[FOM] Paradoxicality and normal-form proofs

[Sean C Stidd]

>"This sentence is true"
>
>which, intuitively, I would just classify as true with no further 
>complication.

For what it's worth, I find that sentence just as surprising as "This
sentence is false", albeit in a different way.

In the case of "this sentence is false", we see that the sentence is in a
certain sense self-refuting or paradoxical. Interpreting it as true has
the consequence that it is false, and vice-versa. (Assuming a naive
interpretation of 'true', 'false' as predicates in the One Language,
etc.) But we at least have some structure to work with, even if it is
'contradictory structure', the structure of a paradox.

On the other hand, in the case of "This sentence is true", we don't even
seem to have that. 'This sentence is true' can be interpreted as true or
false without running into any trouble in either case, at least if one
assumes no extra theoretical apparatus. If I interpret it as true, that
is consistent with the sentence's meaning. But if I interpret it as
false, that is also consistent: it says it's true, sure, but it says so
falsely, and this is consistent with its being a false sentence. 

And there appears to be nothing else to go on here, unless it's some
pragmatic principle about charity etc., which ought to have no role
whatever in deciding this kind of question.

In other words, bivalence demands that the sentence be either true or
false, but there seems to be no human way of knowing which it is that
could be derived from this sentence itself.  Assuming that 'true' is a
predicate within the One Language, then, 'This sentence is true' provides
a simple example of a sentence whose truth-value fundamentally outstrips
our ability to verify or falsify it. 

Of course one needn't think of the sentence this way, and in addition it
may be that our overall theory of logic, semantics, pragmatics or
whatever else gives us a way to decide the truth-value of this sentence
based on a general framework derived from other sentences. But I think
it's a mistake to say that there's no puzzle or complication in this
sentence at all when it's considered in 'natural' or 'intuitive' terms,
prior to logical or semantic analysis. Taken on its own it provides an
example of a sentence which is unverifiable-in-principle, and that is
something that very many people have found surprising, though perhaps
such people are fewer today then they once were.