Introductory LogicPHI 120

Presentation: “Theorems"

Changed Presentation

Homework

Homework over Break

(a) S1 - S27, T1 - T4 (from book)

(b) R. Smith Guides (available online) "Proofs without tears" "Proofs with even fewer tears“

(c) Study class presentations

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⊢

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

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)

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)

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

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

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

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)

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

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?

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!!!!

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

Either ->I or RAA

Conclusion not an ->

Too many assumptions!!!!

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

Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption

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

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

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

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

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

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

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

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

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 yet [5]

Which assumption should you discharge first?• 1, 2, or 5

Multiple RAA Problems

Discharge the RAAassumption last

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)

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)

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)

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)

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) ??

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?

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

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 ⊣⊢ ~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)

now [5]

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)

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)

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)

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?

Theorems

Sentences that can be proven from an empty set of premises

Sequents

• A sequent contains three elements

P Q -> P⊢

Sequents

• A sequent contains three elements

P Q -> P⊢

Premises (basic assumptions)

Sequents

• A sequent contains three elements

P ⊢ Q -> P

Turnstile (conclusion indicator)

Sequents

• A sequent contains three elements

P ⊢ Q -> P

Conclusion

Theorems

• A theorem contains only two elements

⊢ P -> (Q -> P)

Turnstile (conclusion indicator)

Theorems

• A theorem contains only two elements

⊢ P -> (Q -> P)

Conclusion

Remember: every proof begins with at least one assumption.

Set of Theorems

T1: P->P⊢ IdentityT2: P v ~P⊢ Excluded MiddleT3: ~(P&~P)⊢ Non-ContradictionT4:* P->(Q->P)⊢ WeakeningT5:* (P->Q) v (Q->P)⊢ Paradox of Material ImplicationT6: P<->~~P⊢ Double NegationT7: (P<->Q)<->(Q<->P)⊢

Weakening

⊢ P -> (Q -> P)

Every proof begins with at least one assumption.

Weakening

⊢ P -> (Q -> P)1 (1) ?? A

Every proof begins with at least one assumption.

Weakening

⊢ P -> (Q -> P)1 (1) ?? A

Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A

Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A

(2)

Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A

Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A

Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A

(3)

Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A1 (3) Q -> P 1 ->I(2)

Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A1 (3) Q -> P 1 ->I(2)

Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A1 (3) Q -> P 1 ->I(2)

Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A1 (3) Q -> P 1 ->I(2)

(2)

Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A1 (3) Q -> P 1 ->I(2) (4) P -> (Q -> P) 3 ->I(1)Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A1 (3) Q -> P 1 ->I(2)

(4) P -> (Q -> P) 3 ->I(1)Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A1 (3) Q -> P 1 ->I(2)

(4) P -> (Q -> P) 3 ->I(1)Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A1 (3) Q -> P 1 ->I(2)

(4) P -> (Q -> P) 3 ->I(1)Strategy of -> I1. Assume the antecedent of the conclusion2. Solve for the consequent3. Apply ->I rule

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A1 (3) Q -> P 1 ->I(2)

(4) P -> (Q -> P) 3 ->I(1)Ø

Weakening

⊢ P -> (Q -> P)1 (1) P A2 (2) Q A1 (3) Q -> P 1 ->I(2)

(4) P -> (Q -> P) 3 ->I(1)

(1) (2)(1)

Theorems

• Same strategy!

• Only two rules discharge assumptions– RAA or ->I• More difficult theorems may use more than a single

strategy– Multiple RAA– Multiple ->I– May include both ->I and RAA strategies

Homework• Problems– S1-27– T1-T4

• Read “theorem” on p. 35

• Study Over Break:– “R. Smith Guides: Proofs with even fewer tears”• available through class web page