Semantics
• In propositional logic, we associate atoms
with propositions about the world.
• We specify the semantics of our logic, giving
it a “meaning”.
• Such an association of atoms with
propositions is called an interpretation.
• Under a given interpretation, atoms have
values – True or False. We are willing to
accept this idealization (otherwise: fuzzy
logic).
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Interpretations
An interpretation of an expression in the predicate
calculus is an assignment that maps
• object constants into objects in the world,
• n-ary function constants into n-ary functions,
• n-ary relation constants into n-ary relations.
2
Interpretations
Example: Blocks world:
B
A
C
Floor
Predicate Calculus
A
B
C
Fl
On
Clear
World
A
B
C
Floor
On = {<B,A>, <A,C>, <C, Floor>}
Clear = {<B>}
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Semantics
Meaning of a sentence is truth value {t, f}
Interpretation is an assignment of truth values to the
propositional variables
²i f [Sentence f is t in interpretation i ]
2i f [Sentence f is f in interpretation i ]
Semantic Rules
²i true
for all i
2i false
for all i [the sentence false has truth value f in all
interpret.]
²i :f
if and only if 2i f
²i f Æ y
if and only if ²i f and ²i y [conjunction]
²i f Ç y
if and only if ²i f or ²i y [disjunction]
²i P
iff i(P) = t
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Terminology
A sentence is valid iff its truth value is t in
all interpretations (² f)
Valid sentences: true, : false, P Ç : P
A sentence is satisfiable iff its truth value
is t in at least one interpretation
Satisfiable sentences: P, true, : P
A sentence is unsatisfiable iff its truth
value is f in all interpretations
Unsatisfiable sentences: P Æ : P, false, : true
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Models and Entailment
Sentences
semantics
 An interpretation i is a model of a
sentence f iff ²i f
entails
semantics
Sentences
 A set of sentences KB entails f iff
subset
Interpretations
every model of KB is also a modelInterpretations
of f
KB = A Æ B
f=B
KB ² f iff ² KB ! f
KB entails f if and only if (KB ! f) is
valid
AÆB²B
AÆB
B
U
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Sentence
ExamplesInterpretation that make
Valid?
sentence’s truth value = f
smoke ! smoke
valid
smoke ! fire
satisfiable,
not valid
smoke = t, fire = f
(s ! f) ! (: s ! : f)
satisfiable,
not valid
s = f, f = t
(s ! f) ! (: f ! :
s)
b Ç d Ç (b ! d)
bÇdÇ:bÇd
valid
smoke Ç :smoke
s ! f = t, : s ! : f = f
valid
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