## AxFx -> ExFx, in the empty domain

Philosophical, mathematical and computational logic, linguistics, formal argument, game theory, fallacies, paradoxes, puzzles and other related issues.

### AxFx -> ExFx, in the empty domain

If 'a' is a name or description that does not refer, then Fa is false and ~(Fa) is true.

In the empty domain, all values of x do not refer.

Therefore..
ExFx is false in the empty domain, and Ex~(Fx) is true.
ie. Fa v Fb v Fc ... is false, and ~(Fa) v ~(Fb) v ~(Fc)...is true.

Therefore..
AxFx is false in the empty domain, and Ax~(Fx) is true.
ie. Fa & Fb & Fc ... is false, and ~(Fa) & ~(Fb) & ~(Fc) ..is true.

That is, (AxFx -> ExFx) is true in the empty domain
Owen
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### Re: AxFx -> ExFx, in the empty domain

Owen » August 10th, 2016, 3:32 pm wrote:ExFx is false in the empty domain, and Ex~(Fx) is true.

That line contains an error, leading to your erroneous conclusion. Do you see the problem?
someguy1
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### Re: AxFx -> ExFx, in the empty domain

someguy1 » Wed Aug 10, 2016 7:34 pm wrote:
Owen » August 10th, 2016, 3:32 pm wrote:ExFx is false in the empty domain, and Ex~(Fx) is true.

That line contains an error, leading to your erroneous conclusion. Do you see the problem?

No! What problem do you see?
Owen
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Posts: 80
Joined: 06 May 2007

### Re: AxFx -> ExFx, in the empty domain

Owen » August 11th, 2016, 2:39 am wrote:
someguy1 » Wed Aug 10, 2016 7:34 pm wrote:
Owen » August 10th, 2016, 3:32 pm wrote:ExFx is false in the empty domain, and Ex~(Fx) is true.

That line contains an error, leading to your erroneous conclusion. Do you see the problem?

No! What problem do you see?

How did you end up with an object in the empty domain?

The negation of $\exists x Fx$ is $\neg \exists x Fx = \forall x (\neg Fx)$

Can you see that in the empty domain, the latter is still true while what you wrote must be false?
someguy1
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 Lomax liked this post

### Re: AxFx -> ExFx, in the empty domain

someguy1:
How did you end up with an object in the empty domain?

The negation of $\exists x Fx$ is $\neg \exists x Fx = \forall x (\neg Fx)$

Can you see that in the empty domain, the latter is still true while what you wrote must be false?[/quote]

I don't agree.

1. If Ex(Fx) is false Ax~(Fx) is true.
2. If Ex~(Fx) is false then Ax(Fx) is true.

3. Ax(Fx) & Ax~(Fx), is true.
4. (Ax(Fx) & Ax~(Fx)) <-> Ax((Fx) & ~(Fx)).
5. Ax((Fx) & ~(Fx)) is a contradiction.
6. Ax(Fx) & Ax~(Fx), is false...contradicting 3.

In the empty domain...
~Ex(Fx) is true.
Ax~(Fx) is true.
Ex~(Fx) is true
~Ax(Fx) is true.

Ax(Fx) -> Ex(Fx) in the empty domain.
Owen
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Posts: 80
Joined: 06 May 2007

### Re: AxFx -> ExFx, in the empty domain

Owen » August 15th, 2016, 2:11 pm wrote:1. If Ex(Fx) is false Ax~(Fx) is true.

Agreed, this is always valid whether the domain is empty or not.

Owen » August 15th, 2016, 2:11 pm wrote:2. If Ex~(Fx) is false then Ax(Fx) is true.

Agreed, this is always valid whether the domain is empty or not.

Owen » August 15th, 2016, 2:11 pm wrote:3. Ax(Fx) & Ax~(Fx), is true.

This is valid only in the empty domain. It's false in any nonempty domain.

Owen » August 15th, 2016, 2:11 pm wrote:4. (Ax(Fx) & Ax~(Fx)) <-> Ax((Fx) & ~(Fx)).

Agreed. In the empty domain they're both true; and in a nonempty domain they're both false.

Owen » August 15th, 2016, 2:11 pm wrote:5. Ax((Fx) & ~(Fx)) is a contradiction.

No, we just agreed in #4 that this is valid in the empty domain. You made that point yourself. Example: Every positive integer in the empty set is both prime and composite. True statement.

Owen » August 15th, 2016, 2:11 pm wrote:6. Ax(Fx) & Ax~(Fx), is false...contradicting 3.

No, we just agreed in #3 this is valid in the empty domain. You made that point yourself. Example: Every positive integer in the empty set is prime AND every positive integer in the empty set is composite. True.

In #3 you say the statement is true and in #6 you say the same statement is false. But it's the exact same statement both times. It's true in the empty domain.

Owen » August 15th, 2016, 2:11 pm wrote:In the empty domain...

Ex~(Fx) is true
.

As a matter of common sense, how can you claim that something exists in the empty domain? Wouldn't such a conclusion cause you to go back and see where you made a mistake?
someguy1
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