SOLUCIÓN EJERCICIOS CON REGLAS DERIVADAS Y TEOREMA DEDUCCIÓN
PRIMERO: |--- Ø p
® q
- -1 (p Ú q)
- 2 Ø p [Premisa Subsidiaria]
- 3 q [TP 1,2]
- 4 Ø p
® q [TD 2-3]
SEGUNDO: |--- r ® Ø q
- -1 Ø r
Ú Ø s
- -2 q ® s
- 3 r [Premisa Subsidiaria]
- 4 Ø s [TP 1,3]
- 5 Ø q [MT 2,4]
- 6 r ®
Ø q [TD 3,5]
TERCERO: |--- C ® Ø D
- -1 B ®
Ø C
- -2 Ø (D Ù
Ø B)
- 3 C [Premisa Subsidiaria]
- 4 Ø B [MT 1,3]
- 5 (Ø D
Ú Ø
Ø B) [DM 2]
- 6 Ø D [TP 5,4]
- 7 C ®
Ø D [TD 3-6]
CUARTO: |--- Ø q ® t
- -1 s ® r
- -2 s Ú p
- -3 p ® q
- -4 r ® t
- 5 Ø q [Premisa subsidiaria]
- 6 Ø p [MT 3,5]
- 7 s [TP 2,6]
- 8 r [MP 1,7]
- 9 t [MP 4,8]
- 10 Ø q
® t [TD 5-9]
QUINTO: |--- p ® (p Ù q)
- -1 r ® t
- -2 t ®
Ø s
- -3 (r ® Ø s) ® q
- 4 p [Premisa Subsidiaria]
- 5 (r ® Ø s)[Sil 1,2]
- 6 q [MP 3,5]
- 7 (p Ù q) [IC 4,6]
- 8 p ® (p Ù q) [TD 4-8]
SEXTO: |--- s ® q
- -1 r ® q
- -2 t ® r
- -3 s ® t
- 4 s [Premisa Subsidiaria]
- 5 s ® r [Sil 3,2]
- 6 r [MP 5,4]
- 7 q [MP 1,6]
- 8 s ® q [TD 4-7]
SEPTIMO: |--- Ø (r Ù s) ® t
- -1 Ø p
- -2 Ø r
® t
- -3 Ø s
® p
- 4 Ø (r Ù s) [Premisa Provisional]
- 5 (Ø r Ú Ø s)[DM 4]
- 6 t Ú p [Dil 5,2,3]
- 7 t [TP 6,1]
- 8 Ø (r Ù s) ® t [TD 4-7]
OCTAVO: |--- (t Ú Ø s) ® r
- -1 Ø r
® q
- -2 t ®
Ø q
- 3 Ø s
® Ø q
- 4 (t Ú Ø s) [Premisa Subsidiaria]
- 5 Ø q [Dil 4,2,3]
- 6 Ø
Ø r [MT 1,5]
- 7 r [DN 6]
- 8 (t Ú Ø s) ® r [TD 4-7]
NOVENO: |--- t ® Ø ( p Ú q)
- -1 Ø s
Ú Ø p
- -2 q ®
Ø r
- -3 t ® (s
Ù r)
- 4 t [Premisa Subsidiaria]
- 5 s Ù r [MP 3,4]
- 6 s [EC 5]
- 7 Ø p [TP 1,6]
- 8 r [EC 5]
- 9 Ø q [MT 2,8]
- 10 Ø (p Ù Ø q) [IC 7,9]
- 11 Ø (p Ú q) [DM 10]
- 12 t ® Ø
( p Ú q) [TD 4-7]
DECIMO: |--- Ø q ® ( t Ù s)
- -1 r ® s
- -2 s ® q
- -3 r Ú (s
Ù t)
- 4 Ø q [Premisa Subsidiaria]
- 5 Ø s [MT 2,4]
- 6 Ø r [MT 1,5]
- 7 s Ù t [TP 3,6]
- 8 t Ù s [CC 7]
- 9 Ø q ® ( t Ù s)[TD 4-11]
UNDECIMO: |--- (pÙ q) ® (s Ù t)
- -1 r Ú s
- -2 Ø
t ® Ø p
- -3 r ®
Ø q
- 4 p Ù q [Premisa Subsidiaria]
- 5 q [EC 4]
- 6 Ø r [MT 3,5]
- 7 s [TP 1,6]
- 8 p [EC 4]
- 9 Ø
Ø t [MT 2,8]
- 10 t [DN 9]
- 11 s Ù t [IC 7,10]
- 12 (pÙ q) ® (s Ù t)[TD 4-11]
DUODECIMO: |--- s ® ( p Ú q)
- -1 s ® t
- -2 r ® p
- -3 t ® r
- 4 s [Premisa Subsidiaria]
- 5 s ® r [Sil 1,3]
- 6 r [MT 5,4]
- 7 p [MP 2,6]
- 8 p Ú q [ID 7]
- 9 s ® ( p Ú q) [TD 4-8]
