FOM: Anaphoric truth and living well

[Neil Tennant]

On Fri, 30 Aug 2002, Sandy Hodges wrote:

> With regards to Neil Tennant's suggestion that we can live well if we
> stop worrying about paradoxes, I have always assumed that the good life
> consists in more than the occasional utterance of theorems.    Indeed I
> think it will require, among other things,  the anaphoric use of
> "true."    Posit that Neil hears Socrates say:
> 
>     The sentence Plato spoke yesterday was not true.
> 
> If, because of the anaphoric use of "true," Neil treats what Socrates
> said as incomprehensible, then he won't understand much of what he
> hears.  Which is hardly living well.   (What Socrates said has no
> "normal form.")

I never claimed to be able to do without the truth-predicate and the
T-scheme.

Somewhere along the line, Sandy Hodges has misunderstood my proposal.
Perhaps I did not state it as well as I should have. The normal form in
question would be that enjoyed by any warrant, or proof, of the assertion.
Here the assertion is 

(A) "The sentence Plato spoke yesterday is not true."

Sandy does not tell us what exactly the sentence is that Plato spoke
yesterday. Let's suppose it was "Grass is purple." So we have as a
contextual axiom that

    "Grass is purple" = the sentence Plato spoke yesterday.

Let's abbreviate this as
    _______________________________________
    "Purple(grass)" = the s(Spoke(Plato,s))

We can use (2) as a starting point in any proof we might construct.
We can also use the following analytic inference rule:

	Purple(t)   Green(t)
	____________________
		#

which registers that purple and green are contraries. (Here # is the
absurdity sign.)  Finally, we can take as a contextual starting point the
further fact that grass is green, i.e.
	____________
	Green(grass)

We also have the rule of disquotation for the truth-predicate:

	True("P")
	_________
	    P

where the quotation marks have to be understood as so-called
corner-quotes.

Here now is a normal-form proof of (A):

   (1)__________________________ _______________________________________
      True(the s(Spoke(Plato,s)) "Purple(grass)" = the s(Spoke(Plato,s))
      __________________________________________________________________
	True("Purple(grass)")
	_____________________   ____________
	  Purple(grass)		Green(grass)
	  __________________________________
			   #
	   __________________________________(1)
	    not-[True(the s(Spoke(Plato,s))]      , i.e. (A)

This normal-form proof of (A) establishes it uncontrversially as a truth,
modulo the contextual axioms about the actual color of grass and what it
was that Plato actually said. There is nothing at all paradoxical in this
case, because of the choice I made for Plato's actual utterance (namely,
that grass is purple).

My point is that it will be impossible to find a normal-form proof of # in
a paradoxical case, such as the one where the sentence that Plato spoke
yesterday was "Socrates' sole statement tomorrow will be true" (remember
that the statement of Socrates that is in question here is (A)).

When a statement is genuinely or well-groundedly true, there will be a
normal-form proof of it. When it is genuinely or well-groundedly false,
there will be a normal-form disproof of it (i.e. a proof of # from it as
an assumption).

But when a statement P is paradoxical (whether on its own, such as The
Liar, or in combination with other statements X, which could even be
contingent truths) then there is no normal-form disproof of XU{P}.

I challenge Sandy to produce a counterexample to this generalization. I do
not believe he will succeed. For a counterexample would take the form of a
proof Pi, say, in normal form, of # from the assumptions XU{P}. But that
is exactly what is necessary and sufficient for P to be proved false,
given the truths X. 

Now, if Sandy were to object, by saying that he could still show P to be
paradoxical, by similarly showing P to be true, then he would have to do
this by furnishing a normal-form proof Sigma, say, of P from X. But then
we would be able to put Pi and Sigma together as follows:

	     X
	   Sigma
	X , [P]
 	  Pi
	  #

and normalize the result, in order to obtain a normal-form proof of # from
X alone. But then X could not have been a set of (even contingently) true
side-assumptions---contrary to what was assumed at the outset.

Note, by the way, that in the absence of a specification of what Plato's
statement actually was, no paradox arises. Thus Sandy was suppressing a
necessary premiss, concerning what Plato actually said, when he wrote

> But if on the other hand Neil concludes the following (which we can call
> sentence A):
> 
>     The sentence Socrates spoke is true if, and only if,
>     the sentence Plato spoke yesterday was not true.
> 
> (and draws similar conclusions on other occasions) then he is led to a
> contradiction.    

Summary: the normal forms I was talking about were normal forms of proofs
and of disproofs. I claim that the truth of P consists in the existence of
a normal-form proof of P, and the falsity of P consists in the existence
of a normal-form disproof of P (both, possibly, modulo what I have been
calling contextual axioms). In such proofs and disproofs one can use (as I
did above) analytic rules of inference about color contrarieties; axioms
registering singular facts that are not in contention (such as the
identity of a person's utterance); etc.

I claim that paradoxicality consists in THERE NOT BEING any normal-form
proof, or normal-form disproof, of the statement in question (again,
modulo possible contextual axioms as side-assumptions).

Thus paradoxes can be ignored as harmless, since they do not establish
anything. They are a byproduct of language whirring around, without
engaging any semantic gears. They will not lead anyone astray, or into
error, provided only that the thinker keeps an eye on the normal-form
proofs and disproofs by means of which one's stock of genuine knowledge is
increased.

I propose further, as a syntactic reduction of the notion of
paradoxicality (which has had, after all, endless *semantic*
characterizations), the following:

   P is paradoxical (modulo truths X)
	if and only if
   any disproof Pi of P using assumptions X cannot be normalized--
   that is, any reduction-sequence of proofs beginning with Pi has
   the form of a non-terminating loop (as in the case of The Liar), or
   infinitely progressing "spiral" (as in the case of Yablo's paradox).

Anyone interested in further details about all this could look at my two
papers

`Proof and paradox', Dialectica 36, 1982, pp. 265-296; and
`On Paradox without Self-Reference', Analysis 55, 1995, pp. 199-207.

Neil Tennant