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

${\forall x (P x \lor Q x), \neg Q a} ⊢ P a$

1.	$\forall x (P x \lor Q x)$	Premissa
2.	$\neg Q a$	Premissa
3.	$P a \lor Q a$	1 $ℰ \forall$
4.	$P a$	3,2 SD

${\forall x (A x \land B x), (C x \land D x)} ⊢ (A x \land C x)$

1.	$\forall x (A x \land B x)$	Premissa
2.	$\forall x (B x \land C x)$	Premissa
3.	$A d \land B d$	1 $ℰ \forall$
4.	$C d \land D d$	2 $ℰ \forall$
5.	$A d$	3 S
6.	$C d$	4 S
7.	$A d \land C d$	5,6 C
8.	$\forall x (A x \land C x)$	7 $ℐ \forall$

${\forall x (A x \to B x), A l} ⊢ \exists x B x$

1.	$\forall x (A x \to B x)$	Premissa
2.	$A l$	Premissa
3.	$A l \to B l$	1 $ℰ \forall$
4.	$B l$	3,2 MP
5.	$\exists x B x$	5 $ℐ \exists$

$\exists x (P x \land Q x) ⊢ \exists x P x \land \exists x Q x$

1.		$\exists x (P x \land Q x)$		Premissa

2.	$P a \land Q a$	Hipótese para $ℰ \exists$
3.	$P a$	2 S
4.	$Q a$	2 S
5.	$\exists x P x$	3 $ℐ \exists$
6.	$\exists x Q x$	4 $ℐ \exists$
7.	$\exists x P x \land \exists x Q x$	5,6 C

8.

\exists x P x \land \exists x Q x

1,2-7

ℰ \exists

${\exists x P x, \forall x Q x} ⊢ \exists x (P x \land Q x)$

1.	$\exists x P x$	Premissa
2.	$\forall x Q x$	Premissa

3.	$P a$	Hipótese para $ℰ \exists$
4.	$Q a$	2 $ℰ \forall$
5.	$P a \land Q a$	3,4 C
6.	$\exists x (P x \land Q x)$	5 $ℐ \exists$

7.

\exists x (P x \land Q x)

1,3-6

ℰ \exists

Exercícios de teoremas

$⊢ \exists x P x \to \neg \forall x \neg P x$

1.			$\exists x P x$		Hipótese

2.	$\forall x \neg P x$	Hipótese
3.	$\neg \exists x P x$	2 IQ
4.	$\exists x P x \land \neg \exists x P x$	1,3 C

5.

\neg \forall x \neg P x

2,4 RAA

6.

\exists x P x \to \neg \forall x \neg P x

1,5 RPC

$⊢ \forall x (P x \to Q) \to \exists x (P x \to Q)$

1.	$\forall x (P x \to Q)$	Hipótese

2.	$P a \to Q$	1 $ℰ \forall$
3.	$\exists x (P x \to Q)$	2 $ℐ \exists$

4.

\forall x (P x \to Q) \to \exists x (P x \to Q)

1,3 RPC

$⊢ \exists x \exists y P x y \leftrightarrow \exists y \exists x P x y$

01.			$\exists x \exists y P x y$		Hipótese

02.				$\exists y P x y$		Hipótese para $ℰ \exists$

03.	$P a b$	Hipótese para $ℰ \exists$
04.	$\exists x P x b$	3 $ℐ \exists$
05.	$\exists y \exists x P x y$	4 $ℐ \exists$

06.				$\exists y \exists x P x y$		2,3-5 $ℰ \exists$

07.			$\exists y \exists x P x y$		1,2-6 $ℰ \exists$

08.		$\exists y \exists x P x y \to \exists y \exists x P x y$		1,7 RPC

09.			$\exists x \exists y P x y$		Hipótese

10.				$\exists x P x a$		Hipótese para $ℰ \exists$

11.	$P b a$	Hipótese para $ℰ \exists$
12.	$\exists y P y b$	11 $ℐ \exists$
13.	$\exists x \exists y P x y$	12 $ℐ \exists$

14.				$\exists x \exists y P x y$		10,11-13 $ℰ \exists$

15.			$\exists x \exists y P x y$		9,10-14 $ℰ \exists$

08.		$\exists y \exists x P x y \to \exists x \exists y P x y$		1,7 RPC

17.

\exists x \exists y P x y \leftrightarrow \exists y \exists x P x y

8,16 CB

$⊢ (\forall x P x \land \forall x Q x) \leftrightarrow \forall x (P x \land Q x)$

01.	$\forall x P x \land \forall x Q x$	Hipótese

02.	$\forall x P x$	1 S
03.	$\forall x Q x$	1 S
04.	$P a$	2 $ℰ \forall$
05.	$Q a$	3 $ℰ \forall$
06.	$P a \land Q a$	4,5 C
07.	$\forall x (P x \land Q x)$	2 $ℐ \forall$

08.

