Wednesday, June 24, 2009

Getting rid of "is true"

It is always tempting somehow to get rid of the predicate "is true" in order to escape the Liar Paradox on the cheap. I think there is no cheap escape (though I do think there is an escape). To that end, it's worth thinking through ranges of predicates other than "is true" which generate liar-type paradoxes. Some of these are expressly semantic, like "refers to x": We can redefine "is true" in terms of "refers" for instance as follows: "'s' is true iff 'the number 1 if s and the number 0 otherwise' refers to the number 1". Others are more everyday. For instance, consider the predicate "is reliable" as applied to a person. In the sense I am interested, "x is reliable" if and only if most of x's statements are true. But of course, "is reliable" is all we need for a liar paradox. We just imagine a possible world where most of George's statements are logically equivalent to "I am not reliable". Or, for a quite different case (not mine), take "is satisfied" as a predicate of desires, and imagine someone who desires not to have any satisfied desires: is that desire satisfied or not?

It is not plausible that one could not only get rid of "is true" but also of "refers to", "is reliable", "is satisfied" and the rest of the plethora of concepts each of which seems sufficient to generate a liar-type paradox.


Ezra said...

If I am not mistaken Grelling's paradox does the same kind of thing with a single word:

(1) x is heterological iff x does not describe itself.
(2) Is 'heterological' heterological?

Alexander R Pruss said...

Yes, "describes" is another one of those words that will do the trick here.