[FOM] Re: Consistency and completeness in natural language

[Dean Buckner]

I agree with Hartley (at least, if he is saying what I think he is saying).
Some possibly related points.

Nominalism in the broadest and most characteristic sense, is the doctrine
that certain things we can express but not "nominalise" i.e. not construct a
noun phrase that allows us to talk about what we have expressed.  This has
obvious resemblances to the Tractarian doctrine, about what can be shown but
not said.  Failure to recognise this led to Russell's problem about the
"relating relation" that connects the terms of a proposition into a
judgment.  He thought a relating relation is the same entity as its
nominalization, "that grass is green" has the same meaning as "grass is
green", and was puzzled that it hasn't. See Bertrand Russell, Principles of
Mathematics (New York: Norton, 1937), § 38, § 52, § 54, § 478).  For an
interesting discussion of this problem in connexion with Jean Buridan's
theory of the proposition see "John Buridan and the Force-Content
Distinction" on Gyula Klima's website.

For an interesting discussion of the connection between the Tractatus and
Skolem's paradox, see Adrian Moore's paper "Set Theory, Skolem's Paradox and
the Tractatus", Analysis 1985, 45.  Adrian has also written an entertaining
and accessible book on the Infinite called, remarkably "The Infinite"
(Routledge, 1990) which I also recommend.  " There is no particular point of
view in the world which can be spoken of as here: our point of view is a
limit of the world.  That is, there is no particular set in the hierarchy of
sets which can be spoken of as the intended range of the quantifiers: they
are intended to range over the whole hierarchy (though not even this can
properly be said)".

On proving that something is true.  Peter Geach has argued that "when I use
the term "proposition" .  I mean a form of words in which something is
propounded, put forward for consideration; it is surely clear that what is
then put forward neither is ipso facto asserted nor gets altered in content
by being asserted".  We could also say, by "proof", we mean a form of words
in which something is propounded in such a way as it is shown to be true.
It is clear that what is proved is not "ipso facto" proved, nor gets altered
in content by being proved.  The proof itself is not part of what is shown
to be true in the proof.

We can show that something is true, but cannot put in words the showing
itself.  Similarly for "is provable".  That something is provable, as with
something being "sayable", is not part of what is to be proved.   Are there
some true things that we cannot prove to be true?  Well yes, if there are
some true things that are so complicated they cannot be stated in a way that
is any less complex than they are themselves.  For example the London
telephone directory.

Compare "It is obvious that p".  This is best shown by saying no more than p
itself.  The art of proving something is often no more than working out what
needs to be proved or shown, and just saying it.  Some people take ages to
explain some obvious fact because unable to restrict themselves to that
fact.  The art of proof, is of stating exactly what is to be proved.

I presume everyone here already knows the joke about the professor who says
"that is obvious . or is it?" and goes away and comes back 20 minutes later
saying "yes, it's obvious".