SOLUCIONARIO DE LÓGICA Página 6 1.
a) – Si A, entonces B . – No A. – Luego no B . b) – Todo Todoss los A son B . – x es A. – Luego x es B . c) – Tod Todas as las las A son B . – Algunas A son C . – Luego algunos C son B . d) – A o B . – Si B, entonces C y D. – Si C y D, entonces A. – Luego A.
Página 8 2.
a) p ì q
q 8 p b) ¬ q 8 r 5 ¬ q c) r 5 q 5 r d) ¬ q 5 q î p e) ¬ q î r ì q f ) ¬ r ì g) Ä 1. p 8 q Ä 2. ¬ q |Ä 3. ¬ p î r
Página 12 3.
a) – «Sube «Suben n los los sala salario rios»: s»: r – «Aumenta la inflación»: s – «Disminuye la exportación»: t r 8 s Ä 1. r 8 s 8 t Ä 2. s 8 Ä 3. r Ä 4. s |Ä 5. t
MP 1, 3 MP 2, 4
b) – «La metafísica contiene la solución a todos los problemas de la existencia»: r – «Soy un vampiro loco»: s Ä 1. p ID 1 |Ä 2. p î q Solucionario de lógica
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4.
– «El profeso profesorr debe enseñarn enseñarnos os lógica»: lógica»: r – «El profesor profesor es compet competente»: ente»: p – «El profesor profesor quiere quiere hacernos hacernos compete competentes»: ntes»: q Ä 1. p 8 (q 8 q 8 r ) Ä 2. p 8 q Ä 3. p q 8 r 4. q 8 MP 1, 3 5. q MP 2, 3 |Ä 6. r MP 4, 5
Página 13 5.
a) – «Vigi «Vigilo lo mi mi incons inconscient ciente»: e»: p – «Sueño con las vacaciones»: q – «Me despierto triste»: r Ä 1. ¬ p 8 q Ä 2. q 8 q 8 r Ä 3. ¬ p Ä 4. q Ä 5. r |Ä 6. ¬ p 8 r
MP 1, 3 MP 2, 4 II 3, 5
b) – «Sócrates «Sócrates renunci renunciaa a la lógica»: lógica»: p – «Sócrates renuncia a razonar correctamente»: correctamente»: q – «Sócrates admite la lógica»: r – «Sócrates pierde su apuesta con Platón»: s Ä 1. p 8 q r 8 s Ä 2. r 8 Ä 3. p î r Ä 4. p MP 1, 4 Ä 5. q q î s ID 5 Ä 6. q î Ä 7. r MP 2, 7 Ä 8. s q î s ID 8 Ä 9. q î q î s ED 4-6, 7-9 |Ä 10. q î c) – «Los «Los filósof filósofos os están están locos» locos»:: p – «A los filósofos los internarán en manicomios»: q – «Los filósofos aman la libertad»: r – «Los filósofos intentan escaparse»: s – «Los filósofos actúan sensatamente»: t Ä 1. p 8 q
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Ä 4. p ì r 5. p 6. q 7. r 8. s q ì s 9. q ì Ä 10. t p ì r ) 8 t |Ä 11. ( p d) – «Es necesari necesarioo tomar el el poder»: poder»: p – «Hace falta tener mayoría»: q – «Se quiere cambiar el mundo»: r Ä 1. p 8 q r 8 q ) Ä 2. ¬ (r 8 r 8 p Ä 3. r 8 – 4. r 5. p – 6. q r 8 q 7. r 8 r 8 q ) ì ¬ (r 8 r 8 q ) Ä 8. (r 8 r 8 p) |Ä 9. ¬ (r 8 6.
7.
