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Logic11 - Introduction to Logic Lecture 11 Brian Weatherson...

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Introduction to Logic Lecture 11 Brian Weatherson, Department of Philosophy October 7, 2009 Logic 201 (Section 5) Lecture 11 1
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Rule Revision Logic 201 (Section 5) Lecture 11 2
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-Elimination From A B infer A . We cite the line where A B occurs. Logic 201 (Section 5) Lecture 11 3
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-Elimination Also, from A B we can infer B . The same line is cited. Logic 201 (Section 5) Lecture 11 4
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-Introduction From A and B , infer A B . We cite the lines where A and B occur. Logic 201 (Section 5) Lecture 11 5
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-Introduction From A , infer A B . We cite the line where A occurs. Logic 201 (Section 5) Lecture 11 6
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-Introduction Like with -elimination, there are two versions of this rule. You can also use it to get from B , to A B . In that case, we cite the line where B occurs. Logic 201 (Section 5) Lecture 11 7
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-elimination When using -elimination, we need, and cite, three things. 1 The original disjunction. 2 The subproof from A to C . 3 The subproof from B to C . Logic 201 (Section 5) Lecture 11 8
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¬ -elimination From ¬¬ A infer A . Logic 201 (Section 5) Lecture 11 9
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-introduction From A and ¬ A infer . In general, whenever you have two contradictory sentences, you can infer . Logic 201 (Section 5) Lecture 11 10
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¬ -introduction From subproof that starts with A , ends with , infer ¬ A . In general we’ll prove negated sentences this way. Note again that we cite a proof (see the hyphen in the justification of line 4). Logic 201 (Section 5) Lecture 11 11
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-elimination So -elimination rule says, from infer whatever you like. This will help shorten some otherwise tricky proofs we’ll later see. Logic 201 (Section 5) Lecture 11 12
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Examples Logic 201 (Section 5) Lecture 11 13
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Strategy We’ll go over two of the examples from last time, spending a bit more time on how the proofs were put together. We’ll start with A B C ( A C ) B Logic 201 (Section 5) Lecture 11 14
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A B , C , therefore ( A C ) B 1 A B 2 C 3 A 4 A C -intro: 2, 3 5 ( A C ) B -intro: 4 6 B 7 ( A C ) B -intro: 6 8 ( A C ) B -elim: 1, 3-5, 6-7 We start with the disjunction in premise one, then show that the conclusion follows from each disjunct. That sets up a use of -elimination in the final line. Logic 201 (Section 5) Lecture 11 15
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A B , C , therefore ( A C ) B If we were putting this together, the first thing we’d do is notice that we have a disjunctive premise, so we will eventually have to use -elimination. Let’s set that up: 1 A B 2 C 3 A 4 Rule? 5 ( A C ) B Rule? 6 B 7 ( A C ) B Rule? 8 ( A C ) B -elim: 1, 3-5, 6-7 Logic 201 (Section 5) Lecture 11 16
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A B , C , therefore ( A C ) B The path from A to ( A C ) B goes via A C . 1 A B 2 C 3 A 4 A C -intro: 2, 3 5 ( A C ) B -intro: 4 6 B 7 ( A C ) B Rule?
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