# Lógica/Cálculo Quantificacional Clássico/Dedução Natural no CQC/Resolução dos exercícios

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${\displaystyle \left\{\forall x\left(Px\lor Qx\right),\neg Qa\right\}\vdash Pa}$

 1 ${\displaystyle \forall x\left(Px\lor Qx\right)}$ Premissa 2 ${\displaystyle \neg Qa\,\!}$ Premissa 3 ${\displaystyle Pa\lor Qa\,\!}$ 1 ${\displaystyle {\mathcal {E}}\forall }$ 4 ${\displaystyle Pa\,\!}$ 3,2 SD

${\displaystyle \left\{\forall x\left(Ax\land Bx\right),\left(Cx\land Dx\right)\right\}\vdash \left(Ax\land Cx\right)}$

 1 ${\displaystyle \forall x\left(Ax\land Bx\right)}$ Premissa 2 ${\displaystyle \forall x\left(Bx\land Cx\right)}$ Premissa 3 ${\displaystyle Ad\land Bd\,\!}$ 1 ${\displaystyle {\mathcal {E}}\forall }$ 4 ${\displaystyle Cd\land Dd\,\!}$ 2 ${\displaystyle {\mathcal {E}}\forall }$ 5 ${\displaystyle Ad\,\!}$ 3 S 6 ${\displaystyle Cd\,\!}$ 4 S 7 ${\displaystyle Ad\land Cd\,\!}$ 5,6 C 8 ${\displaystyle \forall x\left(Ax\land Cx\right)\,\!}$ 7 ${\displaystyle {\mathcal {I}}\forall }$

${\displaystyle \left\{\forall x\left(Ax\to Bx\right),Al\right\}\vdash \exists xBx}$

 1 ${\displaystyle \forall x\left(Ax\to Bx\right)}$ Premissa 2 ${\displaystyle Al\,\!}$ Premissa 3 ${\displaystyle Al\to Bl\,\!}$ 1 ${\displaystyle {\mathcal {E}}\forall }$ 4 ${\displaystyle Bl\,\!}$ 3,2 MP 5 ${\displaystyle \exists xBx\,\!}$ 5 ${\displaystyle {\mathcal {I}}\exists }$

${\displaystyle \exists x\left(Px\land Qx\right)\vdash \exists xPx\land \exists xQx}$

 1. ${\displaystyle \exists x\left(Px\land Qx\right)\,\!}$ Premissa
 2 ${\displaystyle Pa\land Qa\,\!}$ Hipótese para ${\displaystyle {\mathcal {E}}\exists }$ 3 ${\displaystyle Pa\,\!}$ 2 S 4 ${\displaystyle Qa\,\!}$ 2 S 5 ${\displaystyle \exists xPx\,\!}$ 3 ${\displaystyle {\mathcal {I}}\exists }$ 6 ${\displaystyle \exists xQx\,\!}$ 4 ${\displaystyle {\mathcal {I}}\exists }$ 7 ${\displaystyle \exists xPx\land \exists xQx\,\!}$ 5,6 C
 8 ${\displaystyle \exists xPx\land \exists xQx\,\!}$ 1,2-7 ${\displaystyle {\mathcal {E}}\exists }$

${\displaystyle \left\{\exists xPx,\forall xQx\right\}\vdash \exists x\left(Px\land Qx\right)}$

 1. ${\displaystyle \exists xPx\,\!}$ Premissa 2. ${\displaystyle \forall xQx\,\!}$ Premissa
 3 ${\displaystyle Pa\,\!}$ Hipótese para ${\displaystyle {\mathcal {E}}\exists }$ 4 ${\displaystyle Qa\,\!}$ 2 ${\displaystyle {\mathcal {E}}\forall }$ 5 ${\displaystyle Pa\land Qa\,\!}$ 3,4 C 6 ${\displaystyle \exists x\left(Px\land Qx\right)}$ 5 ${\displaystyle {\mathcal {I}}\exists }$
 7 ${\displaystyle \exists x\left(Px\land Qx\right)}$ 1,3-6 ${\displaystyle {\mathcal {E}}\exists }$

