[Guest]

consider all sentences that refer to themselves. for example, "this is a true statement." let's dub all such sentences hetrogenous. any sentence that does not refer to itself is homogenous.

was that statement a homogenous or hetrogenous one?

[Guest]

Its hetrogenous as it refers to itself too. Hetrogenous/Homogenous is only a nomenclature used

[Guest]

are you sure deegee?

let's shorten it to just "any sentence that does not refer to itself."

now, we have two groups of statements.

group A = any sentence that refers to itself.

group ~A = any sentence that does not refer to itself.

now we are asking, does the statement that describes the entire group ~A fit into A?

basically it boils down to this, either ~A = ~(~A) or A = ~A.

[Guest]

"Many sentences do not refer to themselves." - this is not self-referential. It is an apparently simple statement of 'fact', like "Grass is green."

But let's dig deeper, as you invite us to ...

"All sentences that do not refer to themselves are paradoxes." - if this sentence refers to the class of sentences that includes itself, then it represents a paradox. If it does not refer to the class of sentences of which it is a part, then it is a true statement, and therefore a paradox. Ergo, Paradox is the only truth.

This becomes understandable when you examine more closely the (apparently) non-self-referential statement: "Grass is green." It seems like a simple 'fact'. But ... Does the statement exist? Does it exist in a universe in which it is valid? Does it refer to that universe? If not, of what value is it? But if so, then it refers to a class of entities (including sentences) that includes itself.

Read my signature: Reality is subverted by facts.

[Guest]

It can't be classified under either, thus having a class of its own.

[Guest]

I don't easily associate the homo/hetero-geneous words with the meanings defined here, so I will temporarily use "self-referential" and "non-self-referential" (creative, I know).

Question: Is the following sentence self-referential or not?

"Any sentence that does not refer to itself is non-self-referential."

This sentence does not reference itself directly.

I believe that phillip meant for reference to also be indirect.

Our sentence in question, let us call A.

If A is one of the sentences that does not refer to itself, then it is indirectly referenced in A and defined in this sentence to be non-self-referential. However it was indirectly referenced, which would make the precondition in the last sentence false. If that precondition is false, then it is true that A is one of the sentences that refers to itself. If A refers to itself, then there exists a reference to A within A. All references within A are sentences of the type that do not refer to themselves. Therefore if A is referenced within itself, then A must be a sentence of the type that does not reference itself. Likewise, if A is not referenced within itself, then it is indirectly referenced in A.

This is a paradox, equivalent to the assertion:

Some condition C is true if and only if it is not true (C XNOR ~C = FALSE always)

This logical statement is called a contradiction and is always false, it is the opposite of a tautology.

[Guest]

Well, the statement refers to both groups,thus refering to itself (as it must be in one of the groups), making it hetrogenous.

Edit: 500 posts!

Edited August 17, 2010 by Blablah99

[bonanova]

consider all sentences that refer to themselves. for example, "this is a true statement." let's dub all such sentences hetrogenous. any sentence that does not refer to itself is homogenous.

was that statement a homogenous or hetrogenous one?

Homogeneous.

It describes a class of sentences. The sentence itself belongs to that class. But we make that judgment. The statement does not say what type it is. This falls short of being self-referential.

You'd need something like "This statement is homogeneous." to have a contradiction.