SYMBOLIC LOGIC UNIT 7: THE PROOF METHOD: 8 BASIC INFERENCE RULES - - PowerPoint PPT Presentation

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symbolic logic unit 7 the proof method 8 basic inference

SYMBOLIC LOGIC UNIT 7: THE PROOF METHOD: 8 BASIC INFERENCE RULES - - PowerPoint PPT Presentation

Apr 21, 2023 228 likes •612 views

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.

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

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Constants vs. Variables p ∨ q A ∨ B

Constants are abbreviations of specific sentences with determinate meanings. Variables are placeholders for any formula whatever, including both simple and complex formulas.

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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

ccurrences 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.

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SYMBOLIC LOGIC UNIT 8:   REPLACEMENT RULES

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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)

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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)

1. ~(A ∨ B) Pr. /∴ ~B Unit 8, #5b

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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)

1. ~(A ⊃ B) Pr. /∴ ~B Unit 8, #5c

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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)

1. A ⊃ B Pr.

2. A ⊃ C Pr. /∴ A ⊃ (B • C)

Unit 8, #5f

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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)

1. A ⊃ C Pr. /∴ (A • B) ⊃ C Unit 8, #5m

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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)

1. ~((A ∨ B) ∨ (C ∨ D)) Pr. /∴ ~D Unit 8, #5n

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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)

1. A Pr.

2. ~B Pr. /∴ ~(A ≡ B)

Unit 8, #5o

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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)

1. ~A ⊃ A Pr. /∴ A Unit 8, #5p

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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)

1. (P • G) ⊃ R Pr.

2. (R • S) ⊃ T Pr.
3. P • S Pr.
4. G ∨ R Pr. /∴ R ∨ T

Unit 8, #6k

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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)

1. (~A ∨ ~B) ⊃ ~C Pr.

2. (A ⊃ F) Pr.
3. ~F ≡ (D • ~E) Pr.
4. ~(D ⊃ H) Pr.
5. E ⊃ H Pr. /∴ ~C

Unit 8, #7a

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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)

1. ~P ≡ ~(Q ⊃ R) Pr.

2. ~(P ∨ (S ∨ T)) Pr.
3. Z ⊃ W Pr.
4. ~(R ∨ T) ⊃ ~(S • W) Pr. /∴ ~Z

Unit 8, #7f

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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)

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SYMBOLIC LOGIC UNIT 9:   CONDITIONAL PROOF AND INDIRECT PROOF

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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.

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

4b

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

5b

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1

A ≡ ~(B ∨ C) Pr. 25

2

B ≡ (D • ~E) Pr. 26

3

~(E •A) Pr /∴ A ⊃ 27

4

28

5

29

6

30

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

6d

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1

25

2

26

3

27

4

28

5

29

6

30

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

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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.

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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.

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1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Unit 9, Exercise 7: Construct a proof of the following theorems.

m. p ≡ (p ∨ (q • ~q))

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1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Unit 9, Exercise 7: Construct a proof of the following theorems.

n. p ≡ (p • (q ∨ ~q))