introductory logic phi 120
DESCRIPTION
Presentation: “ Theorems ". Introductory Logic PHI 120. Changed Presentation. Homework. Homework over Break ( a) S1 - S27, T1 - T4 (from book ) R . Smith Guides (available online) " Proofs without tears " " Proofs with even fewer tears “ Study class presentations. - PowerPoint PPT PresentationTRANSCRIPT
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