Last Update: 27 September 2010
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**§1.4: Nested Quantifiers (cont'd):****Uniqueness:**"There is exactly one dog" can be represented as:

- ∃

*x*[Dog(*x*) ∧ ∀*y*[Dog(*y*) →*y*=*x*]]- I.e., There is (at least) one thing in the domain
that is a dog,

and, any (other) dog in the domain is identical to that one.- Sometimes, logicians abbreviate this (with the "E-shriek!"
quantifier) as:
- ∃!

*x*Dog(*x*)(read "There is a

*unique x*in the domain such that*x*is a dog".)

- Sometimes, logicians abbreviate this (with the "E-shriek!"
quantifier) as:
- Some analogies with computer programs:
- The first "∃" is an existential quantifier.
- It can be read "there is something (call it
*x*) in the domain such that…"

- It can be read "there is something (call it
- The first "
*x*" is an existentially quantified (or "bound") variable.- Think of it as being like a variable-declaration in a programming language.

- The scope of "∃
*x*" is [Dog(*x*) ∧ ∀*y*[Dog(*y*) →*y*=*x*]]- Think of as being like the body of a procedure.

- The "∀" is a nested universal quantifier.
- Think of it as being like the beginning of a nested procedure.

- The "
*y*" is a universally quantified (or "bound") variable.- Think of it as being like a local variable-declaration in the nested procedure.

- The scope of "∀
*y*" is [Dog(*y*) →*y*=*x*]- Think of it as being like the body of the nested procedure.

- All 3 occurrences of "
*x*" are said to be**bound occurrences of***x*in "∃*x*[…]"- Think of them as being variables that are "local" to the "procedure" that is the scope of the existential quantifier.

- All 3 occurrences of
*y*are**bound occurrences of***y*in "∀*y*[…]"- Think of them as being variables that are "local" to the "nested procedure" that is the scope of the universal quantifier.

- The rightmost occurrence of
*x*(i.e., the one in the equality proposition "*y*=*x*")

is said to be a*free*occurrence of*x*in*"∀**y*[…]"- Think of it as being a variable that is
*global*to the "nested procedure" that is the scope of the universal quantifier.- In general, a variable is:
- "bound" if it's "local"
- "free" if it's "global".

- In general, a variable is:

- Think of it as being a variable that is

And now for something apparently completely different…

I didn't get a chance to cover this in today's lecture,

and I think I won't spend any lecture time on it (in order to move on with other topics),

but I thought you might find this interesting. - The first "∃" is an existential quantifier.
- Is the present King of France bald?
- Ask your friends; see what they say.
- It's a trick question: There is no present King of France!
- OK…so…is "The present King of France is bald"
true, or is it false?
- Surely, it's got to be one or the
other!

- Here's
how Bertrand Russell analyzed it:
- ∃
*x*[IsPresentlyKingOfFrance(*x*) ∧ ∀*y*[IsPresentlyKingOfFrance(*y*) →*y*=*x*] ∧ IsBald(*x*)] - I.e., there is someone who is presently King of France,

and there's only one (see IA, above!)

and he's bald. - On this analysis, the sentence is false,

because there is no present King of France. - But if it's false, how do we say that the present King of
France is not bald?
- There are two ways to negate the English sentence:
- It's not the case that the present King of France is bald.
- The present King of France is not bald.

- On the first way, we just put "¬" in front of the whole
existentially quantified proposition
- That negation will be true, because the existentially quantified proposition was false.

- On the second way, we put the "¬" in front of "IsBald(
*x*)"- Now the negation is also false!

—again, because there's no present King of France!

- Now the negation is also false!

- There are two ways to negate the English sentence:

- ∃

- I.e., There is (at least) one thing in the domain
that is a dog,
**Peano's Axioms:**- More examples of nested quantifiers, here used to axiomatically characterize ("define") the natural numbers.
- For Giuseppe Peano's biography, see here and here.
- For more on Peano's axioms, see this course's webpages on:
- Peano's Axioms:
- Syntax and semantics of our representation:
**Let N(***x*) = "*x*is a natural number".- (i.e.,

*x*∈ {0,1,2,3,…}

or:*x*is one of: 0 ⇒ 1 ⇒ 2 ⇒ 3 ⇒ …)**Let S(***x*,*y*) = "*x*is succeeded b*y**y*" or "*y*is a successor of*x*".- (i.e.,

*y*is a very next number after*x*) **Axiom 1: 0 is a natural number**- (or: There is a natural number; call it "0")

In FOL: N(0)

**Axiom 2: Ever***y*natural number has a unique successor that is a natural number.In FOL: ∀

*x*[N(*x*) → ∃!*y*[N(*y*) ∧ S(*x*,*y*)]]**Axiom 3: No natural number is its own successor.**In FOL: ∀

*x*∀*y*[(N(*x*) ∧ N(*y*) ∧ S(*x*,*y*)) →*x*≠*y*]**Axiom 4: 0 is not the successor of an***y*natural number.- (i.e., no natural number has 0 as its successor;

i.e., it all starts with 0)In FOL: ¬∃

*x*[N(*x*) ∧ S(*x*,0)]**Axiom 5: If (what appear to be) "two" natural numbers have the same successor, then "they" are the same.**In FOL: ∀

*x*∀*y*[(N(*x*) ∧ N(*y*)) → (∃*z*[N(*z*) ∧ S(*x*,*z*) ∧ S(*y*,*z*)] →*x*=*y*)]**Axiom 6: The Principle of Mathematical Induction**(later…)

- Syntax and semantics of our representation:

Text copyright © 2010 by William J. Rapaport (rapaport@buffalo.edu)

http://www.cse.buffalo.edu/~rapaport/191/F10/lecturenotes-20100927.html-20100927