Logic13

Logic13 - Introduction to Logic Lecture 13 Brian Weatherson...

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Introduction to Logic Lecture 13 Brian Weatherson, Department of Philosophy October 14, 2009 Logic 201 (Section 5) Lecture 13 1 / 46

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Conditionals and Proofs Logic 201 (Section 5) Lecture 13 2 / 46
Modus Ponens If we have a conditional as a premise, it is easy to see what we can prove with it. From A and A B you can infer B . The Latin name for this rule is Modus Ponens. We’ll also call it -elimination. We’ve been tacitly using this rule quite a bit in our informal arguments so far in the course, and now we have a formalisation of it. Logic 201 (Section 5) Lecture 13 3 / 46

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Conditional Proof The way we prove conditionals is also very intuitive, though a little harder to apply. You can infer A B from a subproof that starts with A and ends with B . That is, to prove A B , you show that B follows from A . In informal proofs, you show that you can get B from a supposition of A , and inferring A B discharges the supposition that A . The idea here is quite natural. The problem is that in practice subproofs are hard, especially when we have to embed them. Like the previous rule, this has two names. -introduction; and Conditional Proof Logic 201 (Section 5) Lecture 13 4 / 46
An Example Here’s an example from the book of the rules in use. They’re trying to show that Tet ( a ) Tet ( c ) follows from Tet ( a ) Tet ( b ) and Tet ( b ) Tet ( c ) . We are given, as premises, Tet ( a ) Tet ( b ) and Tet ( b ) Tet ( c ) . We want to prove Tet ( a ) Tet ( c ) . With an eye toward using conditional proof, let us assume, in addition to our premises, that Tet ( a ) is true. Then, by applying modus ponens using our first premise, we can conclude Tet ( b ) . Using modus ponens again, this time with our second premise, we get Tet ( c ) . So we have established the consequent, Tet ( c ) , of our desired conditional on the basis of our assumption of Tet ( a ) . But then the rule of conditional proof assures us that Tet ( a ) Tet ( c ) follows from the initial premises alone. Logic 201 (Section 5) Lecture 13 5 / 46

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We can do the same proof in Fitch notation. 1 Tet ( a ) Tet ( b ) 2 Tet ( b ) Tet ( c ) 3 Tet ( a ) 4 Tet ( b ) -elim: 1, 3 5 Tet ( c ) -elim: 2, 4 6 Tet ( a ) Tet ( c ) -intro: 3-5 There are several things to note about the last line. 1 We cite the subproof that justifies the conditional. 2 The conditional itself is outside that subproof. 3 The conditional’s antecedent is the first line of the subproof, and its consequent is the last line. Logic 201 (Section 5) Lecture 13 6 / 46
If n 2 is even, then n is even A trickier example from the book. The method of conditional proof tells us that we can proceed by assuming Even ( n 2 ) and proving Even ( n ) . So assume that n 2 is even. To prove that n is even, we will use proof by contradiction. Thus, assume that n is not even, that is, that it is odd. Then we can express n as 2 m + 1, for some m. But then we see that: n 2 = ( 2 m + 1 ) 2 = 4 m 2 + 4 m + 1 = 2 ( 2 m 2 + 2 m ) + 1 But this shows that n 2 is odd, contradicting our first assumption.

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