Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 231 Dodec(e) Proof 6.122 ¬Small(e)3 ¬Dodec(e) ᴠ Dodec(f) ᴠ Small(e)45 Is it valid?67 What do you have to work with?8910 1112 Dodec(f)1 Dodec(e) Proof 6.122 ¬Small(e)3 ¬Dodec(e) ᴠ Dodec(f) ᴠ Small(e)4 ¬Dodec(e) 3 ‘cases’56 Dodec(f)7 Small(e)89 Dodec(f)10 Dodec(f)11 Dodec(f) 12 Dodec(f) same conclusion as for all subproofs1 Dodec(e) Proof 6.122 ¬Small(e)3 ¬Dodec(e) ᴠ Dodec(f) ᴠ Small(e)4 ¬Dodec(e)567 Small(e)8910 Dodec(f)11 12 Dodec(f)1 Dodec(e) Proof 6.122 ¬Small(e)3 ¬Dodec(e) ᴠ Dodec(f) ᴠ Small(e)4 ¬Dodec(e)5 ⊥ ⊥ Intro: 1,467 Small(e)8910 Dodec(f)11 12 Dodec(f)1 Dodec(e) Proof 6.122 ¬Small(e)3 ¬Dodec(e) ᴠ Dodec(f) ᴠ Small(e)4 ¬Dodec(e)5 ⊥ ⊥ Intro: 1,46 Dodec(f) ⊥ Elim: 57 Small(e)8910 Dodec(f)11 12 Dodec(f)1 Dodec(e) Proof 6.122 ¬Small(e)3 ¬Dodec(e) ᴠ Dodec(f) ᴠ Small(e)4 ¬Dodec(e)5 ⊥ ⊥ Intro: 1,46 Dodec(f) ⊥ Elim: 57 Small(e)8910 Dodec(f)11 12 Dodec(f)1 Dodec(e) Proof 6.122 ¬Small(e)3 ¬Dodec(e) ᴠ Dodec(f) ᴠ Small(e)4 ¬Dodec(e)5 ⊥ ⊥ Intro: 1,46 Dodec(f) ⊥ Elim: 57 Small(e)8 ⊥ ⊥ Intro: 2,7910 Dodec(f)11 12 Dodec(f)1 Dodec(e) Proof 6.122 ¬Small(e)3 ¬Dodec(e) ᴠ Dodec(f) ᴠ Small(e)4 ¬Dodec(e)5 ⊥ ⊥ Intro: 1,46 Dodec(f) ⊥ Elim: 57 Small(e)8 ⊥ ⊥ Intro: 2,79 Dodec(f) ⊥ Elim: 810 Dodec(f)11 12 Dodec(f)1 Dodec(e) Proof 6.122 ¬Small(e)3 ¬Dodec(e) ᴠ Dodec(f) ᴠ Small(e)4 ¬Dodec(e)5 ⊥ ⊥ Intro: 1,46 Dodec(f) ⊥ Elim: 57 Small(e)8 ⊥ ⊥ Intro: 2,79 Dodec(f) ⊥ Elim: 810 Dodec(f)11 12 Dodec(f)1 Dodec(e) Proof 6.122 ¬Small(e)3 ¬Dodec(e) ᴠ Dodec(f) ᴠ Small(e)4 ¬Dodec(e)5 ⊥ ⊥ Intro: 1,46 Dodec(f) ⊥ Elim: 57 Small(e)8 ⊥ ⊥ Intro: 2,79 Dodec(f) ⊥ Elim: 810 Dodec(f)11 Dodec(f) Reit: 1012 Dodec(f)1 Dodec(e) Proof 6.122 ¬Small(e)3 ¬Dodec(e) ᴠ Dodec(f) ᴠ Small(e)4 ¬Dodec(e)5 ⊥ ⊥ Intro: 1,46 Dodec(f) ⊥ Elim: 57 Small(e)8 ⊥ ⊥ Intro: 2,79 Dodec(f) ⊥ Elim: 810 Dodec(f)11 Dodec(f) Reit: 1012 Dodec(f) ᴠ Elim: 3, 7-9, 10-11How not to screw up using subproofs:You can insert anything as the premise of a subproof, but be sure it will be useful to you in terms of some rule strategy.You can only draw material out of a subproof and use it back at a more basic argument level if you have a rule that allows you to! Closely related to the above, once a subproof has been discharged (ended), it is only the subproof as a whole that is available to justify some later step.Strategies for tackling proofs:Take the time to understand the meanings of the sentences in the proof, even if you’ve already been told the argument is valid.Try first writing out an informal proof.Work backwards. (Start with the conclusion—what additional sentence or sentences would allow you to infer the conclusion? Insert such sentences into your proof toward the bottom, listing the rules, then try to go back and prove these intermediate steps.)1 ¬P ᴠ ¬Q Prove another DeMorgan’s Law(from p. 171 of textbook)¬(P ᴧ Q)1 ¬P ᴠ ¬Q2 P ᴧ Q¬(P ᴧ Q)1 ¬P ᴠ ¬Q2 P ᴧ Q3 ¬P¬(P ᴧ Q)1 ¬P ᴠ ¬Q2 P ᴧ Q3 ¬P4 P ᴧ Elim: 2¬(P ᴧ Q)1 ¬P ᴠ ¬Q2 P ᴧ Q3 ¬P4 P ᴧ Elim: 25 Intro: 3,4¬(P ᴧ Q)1 ¬P ᴠ ¬Q2 P ᴧ Q3 ¬P4 P ᴧ Elim: 25 Intro: 3,46 ¬Q¬(P ᴧ Q)1 ¬P ᴠ ¬Q2 P ᴧ Q3 ¬P4 P ᴧ Elim: 25 Intro: 3,46 ¬Q7 Q ᴧ Elim: 2¬(P ᴧ Q)1 ¬P ᴠ ¬Q2 P ᴧ Q3 ¬P4 P ᴧ Elim: 25 Intro: 3,46 ¬Q7 Q ᴧ Elim: 28 Intro: 6,7¬(P ᴧ Q)1 ¬P ᴠ ¬Q2 P ᴧ Q3 ¬P4 P ᴧ Elim: 25 Intro: 3,46 ¬Q7 Q ᴧ Elim: 28 Intro: 6,79 ᴠ Elim: 1, 3-5, 6-8¬(P ᴧ Q)1 ¬P ᴠ ¬Q2 P ᴧ Q3 ¬P4 P ᴧ Elim: 25 Intro: 3,46 ¬Q7 Q ᴧ Elim: 28 Intro: 6,79 ᴠ Elim: 1, 3-5, 6-810 ¬(P ᴧ Q) ¬ Intro:
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