Propositional Logic Predicate Logic

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Propositional LogicPredicate Logic
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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.
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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
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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
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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.
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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
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Proof by refutation
Add the negation of the statement to be proved
P4 v ?P6 P6
P4 ?P4
contradiction
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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 Existen
tial quantifier ? X likes ( mary , X) there is
something which Mary likes.
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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)
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Knowledge representation using predicate logic
Some books are interesting. ? x (book(x) ?
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))
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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)) pare
nt(X,Y)- father(X,Y). parent(X,Y)-
mother(X,Y). siblings(X,Y)- parent(Z,X),parent(Z,
Y).