11-LogImplDeriv

11-LogImplDeriv - Discussion#11 Logical Implications and...

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Discussion #11 Chapter 1, Section 6.1-5 1/16 Discussion #11 Logical Implications and Derivations
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Discussion #11 Chapter 1, Section 6.1-5 2/16 Topics Sound Arguments Derivations Proofs Inference rules Deduction
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Discussion #11 Chapter 1, Section 6.1-5 3/16 Sound Arguments A logical argument is sound if when all premises are true, the conclusion must also be true. Observe that if A represents the premises and C represents the conclusion, a logical argument is sound if A C is a tautology. A is an expression that can always be in CNF, in which case each term in the conjunction is called a premise, which by itself is true. – A = P 1 P 2 P n or just P 1 , P 2 , … P n . Thus, what we wish to show is that P 1 P 2 P n C is a tautology. – Note that if any one of the P i ’s is false, the implication is true; thus, we only have to worry about the case when all P i ’s are true. This idea is called the discharge rule because we can discharge the case in which the left-hand side is false.
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Discussion #11 Chapter 1, Section 6.1-5 4/16 Sound Argument Showing that A C is a tautology constitutes a sound logical argument. • If |= (A 1 A 2 A n C), and A i = T for all i, then C must necessarily be true. A C A C T ? T C = T is the only possibility for the conclusion!
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Discussion #11 Chapter 1, Section 6.1-5 5/16 Deductions and Formal Proofs A deduction is a sequence of logic statements, each of which is known or assumed to be true.
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11-LogImplDeriv - Discussion#11 Logical Implications and...

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