introductory logic phi 120
DESCRIPTION
Presentation: “ Double Turnstile Problems ". Introductory Logic PHI 120. Homework. Proofs: 1.5.1 (A/H, p.29-30) S21 – S24 (v ->) S25 – S27 (the dilemmas) S44 (Imp/Exp) External Web Pages: “ R. Smith Guide: Proofs without tears ” available through class web page. ->I and RAA. - PowerPoint PPT PresentationTRANSCRIPT
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Introductory LogicPHI 120
Presentation: “Double Turnstile Problems"
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Homework• Proofs: 1.5.1 (A/H, p.29-30)
– S21 – S24 (v ->)
– S25 – S27 (the dilemmas)
– S44 (Imp/Exp)
• External Web Pages: – “R. Smith Guide: Proofs without tears”• available through class web page
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->I and RAA
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Internalize These Strategies
->I1. Assume antecedent of
the conclusion
2. Solve for the consequent
3. Apply ->I rule
RAA1. Assume the denial of
what you’re solving for
2. Derive a contradiction
3. Apply RAA rule
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P v Q ~P -> Q⊣⊢Double Turnstile Problems
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P v Q ⊣⊢ ~P -> Q
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢ ~P -> Q P v Q⊢
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢ ~P -> Q P v Q⊢
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A
(2)
~P -> Q P v Q⊢
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P v Q ⊣⊢ ~P -> QP v Q ~P ⊢ -> Q1 (1) P v Q A
(2) ??
~P -> Q P v Q⊢
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P v Q ⊣⊢ ~P -> QP v Q ~P ⊢ -> Q1 (1) P v Q A
(2) ??
~P -> Q P v Q⊢
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P v Q ⊣⊢ ~P -> QP v Q ⊢ ~P -> Q1 (1) P v Q A
(2) ??
~P -> Q P v Q⊢
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A
(2) ??
~P -> Q P v Q⊢
Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.
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P v Q ⊣⊢ ~P -> QP v Q ⊢ ~P -> Q1 (1) P v Q A2 (2) ~P A
~P -> Q P v Q⊢
Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.
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P v Q ⊣⊢ ~P -> QP v Q ~P⊢ -> Q1 (1) P v Q A2 (2) ~P A
~P -> Q P v Q⊢We now have too many assumptions!
Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.
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P v Q ⊣⊢ ~P -> QP v Q ~P -> ⊢ Q1 (1) P v Q A2 (2) ~P A
(3) ??
~P -> Q P v Q⊢
Phase II: Solve for consequent
Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A
(3) ??
~P -> Q P v Q⊢
Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A
(3) Q 1,2 vE
~P -> Q P v Q⊢
Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE
~P -> Q P v Q⊢
Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE
(4) ??
~P -> Q P v Q⊢
Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.
Phase III: Apply ->I rule
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE
(4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE
(4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE
(4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢
1. Is the final line the main conclusion?2. Are the assumptions correct on this final line?
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A
(2)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P ⊢ v Q1 (1) ~P -> Q A
(2)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A
(2) ??
Look at the premise in relation to the conclusion?
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q ⊢ P v Q1 (1) ~P -> Q A
(2) ??
Look at the premise in relation to the conclusion?
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A
(2) A
Assume what?
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A
The antecedent of (1)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A
(3)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A
(3) Q 1,2 ->E
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A
(3) Q 1,2 ->E
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q ⊢ P v Q1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E
(4) ??
Make the wedge (i.e., the conclusion)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI
1. Is the final line the main conclusion?2. Are the assumptions correct on this final line?
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI
Too many assumptions!!!!
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI
To discharge assumptions:->I or RAA?
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI
(5) A
Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI
(5) ~(P v Q) A
Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A
Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A
Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A
(6)Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A
(6) 4,5 RAA(?) Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A
(6) 4,5 RAA(?) Which assumption should you discharge first?• 1, 2, or 5
Assumptions
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A
(6) 4,5 RAA(?)
not [1]
Which assumption should you discharge first?• 1, 2, or 5
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A
(6) 4,5 RAA(?) not [5]
Which assumption should you discharge first?• 1, 2, or 5
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A
(6) 4,5 RAA(2)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A
(6) P 4,5 RAA(2)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2)
(7) ??
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vILook at your assumptions
1. Is the final line the main conclusion?2. Are the assumptions correct on this final line?
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI
(8) ??
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI
(8) 5,7 RAA(5)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI
(8) P v Q 5,7 RAA(5)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI
(8) P v Q 5,7 RAA(5)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI1 (8) P v Q 5,7 RAA(5)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI1 (8) P v Q 5,7 RAA(5)
1. Is the final line the main conclusion?2. Are the assumptions correct on this final line?
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI1 (8) P v Q 5,7 RAA(5)
Typical Structure for multiple RAA
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI1 (8) P v Q 5,7 RAA(5)
m,n RAA (k)
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P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)
~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI1 (8) P v Q 5,7 RAA(5)
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Homework• Proofs: 1.5.1 (A/H, p.29-30)
– S21 – S24 (v ->)
– S25 – S27 (the dilemmas)
– S44 (Imp/Exp)
• External Web Pages: “R. Smith Guides”– available through class web page