### Exercícios de teoremas

${\displaystyle \vdash \exists xPx\to \neg \forall x\neg Px}$

 1. ${\displaystyle \exists xPx\,\!}$ Hipótese
 2 ${\displaystyle \forall x\neg Px\,\!}$ Hipótese 3 ${\displaystyle \neg \exists xPx\,\!}$ 2 IQ 4 ${\displaystyle \exists xPx\land \neg \exists xPx\,\!}$ 1,3 C
 5 ${\displaystyle \neg \forall x\neg Px\,\!}$ 2,4 RAA
 6 ${\displaystyle \exists xPx\to \neg \forall x\neg Px}$ 1,5 RPC

${\displaystyle \vdash \forall x\left(Px\to Q\right)\to \exists x\left(Px\to Q\right)}$

 1. ${\displaystyle \forall x\left(Px\to Q\right)\,\!}$ Hipótese 2. ${\displaystyle Pa\to Q\,\!}$ 1 ${\displaystyle {\mathcal {E}}\forall }$ 3. ${\displaystyle \exists x\left(Px\to Q\right)\,\!}$ 2 ${\displaystyle {\mathcal {I}}\exists }$
 4 ${\displaystyle \forall x\left(Px\to Q\right)\to \exists x\left(Px\to Q\right)}$ 1,3 RPC

${\displaystyle \vdash \exists x\exists yPxy\leftrightarrow \exists y\exists xPxy}$

 01. ${\displaystyle \exists x\exists yPxy\,\!}$ Hipótese
 02. ${\displaystyle \exists yPxy\,\!}$ Hipótese para ${\displaystyle {\mathcal {E}}\exists }$
 03. ${\displaystyle Pab\,\!}$ Hipótese para ${\displaystyle {\mathcal {E}}\exists }$ 04. ${\displaystyle \exists xPxb\,\!}$ 3 ${\displaystyle {\mathcal {I}}\exists }$ 05. ${\displaystyle \exists y\exists xPxy\,\!}$ 4 ${\displaystyle {\mathcal {I}}\exists }$
 06. ${\displaystyle \exists y\exists xPxy\,\!}$ 2,3-5 ${\displaystyle {\mathcal {E}}\exists }$
 07. ${\displaystyle \exists y\exists xPxy\,\!\,\!}$ 1,2-6 ${\displaystyle {\mathcal {E}}\exists }$
 08. ${\displaystyle \exists y\exists xPxy\to \exists y\exists xPxy\,\!}$ 1,7 RPC

 09. ${\displaystyle \exists x\exists yPxy\,\!}$ Hipótese
 10. ${\displaystyle \exists xPxa\,\!}$ Hipótese para ${\displaystyle {\mathcal {E}}\exists }$
 11. ${\displaystyle Pba\,\!}$ Hipótese para ${\displaystyle {\mathcal {E}}\exists }$ 12. ${\displaystyle \exists yPyb\,\!}$ 11 ${\displaystyle {\mathcal {I}}\exists }$ 13. ${\displaystyle \exists x\exists yPxy\,\!}$ 12 ${\displaystyle {\mathcal {I}}\exists }$
 14. ${\displaystyle \exists x\exists yPxy\,\!}$ 10,11-13 ${\displaystyle {\mathcal {E}}\exists }$
 15. ${\displaystyle \exists x\exists yPxy\,\!\,\!}$ 9,10-14 ${\displaystyle {\mathcal {E}}\exists }$
 08. ${\displaystyle \exists y\exists xPxy\to \exists x\exists yPxy\,\!}$ 1,7 RPC
 17 ${\displaystyle \exists x\exists yPxy\leftrightarrow \exists y\exists xPxy\,\!}$ 8,16 CB

${\displaystyle \vdash \left(\forall xPx\land \forall xQx\right)\leftrightarrow \forall x\left(Px\land Qx\right)}$

