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Semantics

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 .

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Semantics

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  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 A • Example: Blocks world: 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 • ²if [Sentence f is t in interpretation i ] • 2if[Sentence f is f in interpretation i ] • Semantic Rules • ²itrue for all i • 2ifalse for all i [the sentence false has truth value f in all interpret.] • ²i:f if and only if 2if • ²ifÆy if and only if ²if and ²iy [conjunction] • ²ifÇy if and only if ²if or ²iy [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. Models and Entailment entails Sentences Sentences • An interpretation i is a model of a sentence f iff ²if • A set of sentences KB entailsf iff every model of KB is also a model of f semantics semantics subset Interpretations Interpretations KB = A Æ B f = B KB ²f 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 satisfiable, not valid smoke ! fire 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

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