luvluv

Further proof that bd-from-kg is the smartest man alive.

Scrambles

bd-from-kg,

This thread is not mine. I think your post was actually directed at Sodium. The one post I have made in this thread was critical of the topic post.



Scrambles

bd-from-kg

Scrambles:

Quite right. My bad.

Autonemesis

The OP just restates the well-known Liar Paradox. You don't even need self-reference to generate it:

God knows the following statement is true.
God knows the previous statement is false.

luvluv

Jobar:

It was kind of a joke. But as bd-from-kg said (from what I understand) a Godel-style paradox can be settled by getting outside the system so to speak. So anyone but God could solve a Godel-style paradox using God's name. So even if God can't solve one, Jesus could, and so according to Trinitarian theology God would have solved the paradox, Jesus being God and all.

I'll tell ya... every now and again... being three persons is SWWEEEET.

lazcatluc

Let us supose that George Bush exists!

Set St= the set of all the true statements that George Bush knows

St={P|George Bush knows that P is true}
Sf={P|George Bush knows that P is false}

Def: S=St U Sf

statement P="George Bush knows that this statement is not in S"

So, what could GB think about this statement?

1. GB doesn't know if P is true => P is not in S => GB knows that P is not in S => P is true => GB knows that P is true. But this contradicts the premise.

2. GB knows that P is true => P is true => P is not in S => P is not in St => GB doesn't know that P is true. But this contradicts the premise

3. GB knows that P is false => GB doesn't know if P is in S or not (from def of statement P)|
3. => P is in Sf => P is in S => GB knows P is in S | These 2 are contradictory


Therefore, by reductio ad absurdum, George Bush doesn't exist!

sodium

Only if you assume that GB must be omniscient, which is I think what you're doing in your P is true => GB knows that P is true step.

Note that I've switched to using "believes" instead of "knows" because this is a less philosophically complicated idea.

GB cannot consistently believe the statement to be true or false. So, if we can assume he is consistent, then we know that the statement is false, although he does not. This is possible, as he is not omniscient.

Of course, we don't really know that he is consistent, so he may very well decide that the statement is false, in which case it will be true. But there's no paradox. He's just made an error. Once again, he isn't omniscient.

lazcatluc

You don't need GB to be omniscient in order for him to understand a simple deduction. The fact that P is true is a direct consequence of the premise; so, just like you and I know that P is true under that premise, so does GB...

lazcatluc

Let us suppose that X is a smart person reading this post.

Statement P:="X, after reading this post, will believe P is false"

1. if X, after reading this post, will believe that P is false then P is true (from the def of P). But, of course, X already knows this and, being a smart person, will not believe that P is false.

2. if X, after reading this post, will not believe that P is false then P is false (from the def of P). But X already knows this and, being a smart person, must believe that P is false.

Both 1 and 2 are contradictory. Therefore, by reductio ad absurdum, no smart person will ever read this post (Let's see who quotes me on this )

bd-from-kg

sodium:

The same kind of argument that lazcatluc gave can be applied to your “quining” example:

(+)God believes "God believes Q is false" when Quined is false

Let’s compare:

(++) I believe “I believe Q is false” when Quined is false.

Let’s assume that (++) expresses a proposition. If I believe that it’s false, it’s true. And if I believe that it’s true, it’s false. And I’ve figured this out, so I can’t possibly believe that it’s true or false. Which means that it’s false.

So if this statement expresses a proposition, it must be false, I can figure out that it must be false, but I can’t believe that it’s false.

Needless to say, this is ridiculous. The only rational conclusion is that (++) does not express a proposition, which means that it’s nonsense. But if (++) is nonsense, so is (+).

At some point a rational person might begin to suspect that a style of argument that repeatedly produces absurd results may be defective...

On a more serious note:

Russell and Whitehead dealt with a similar problem in set theory (Russell’s Paradox) by defining a hierarchy of sets. “Level 0” sets could contain no sets; “Level 1” sets could contain “Level 0” sets but not “Level 1” ones; “Level 2” sets could contain sets of Level 0 or 1, etc. (right through the infinite ordinals). A corresponding proposal to avoid the kinds of paradox discussed here is to define “Level 0” statements, which cannot reference statements at all, “Level 1” statements which can reference Level 0 statements but not others, etc. Any statements that cannot be assigned to some level are out of court (i.e., are not assigned a meaning). (This is just an extension of the concept of statements, meta-statements, etc.) This prevents both direct and indirect self-reference and other situations that can lead to paradoxes.

luvluv:

A Godel-style paradox cannot be resolved by “getting outside” the system. This is how the paradox is generated! (Of course, I should point out once again that sodium’s arguments are not, by any stretch of imagination, “Godel-style”. ) But Godel-type paradoxes cannot be applied to God because the Godel procedure, by its very nature, requires that the system in question be finitary, and God is not finitary.