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# Logic14 - Introduction to Logic Lecture 14 Brian Weatherson...

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Introduction to Logic Lecture 14 Brian Weatherson, Department of Philosophy October 19, 2009 Logic 201 (Section 5) Lecture 14 1 / 90

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Revision Logic 201 (Section 5) Lecture 14 2 / 90
Revision The plan for today is to work through questions 8.44-8.51 from the textbook as revision for the test. (We won’t have time to get to 8.52 and 8.53) The test will consist of some translation questions, and some questions where you have to judge whether an argument is valid, come up with a proof if it is, and a countermodel if it is not. Logic 201 (Section 5) Lecture 14 3 / 90

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Rules You will not be allowed to have notes, calculators, phones etc in use during the test. For most of the proofs, you will not be allowed to use the TautCon and AnaCon rules. For some questions, you will be allowed to use TautCon under two conditions 1 You use at most one input sentence; 2 As well as the proof, you turn in a truth table showing that the output sentence is a consequence of the input sentence, or is a tautology if there is no input sentence. Logic 201 (Section 5) Lecture 14 4 / 90
Question 8.44 Logic 201 (Section 5) Lecture 14 5 / 90

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Question 8.44 Is this argument valid? Adjoins ( a , b ) Adjoins ( b , c ) SameRow ( a , c ) a c Logic 201 (Section 5) Lecture 14 6 / 90
Question 8.44 It is invalid if there is a world where the premises are true and the conclusion false. That is, if there is a world where: 1 Adjoins ( a , b ) Adjoins ( b , c ) is true, so a adjoins b and b adjoins c . 2 SameRow ( a , c ) is true, so a and c are in the same row. 3 a c is false, so a is identical to c Logic 201 (Section 5) Lecture 14 7 / 90

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Question 8.44 If a and c are identical, then the first two premises reduce to: 1 Adjoins ( a , b ) Adjoins ( b , a ) ; and 2 SameRow ( a , a ) And it is easy to make those true by just putting a and b next to one another. Logic 201 (Section 5) Lecture 14 8 / 90
Question 8.44 In this model we have Adjoins ( a , b ) Adjoins ( b , a ) and SameRow ( a , a ) true and a c false, so the argument is invalid. Logic 201 (Section 5) Lecture 14 9 / 90

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Question 8.45 Logic 201 (Section 5) Lecture 14 10 / 90
Question 8.45 Here is the question: ¬ ( Cube ( b ) b = c ) Cube ( c ) There is no premise, so the issue is whether the conclusion is a logical truth. Logic 201 (Section 5) Lecture 14 11 / 90

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A truth table says that the conclusion is not a tautology. Logic 201 (Section 5) Lecture 14 12 / 90
But look at the only row where it is false. We have Cube ( b ) true, b = c true and Cube ( c ) false. Logic 201 (Section 5) Lecture 14 13 / 90

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That isn’t possible. If Cube ( b ) and b = c are true, then Cube ( c ) must be too. So there’s no possibility where the conclusion is false, so it’s a logical truth. Logic 201 (Section 5) Lecture 14 14 / 90
Question 8.45 So we’re going to have to prove the sentence is true. It’s a disjunction, a sentence of the form X Y , and there are three ways to prove a disjunction. 1 Prove X and use -introduction to get X Y .

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Logic14 - Introduction to Logic Lecture 14 Brian Weatherson...

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