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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.
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Presentation: “Double Turnstile Problems" Introductory LogicPHI 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
Internalize These Strategies ->I • Assume antecedent of the conclusion • Solve for the consequent • Apply ->I rule RAA • Assume the denial of what you’re solving for • Derive a contradiction • Apply RAA rule
Double Turnstile Problems P v Q ⊣⊢ ~P -> Q
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q ~P -> Q ⊢ P v Q
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q ~P -> Q ⊢ P v Q
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A (2) ~P -> Q ⊢ P v Q
P v Q ⊣⊢ ~P -> Q P v Q⊢ ~P -> Q • (1) P v Q A (2) ?? ~P -> Q ⊢ P v Q
P v Q ⊣⊢ ~P -> Q P v Q⊢ ~P ->Q • (1) P v Q A (2) ?? ~P -> Q ⊢ P v Q
P v Q ⊣⊢ ~P -> Q P v Q⊢ ~P-> Q • (1) P v Q A (2) ?? ~P -> Q ⊢ P v Q
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A (2) ?? ~P -> Q ⊢ P v Q Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A 2 (2) ~PA ~P -> Q ⊢ P v Q Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P-> Q • (1) P v Q A 2 (2) ~P A ~P -> Q ⊢ P v Q We now have too many assumptions! Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A (3) ?? ~P -> Q ⊢ P v Q Phase II: Solve for consequent Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A (3) ?? ~P -> Q ⊢ P v Q Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A (3) Q1,2vE ~P -> Q ⊢ P v Q Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE ~P -> Q ⊢ P v Q Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) ?? ~P -> Q ⊢ P v Q Phase III: Apply ->I rule Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) ~P -> Q3 ->I(2) ~P -> Q ⊢ P v Q
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q Is the final line the main conclusion? Are the assumptions correct on this final line?
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1 (1) ~P -> Q A
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A (2)
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A (2)
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A (2) ?? Look at the premise in relation to the conclusion?
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A (2) ?? Look at the premise in relation to the conclusion?
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A (2) A Assume what?
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A 2 (2) ~P A The antecedent of (1)
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A • (2) ~P A (3)
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A • (2) ~P A (3) Q 1,2 ->E
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A • (2) ~P A (3) Q 1,2 ->E
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1 (1) ~P -> Q A 2 (2) ~P A 1,2 (3) Q 1,2 ->E
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E (4) ?? Make the wedge (i.e., the conclusion)
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI Is the final line the main conclusion? Are the assumptions correct on this final line?
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI Too many assumptions!!!!
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI To discharge assumptions: ->I or RAA?
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI (5) A Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI (5) ~(P v Q)A Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI 5 (5) ~(P v Q)A Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI 5 (5) ~(P v Q) A Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI • (5) ~(P v Q) A (6) Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1 (1) ~P -> Q A 2 (2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI 5 (5) ~(P v Q) A (6) 4,5 RAA(?) Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1 (1) ~P -> Q A 2 (2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI 5 (5) ~(P v Q) A (6) 4,5 RAA(?) Assumptions • Which assumption should you discharge first? • 1, 2, or 5
P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2 (2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI 5 (5) ~(P v Q) A (6) 4,5 RAA(?) not [1] • Which assumption should you discharge first? • 1, 2, or 5