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MIT24_241F09_handout04 - ⊃ B ∨ C Argument ± B C ∼...

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Logic I Fall 2009 Session 4 Handout Using shortened truth-tables (STTs) to determine the TV of a compound given TVs for the atomic components. A B C (A B) (C A) F T T Building STTs in reverse to determine whether a compound sentence can possibly have a certain TV. J K J & ( K ∨ ∼ J) P is truth-functionally true i ff . . . Can be proven by. . . Can be refuted by. . . P is truth-functionally false i ff . . . Can by proven by. . . Can be refuted by. . . Notions and notation to know Curly braces for sets: { A, B, { C } , { C, D } , . . . } Set union. { A, B, C } { B, C, D } = Variables for sets of sentences: Γ n The empty set: or { } . Unit / singleton set Γ is truth-functionally consistent i ff . . . Can be established by. . . Inconsistency of Γ can be established by. . . A set Γ of SL sentences truth-functionally entails a sentence P i ff no TVA makes every member of Γ true but P is false. In other words, P is true on every TVA that makes all members of Γ true. Notation: Γ 1 P. Also, note: Γ abbreviates Γ .
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An argument in SL is a set of SL sentences with one designated as the conclusion and the rest designated as premises. E.g.: A ( B
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Unformatted text preview: ⊃ ( B ∨ C ) Argument ( ± ): B ≡ C ∼ B ∼ A P 1 . . . • P n is truth-functionally valid iF no TVA makes P 1 , . . . P n true and C false. C • We can connect truth-functional entailment with truth-functional validity: In the definition of truth-functional entailment, let Γ be { P 1 , . . . P n } , and let P be C. P 1 . . • Thus, P . n is valid iF { P 1 , . . . P n } truth-functionally entails C. C • Prove that Argument ( ± ) is valid and that its premises truth-functionally entail its conclusion by means of a truth-table. A B C MIT OpenCourseWare http://ocw.mit.edu 24.241 Logic I Fall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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