– «El Racing ganará la Liga»: p – «Pierdo una apuesta»: r – «Me voy de viaje»: s – «Me quedo sin ahorros»: t Ä 1. ¬ p 8 r r î s ) 8 t Ä 2. (r î Ä 3. ¬ t Ä 4. ¬ p Ä 5. r r î s Ä 6. r î Ä 7. t t ì ¬ t Ä 8. t ì Ä 9. ¬ ¬ p |Ä 10. p a) Ä 1. p 8 q Ä 2. p ì ¬ q 3. p 4. q 5. ¬ q q ì ¬ q Ä 6. q ì
EC 4 MP 1, 5 EC 4 MP 2, 7 IC 6, 8 MP 3, 9 II 4-10
MP 3, 4 MP 1, 5 II 4-6 IC 2, 7 IN 3-8
MP 1, 4 ID 5 MP 2, 6 IC 3, 7 IN 4-8 EN 9
EC 2 MP 1, 3 EC 2 IC 4, 5
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b) Ä 1. p 8 q Ä 2. ¬ q – 3. p 4. q q ì ¬ q – 5. q ì Ä 6. ¬ p q 8 ¬ p |Ä 7. ¬ q 8
MP 1, 3 IC 2, 4 IN 3-5 II 2-6
q 8 r ) c) Ä 1. p 8 (q 8 Ä 2. p ì q 3. p q 8 r 4. q 8 5. q Ä 6. r p ì q ) 8 r |Ä 7. ( p
EC 2 MP 1, 3 EC 2 MP 4, 5 II 2-6
Página 16 8.
1) Silogismo Silogismo hipotét hipotético ico (SIL): (SIL): A 8 B B 8 B 8 C ÄÄÄ A 8 C Ä 1. A 8 B Ä 2. B 8 B 8 C Ä 3. A Ä 4. B Ä 5. C |Ä 6. A 8 C 2) Identi Identidad dad (IDE (IDE): ): A = A Ä 1. A Ä 2. ¬ A Ä 3. A ì ¬ A Ä 4. ¬ ¬ A |Ä 5. 5. A 3) Mutació Mutación n de premisa premisa (MUT) A 8 (B 8 B 8 C ) ––––––––––– B 8 B 8 ( A A 8 C ) B 8 C ) Ä 1. A 8 (B 8
MP 1, 3 MP 2, 4 II 3-5
IC 1, 2 IN 2-3 EN 4
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4) Carga Carga de premi premisa sa (CPr) (CPr) A ––––– B 8 B 8 A Ä 1. A Ä 2. B – 3. ¬ A – 4. A ì ¬ A 5. ¬ ¬ A Ä 6. A |Ä 7. B 8 B 8 A 5) Conmutativa Conmutativa del conjunt conjuntor or (CC) (CC) A ì B ===== B ì B ì A - Ä 1. A ì B 2. A 3. B |Ä 4. B ì B ì A - Ä 1. B ì B ì A 2. B 3. A |Ä 4. A ì B 6) Conmutativa Conmutativa del disyunto disyuntorr (CD) A î B ===== B î B î A - Ä 1. A î B Ä 2. A B î A Ä 3. B î Ä 4. B B î A Ä 5. B î B î A |Ä 6. B î B î A - Ä 1. B î Ä 2. B Ä 3. A î B Ä 4. A Ä 5. A î B |Ä 6. A î B 7) Asociativa Asociativa del del conjuntor conjuntor (AC) (AC) A ì B ) ì C ( A =========
IC 1, 3 IN 3-4 EN 5 II 2-6
EC 1 EC 1 IC 2, 3 EC 1 EC 1 IC 2, 3
ID 2 ID 4 ED 1, 2-3, 4-5
ID 2 ID 4 ED 1, 2-3, 4-5
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B ì C 6. B ì B ì C ) |Ä 7. A ì (B ì B ì C ) - Ä 1. A ì (B ì 2. A B ì C 3. B ì 4. B 5. C 6. A ì B A ì B ) ì C |Ä 7. ( A 8) Asociativa Asociativa del disyunto disyuntorr (AD) A î B ) î C ( A ========= A î (B î B î C ) A î B ) î C - Ä 1. ( A Ä 2. A î B – 3. A B î C ) – 4. A î (B î – 5. B B î C 6. B î B î C ) – 7. A î (B î B î C ) Ä 8. A î (B î –––9. –– 9. C B î C 10. B î B î C ) ––11. A î (B î B î C ) |Ä 12. A î (B î B î C ) - Ä 1. A î (B î – 2. A 3. A î B A î B ) î C – 4. 