1. Propositional Logic Predicate Logic

2. Review of Propositional Logic
Propositional variables
Propositional constants: T, F
Logical connectives
Let P: Today is Sunday
Q: We have guests
P  Q : Today is Sunday and We have guests
P  Q : Today is Sunday or We have guests
P : Today is not Sunday.
P  Q : if today is Sunday then we have guests.
P  Q : Today is Sunday if and only if we have guests.

3. Semantics Propositions and expressions have
truth values: can be true or false.
The truth value of an expression is determined by truth tables, e.g.:
P Q P v Q
T T T
T F T
F T T
F F F

4. Propositional Proof Theory
Propositional proof theory: A set of axioms (logical identities) and inference rules used to manipulate expressions and to obtain true expressions out of other true expressions.
Inference rules: give a mechanical procedure to obtain true expressions out of other true expressions.
Modus ponens: P, P  Q |= Q

5. Resolution A  B, B  C |= A  C compare with: B, B  C |= C
(B C = B  C)
If B is true, then C must be true, if B is false than A must be true.
This means that WHENEVER the conjunction
(A  B)  (B  C) is true, A  C is also true.

6. How resolution works
Eliminate opposite literals and rewrite two expressions as one
P1 v P2 P2
P1 P1 v P3 v P4
P3 v P4
This is the conclusion

7. Proof by refutation Add the negation of the statement to be proved
P4 v P6 P6
P4 P4
contradiction

8. Predicate Logic
Represents properties and relations by using predicates with arguments.
likes(mary,apples)  likes(mary,grapes)
Quantifiers: indicate the scope of the predicate
Universal quantifier:
 X likes ( mary , X) Mary likes everything
Existential quantifier
 X likes ( mary , X)
there is something which Mary likes.

9. Proof Theory for Predicate Logic
Based on the resolution procedure.
Unification: matching predicates with variables to atomic sentences (matching variables to constants.)
Example: Is Socrates mortal?
 X (human(X)  mortal(X))
human(socrates)
human(plato)
alien(spock)

10. Knowledge representation using predicate logic
Some books are interesting.
 x (book(x) Λ interesting(x))
Anybody that has a friend is not lonely
(If someone has a friend, they are not lonely)
x ( y friend(x,y)  ~lonely(x))

11. Prolog example
Predicate logic representation is used only on paper. Computational implementation: logic programming languages
In Logic and in Prolog:
 X  Y (father(X,Y) v mother(X,Y)  parent(X,Y))
 X  Y( ( Z(parent(Z,X)  parent(Z,Y))  siblings(X,Y))
parent(X,Y):- father(X,Y).
parent(X,Y):- mother(X,Y).
siblings(X,Y):- parent(Z,X),parent(Z,Y).