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IMPORTANT!!

IMPORTANT!!. As one member of our class recognized, there is a major mistake on page 180 of the text where the rule schemas for SD are laid out. It symbolizes a rule it calls E2 – there is no such rule! – as P  Q Q P Such a rule is NOT truth preserving and not in SD. IMPORTANT!!.

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IMPORTANT!!

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  1. IMPORTANT!! As one member of our class recognized, there is a major mistake on page 180 of the text where the rule schemas for SD are laid out. It symbolizes a rule it calls E2 – there is no such rule! – as P  Q Q P Such a rule is NOT truth preserving and not in SD

  2. IMPORTANT!! There is only one rule for eliminating the horseshoe (E). And it is symbolized properly inside the front cover of the text and used throughout the chapter. P  Q P Q

  3. Less important… I did not notice that this new edition has us add the relevant rule following an auxiliary assumption that starts a subderivation. This is useful when you’re trying to go back to fill in line numbers especially if the derivation contains a lot of subderivations and auxiliary assumptions.

  4. Proving SD notions • Using derivations to prove that a sentence of SL is a theorem in SD a sentence Pis derivable in SD from a set  of sentences of SL an argument of SL is valid in SD a set of sentences of SL is inconsistent in SD sentences P and Q are equivalent in SD

  5. Show that⊦ A  (B  A) A A/I B A/I A 1 R B  A I A  (B  A) I

  6. Can we show that⊦ A  (B  C) A A/I ----------- B A/I ----- C B  C I A  (B  C) I

  7. An argument is valid in SD IFF its conclusion is derivable from the set consisting of its premises Show that the following argument is valid in SD: ~A v ~B A ----------- ~B

  8. 1 ~A v ~B A 2 A A 3 ~A A/vE ~B ~B A/vE ~B ~B vE

  9. 1 ~A v ~B A 2 A A 3 ~A A/vE 4 B A/~I ~B ~I ~B A/vE ~B R ~B vE

  10. 1 ~A v ~B A 2 A A 3 ~A A/vE 4 B A/~I 5 A 2R 6 ~A 3R 7 ~B 4-6 ~I 8 ~B A/vE 9 ~B 8R 10 ~B 1, 3-7, 8-9 vE

  11. There’s more than one way to derive a sentence, but some are easier… 1 ~A v ~B A 2 A A ~B how about ~I?

  12. There’s more than one way to derive a sentence, but some are easier… 1 ~A v ~B A 2 A A 3 B A/~I 4 A 2 R ~A ~B how about ~I?

  13. 1 ~A v ~B A 2 A A 3 B A/~I 4 A 2 R 5 ~A A/vE 6 ~A 5R 7 ~B A/vE 8 A A/~I 9 B 3 R 10 ~B 7R 11 ~A 8-10 ~I 12 ~A 5-6, 7-11 vE 13 ~B 3-12 ~I

  14. One special case of validity… Show that the following argument is valid in SD: A  B A  ~B A ----------- M  R

  15. Special cases… 1 A  B A 2 A  ~B A 3 A A M  R

  16. Special cases… 1 A  B A 2 A  ~B A 3 A 4 ~(M  R) A/~E B ~B M  R ~E

  17. Special cases… 1 A  B A 2 A  ~B A 3 A 4 ~(M  R) A/~E 5 B 1, 3  E 6 ~B 2, 3  E 7 M  R 4-6, ~E

  18. P and Q are equivalent in SD IFF Q is derivable in SD from {P} and P is derivable in SD from {Q} Show that the following pair of sentences is equivalent in SD: A ~~A So we need 2 derivations

  19. Demonstrating equivalence 1 A A 2 ~A A/~I 3 ~A 2R 4 A 1R 5 ~~A 2-4,~I

  20. Demonstrating equivalence 1 ~~A A A

  21. Demonstrating equivalence 1 ~~A A 2 ~A A/~E 3 ~A 2 R 4 ~~A 1 R 5 A 2-4 ~E

  22. Demonstrating that a set is inconsistent in SD • A set  is inconsistent in SD IFF there is some sentence P such that both P and ~P are derivable from . • A set  is consistent in SD IFF there is no sentence P such that both P and ~P are derivable from 

  23. Show that {A  B, B  ~A, A} is inconsistent in SD 1 A  B A 2 B  ~A A 3 A A 4 A 3R ~A

  24. Show that {A  B, B  ~A, A} is inconsistent in SD 1 A  B A 2 B  ~A A 3 A A 4 A 3R 5 B 1, 4  E 6 ~A 2, 5 E

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