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Artificial Intelligence
SCSB3553
Knowledge Representation and Search
Propositional Logic
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Outline
Formalize Knowledge using Logic
Syntax and Semantic for Propositional
Logic
Compounding Propositions using Truth
Tables
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Formalize Knowledge using Logic
What is Logic?
A system of symbolic logic using symbolsto stand for whole
propositionsand logical connectives
Proposition: a statement or assertion that expresses a judgment
or opinion
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Formalize Knowledge using Logic
Problem
Solution
True
False
Proposition
LogicPredicate
Logic
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Logic
Formalize Knowledge using Logic
Syntax
Syntax
Syntax
Semantics
Problem
Logic- has a well defined syntax,semanticsand proof theory
Syntax- a logic defines thesyntactically acceptable objects
of the logic (well-formed formulae)
Semantics- a logic associate eachformula with a meaning
Proof Theory - concerned withmanipulating formulae according
to certain rules
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Formalize Knowledge using Logic
Proposition LogicTrue
False
Logical Connectivity
Syntax
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Formalize Knowledge using Logic
What is a Proposition?
Statement 1 Statement 2
Letpbe gene A is cancerous Let qbe gene A is healthy
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Formalize Knowledge using Logic
What is a Connective?
combination of propositions to formcompound propositions
AND Conjunction (&& or .)
OR Disjunction (|| or +)
NOT Negation (~)
IF!
THEN Implication (!)
IF AND ONLY IF Equivalent ()
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Syntax for Propositional Logic
The proposition variable S1, andS2are statements
IfSis a sentence, Sis not a sentence (negation)
IfS1andS2are statements, S1 ^ S2 is a statement
(conjunction)
IfS1andS2are statements, S1v S2is a statement (disjunction)
IfS1andS2are statements, S1!S2is a statement (implication)
IfS1andS2are statements, S1S2is a statement
(equivalence)
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Each model species true/false for eachproposition symbol
E.g. P Q Rtrue/false true/false true/false
(With these symbols,!""#$
possiblemodels, can be enumerated automatically)
Rules for evaluating truth with respect to amodel are determined
by truth tables
Semantic for Propositional Logic
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Truth Table for Compound
Propositions - AND
The conjunction pAND q,writtenp ^ q, of
two propositions is true when bothpand q
are true, false otherwise.
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Truth Table for Compound
Propositions - OR
The disjunction pOR q,writtenp vq, of two
propositions is true when p or q (or both) are
true, false otherwise.
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Truth Table for Compound
Propositions - NOT
The negation NOTp of a proposition (or
p) is true whenpis false and is false
otherwise.pmay be read that it is false that
p.
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Truth Table for Compound
Propositions IF!THEN
The implication IFpTHEN q,writtenp!q,
of two propositions is true when eitherpis
false or qis true, and false otherwise.
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Truth Table for Compound
Propositions IF AND ONLY IF
The equivalence IF AND ONLY IFpTHEN
q,writtenp q, of two propositions is true
when bothpand qare true or when bothp
and qare false, and false otherwise.
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Exercise
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Exercise
p..(my breakfast is) eggs.
q..(my breakfast is) cereal.
r..(my breakfast is) toast.
The statement my breakfast is
either eggs or cereal, and toast
may be written in symbolic form
as ???
Thus, this logic tells us thatThe compound proposition is true
if
1.
I eat cereal, eggs and toast; or2. I eat cereal and toast;
or
3. I eat eggs and toast.
