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Unformatted text preview: Chapter 5: Some Extensional Many Valued Semantics First many valued logic (defined semantically only) was formulated by ˆLukasiewicz in 1920. We present here five 3valued logics seman tics that are named after their authors: ˆLukasiewicz, Kleene, Heyting , and Bochvar . Three valued logics , when defined semanti cally, enlist a third logical value ⊥ , or m in Bochvar semantics.. We assume that the third value is interme diate between truth and falsity, i.e. that F < ⊥ < T , or F < m < T . 1 All of presented here semantics take T as des ignated value, i.e. the value that defines the notion of satisfiability and tautology. The third value ⊥ corresponds to some no tion of incomplete information , or inconsis tent information or undefined or unknown . Historically all these semantics were are called logics, we use the name logic for them, instead saying each time ”logic defined se mantically”, or ”semantics for a given logic”. 2 ˆLukasiewicz Logic ˆL: Motivation ˆLukasiewicz developed his semantics (called logic ) to deal with future contingent state ments. Contingent statements are not just neither true nor false but are indeterminate in some metaphysical sense. It is not only that we do not know their truth value but rather that they do not possess one. 3 The Language : L = L {¬ , ⇒ , ∪ , ∩} . Logical Connectives are the following oper ations in the set { F, ⊥ ,T } . For any a,b ∈ { F, ⊥ ,T } , ¬ ⊥ = ⊥ , ¬ F = T, ¬ T = F, a ∪ b = max { a,b } , a ∩ b = min { a,b } , a ⇒ b = ( ¬ a ∪ b if a > b T otherwise 4 ˆL 3valued truth tables ˆL Negation ¬ F ⊥ T T ⊥ F ˆL Conjunction ∩ F ⊥ T F F F F ⊥ F ⊥ ⊥ T F ⊥ T ˆL Disjunction ∪ F ⊥ T F F ⊥ T ⊥ ⊥ ⊥ T T T T T ˆLImplication ⇒ F ⊥ T F T T T ⊥ ⊥ T T T F ⊥ T 5 A truth assignment is any function v : V AR→ { F, ⊥ ,T } Extension of v to the set F of all formulas: v * : F → { F, ⊥ ,T } . is defined by the induction on the degree of formulas as follows. v * ( a ) = v ( a ) for a ∈ V AR , v * ( ¬ A ) = ¬ v * ( A ), v * ( A ∩ B ) = ( v * ( A ) ∩ v * ( B )), v * ( A ∪ B ) = ( v * ( A ) ∪ v * ( B )), v * ( A ⇒ B ) = ( v * ( A ) ⇒ v * ( B ))....
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 Spring '08
 Bachmair,L
 Computer Science, Logic, Logical connective, Truth value, Intuitionistic Logic, Lukasiewicz, 3element Heyting

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