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Introduction to Logic Lecture 9 Brian Weatherson, Department of Philosophy September 30, 2009 Logic 201 (Section 5) Lecture 9 1 / 30 Informal Proofs Logic 201 (Section 5) Lecture 9 2 / 30 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 9 3 / 30 Formal and Informal Proofs We write formal proofs out using the Fitch notation. We write informal proofs in English, either in sentences or in bullet points. In these slides I’ll often use bullet points, or even numbered bullet points, to make it easier to see what I’m referring to. There are a lot of different ways of doing formal proofs. We are using the Fitch system because it looks the most like reasoning in English, especially when the English version is in numbered sentences. Logic 201 (Section 5) Lecture 9 4 / 30 Follows Directly One of our conditions on a proof was: Each statement follows directly from earlier statements. But what does it mean to say that something ‘follows directly’? Logic 201 (Section 5) Lecture 9 5 / 30 Follows Directly In practice, what follows directly depends a lot on what your audience is. Some things that might be obvious in an engineering lab might need to be spelled out in a law court, and vice versa. In intro logic, at least at the start, we spell out everything as clearly as we can. As we move through the course, we’ll get more and more liberal about what steps we allow. Logic 201 (Section 5) Lecture 9 6 / 30 Starting Steps But even in intro logic, there are a few steps that are too obvious to need more justification. These include: From P ∧ Q, infer P. From P ∧ Q, infer Q. From P and Q, infer P ∧ Q. From P, infer P ∨ Q. From Q, infer P ∨ Q. Logic 201 (Section 5) Lecture 9 7 / 30 Proof by Cases Logic 201 (Section 5) Lecture 9 8 / 30 Suppositions It’s time to complicate a little our picture of what goes on in a proof. Consider this situation: A, B and C are baseball teams. Each of them has won 90 games this season, and no other team has won more games. C has played all its games for the season. A and B have one game left, and they will be playing each other. In baseball, every game must end with one or other team winning. How might we prove that at least one team will end up with more wins than C? Logic 201 (Section 5) Lecture 9 9 / 30 Here is one way the proof might go. 1 A and B are playing against each other. 2 So one of A and B will win that game. 3 Suppose A wins the game. 4 Then A will have 91 wins. 5 So A will have one more win than C. 6 So a team has more wins than C. 7 Suppose B wins the game. 8 Then B will have 91 wins. 9 So B will have one more win than C. 10 So a team has more wins than C. 11 So, one way or the other, a team will end up with more wins than C. Logic 201 (Section 5) Lecture 9 10 / 30 Here is one way the proof might go. 1 A and B are playing against each other. 2 So one of A and B will win that game. 3 Suppose A wins the game. 4 Then A will have 91 wins. 5 So A will have one more win than C. 6 So a team has more wins than C. 7 Suppose B wins the game. 8 Then B will have 91 wins. 9 So B will have one more win than C. 10 So a team has more wins than C. 11 So, one way or the other, a team will end up with more wins than C. That’s a good proof, but what is going on at lines 3 and 7? Logic 201 (Section 5) Lecture 9 10 / 30 Suppositions The answer is that we are supposing something. A supposition is not something that follows from the premises. So it is not something that we are claiming must be true. But it is something that helps to structure the proof   ::::::::::::::::::::::::::::::::::::::::CONTENT CONTINUED IN THE ATTACHMENT::::::::::::::::::::::::::::::::::::::::::::::::::: [Show More]

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