 01. ${\displaystyle \forall xPx\land \forall xQx\,\!}$ Hipótese 02. ${\displaystyle \forall xPx\,\!}$ 1 S 03. ${\displaystyle \forall xQx\,\!}$ 1 S 04. ${\displaystyle Pa\,\!}$ 2 ${\displaystyle {\mathcal {E}}\forall }$ 05. ${\displaystyle Qa\,\!}$ 3 ${\displaystyle {\mathcal {E}}\forall }$ 06. ${\displaystyle Pa\land Qa\,\!}$ 4,5 C 07. ${\displaystyle \forall x\left(Px\land Qx\right)\,\!}$ 2 ${\displaystyle {\mathcal {I}}\forall }$
 8 ${\displaystyle \left(\forall xPx\land \forall xQx\right)\to \forall x\left(Px\land Qx\right)\,\!}$ 1,7 RPC
 09. ${\displaystyle \forall x\left(Px\land Qx\right)\,\!}$ Hipótese 10. ${\displaystyle Pa\land Qa\,\!}$ 9 ${\displaystyle {\mathcal {E}}\forall }$ 11. ${\displaystyle Pa\,\!}$ 10 S 12. ${\displaystyle \forall xPx\,\!}$ 11 ${\displaystyle {\mathcal {I}}\forall }$ 13. ${\displaystyle Qa\,\!}$ 10 S 14. ${\displaystyle \forall xQx\,\!}$ 13 ${\displaystyle {\mathcal {I}}\forall }$ 15. ${\displaystyle \forall xQx\,\!}$ 13 ${\displaystyle {\mathcal {I}}\forall }$
 16 ${\displaystyle \forall x\left(Px\land Qx\right)\to \left(\forall xPx\land \forall xQx\right)\,\!}$ 4,5 RPC 17 ${\displaystyle \left(\forall xPx\land \forall xQx\right)\leftrightarrow \forall x\left(Px\land Qx\right)}$ 8,16 CB

${\displaystyle \vdash \left(P\land \exists xQx\right)\leftrightarrow \exists x\left(P\land Qx\right)}$

 1 ${\displaystyle P\land \exists xQx}$ Hipótese 2 ${\displaystyle \exists xQx\,\!}$ 1 S
 3 ${\displaystyle Qa\,\!}$ Hipótese para ${\displaystyle {\mathcal {E}}\exists }$ 4 ${\displaystyle P\,\!}$ 1 S 5 ${\displaystyle P\land Qa\,\!}$ 4,3 C 6 ${\displaystyle \exists x\left(P\land Qa\right)\,\!}$ 5 ${\displaystyle {\mathcal {I}}\exists }$
 7 ${\displaystyle \exists x\left(P\land Qa\right)\,\!}$ 2,3-6 ${\displaystyle {\mathcal {E}}\exists }$
 8 ${\displaystyle \left(P\land \exists xQx\right)\to \exists x\left(P\land Qx\right)}$ 1,7 RPC
 09. ${\displaystyle \exists x\left(P\land Qx\right)}$ Hipótese
 10 ${\displaystyle P\land Qa\,\!}$ Hipótese para ${\displaystyle {\mathcal {E}}\exists }$ 11 ${\displaystyle Qa\,\!}$ 10 S 12 ${\displaystyle \exists xQx\,\!}$ 11 ${\displaystyle {\mathcal {I}}\exists }$ 13 ${\displaystyle P\,\!}$ 10 S 14 ${\displaystyle P\land \exists xQx\,\!}$ 12,13 C
 15 ${\displaystyle P\land \exists xQx\,\!}$ 9,10-14 ${\displaystyle {\mathcal {E}}\exists }$
 16 ${\displaystyle \exists x\left(P\land Qx\right)\to \left(P\land \exists xQx\right)}$ 9,15 RPC 17 ${\displaystyle \left(P\land \exists xQx\right)\leftrightarrow \exists x\left(P\land Qx\right)}$ 8,16 CB

${\displaystyle \vdash \left(P\lor \forall xQx\right)\leftrightarrow \forall x(P\lor Qx)}$