DECIMOTERCERO: |--- Ø ( p Ú r) ® t)
- -1 q ® p
- -2 t Ú s
- -3 q Ú
Ø s
- 4 Ø ( p Ú r) [Premisa Subsidiaria]
- 5 (Ø p
Ù Ø r)[DM 4]
- 6 Ø p [EC 5]
- 7 Ø q [MT 1,6]
- 8 Ø s [TP 3,7]
- 9 t [TP 2,8]
- 10 Ø ( p Ú r) ® t)[TD 4-9]
DECIMOCUARTO: |--- E ® K
- -1 (E Ú F)
® G
- -2 J ®
Ø G
- -3 J Ú K
- 4 E [Premisa Subsidiaria]
- 5 E Ú F [ID 4]
- 6 G [MP 1,5]
- 7 Ø J [MT 2,6]
- 8 K [TP 3,7]
- 10 E ® K [TD 4-8]
DECIMOQUINTO: |--- (Ø x
®
Ø s)
- -1 p Ú t
- -2 ( r Ú s )
®
(t Ú w)
- -3 Ø q
Ú s
- -4 p ® r
- -5 t ® r
- -6 t ® x
- -7 w ®
Ø
(r Ú s)
- 8 Ø x [Premisa subsidiaria]
- 9 Ø t [MT 6,8]
- 10 r Ú x [Dil 1,4,5]
- 11 r [TP 10,8]
- 12 r Ú s [ID 11]
- 13 t Ú u [MP 2,12]
- 14 w [TP 13,9]
- 15 Ø
( r Ú s)
- 16 Ø r
Ù
Ø s
- 17 Ø s [EC 16]
- 18 (Ø x
®
Ø s) [TD 8-17]
DECIMOSEXTO: |--- Ø p
® q
- -1 r Ù
Ø s
- -2 t ®
(v Ú w)
- -3 v ® p
- -4 w ® q
- -5 r ® t
- 6 Ø p [Premisa Subsidiaria]
- 7 Ø v [MT 3,6]
- 8 r ®
v Ú u [Sil 5,2]
- 9 r [EC 1]
- 10 v Ú w [MP 8,9]
- 11 w [TP 10,7]
- 12 q [MP 4,11]
- 13 Ø p
® q [TD 6-12]
DECIMOSEPTIMO: |--- p ® s
- -1 Ø q
®
Ø p
- -2 p ®
(q ® r)
- -3 Ø
( r ® s)
®
Ø q
- 4 p [Premisa subsidiaria]
- 5 q ® r [MP 2,4]
- 6 Ø
Ø q [MT 1,4]
- 7 q [DN 6]
- 8 Ø
Ø
(r ® s) [MT 3,7]
- 9 r ® s [DN 8]
- 10 r [MP 5,7]
- 11 s [MP 9,10]
- 12 p ® s [TD 4-11]
DECIMOOCTAVO: |--- Ø
(Ø p
Ú
Ø q)
®
(r Ús)
- -1 p ® r
- -2 Ø
(Ø p
Ú
Ø q)
®
(r Ús)
- 3 Ø
(Ø p
Ú
Ø q) [Premisa subsidiaria]
- 4 p Ù q [DM 3]
- 5 p [EC 4]
- 6 r [MP 1,5]
- 7 r Ú s [ID 6]
- Ø
(Ø p
Ú
Ø q)
®
(r Ús)
DECIMONOVENO: |--- t ®
Ø
( p Ú q)
- -1 Ø s
Ú
Ø p
- -2 q ® r
- -3 t ®
(s Ù r)
- 4 t [Premisa Subsidiaria]
- 5 (s Ù r) [MP 4,3]
- 6 r [EC 5]
- 7 Ø q [MT 2,6]
- 8 s [EC 5]
- 9 Ø p [TP 1,8]
- 10 Ø p
Ù
Ø q [IC 9,7]
- 11 Ø
( p Ú q) [DM 10]
- 12 t ®
Ø
( p Ú q) [TD 4-11]
VEINTE: |--- s ® t
- -1 (p Ú
Ø q)
® r
- -2 Ø p
® t
- -3 s ®
Ø q
- -4 r ® q
- 5 s [Premisa subsidaria]
- 6 Ø q [MP 3,5]
- 7 Ø r [MT 4,6]
- 8 Ø
(p Ú
Ø q) [MT 1,7]
- 9 Ø p
Ù
Ø
Ø q [DM 8]
- 10 Ø p [EC 9]
- 11 t [MP 2,10]
- 12 s ® t [TD 5-11]
VEINTEIUNO: |--- p ®
( r ® t)
- -1 (p Ú q)
®
(r Ú s)
® t
- 2 p [Premisa Subsidiaria]
- 3 r [Premisa Subsidiaria]
- 4 p Ú q [ID 2]
- 5 (r Ú s)
® t [MP 1,4]
- 6 r Ú s [ID 3]
- 7 t [MP 5,6]
- 8 r ® t [TD 3,7]
- 9 p ®
( r ® t) [TD 2-8]
Reglas derivadas y Teorema Deducción