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).

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.

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>}

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

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

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

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