Mathematics Stack Exchange
Mathematics Stack Exchange

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.

1906 questions
50
votes
15 answers

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

Why is this true? $\exists x\,\big(P(x) \rightarrow \forall y\:P(y)\big)$
Mats
  • 887
44
votes
6 answers

"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
  • 579
39
votes
5 answers

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
  • 659
37
votes
4 answers

Difference between "for any" and "for all"?

Though searching for previous questions returns thousands of results for the query "for any" "for all", none specifically address the following query: I'm reading a textbook in which one definition requires that some condition holds for any $x,\ x'…
Bernd
  • 869
33
votes
8 answers

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.
user62287
  • 415
27
votes
3 answers

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
  • 985
24
votes
1 answer

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…
yungtoad
  • 352
22
votes
3 answers

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
  • 69,385
21
votes
7 answers

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…
Joshua Taylor
  • 2,551
21
votes
6 answers

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 $\forall x$?
Ned Stack
  • 245
19
votes
3 answers

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
  • 5,375
18
votes
7 answers

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
  • 373
16
votes
1 answer

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
  • 10,157
14
votes
4 answers

Write ‘There is exactly 1 person…’ without the uniqueness quantifier

During a lecture today the prof. posed the question of how we could write "There is exactly one person whom everybody loves." without using the uniqueness quantifier. The first part we wrote as a logical expression was "There is one person whom…
BustedZen
  • 185
14
votes
10 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…
Benzne_O
  • 857
1
2 3
…
99 100