1. CS 1502 Formal Methods in Computer Science
Lecture Notes 13
Equivalences, Arguments, and Proofs involving Quantifiers

2. Propositional Logic Tautology Tautological Consequence
Tautological Equivalence
Based on the
truth-functional
Connectives

3. First-Order Logic
Takes into consideration all of the truth-functional connectives (     ), the identity symbol (=), and the quantifiers (x y).

4. First-Order Logic FO Validity: a sentence that can’t be false
FO Consequence: applies to an argument whose conclusion can’t be made false when all of its premises are true.
FO Equivalence applies to a pair of sentences that, in all possible circumstances, have the same truth values

5. Facts All tautological consequences are FO Consequences.
All tautological equivalencies are FO Equivalencies.

6. FO Consequence x [P(x)  Q(x)] Q(b) P(b) x [Tet(x)  Large(x)]
 C is not a
tautological
consequence
of A and  B
x [P(x)  Q(x)]
Q(b)
P(b) Q P b
x [Tet(x)  Large(x)]
Large(b)
Tet(b)
A  B
 C

7. Replacement Method
This method is used to determine if a sentence is an FO Validity and if an argument is an FO Consequence.

8. Replacement Method
Replace all predicates in the sentence or in the argument with symbolic ones making sure that if a predicate appears more than once it is replaced with the same symbolic name.
See if you can describe a circumstance where the sentence is false, if this is impossible then the sentence is a FO Validity.
See if you can describe a circumstance where the conclusion is false and the premises are all true. If this is impossible, then the conclusion is an FO Consequence of its premises.

9. DeMorgan’s Laws for Quantifiers
x P(x)  x [P(x)] Nobody is P. Everyone is not P.
x P(x)  x [P(x)] It is not the case that everyone is P. Somebody is not P. P P

10. Aristotelian Forms Revisited
Negate: All P’s are Q’s. ~all x (P(x)  Q(x))  ~all x (~P(x) v Q(x))  exist x (~(~P(x) v Q(x))) 
exist x (P(x) ^ ~Q(x))
Some P’s are not Q’s

11. A Special Form and its Equivalent
Only Q’s are P’s
All P’s are Q’s P Q

13. Other Equivalences and Non-Equivalences (which are which?)
x [P(x)  Q(x)]  x P(x)  x Q(x)
x [P(x)  Q(x)]  x P(x)  x Q(x)
x [P(x)  Q(x)]  x P(x)  x Q(x)
x [P(x)  Q(x)]  x P(x)  x Q(x)

14. Other Equivalences x P  P, where x is not free in P
x [P  Q(x)]  P  x Q(x)
x [P  Q(x)]  P  x Q(x)
x P(x)  y P(y)
 x P(x)   y P(y)

15. Proofs Involving Quantifiers
Universal Elimination
x S(x) …
S(c)  Elim

16. Example
Prove x Cube(x) x Large(x) Large(d)  Cube(d)]

18. Proofs Involving Quantifiers
Universal Introduction c …
S(c) x S(x)  Intro
Assume c is an
arbitrary element
in the domain of
discourse.

19. Example
Prove x Cube(x) x Large(x) x [Large(x)  Cube(x)]

21. Proofs Involving Quantifiers
Existential Introduction
S(c) …
x S(x)  Intro

22. Example
Prove Cube(e) Large(e)  LeftOf(e,a) x [Cube(x)  LeftOf(x,a)]

24. Proofs Involving Quantifiers
Existential Elimination x S(x)
c S(c) …
Q Q  Elim
Symbol c cannot
appear outside this
subproof!
Since there exists an
x such that S(x),
let c designate
this object.

25. Example
Prove x Large(x) x Cube(x) x [Large(x)  Cube(x)]

27. General Conditional Proof
Universal Introduction
c P(c) …
Q(c) x [P(x)  Q(x)]  Intro
Assume c is an
arbitrary element
in the domain of
Discourse and
assume P(c)

28. Example
Prove x [P(x)  Q(x)] z [Q(z)  R(z)] x [P(x)  R(x)]