[FOM] Consistency and Completeness in Natural Language

[Dean Buckner]

Concerning the paper by Tennant referenced in current FOM postings
("Deflationism and the Gödel Phenomena",Mind 443, pp.551-582), there is a
difficulty running all through it that worries me.  It is the same
difficulty as with Tarksi's own formulation of the semantic theory, where he
says

 (T)  S is true iff p

and then says that "p", in the formula, "stands for" a sentence [1].  But he
surely means that it *is* a sentence - a one-word sentence like the Latin
"ambulo".  For suppose that "Fred" stood for the sentence. Then Tarksi
would have had to write

 (T)  S is true iff Fred

which is nonsense.  I believe Tennant has made a similar mistake, for
example when he writes

    If _psi is a primitive recursive sentence and _psi is provable-in-S then
    psi

and then says "psi is a sentence of L", i.e. "psi" designates a certain
sentence.  But the formula only makes sense if "psi" rather than psi is the
sentence - a one-word sentence, like Tarski's "p".

This is not just extreme pedantry, even if it seems like that. We are
talking about the difference between "asserting that some sentence phi is
true", where "phi" is a variable standing in for the name of some sentence,
and "asserting that it is true that phi", where "phi" is not a variable at
all, _but is itself a sentence_.

We can't use the name of a sentence simpliciter within another sentence to
assert what the named sentence asserts.   We need an operator of some sort.
If "Fred" designates the sentence "grass is red", what Fred says is true iff
what "grass is red" says is true, i.e. what Fred says is true iff grass is
red.
But we need the operator, "what -- says is true", where the blank is filled
by the name ("Fred") of a sentence.  So Tennant's argument that we can get
by with a "thin" notion of truth doesn't really get off the ground.

This is connected with a point I've made a couple of times on FOM before
(but without discernible effect) namely that the truth operator and
assertoric force are one and the same thing and are embedded in the main
verb of any sentence.  They are what makes the main verb a verb.  Add the
"that" operator to the sentence "grass is red", and you remove this
assertoric part.  You get something "that grass is red" that kind of names
what the sentence expresses, but without actually expressing it, so you can
utter "Jake thinks that grass is red" without saying anything untrue.  Add
the operator "it is true", to get " it is true that grass is red", and you
put the verb back again, giving something like the original "grass is red".

We need to give up once and for all the idea that variables can stand in for
the semantic value of a sentence (though perhaps they can stand in for what
sentences express).  This is the "ineffability of semantics".  Since the
semantic value of a sentence includes the content added by the verb itself,
the "assertoric" force of the sentence, this alone we cannot name.  It is
there, but unnameable, because a name cannot (for obvious reasons) stand in
for a verb.

I don't know how this affects Neil's point about Goedel.  I suspect it does,
for reasons connected with Hartley's point.  But enough for now.

Dean




[1] Tarski, A., "The Semantic Conception of Truth and the Foundation of
Semantics", Philosophy and Phemenological Research, 4 (1944) pp. 341-375,
section 9.  To be sure, in an earlier section (4) he says that 'p' can be
*replaced* by any sentence of the language.  Yet in section 7 he argues that
we can "replace" a sentence with the letter 's', then writes

    's' is identical with the sentence printed in this paper on p. 52 ...

as though the sentence were just the letter 's'.

Dean Buckner
London
ENGLAND

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