4. ( A – ––5. B î C ––5. B î –6. B 7. A î B A î B ) î C –8. ( A ––9. C – 10. ( A A î B ) î C –– 11. ( A A î B ) î C A î B ) î C |Ä 12. ( A 9) Distributiv Distributivaa del conjunto conjuntorr (DC) A ì (B î B î C ) ============== A ì B ) î ( A A ì C ) ( A B î C ) - Ä 1. A ì (B î
IC 3, 5 IC 4, 6 EC 1 EC 1 EC 3 EC 3 IC 2, 4 IC 5, 6
ID 3 ID 5 ID 6 ED 2, 3-4, 5-7 ID 8 ID 9 ED 1, 2-8, 9-11
ID 2 ID 3
ID 6 ID 7 ID 9 ED 5, 6-8, 9-10 ED 1, 2-4, 5-11
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Ä 7. C 8. A ì C A ì B ) î ( A A ì C ) Ä 9. ( A A ì B ) î ( A A ì C ) |Ä 10. ( A A ì B ) î ( A A ì C ) - Ä 1. ( A Ä 2. A ì B 3. A 4. B B î C 5. B î B î C ) Ä 6. A ì (B î Ä 7. A ì C 8. A 9. C B î C 10. B î B î C ) – 11. A ì (B î B î C ) 12. A ì (B î 10) Distributi Distributiva va del disyuntor disyuntor (DD) (DD) A î (B ì B ì C ) ============= A î B ) ì ( A A î C ) ( A B ì C ) - Ä 1. A î (B ì 2. A Ä 2. 3. A î B 4. A î C A î B ) ì ( A A î C ) 5. ( A Ä 5. B ì C ––6. B ì 7. B 8. C 9. A î B 10. A î C A î B ) ì ( A A î C ) ––11. 11 . ( A A î B ) ì ( A A î C ) |Ä 12. ( A A î B ) ì ( A A î C ) - Ä 1. ( A
IC 2, 7 ID 8 ED 3, 4-6, 7-9
EC 2 EC 2 ID 4 IC 3, 5 EC 7 EC 7 ID 9 IC 8, 10 ED 1, 2-6, 7-11
ID 2 ID 2 IC 3, 4 EC 6 EC 6 ID 7 ID 8 IC 9, 10 ED 1, 2-5, 6-11
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11) Idempoten Idempotencia cia del conjun conjuntor tor (IdC) (IdC) A ì A ===== A - Ä 1. A ì A EC 1 - Ä 2. A - Ä 1. A Ä 2. ¬ A IC 1, 2 Ä 3. A ì ¬ A 4. ¬ ¬ A IN 2-3 5. A EN 4 IC 1, 5 |Ä 6. A ì A 12) Idempoten Idempotencia cia del disyun disyuntor tor (IdD) (IdD) A î A ====== A - Ä 1. A î A Ä 2. A – 3. ¬ A – 4. A ì ¬ A IC 2, 3 ¬¬ A 5. ¬¬ A IN 3-4 Ä 6. A EN 5 –– 7. A – 8. ¬ A – 9. A ì ¬ A IC 7, 8 ¬¬ A 10. ¬¬ A IN 8, 9 – 11. A EN 10 |Ä 12. 12. A ED 1, 2-6, 7-11 - Ä 1. 1. A |Ä 2. 2. A î A ID 1 13) Absorci Absorción ón del conjunto conjuntorr (AbsC) (AbsC) A ì ( A A î B ) ========= A
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– 7. A ì B – 8. A |Ä 9. A - Ä 1. A A ì B ) |Ä 2. A î ( A 15a) Contraposición 1 (CP 1) A 8 B ––––––––– B 8 ¬ A ¬ B 8 Ä 1. A 8 B Ä 2. ¬ B – 3. A 4. B B ì ¬ B – 5. B ì Ä 6. ¬ A B 8 ¬ A |Ä 7. ¬ B 8 15b) Contraposición 2 (CP 2) A 8 ¬ B ÄÄÄÄ B 8 B 8 ¬ A Ä 1. A 8 ¬ B Ä 2. B – 3. A 4. ¬ B B ì ¬ B – 5. B ì Ä 6. ¬ A B 8 ¬ A |Ä 7. B 8 15c) Contraposición 3 (CP 3) ¬ A 8 B ÄÄÄÄ B 8 A ¬ B 8 Ä 1. ¬ A 8 B Ä 2. ¬ B
