symbolic logic unit 7 the proof method 8 basic inference
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SYMBOLIC LOGIC UNIT 7: THE PROOF METHOD: 8 BASIC INFERENCE RULES Constants vs. Variables A B p q Constants are Variables are abbreviations of placeholders for any specific sentences with formula whatever, determinate meanings.

  1. SYMBOLIC LOGIC UNIT 7: 
 THE PROOF METHOD: 8 BASIC INFERENCE RULES

  2. Constants vs. Variables A ∨ B p ∨ q Constants are Variables are abbreviations of placeholders for any specific sentences with formula whatever, determinate meanings. including both simple and complex formulas.

  3. Substitution Instances A substitution instance (s.i.) of a statement form is a statement obtained by substituting (uniformly) some statement for each variable in the statement form. We must substitute the same statement for repeated occurrences of the same variable, and we may substitute the same statement for different variables. Thus both A v B and A ∨ A are s.i.’s of p ∨ q, but A ∨ B is not an s.i. of p ∨ p.

  12. SYMBOLIC LOGIC UNIT 8: 
 REPLACEMENT RULES

  13. DN: p :: ~~p Dup: p :: p ∨ p 
 p :: p•p Comm: p ∨ q :: q ∨ p 
 p•q :: q•p Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  14. Unit 8, #5b DN: p :: ~~p 1. ~(A ∨ B) Pr. / ∴ ~B Dup: p :: p ∨ p 
 p :: p•p Comm: p ∨ q :: q ∨ p 
 p•q :: q•p Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  15. Unit 8, #5c DN: p :: ~~p 1. ~(A ⊃ B) Pr. / ∴ ~B Dup: p :: p ∨ p 
 p :: p•p Comm: p ∨ q :: q ∨ p 
 p•q :: q•p Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  16. Unit 8, #5f DN: p :: ~~p 1. A ⊃ B Pr. Dup: p :: p ∨ p 
 p :: p•p 2. A ⊃ C Pr. / ∴ A ⊃ (B • C) Comm: p ∨ q :: q ∨ p 
 p•q :: q•p Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  17. Unit 8, #5m DN: p :: ~~p 1. A ⊃ C Pr. / ∴ (A • B) ⊃ C Dup: p :: p ∨ p 
 p :: p•p Comm: p ∨ q :: q ∨ p 
 p•q :: q•p Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  18. Unit 8, #5n DN: p :: ~~p 1. ~((A ∨ B) ∨ (C ∨ D)) Pr. / ∴ ~D Dup: p :: p ∨ p 
 p :: p•p Comm: p ∨ q :: q ∨ p 
 p•q :: q•p Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  19. Unit 8, #5o DN: p :: ~~p 1. A Pr. Dup: p :: p ∨ p 
 2. ~B Pr. / ∴ ~(A ≡ B) p :: p•p Comm: p ∨ q :: q ∨ p 
 p•q :: q•p Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  20. Unit 8, #5p DN: p :: ~~p 1. ~A ⊃ A Pr. / ∴ A Dup: p :: p ∨ p 
 p :: p•p Comm: p ∨ q :: q ∨ p 
 p•q :: q•p Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  21. Unit 8, #6k DN: p :: ~~p 1. (P • G) ⊃ R Pr. Dup: p :: p ∨ p 
 p :: p•p 2. (R • S) ⊃ T Pr. 3. P • S Pr. Comm: p ∨ q :: q ∨ p 
 p•q :: q•p 4. G ∨ R Pr. / ∴ R ∨ T Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  22. Unit 8, #7a DN: p :: ~~p 1. (~A ∨ ~B) ⊃ ~C Pr. Dup: p :: p ∨ p 
 2. (A ⊃ F) Pr. p :: p•p 3. ~F ≡ (D • ~E) Pr. Comm: p ∨ q :: q ∨ p 
 p•q :: q•p 4. ~(D ⊃ H) Pr. Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) 5. E ⊃ H Pr. / ∴ ~C (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  23. Unit 8, #7f DN: p :: ~~p 1. ~P ≡ ~(Q ⊃ R) Pr. Dup: p :: p ∨ p 
 p :: p•p 2. ~(P ∨ (S ∨ T)) Pr. Comm: p ∨ q :: q ∨ p 
 3. Z ⊃ W Pr. p•q :: q•p 4. ~(R ∨ T) ⊃ ~(S • W) Pr. / ∴ ~Z Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  24. DN: p :: ~~p Dup: p :: p ∨ p 
 p :: p•p Comm: p ∨ q :: q ∨ p 
 p•q :: q•p Assoc: (p ∨ q) ∨ r :: p ∨ (q ∨ r) (p•q)•r :: p•(q•r) Contra: p ⊃ q :: ~q ⊃ ~p DeM: ~(p ∨ q) :: ~p•~q ~(p•q) :: ~p ∨ ~q BE: p ≡ q :: (p ⊃ q)•(q ⊃ p) CE: p ⊃ q :: ~p ∨ q Dist: p•(q ∨ r) :: (p•q) ∨ (p•r) p ∨ (q•r) :: (p ∨ q)•(p ∨ r) Exp: (p•q) ⊃ r :: p ⊃ (q ⊃ r)

  25. SYMBOLIC LOGIC UNIT 9: 
 CONDITIONAL PROOF AND INDIRECT PROOF

  29. subproof A section of a proof starting with an assumption and finishing when the assumption is discharged. three rules about subproofs 1. All assumptions must be discharged before the end of a proof. 2. Once a subproof is finished (after the assumption is discharged), none of the lines of the subproof may be used in later justifications. 3. Subproof lines can’t cross: when more than one subproof is happening at the same time, the most recent assumption must be discharged first.

  30. 4b 1. (~A ∨ ~B) ⊃ ~C Pr. / ∴ C ⊃ A 2. 3. 4. 5. 6. 7. 8. 9. 10.

  31. 5b 1. A • ~B Pr. / ∴ ~(A ≡ B) 2. 3. 4. 5. 6. 7. 8. 9. 10.

  32. 6d A ≡ ~(B ∨ C) Pr. 25 1 B ≡ (D • ~E) Pr. 26 2 ~(E •A) Pr / ∴ A ⊃ 27 3 28 4 29 5 30 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

  33. 25 1 26 2 27 3 28 4 29 5 30 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

  34. Unit 9, Exercise 7: Construct a proof of the following theorems. h. (p ⊃ (~p ⊃ q)) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

  35. Unit 9, Exercise 7: Construct a proof of the following theorems. b. (p ⊃ (p • q)) ∨ (q ⊃ (p • q)) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

  36. Unit 9, Exercise 7: Construct a proof of the following theorems. m. p ≡ (p ∨ (q • ~q)) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

  37. Unit 9, Exercise 7: Construct a proof of the following theorems. n. p ≡ (p • (q ∨ ~q)) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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