exercise12

# exercise12 - 6| = a ⇒ ¬ b ∩ a ⇒ ¬ b ⇒ a ∪ b...

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CSE541 EXERCISE 12 Chapters 10, 11, 12 Read and learn all examples and exercises in the chapters as well! QUESTION 1 Let GL be the Gentzen style proof system for classical logic deﬁned in chapter 11. Prove, by constructing a proper decomposition tree that (1) GL (( ¬ a b ) ( ¬ b a )). (2) 6 ‘ GL (( a b ) ( ¬ b a )) . QUESTION 2 Show that tree below do not constitute a proof in GL deﬁned in chapter 10? Provide a correct proof. T A -→ ¬¬ (( ¬ a b ) ( ¬ b a )) | ( → ¬ ) ¬ (( ¬ a b ) ( ¬ b a )) -→ | ( ¬ → ) -→ (( ¬ a b ) ( ¬ b a )) | ( →⇒ ) ( ¬ a b ) -→ ( ¬ b a ) | ( →⇒ ) ( ¬ a b ) , ¬ b -→ a | ( ¬ → ) ( ¬ a b ) -→ b,a ^ ( ⇒-→ ) -→ ¬ a,b,a | ( → ¬ ) a -→ b,a axiom b -→ b,a axiom 1

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QUESTION 3 Let GL be the Gentzen style proof system for classical logic deﬁned in chapter 11. Prove, by constructing a counter-model deﬁned by a proper decomposition tree that
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Unformatted text preview: 6| = (( a ⇒ ( ¬ b ∩ a )) ⇒ ( ¬ b ⇒ ( a ∪ b ))) . QUESTION 4 Let LI be the Gentzen system for intuitionistic logic as deﬁned in chapter 11. Show that ‘ LI ¬¬ (( ¬ a ⇒ b ) ⇒ ( ¬ b ⇒ a )) . QUESTION 5 We know that the formulas below are not Intuitionistic Tautologies. Verify whether H seman-tics (chapter 5) provides a counter-model for them. (( a ⇒ b ) ⇒ ( ¬ a ∪ b )) , (( ¬ a ⇒ ¬ b ) ⇒ ( b ⇒ a )) . QUESTION 6 Show that ‘ LI ¬¬ (( ¬ a ⇒ ¬ b ) ⇒ ( b ⇒ a )) QUESTION 7 Use the heuristic method deﬁned in chapter 12 to prove that 6 ‘ LI (( ¬ a ⇒ b ) ⇒ ( ¬ b ⇒ a )) . 2...
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exercise12 - 6| = a ⇒ ¬ b ∩ a ⇒ ¬ b ⇒ a ∪ b...

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