-
EC 7 ED 1, 2-6, 7-8 ID 1
MP 1, 3 IC 2, 4 IN 3-5 II 2-6
MP 1, 3 IC 2, 4 IN 3-5 II 2-6
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Ä 1. ¬ A 8 ¬ B Ä 2. B – 3. ¬ A 4. ¬ B MP 1, 3 B ì ¬ B – 5. B ì IC 2, 4 6. ¬ ¬ A IN 3-5 EN 6 Ä 7. A B 8 A II 2-7 |Ä 8. B 8 16) Modus tollens (MT) A 8 B ¬ B ––––– ¬ A Ä 1. A 8 B Ä 2. ¬ B Ä 3. A 4. B MP 1, 3 Ä 5. B ì B ì ¬ B IC 2, 4 |Ä 6. ¬ A IN 3-5 17) Introducción del doble negador (IDN) A ––––– ¬ ¬ A ¬ A Ä 1. A Ä 2. ¬ A IC 1, 2 Ä 3. A ì ¬ A IN 2-3 |Ä 4. ¬ ¬ A 18) De una contradicción se sigue cualquier cosa (ECQ) A ì ¬ A ÄÄÄ B Ä 1. A ì ¬ A Ä 2. ¬ B 3. A EC 1
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20) Exportación (EXP) A ì B ) 8 C ( A –––––––––– A 8 (B 8 B 8 C ) Ä 1. ( A A ì B ) 8 C Ä 2. A – 3. B 4. A ì B – 5. C Ä 6. B 8 B 8 C |Ä 7. A 8 (B 8 B 8 C ) 21a) Silogismo disyuntivo (SD 1) A î B ¬ B ÄÄ A Ä 1. A î B Ä 2. ¬ B Ä 3. A – 4. ¬ A – 5. A ì ¬ A 6. ¬ ¬ A Ä 7. A ––– 8. B –– 9. ¬ A B ì ¬ B – 10. B ì 11. ¬ ¬ A –– 12. A |Ä 13. A 21b) Silogismo disyuntivo (SD 2) A î B ¬ A –––
IC 2, 3 MP 1, 4 II 3-5 II 2-6
IC 3, 4 IN 4-5 EN 6
IC 2, 8 IN 9-10 EN 11 ED 1, 3-7, 8-12
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22a) Dilemas (DIL 1) A î B A 8 C B 8 B 8 C ––––– C Ä 1. A î B Ä 2. A 8 C B 8 C Ä 3. B 8 Ä 4. A Ä 5. C Ä 6. B Ä 7. C |Ä 8. C 22b) Dilemas (DIL 2) A î B A 8 C B 8 B 8 D –––––– C î C î D Ä 1. A î B Ä 2. A 8 C Ä 3. B 8 B 8 D 4. A
MP 2, 4 MP 3, 6 ED 1, 4-5, 6-7
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––– 9. ¬ B – 10. C 11. B B ì ¬ B – 12. B ì –– 13. ¬ C |Ä 14. ¬ C 22d) Dilemas (DIL 4) ¬ A î ¬ B C 8 C 8 A D 8 B ––––––––– C î ¬ D ¬ C î Ä 1. ¬ A î ¬ B C 8 A Ä 2. C 8 Ä 3. D 8 B –– 4. ¬ A – 5. C 6. A – 7. A ì¬ A ì¬ A 8. ¬ C C î ¬ D –– 9. ¬ C î Ä 10. ¬ B –11. D 12. B
MP 3, 10 IC 9, 11 IN 10-12 ED 1, 4-8, 9-13
MP 2, 5 IC 4, 6 IN 5-7 ID 8
MP 3, 11
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23b) Definición del implicador (DI 2) A 8 B ========= ¬ A î B - Ä 1. A 8 B ––– 2. ¬ (¬ A î B ) ––– 3. A – 4. ¬ B 5. B B ì ¬ B – 6. B ì 7. ¬ ¬ B 8. B 9. ¬ A î B ––10. ¬ (¬ A î B ) ì (¬ A î B ) 11. ¬ A 12. ¬ A î B –– 13. ¬ (¬ A î B ) ì (¬ A î B ) 14. ¬ ¬ (¬ A î B ) |Ä 15. ¬ A î B - Ä 1. ¬ A î B ––– 2. ¬ A Ä 3. A – 4. ¬ B – 5. A ì ¬ A
MP 1, 3 IC 4, 5 IN 4-6 EN 7 ID 8 IC 2, 9 IN 3-10 ID 11 IC 2, 12 IN 2-13 EN 14