 01. ${\displaystyle P\lor \forall xQx\,\!}$ Hipótese
 02. ${\displaystyle \neg \forall x\left(P\lor Qx\right)\,\!}$ Hipótese 03. ${\displaystyle \exists x\neg \left(P\lor Qx\right)\,\!}$ 2 IQ
 04. ${\displaystyle \neg \left(P\lor Qa\right)\,\!}$ Hipótese para ${\displaystyle {\mathcal {E}}\exists }$ 05. ${\displaystyle \neg P\land \neg Qa\,\!}$ 4 DM 06. ${\displaystyle \neg Qa\,\!}$ 5 S 07. ${\displaystyle \neg P\,\!}$ 5 S 08. ${\displaystyle \forall xQx\,\!}$ 1,7 SD 09. ${\displaystyle \exists x\neg Qx\,\!}$ 6 ${\displaystyle {\mathcal {I}}\exists }$ 10. ${\displaystyle \neg \forall xQx\,\!}$ 9 IQ 11. ${\displaystyle \forall xQx\land \neg \forall xQx\,\!}$ 8,10 C
 12. ${\displaystyle \forall xQx\land \neg \forall xQx\,\!}$ 3,4-11 ${\displaystyle {\mathcal {E}}\exists }$
 13. ${\displaystyle \neg \neg \forall x\left(P\lor Qx\right)\,\!}$ 2,17 RAA 14. ${\displaystyle \exists x\left(Px\lor Qx\right)\,\!}$ 13 DN
 15 ${\displaystyle \left(P\lor \forall xQx\right)\to \forall x\left(P\lor Qx\right)}$ 1,19 RPC
 16. ${\displaystyle \forall x\left(P\lor Qx\right)\,\!}$ Hipótese 17. ${\displaystyle P\lor Qa\,\!}$ 16 ${\displaystyle {\mathcal {E}}\forall }$
 18. ${\displaystyle \neg \left(P\lor \forall xQx\right)\,\!}$ Hipótese 19. ${\displaystyle \neg P\land \neg \forall xQx\,\!}$ 18 DM 20. ${\displaystyle \neg P\,\!}$ 19 S 21. ${\displaystyle Qa\,\!}$ 17,20 SD 22. ${\displaystyle \forall Qx\,\!}$ 21 ${\displaystyle {\mathcal {I}}\forall }$ 23. ${\displaystyle \neg \forall Qx\,\!}$ 19 S 24. ${\displaystyle \forall Qx\land \neg \forall Qx\,\!}$ 22,23 C
 25 ${\displaystyle \neg \neg \left(P\lor \forall xQx\right)\,\!}$ 18,24 RAA 26 ${\displaystyle \left(P\lor \forall xQx\right)\,\!}$ 25 DN
 27 ${\displaystyle \forall x\left(P\lor Qx\right)\to \left(P\lor \forall xQx\right)\,\!}$ 16,26 RPC 28 ${\displaystyle \left(P\lor \forall xQx\right)\leftrightarrow \forall x(P\lor Qx)\,\!}$ 15,27 CB

${\displaystyle \vdash \exists x(Px\rightarrow Q)\to (\forall xPx\rightarrow Q)}$

 1. ${\displaystyle \exists x(Px\rightarrow Q)\,\!}$ Hipótese
 2. ${\displaystyle \forall xPx\,\!}$ Hipótese 3. ${\displaystyle Pa\,\!}$ 2 ${\displaystyle {\mathcal {E}}\forall }$
 4. ${\displaystyle Pa\to Q\,\!}$ Hipótese para ${\displaystyle {\mathcal {E}}\exists }$ 5. ${\displaystyle Q\,\!}$ 4,3 MP
 6. ${\displaystyle Q\,\!}$ 1,4-5 ${\displaystyle {\mathcal {E}}\exists }$
 7. ${\displaystyle \forall xPx\to Q\,\!}$ 2,6 RPC
 8 ${\displaystyle \exists x(Px\rightarrow Q)\to (\forall xPx\rightarrow Q)\,\!}$ 1,7 RPC