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Problema de lógica propuesto en clase y en sci.logic
Demostrar que (p <-> q) -> r <-> (p & q -> r) &
( ~p & ~q -> r)
Soluciones propuestas en sci.logic
Prueba 1
p q r ((p <-> q) -> r)
-----------------------
v v v v v v v v
v v F v v v F F
v F v v F F v v
v F F v F F v F
F v v F F v v v
F v F F F v v F
F F v F v F v v
F F F F v F F F
(1)
p q r (((p & q) -> r) & ((~p & ~q)
-> r))
---------------------------------------------
v v v v v v v v v
F F F v v
v v F v v v F F F
F F F v F
v F v v F F v v v
F F v v v
v F F v F F v F v
F F v v F
F v v F F v v v v
v F F v v
F v F F F v v F v
v F F v F
F F v F F F v v v
v v v v v
F F F F F F v F F
v v v F F
(2)
(1) = (2) => (p <-> q) -> r <-> (p & q -> r) &
( ~p & ~q -> r)
Prueba 2:
a) |- ((P <-> Q) -> R) -> (P & Q -> R) & (~P
& ~Q -> R)
b) |- (P & Q -> R) & (~P & ~Q -> R) -> ((P <->
Q) -> R).
c) |- ((P <-> Q) -> R) <-> (P & Q -> R) &
(~P & ~Q -> R)
a) |- ((P <-> Q) -> R) -> (P & Q -> R) & (~P &
~Q -> R)
1 (1) (P <-> Q) -> R)
A
2 (2) P & Q
A
2 (3) P
2 &E
2 (4) Q
2 &E
2 (5) Q -> P
3 SI(S) "TC"
2 (6) P -> Q
4 SI(S) "TC"
2 (7) P -> Q & Q -> P
5,6 &I
2 (8) P <-> Q
7 Df. <->
1,2 (9) R
1,8 MPP
1 (10) P & Q -> R
2,9 CP
11 (11) ~P & ~Q
A
11 (12) ~P
11 &E
11 (13) ~Q
11 &E
11 (14) P -> Q
12 SI(S) "FA"
11 (15) Q -> P
13 SI(S) "FA"
11 (16) P -> Q & Q -> P
14,15 &I
11 (17) P <-> Q
16 Df. <->
1,11 (18) R
1,17 MPP
1 (19) ~P & ~Q -> R
11,18 CP
1 (20) (P & Q -> R) & (~P & ~Q -> R)
10,19 &I
(21) ((P <-> Q) -> R) ->
(P & Q -> R)
& (~P & ~Q -> R) 1,20 CP
b) |- (P & Q -> R) & (~P & ~Q -> R) -> ((P <->
Q) -> R)
1 (1) (P & Q -> R) & (~P &
~Q -> R) A
2 (2) (P <-> Q)
A
1 (3) P & Q -> R
1 &E
1 (4) ~P & ~Q -> R
1 &E
2 (5) (P -> Q) & (Q -> P)
2 Df. <->
2 (6) P -> Q
5 &E
2 (7) Q -> P
5 &E
(8) P v ~P
TI(S) "TND"
9 (9) P
A
2,9 (10) Q
6,9 MPP
2,9 (11) P & Q
9,10 &I
1,2,9 (12) R
3,11 MPP
13 (13) ~P
A
2,13 (14) ~Q
7,13 MTT
2,13 (15) ~P & ~Q
13,14 &I
1,2,13 (16) R
4,15 MPP
1,2 (17) R
8,9,12,13,16 vE
1 (18) (P <-> Q) -> R
2,17 CP
(19) (P & Q -> R) & (~P &
~Q -> R)
->
((P <-> Q) -> R)
1,18 CP
c) |- ((P <-> Q) -> R) <-> (P & Q -> R) & (~P &
~Q -> R)
(1) ((P <-> Q) -> R) -> (P & Q -> R) & (~P & ~Q
-> R) TI (a)
(2) (P & Q -> R) & (~P & ~Q -> R) -> ((P <-> Q)
-> R) TI (b)
(3) (((P <-> Q) -> R) -> (P & Q -> R) & (~P &
~Q -> R))
& ((P & Q -> R) & (~P & ~Q -> R) -> ((P <->
Q) -> R)) 1,2 &I
(4) ((P <-> Q) -> R) <-> (P & Q -> R) & (~P &
~Q -> R) 3 Df. <->
Prueba 3
1 (1) (P & Q -> R) & (~P &
~Q -> R) A
2 (2) (P <-> Q)
A
1 (3) (P & Q -> R)
1 &E
1 (4) (~P & ~Q -> R)
1 &E
2 (5) (P & Q) v (~P & ~Q)
2 SI(S) "Equiv"
6 (6) P & Q
A
1,6 (7) R
3,6 MPP
8 (8) ~P & ~Q
A
1,8 (9) R
4,8 MPP
1,2 (10) R
5,6,7,8,9 vE
1 (11) (P <-> Q) -> R
2,10 CP
(12) (P & Q -> R) & (~P &
~Q -> R)
-> ((P <-> Q) -> R)
1,11 CP
Now IF "Equiv" were actually a primitive _rule of derivation_ of our system,
we
could write down the 12-step proof (only consisting of "atomic steps"):
1 (1) (P & Q -> R) & (~P &
~Q -> R) A
2 (2) (P <-> Q)
A
1 (3) (P & Q -> R)
1 &E
1 (4) (~P & ~Q -> R)
1 &E
2 (5) (P & Q) v (~P & ~Q)
2 Equiv
6 (6) P & Q
A
1,6 (7) R
3,6 MPP
8 (8) ~P & ~Q
A
1,8 (9) R
4,8 MPP
1,2 (10) R
5,6,7,8,9 vE
1 (11) (P <-> Q) -> R
2,10 CP
(12) (P & Q -> R) & (~P &
~Q -> R)
-> ((P <-> Q) -> R)
1,11 CP
Prueba 4 (incompleta)
----1 (p <-> q) -> r
| --2 p & q
| | 3 p <-> q (porque (p<->q) <-> (p
& q) |(~p & ~q) )
| --4 r (por 1)
| 5 (p & q) -> r
| --6 ~p & ~q
| | 7 p <->q
| --8 r
| 9 (~p & ~q) ->r
---10 ((p & q) -> r) & ((~p & ~q) -> r)
11 ((p <-> q) -> r) -> (((p &
q) -> r) & ((~p & ~q) -> r))
-----12 ((p & q) -> r) & ((~p & ~q) -> r)
| ---13 (p <-> q)
| | 14 (p & q) | (~p & ~q)
| | -15 p & q
| | |16 (p & q) -> r
| | -17 r
| | -18 ~p & ~q
| | |19 (~p & ~q) -> r
| | -20 r
| |--21 r
|----22 (p <-> q) -> r
Prueba 5 (incompleta)
1. (p <-> q) -> r
2. ~r -> ~(p <-> q)
3. ~r -> ~((p & q) V (~p & ~q))
4. ~r -> (~(p & q) & ~(~p & ~q))
5. (~r -> ~(p & q)) & (~r -> ~(~p & ~q))
6. ((p & q) -> r) & ((~p & ~q) -> r)
Supuestos aplicables:
p = Iraq had weapons of mass destruction
q = U.S. Administration is trustworthy
r = U.S. invasion of Iraq was justified
Bibliografía propuesta:
Pudlak, P.: The Lengths of Proofs,
in Buss, S.R. ed., Handbook of Proof
Theory, Elsevier Science, 1998
Bünning, H.K. and Lettmann, T.:
Propositional Logic Deduction and
Algorithms, Cambridge University
Press, 1999
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