IC 2, 3
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- Ä 1. ¬ ( A A 8 ¬ B ) A ì B ) ––– 2. ¬ ( A Ä 3. A – 4. B 5. A ì B A ì B ) ì ( A A ì B ) – 6. ¬ ( A Ä 7. ¬ B 8. A 8 ¬ B A 8 ¬ B ) ì ( A A 8 ¬ B ) ––– 9. ¬ ( A A ì B ) 10. ¬ ¬ ( A |Ä 11. A ì B 24b) Definición del conjuntor (DfC 2) A ì B =========== ¬ (¬ A (¬ A î ¬ B ) - Ä 1. A ì B ––– ––– 2. ¬ A î ¬ B Ä 3. ¬ A – 4. ¬ A î ¬ B 5. A – 6. A ì ¬ A Ä 7. ¬ (¬ A î ¬ B ) –––8. ¬ B B –– 9. A
IC 3, 4 IC 2, 5 IN 4-6 II 3-7 IC 1, 9 IN 2-9 EN 10
EC 1 IC 3, 5 IN 4-6
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- Ä 1. A î B –––– 2. ¬ A Ä 3. A – 4. ¬ B ¬ A – 5. A ì ¬ A 6. ¬ ¬ B Ä 7. B ––– 8. B –– 9. ¬ B B ì ¬ B – 10. B ì 11. ¬ ¬ B Ä 12. 12. B ––––13. B |Ä 14. ¬ A 8 B - Ä 1. ¬ A 8 B A î B ) Ä 2. ¬ ( A – 3. ¬ A 4. B 5. A î B A î B ) ì ( A A î B ) – 6. ¬ ( A 7. ¬ ¬ A 8. A 9. A î B A B ) ( A A B ) 10 ( A
IC 2, 3 IN 4-5 EN 6
IC 8, 9 IN 9-10 EN 11 ED 1, 3-7, 8-12 II 2-13
MP 1, 3 ID 4 IC 2, 5 IN 3-6 EN 7 ID 8 IC 2, 9
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- Ä 1. ¬ (¬ A ì ¬ B ) A î B ) ––– 2. ¬ ( A Ä 3. ¬ A – 4. ¬ B 5. ¬ A ì ¬ B – 6. ¬ (¬ A ì ¬ B ) ì (¬ A ì ¬ B ) 7. ¬ ¬ B 8. B 9. A î B A î B ) ì ( A A î B ) – 10. ¬ ( A 11. ¬ ¬ A 12. A 13. A î B A î B ) ì ( A A î B ) –– 14. ¬ ( A
IC 3, 4 IC 1, 5 IN 4-6 EN 7 ID 8 IC 2, 9 IN 3-10 EN 11 ID 12 IC 2, 13
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26b) Leyes De Morgan (DM 2) A î B ) ¬ ( A =========== ¬ A ì ¬ B A î B ) - Ä 1. ¬ ( A Ä 2. A 3. A î B A î B ) ì ( A A î B ) Ä 4. ¬ ( A 5. ¬ A Ä 6. B 7. A î B A î B ) ì ( A A î B ) Ä 8. ¬ ( A 9. ¬ B |Ä 10. ¬ A ì ¬ B
ID 2 IC 1, 3 IN 2-4 ID 6 IC 1, 7 IN 6-8 IC 5, 9
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q 8 p) c) (¬ p 8 q ) ì (¬ q 8 p q ¬ p ¬ q ¬ p 8 q ¬ q 8 q 8 p V V F F V V V F F V V V F V V F V V F F V V F F Es una proposición contingente. p ì q 8 q 8 r ) 8 ( p p 8 ¬ q î q î r ) d) ( p p q r ¬ ¬ q p ì q ¬ q î q î r p ì q 8 q 8 r V V V F V V V V V F F V F F V F V V F V V V F F V F V V
q 8 p) (¬ p 8 q ) ì (¬ q 8
V V V F
p 8 ¬ q î q î r ( p p ì q 8 q 8 r ) 8 ( p p 8 ¬ q î q î r ) V F V V
V V
V V
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Página 23 10.
a) – Todos los franceses son europeos. Franceses
Europeos
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b) – Todos los filósofos son platónicos. Platónicos
Filósofos