(\forall x P x \land \forall x Q x) \to \forall x (P x \land Q x)

1,7 RPC

09.	$\forall x (P x \land Q x)$	Hipótese

10.	$P a \land Q a$	9 $ℰ \forall$
11.	$P a$	10 S
12.	$\forall x P x$	11 $ℐ \forall$
13.	$Q a$	10 S
14.	$\forall x Q x$	13 $ℐ \forall$
15.	$\forall x Q x$	13 $ℐ \forall$

16.		$\forall x (P x \land Q x) \to (\forall x P x \land \forall x Q x)$		4,5 RPC
17.		$(\forall x P x \land \forall x Q x) \leftrightarrow \forall x (P x \land Q x)$		8,16 CB

$⊢ (P \land \exists x Q x) \leftrightarrow \exists x (P \land Q x)$

01.			$P \land \exists x Q x$		Hipótese
02.			$\exists x Q x$		1 S

03.	$Q a$	Hipótese para $ℰ \exists$
04.	$P$	1 S
05.	$P \land Q a$	4,3 C
06.	$\exists x (P \land Q a)$	5 $ℐ \exists$

07.

\exists x (P \land Q a)

2,3-6

ℰ \exists

08.

(P \land \exists x Q x) \to \exists x (P \land Q x)

1,7 RPC

09.			$\exists x (P \land Q x)$		Hipótese

10.	$P \land Q a$	Hipótese para $ℰ \exists$
11.	$Q a$	10 S
12.	$\exists x Q x$	11 $ℐ \exists$
13.	$P$	10 S
14.	$P \land \exists x Q x$	12,13 C

15.

P \land \exists x Q x

9,10-14

ℰ \exists

16.		$\exists x (P \land Q x) \to (P \land \exists x Q x)$		9,15 RPC
17.		$(P \land \exists x Q x) \leftrightarrow \exists x (P \land Q x)$		8,16 CB

$⊢ (P \lor \forall x Q x) \leftrightarrow \forall x (P \lor Q x)$

01.			$P \lor \forall x Q x$		Hipótese

02.	$\neg \forall x (P \lor Q x)$	Hipótese
03.	$\exists x \neg (P \lor Q x)$	2 IQ

04.	$\neg (P \lor Q a)$	Hipótese para $ℰ \exists$
05.	$\neg P \land \neg Q a$	4 DM
06.	$\neg Q a$	5 S
07.	$\neg P$	5 S
08.	$\forall x Q x$	1,7 SD
09.	$\exists x \neg Q x$	6 $ℐ \exists$
10.	$\neg \forall x Q x$	9 IQ
11.	$\forall x Q x \land \neg \forall x Q x$	8,10 C

12.				$\forall x Q x \land \neg \forall x Q x$		3,4-11 $ℰ \exists$

13.	$\neg \neg \forall x (P \lor Q x)$	2,17 RAA
14.	$\exists x (P x \lor Q x)$	13 DN

15.

(P \lor \forall x Q x) \to \forall x (P \lor Q x)

1,19 RPC

16.	$\forall x (P \lor Q x)$	Hipótese
17.	$P \lor Q a$	16 $ℰ \forall$

18.	$\neg (P \lor \forall x Q x)$	Hipótese
19.	$\neg P \land \neg \forall x Q x$	18 DM
20.	$\neg P$	19 S
21.	$Q a$	17,20 SD
22.	$\forall Q x$	21 $ℐ \forall$
23.	$\neg \forall Q x$	19 S
24.	$\forall Q x \land \neg \forall Q x$	22,23 C

25.			$\neg \neg (P \lor \forall x Q x)$		18,24 RAA
26.			$(P \lor \forall x Q x)$		25 DN

27.		$\forall x (P \lor Q x) \to (P \lor \forall x Q x)$		16,26 RPC
28.		$(P \lor \forall x Q x) \leftrightarrow \forall x (P \lor Q x)$		15,27 CB

$⊢ \exists x (P x \to Q) \to (\forall x P x \to Q)$

1.			$\exists x (P x \to Q)$		Hipótese

2.	$\forall x P x$	Hipótese
3.	$P a$	2 $ℰ \forall$

4.	$P a \to Q$		Hipótese para $ℰ \exists$
5.	$Q$	4,3 MP

6.				$Q$		1,4-5 $ℰ \exists$

7.			$\forall x P x \to Q$		2,6 RPC

8.

\exists x (P x \to Q) \to (\forall x P x \to Q)

1,7 RPC

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

Exercícios de teoremas

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