1. 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).

2. 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.

3. Interpretations B Example: Blocks world: A C Floor
Predicate Calculus World
A A
B B
C C
Fl Floor
On On = {<B,A>, <A,C>, <C, Floor>}
Clear Clear = {<B>}

4. 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

5. 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

6. KB entails f if and only if (KB ! f) is valid
Models and Entailment
entails
Sentences
Sentences
An interpretation i is a model of a sentence f iff ²i f
A set of sentences KB entails f iff every model of KB is also a model of f
semantics
semantics
subset
Interpretations
Interpretations
KB = A Æ B
f = B
KB ² f iff ² KB ! f
KB entails f if and only if (KB ! f) is valid
A Æ B B U
A Æ B ² B

7. Examples Interpretation that make sentence’s truth value = f Sentence
Valid?
smoke ! smoke
valid
smoke Ç :smoke
smoke ! fire
satisfiable, not valid
smoke = t, fire = f
satisfiable, not valid
s = f, f = t
s ! f = t, : s ! : f = f
(s ! f) ! (: s ! : f)
valid
(s ! f) ! (: f ! : s)
b Ç d Ç (b ! d)
valid
b Ç d Ç : b Ç d