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Logic10 - Introduction to Logic Lecture 10 Brian Weatherson...

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Introduction to Logic Lecture 10 Brian Weatherson, Department of Philosophy October 5, 2009 Logic 201 (Section 5) Lecture 10 1 / 51
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What are Proofs A proof that an argument is valid is: 1 A sequence of statements; 2 The first statements are the premises of the argument; 3 The last statement is the conclusion of the argument; 4 Each statement follows directly from earlier statements; 5 And we say exactly how each statement follows from earlier statements. Logic 201 (Section 5) Lecture 10 2 / 51
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Formal Proofs Every sentence is in FOL. We have careful restrictions on what counts as an acceptable move. These restrictions are defined syntactically – i.e., in terms of how the sentences are spelled. There are two moves that go along with each connective. These are called, for reasons that might be mysterious at first, introduction and elimination rules. Logic 201 (Section 5) Lecture 10 3 / 51
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Rules for Logic 201 (Section 5) Lecture 10 4 / 51
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-Elimination From A B infer A . We cite the line where A B occurs. Logic 201 (Section 5) Lecture 10 5 / 51
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-Elimination Also, from A B we can infer B . The same line is cited. Logic 201 (Section 5) Lecture 10 6 / 51
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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 10 7 / 51
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-Introduction It doesn’t matter which order in the proof A and B turn up in. B could be before A , and you can still infer A B . Logic 201 (Section 5) Lecture 10 8 / 51
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-Introduction. Also works for longer conjunctions: From A , B , C infer A B C . Logic 201 (Section 5) Lecture 10 9 / 51
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Rules for Logic 201 (Section 5) Lecture 10 10 / 51
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-Introduction From A , infer A B . We cite the line where A occurs. Logic 201 (Section 5) Lecture 10 11 / 51
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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 10 12 / 51
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-elimination -elimination is harder. The idea is that if A implies C , and B implies C , then A B implies C . The reason for that is, if you know A B is true, and you know that whichever one of A and B is true, C must follow, then you know that C is true. So the way the rule goes is that from a line that says A B , and a proof of C from A , and a proof of C from B , we can infer C . Logic 201 (Section 5) Lecture 10 13 / 51
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-elimination Here’s how we apply it.
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