First Order Logic

1
First Order Logic
2
The Limits of Propositional Logic

• Consider the argument
• All College of Creative Studies classes are easy.
• This is a College of Creative Studies class.
• Therefore, this class is easy.
• Translating into propositions, gives the form
• p
• q
• Therefore, r.
• An invalid argument.

3
Declaration subject predicate

• It is a valid argument in the richer First-order
  logic.
• A declarative sentence has a subject a
  predicate.
• The subject is the thing about which an assertion
  is made.
• The predicate asserts that the subject has a
  property.

4
Examples

• Joes serves prime rib. P(Joes) true
• 7 is a prime number. Q(7) true
• Jill is a prime candidate. R(Jill) true
• The variables must be taken from a specified set.
• In these examples, each proposition has a
  different domain (restaurants, integers, people,
  respectively).

5
Using variables

• Consider the argument again
• (x is a CCS class) ? (x is easy)
• This class is a CCS class.
• Therefore, this class is easy.
• x is a CCS class is not a proposition because x
  is unspecified.
• It is called an open proposition.

6
Open Propositions

• An open proposition is a declarative sentence
  that
• contains 1 or more variables
• is not a proposition
• becomes a proposition when its variables are
  replaced by an element from the domain.
• That is, it is a function f Un ? T, F, where n
  is the number of variables, and U is the universe
  of discourse.

7

• x is a CCS class is an open proposition.
• It is a function fUCSB classes ? T, F.
• f(CCS CS2) true, if
• CCS CS2 is a CCS class evaluates to true.
• For x is irrational, is the universe R or the
  human race? When in doubt, specify.
• Notation Denote open propositions with upper
  case letters.

8
Open Proposition Examples

• C(x) x is a CCS class.
• E(x) x is an easy class.
• F(x, y) x y 5
• S(x) x is a senator.
• H(x) x is an honorable person.
• Compose open propositions with logical operators
  (e.g., C(x) ? E(x) ).

9
Quantifiers ? ?

• The universal quantifier, ?, means for all
• For all, x, if x is a senator, then x is not
  honest.
• ?x, S(x) ? H(x).
• The existential quantifier, ?, means there
  exists
• It is not the case that there exists, x such
  that if x is a senator, then x is honest.
• ? x, S(x) ? H(x).

10
First-Order Arguments

• In English
• All CCS classes are easy.
• This is a CCS class.
• Therefore, this class is easy.
• A more compact representation
• ?x, C(x) ? E(x).
• C(CCS CS 2).
• Therefore, E(CCS CS 2).

11
Equivalent Forms

• Below, one means at least one
• all true ?x, P(x) ? ?x, P(x) none false
• all false ?x, P(x) ? ?x, P(x) none true
• not all true ?x, P(x) ? ?x, P(x) one false
• not all false ?x, P(x) ? ?x, P(x) one true

12
Existence Proofs

• Constructive
• Exhibit an x such that P(x) true.
• ?x ?N, x3 - 17x2 6x - 51 -579
• Proof x 8.
• Exhibit an algorithm that generates an x such
  that P(x) true.
• ?x ?N, ?y,((y is prime) ? y ? x)
• Proof Given x, set y to x2.

13
Existence Proofs ...

• Non-constructive
• ?x ?Humans, ?y ?Humans, ( hairs on xs head
  hairs on ys head)
• Proof
• 1. There are lt 4 million hairs on any human head.
• 2. There are gt 4 million people.
• At least 2 people have the same number of hairs
  on their head, but we have not exhibited 2 such
  people.

14
Multiple Quantifiers

• ?x, ?y, x y 5
• ?y, ?x, x y 5
• Are they both true?
• What is the universe of discourse?

15
Multiple Quantifiers
?x, ? y, P(x, y)
?y, ? x, P(x, y)
?y, ?x, P(x, y)
?x, ?y, P(x, y)
?x, ?y, P(x, y)
?y, ?x, P(x, y)
? y, ?x, P(x, y)
? x, ?y, P(x, y)
Legend A
B is valid
16
Characters

• ? ? ? ?
• ? ? ? ? ? ? ?
• ? ? ?
• ? ?
• ? ? ? ? ?
• ? ? ? ? ? ? ? ?