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Lecture 14

Lecture 14. contradiction elimination subproofs strategy and tactics. exams.

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Lecture 14

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  1. Lecture 14 contradiction elimination subproofs strategy and tactics

  2. exams • The large circled number towards the end of your exam is your score out of the total possible number of points (110). So it’s not your percentage. These “raw scores” were correlated with letter grades so as to yield a normal distribution around a roughly B+ average. • The exam and answers to it have been posted to the course website. • The rest of the course will be building on the material covered in this midterm. If you did very poorly on the midterm, you will not be able to catch up by starting with the current material and putting the old stuff aside. • In fact, if you did very poorly, you may be simply unable to catch up and you should consider withdrawing. If you are considering doing so, I recommend that you visit me in my office. Bring your exam.

  3. Contradiction introduction and subproofs • A set of sentences is inconsistent if and only if there is no possible world in which the sentences are all true. • A set of sentences is TT-inconsistent if and only if there is no row of the sentences’ joint truth table in which they’re all true. • If you can derive ⊥ in the main proof, then your premises are inconsistent. • Since we will usually be dealing with consistent premises, then, we will generally use ⊥ within subproofs. • A derivation of ⊥ in a subproof shows only that the assumption of the subproofis inconsistent with the premises.

  4. 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, we can infer ⊥ using Ana Con (provided the sentences are both accessible). • Example: 1. LeftOf(a,b) 2. RightOf(a,b) 3. ⊥ Ana Con: 1,2

  5. i>clicker question • Suppose there is a Tarski’s World in which sentences A and B are both true. Then • Ana Con would allow us to infer ⊥ from A and B. • Ana Con would not allow us to infer ⊥ from A and B. • Not enough info.

  6. Contradiction elimination

  7. Prelude to Contradiction elimination • As strange as it may seem, ifan argument has contradictory premises, then its conclusion is a logical consequence of its premises. … P ¬P Q T F ? F T ? • We want our Fitch rules to capture valid argumentation: • if an argument is valid, then we want our rules to allow us to infer the conclusion from the premises in Fitch. • In this case, that means that when we have contradictory sentences in our proof, we should be able to infer anything!

  8. Contradiction elimination • ⊥Elim: ⊥ . . . . Q

  9. Examples A∨¬(B∨C) B A ¬¬(A∧B) C∨¬A C

  10. i>clicker question • Suppose that, assuming A for a subproof, we derive ¬(a=a). This would allow us to prove, under the main vertical line of the proof, • ¬A • ⊥ • A and B • None of the above

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