btanksley@hifn.com btanksley@hifn.com
Mon, 16 Aug 1999 09:47:54 -0700

```>From: iepos@tunes.org [mailto:iepos@tunes.org]

>i've been doing a bit of thinking about the paradoxes and haven't
>come to any really good answers and am wondering if any of you have.

>one of the most paradoxes occurs when reasoning on a statement
>that says "this statement is not true". One first supposes that it
>is true; then it follows that it is not true. This is a contradiction,
>so the assumption must have been not true. So, the statement is not
>true; but this is precisely what the statement says, so we have
>admitted the statement. So there is an inconsistency in this logic.
>Unfortunately, the inconsistency is not caused merely by the funny
>nature of the English language; the argument can be formalized in
>a fairly simple sound-appearing logic using the Y combinator (or
>lambda term) to achieve the self-reference.

Only if you fail to typecheck your variables.

>I: x -> x
>B: (x -> y) -> (z -> x) -> z -> y
>C: (x -> y -> z) -> y -> x -> z
>W: (x -> x -> y) -> x -> y
>K: x -> y -> x

The trouble is, what are you allowed to substitute for the variables?
Obviously, you can only substitute things which have possible boolean
answers (as an example, you can't set x to any natural number and expect