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Lecture 13

Lecture 13. fitch rules for negation. Disjunction elimination and fitch. Example: Disjunction 1. I>clicker question. Is this disjunction elimination? A. Negation elimination. If a double negation ¬ ¬ P is true, then P must be true too. ¬ Elim : ¬¬P . . . P.

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Lecture 13

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  1. Lecture 13 fitch rules for negation

  2. Disjunction elimination and fitch • Example: Disjunction 1

  3. I>clicker question • Is this disjunction elimination? • A

  4. Negation elimination • If a double negation ¬¬P is true, then P must be true too. • ¬Elim: ¬¬P . . . P

  5. Prelude to Negation Introduction: Proof by contradiction • AKA: Reductio ad absurdum, which is Latin for “reduction to the absurd”. • In this kind of proof, we presuppose something for the sake of argument in order to show that it will lead us into contradiction. • If a presupposition leads us into a contradiction, we can conclude that it’s false.

  6. Reductio example 1 • Some people say that all beliefs are equally true. Suppose that that were the case, and that at least one belief were true. Then all beliefs are true. But I believe that not all beliefs are true. So from our supposition, not all beliefs are true. Therefore, all beliefs are and all beliefs aren’t true. Since the supposition leads to contradiction, we should reject it.

  7. Reductio example 2 • There is no chess computer program that will win every game it plays. To see this, suppose that there were such a program. Then we could run it on two computers, A and B, and have them play each other. Since A is running a program that always wins, it will win the game and B will lose. But since B is running the program, it will win and A will lose. Thus the assumption that the unbeatable chess program is possible leads to contradiction and must be rejected.

  8. Reductio example 3 • The legal example on p. 139 of the text.

  9. Reductio and contradiction • In a reductio proof, we may infer the negation of the supposition if it leads to contradiction. • Q: But what’s a contradiction? • A: A logical impossibility, i.e., a sentence that’s false in all possible worlds. • Examples: • P∧¬P • Tet(a)∧Cube(a) • a≠a

  10. Negation introduction • ‘⊥’ is the contradiction symbol; we will use it to indicate in Fitch that a logical impossibility has been reached. • ¬Intro: P . . . ⊥ ¬P

  11. Contradiction introduction • The logical impossibilities that allow us to introduce ‘⊥’ aren’t single sentences, but rather pairs of sentences of the form P and ¬P. • That is, instead of P∧¬P on one line, we may introduce ‘⊥’ only when we have P on one line of the proof and ¬P on another. • ⊥Intro: P . . . ¬P . . . ⊥

  12. Example DeMorgan’s Laws

  13. Contradiction and ana con • Cube(b) and Tet(b) are mutually inconsistent. • However, since there is no way to get them into the form, P and ¬P, ⊥Intro does not recognize them as inconsistent. • Instead, if a proof has sentences that are inconsistent in virtue of the meanings of the Block Language predicates next to the same vertical line, we can infer ⊥ using Ana Con. • Example: 1. LeftOf(a,b) 2. RightOf(a,b) 3. ⊥ Ana Con: 1,2

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