Can a statement imply itself without being true or false?

[Alkatran]

I've been doing some thinking on self-referencing statements and the problems they imply. For example:
THIS=TRUE is both true and false
THIS=FALSE is neither true nor false
THIS>TRUE is both true and false
THIS>FALSE is neither true nor false
THIS > X implies itself and x (using the fact that THIS = (THIS > X))
etc...

I was wondering if the people here could shoot down this idea:
A self referential statement is true if and only if it implies itself.
THAT(written) = (THAT(value) > THAT(written))

Given this, we would get:
(THIS=TRUE) = (THAT > THAT=TRUE) = TRUE > TRUE = TRUE
(THIS=FALSE) = (THAT > THAT=FALSE) = FALSE
and we wouldn't be able to imply X using (THIS > X) because once we get THIS = (THIS > X) we have change it to THAT = (THAT > (THAT = (THAT > THAT))) before we can evaluate it.

I suppose what I'm looking for here are interesting statements that break this rule. I know it doesn't handle indirect self-reference.

 

[CRGreathouse]

"(THIS=THIS) and false" would be a counterexample, although it doesn't show that your method is wrong (perhaps you need a stronger form of "self-referential" to exclude this).

 

[Hurkyl]

I've been doing some thinking on self-referencing statements and the problems they imply. For example:
THIS=TRUE is both true and false
THIS=FALSE is neither true nor false
THIS>TRUE is both true and false
THIS>FALSE is neither true nor false
THIS > X implies itself and x (using the fact that THIS = (THIS > X))
etc...

You're hitting a seam in a common abuse of language, I think.

Statements don't have inherent truth value -- what you're really saying here is that you can (consistently) label "this = true" with either truth value, and that you cannot (consistently) label "this = false" with either truth value.



Logically, any statement implies itself:

P --> P

is a tautology.