Post on 22-Mar-2016
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Introductory LogicPHI 120
Presentation: "n ->I(m) and m,n RAA(k)“
Homework• Get Proofs handout (online)
1. Identify and Solve first two ->I problems on handout.
2. Solve S14* : ~P -> Q, ~Q P⊢
• Read pp.28-9 "double turnstile“
• Study this presentation at home– esp. S14
• All 10 rules committed to memory!!!
TAs may collect this assignment
The 10 Primitive Rules
• You should have the following in hand:– “The Rules” Handout• See bottom of handout
Two Rules of Importance
• Arrow – Introduction: ->In ->I(m)
• Reductio ad absurdum: RAAm, n RAA(k)
Discharging assumptionn ->I(m)m, n RAA(k)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One premise rule
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two premise rule
Two Rules of Importance
• Arrow – Introduction: ->I
• Reductio ad absurdum: RAA
Discharging assumptionn ->I(m)m, n RAA(k)Strategy
n ->I(m)Arrow - Introduction
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A
(3) ???
n ->I(m)
S16: P -> Q, Q -> R ⊢ P -> R1 (1) P -> Q A2 (2) Q -> R A
(3) ???
“P -> R” is not in the premises. Hence, we have to make it.
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
⊢ P -> R⊢ P -> R⊢ P -> R⊢ P -> R
n ->I(m)
S16: P -> Q, Q -> R ⊢ P -> R1 (1) P -> Q A2 (2) Q -> R A
(3) ???
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R ⊢ P -> R1 (1) P -> Q A2 (2) Q -> R A
(3) ???
possible premise of an ->E
possible premise of an ->E
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A (step 1 in strategy of ->I)
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
Apply ->I Strategy
n ->I(m)
S16: P -> Q, Q -> R ⊢ P -> R1 (1) P -> Q A2 (2) Q -> R A3 (3) P A (step 1 in strategy of ->I)
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R ⊢ P -> R1 (1) P -> Q A2 (2) Q -> R A3 (3) P A
(4)
(1) What kind of statement is “R” (the consequent)?
(2) Where is it located in premises?
Step 2(often more than one line)
Read the problem properly!
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A antecedent of (1)
(4)
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A
(4) 1,3 ->E
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A
(4) Q 1,3 ->E
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E antecedent of (2)
(5)
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E
(5) 2,4 ->E
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E
(5) R 2,4 ->E
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E1,2,3 (5) R 2,4 ->E
Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E1,2,3 (5) R 2,4 ->E
(6) Strategy to make an ->I: 1) Assume the antecedent of conclusion 2) Solve for the consequent (i.e., as a conclusion) 3) Apply ->I rule to generate the conditional
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E1,2,3 (5) R 2,4 ->E
(6) P -> R n ->I(m) Step 3
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E1,2,3 (5) R 2,4 ->E
(6) P -> R 5 ->I(3)
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E1,2,3 (5) R 2,4 ->E
(6) P -> R 5 ->I(3)
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E1,2,3 (5) R 2,4 ->E
(6) P -> R 5 ->I(3)
This must be an assumption
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E1,2,3 (5) R 2,4 ->E
(6) P -> R 5 ->I(3)
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E1,2,3 (5) R 2,4 ->E1,2 (6) P -> R 5 ->I(3)
n ->I(m)
S16: P -> Q, Q -> R P -> R⊢1 (1) P -> Q A2 (2) Q -> R A3 (3) P A1,3 (4) Q 1,3 ->E1,2,3 (5) R 2,4 ->E1,2 (6) P -> R 5 ->I(3)
(i) Is (6) the conclusion of the sequent?
(ii) Are the assumptions correct?
The Two Questions
Any kind of wff(will be the consequent)
Any kind of wff(will be the antecedent)
must be an assumption
n ->I(m)
m,n RAA(k)Reductio ad absurdum
The Key to RAA
– If the proof contains incompatible premises, you are allowed to deny any assumption within the proof.
m, n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of some assumption (k)
Denials
A B
1 (1) P & Q A2 (2) ~P A
(3) ??
TheBasicAssumptions
– If the proof contains incompatible premises, you are allowed to deny any assumption within the proof.
m, n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of some assumption (k)
P & Q, ~P ~R⊢
1 (1) P & Q A2 (2) ~P A
(3) ??
– If the proof contains incompatible premises, you are allowed to deny any assumption within the proof
m, n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of some assumption (k)
P & Q, ~P ~R⊢
Elimination won’t work
Introduction won’t work
RAA
1 (1) P & Q A2 (2) ~P A
(3) ??
Strategy of RAA: 1) Assume the denial of the
conclusion
– If the proof contains incompatible premises, you are allowed to deny any assumption within the proof
m, n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of some assumption (k)
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
Strategy of RAA: 1) Assume the denial of the
conclusion
Strategy of RAA: 1) Assume the denial of the
conclusion1 (1) P & Q A2 (2) ~P A
(3) ??
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
Strategy of RAA: 1) Assume the denial of the
conclusion
Strategy of RAA: 1) Assume the denial of the
conclusion1 (1) P & Q A2 (2) ~P A
(3) A
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
Strategy of RAA: 1) Assume the denial of the
conclusion
Strategy of RAA: 1) Assume the denial of the
conclusion1 (1) P & Q A2 (2) ~P A
(3) R A
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
Strategy of RAA: 1) Assume the denial of the
conclusion
Strategy of RAA: 1) Assume the denial of the
conclusion1 (1) P & Q A2 (2) ~P A3 (3) R A
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
Strategy of RAA: 1) Assume the denial of the
conclusion1 (1) P & Q A2 (2) ~P A3 (3) R A
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
2) Derive a contradiction.
