Sentential Logic

1
Sentential Logic

• Derivations

2
Derivation Rules of SD

• Last day we started to look at the derivation
  rules of the natural deduction system, SD.
• There are 11 rules Reiteration, and an
  introduction and elimination rule for each of the
  connectives.

3
Reiteration (R)

• P
• gt P

4
Conjunction Introduction (I)

• P
• Q
• gt P Q

5
Conjunction Elimination (E)

• P Q P Q
• gt P gt Q

6
Conditional Elimination (ÉE)

• P É Q
• P
• gt Q

7
Example

• Derive U
• H É U Assumption
• S H Assumption
• H 2 E
• U 1,3 ÉE

8
Conditional Introduction (ÉI)

• Conditional Introduction requires the
  introduction of a new mechanism in our derivation
  system subderivation.
• A subderivation is a derivation which is
  conducted within the main derivation.
• It is initiated by introducing a new sentence to
  the derivation, one whose position is not
  justified by being derived from lines previous to
  it.

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Conditional Introduction (ÉI)

• E.g.
• main scope line
• auxiliary assumption
• P
• scope line of subderivation

10
Conditional Introduction (ÉI)

• P
• Q
• gt P É Q

11
Examples

• Derive E v K
• (Q M) É (E v K) Assumption
• M (E v C) Assumption
• Q N Assumption
• Q 3 E
• M 2 E
• Q M 4,5 I
• E v K 1,6 ÉE

12
Examples

• Derive J É T
• J É (S T) Assumption

13
Examples

• Derive J É T
• J É (S T) Assumption
• J Assumption

14
Examples

• Derive J É T
• J É (S T) Assumption
• J Assumption
• S T 1,2 ÉE
• T 3 E
• J É T 2-4 ÉI

15
Examples

• Derive (S B) É N
• S É (L N) Assumption

16
Examples

• Derive (S B) É N
• S É (L N) Assumption
• (S B) É N Goal sentence

17
Examples

• Derive (S B) É N
• S É (L N) Assumption
• S B Assumption
• (S B) É N 2-? ÉI

18
Examples

• Derive (S B) É N
• S É (L N) Assumption
• S B Assumption
• S 2 E
• L N 1,3 ÉE
• N 4 E
• (S B) É N 2-? ÉI

19
Examples

• Derive (S B) É N
• S É (L N) Assumption
• S B Assumption
• S 2 E
• L N 1,2 ÉE
• N 4 E
• (S B) É N 2-5 ÉI

20
Negation Introduction (I)

• Both Negation Introduction and Negation
  Elimination are also known as Reductio Ad
  Absurdum.
• Reductios work by assuming the negation of the
  sentence that you want to prove, and then showing
  that that assumption is false as it leads to a
  contradiction, and hence the original sentence is
  true.

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Negation Introduction (I)

• P
• Q
• Q
• gt P

22
Negation Elimination (E)

• P
• Q
• Q
• gt P

23
Examples

• Derive G
• (G É I) I Assumption
• G Goal sentence

24
Examples

• Derive G
• (G É I) I Assumption
• G Assumption
• try to derive a contradiction
• here
• G Goal sentence

25
Examples

• Derive G
• (G É I) I Assumption
• G Assumption
• G É I 1 E
• I 2,3 ÉE
• I 1 E
• G Goal sentence

26
Examples

• Derive G
• (G É I) I Assumption
• G Assumption
• G É I 1 E
• I 2,3 ÉE
• I 1 E
• G 2-5 I

27
Examples

• Derive P
• (P É L) (L É L) Assumption
• P Goal sentence

28
Examples

• Derive P
• (P É L) (L É L) Assumption
• P Assumption
• P É L 1 E
• L 2,3 ÉE
• L É L 1 E
• L 4,5 ÉE
• P Goal sentence

29
Examples

• Derive P
• (P É L) (L É L) Assumption
• P Assumption
• P É L 1 E
• L 2,3 ÉE
• L É L 1 E
• L 4,5 ÉE
• P 2-6 E

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Disjunction Introduction (vI)

• P P
• gt P v Q gt Q v P

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Disjunction Elimination (vE)

• P v Q
• gt R

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Disjunction Elimination (vE)

• P v Q
• P
• R
• Q
• R
• gt R

33
Examples

• Derive B v (K v G)
• K Assumption
• K v G 1 vI
• B v (K v G) Goal sentence

34
Examples

• Derive B v (K v G)
• K Assumption
• K v G 1 vI
• B v (K v G) 2 vI

35
Examples

• Derive H
• (K v P) É H Assumption
• P Assumption
• K v P 2 vI
• H goal sentence

36
Examples

• Derive H
• (K v P) É H Assumption
• P Assumption
• K v P 2 vI
• H 1,3 ÉE

37
Examples

• Derive X
• E v X Assumption
• E É X Assumption
• X Goal sentence

38
Examples

• Derive X
• E v X Assumption
• E É X Assumption
• E Assumption
• X 2,3 ÉE
• X Assumption
• X 5 R
• X goal sentence

39
Examples

• Derive X
• E v X Assumption
• E É X Assumption
• E Assumption
• X 2,3 ÉE
• X Assumption
• X 5 R
• X 3-4, 5-6 vE

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Biconditional Introduction (I)

• A biconditional could be understood as
• the conjunction of two conditionals
• gt P Q

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Biconditional Introduction (I)

• i.e.,
• (P É Q) (Q É P)
• gt P Q

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Biconditional Introduction (I)

• P
• Q
• Q
• P
• gt P Q

43
Biconditional Elimination (E)

• P Q P Q
• P Q
• gt Q gt P

44
Examples

• Derive Q
• K (E Q) Assumption
• K Assumption
• E Q 1,2 E
• Q 3 E

45
Examples

• Derive R E
• (R É E) (E É R) Assumption
• R E goal sentence

46
Examples

• Derive R E
• (R É E) (E É R) Assumption
• R Assumption
• E sub goal
• E Assumption
• R sub goal
• R E goal sentence

47
Examples

• Derive R E
• (R É E) (E É R) Assumption
• R Assumption
• R É E 1 E
• E sub goal
• E Assumption
• R sub goal
• R E goal sentence

48
Examples

• Derive R E
• (R É E) (E É R) Assumption
• R Assumption
• R É E 1 E
• E 2,3 ÉE
• E Assumption
• R sub goal
• R E goal sentence

49
Examples

• Derive R E
• (R É E) (E É R) Assumption
• R Assumption
• R É E 1 E
• E 2,3 ÉE
• E Assumption
• E É R 1 E
• R 5,6 ÉE
• R E goal sentence

50
Examples

• Derive R E
• (R É E) (E É R) Assumption
• R Assumption
• R É E 1 E
• E 2,3 ÉE
• E Assumption
• E É R 1 E
• R 5,6 ÉE
• R E 2-4, 5-7 I