\wedge\cdots\Leftrightarrow\forall x\in…
M. Logic
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