Strategy of RAA: 1) Assume the denial of the
conclusion1 (1) P & Q A2 (2) ~P A3 (3) R A
(4) ???
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
2) Derive a contradiction.
Strategy of RAA: 1) Assume the denial of the
conclusion1 (1) P & Q A2 (2) ~P A3 (3) R A
(4) 1 &E
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
2) Derive a contradiction.
Strategy of RAA: 1) Assume the denial of the
conclusion1 (1) P & Q A2 (2) ~P A3 (3) R A
(4) P 1 &E
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
2) Derive a contradiction.
Strategy of RAA: 1) Assume the denial of the
conclusion1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
2) Derive a contradiction.
Strategy of RAA: 1) Assume the denial of the
conclusion1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
2) Derive a contradiction.
Strategy of RAA: 1) Assume the denial of the
conclusion1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
P & Q, ~P ~R⊢
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
(5) m, n RAA(k)
P & Q, ~P ~R⊢
Premises: denials of one another
1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
(5) 2, 4 RAA(k)
P & Q, ~P ~R⊢
Conclusion: will be the denial of some assumption (k)
1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
(5) 2, 4 RAA(k)
P & Q, ~P ~R⊢
TheBasicAssumptions
Conclusion: will be the denial of some assumption (k)
1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
(5) 2, 4 RAA(k)
P & Q, ~P ~R⊢
Conclusion: will be the denial of some assumption (k)
1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
(5) 2, 4 RAA(3)
P & Q, ~P ~R⊢
Conclusion: will be the denial of some assumption (k)
1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
(5) ~R 2, 4 RAA(3)
P & Q, ~P ~R⊢
Conclusion: will be the denial of some assumption (k)
1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
(5) ~R 2, 4 RAA(3)
P & Q, ~P ~R⊢
Don't forget to define the assumption set!
1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
(5) ~R 2, 4 RAA(3)
P & Q, ~P ~R⊢
Don't forget to define the assumption set!
1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E
(5) ~R 2, 4 RAA(3)
P & Q, ~P ~R⊢
Don't forget to define the assumption set!
1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E1,2 (5) ~R 2, 4 RAA(3)
P & Q, ~P ~R⊢
Don't forget to define the assumption set!
1 (1) P & Q A2 (2) ~P A3 (3) R A1 (4) P 1 &E1,2 (5) ~R 2, 4 RAA(3)
P & Q, ~P ~R⊢
Any kind of wff(will be the consequent)
Any kind of wff(will be the antecedent)
must be an assumption
n ->I(m)
m,n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of assumption: k
SOLVE S14 FOR HOMEWORKm,n RAA (k)
Homework• Get Proofs handout (online)
1. Identify and Solve first two ->I problems on handout.
2. Solve S14 : ~P -> Q, Q P⊢
• Read pp.28-9 "double turnstile“
• Study this presentation at home– esp. S14
TAs may collect this assignment
m,n RAA(k)
S14: ~P -> Q, ~Q P⊢
m,n RAA(k)
S14: ~P -> Q, ~Q P⊢1 (1) ~P -> Q A2 (2) ~Q A
(3)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
Note: neither introduction nor elimination strategy will work for “P”
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A
(3) (first step of RAA)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A
(3) A (assume)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A
(3) ~P A (denial of conclusion)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A (denial of conclusion)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A
Step Back. Read the premises.
m,n RAA(k)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A (possible –>E)
2 (2) ~Q A3 (3) ~P A
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A (possible –>E)
2 (2) ~Q A3 (3) ~P A (antecedent)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A (possible –>E)
2 (2) ~Q A (denial of consequent)
3 (3) ~P A (antecedent)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A
(4) ??
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A
(4) 1, 3 ->E
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A
(4) Q 1, 3 ->E
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A
(4) Q 1, 3 ->E
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E
(5) ??
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E
(5) ??
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E
(5) 2, 4 RAA(?)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E
(5) 2, 4 RAA(?)
Question: which assumption will we discharge?
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E
(5) 2, 4 RAA(?)
Never discharge your basic premises!
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E
(5) 2, 4 RAA(3)
The sole remaining assumption
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E
(5) ? 2, 4 RAA(3)
Conclusion of RAA: denial of
discharged assumption
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E
(5) P 2, 4 RAA(3)
Conclusion of RAA: denial of
discharged assumption
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E
(5) P 2, 4 RAA(3)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E
(5) P 2, 4 RAA(3)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E1,2 (5) P 2, 4 RAA(3)
Strategy of RAA: 1) Assume the denial of the conclusion 2) Derive a contradiction 3) Use RAA to deny/discharge an assumption
m,n RAA(k)
S14: ~P -> Q, ~Q P⊢1 (1) ~P -> Q A2 (2) ~Q A3 (3) ~P A1,3 (4) Q 1, 3 ->E1,2 (5) P 2, 4 RAA(3)
(i) Is (5) the conclusion of the sequent?(ii) Is (5) derived from the basic assumptions given in the sequent?
The Two Questions
Any kind of wff(will be the consequent)
Any kind of wff(will be the antecedent)
must be an assumption
n ->I(m)
m,n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of assumption: k
Any kind of wff(will be the consequent)
Any kind of wff(will be the antecedent)
must be an assumption
n ->I(m)
m,n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of assumption: k
Strategy
Homework• Get Proofs handout (online)
1. Identify and Solve first two ->I problems on handout.
2. Solve S14* : ~P -> Q, ~Q P⊢
• Read pp.28-9 "double turnstile“
• Study this presentation at home– esp. S14
• All 10 rules committed to memory!!!
TAs may collect this assignment