[FOM] Unrestricted Quantification and Paradox
laureano luna
laureanoluna at yahoo.es
Sat Jun 2 07:59:57 EDT 2007
Im ‘Sets, Properties, and Unrestricted
Quantification’, to be found at
http://seis.bris.ac.uk/~plxol/UQ.pdf, Linnebo proposes
an argument against unrestricted quantification that
some others like Glanzberg endorse and that I sketch
below. This argument has been revisited in the recent
collective work ‘Absolute Generality’, Rayo and
Uzquiano eds. OUP, whose introduction can be found at
http://web.mit.edu/arayo/www/Introduction.pdf
I think the editors have arrived in this topic from
the study of Orayen’s paradox: for all I know they are
both Argentinians, as Orayen was.
Now the argument.
Let’s assume we can quantify over everything. Let L be
a first order language, P a predicative letter of L, M
a metalinguistic predicate; then we can interpret P as
M.
We can define M in the following way:
1. Ax Mx <-> AI (x=I -> !I /- Px)
where ‘AI’ ranges over all interpretations and ‘I /-
phi’ means that I satisfies phi; i.e. x is M iff x is
no interpretation satisfying ‘Px’.
Let I+ be an interpretation that interprets P as M;
so:
2. Ax (I+ /- Px <-> AI (x=I -> !I /- Px))
Since we can quantify over everything, ‘Ax’ can
quantify with no restriction in 3. and in particular
it can quantify over all interpretations. Then it
easily follows that
3. I+ /- PI+ <-> !I+ /- PI+
which is absurd.
Therefore ‘Ax’ cannot quantify with no restriction in
3. and it is not possible to quantify over everything.
As I see it, the argument shows that while we define
an interpretation INT by quantifying over
interpretations, we cannot be quantifying over INT
itself.
It is easy to see that the argument relies on
Thomson’s 1962 theorem:
!Ex Ay Rxy <-> !Ryy
The same theorem shows that if we are defining the set
of all non-self-membered sets, our ‘all’ cannot
encompass the very set we are defining; and that when
we produce a proposition about all propositions not
about themselves, our ‘all’ cannot refer to our
current proposition, etc.
In my opinion, the moral of the whole story is that we
cannot refer by means of an act to the very act we are
accomplishing or to the results of it. In
phenomenological terms: no intentional act is about
itself. I suspect this approaches the very root of
paradox.
Best regards,
Laureano Luna
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