This text has been transcribed by Max Cresswell. Folder 7.
UNIVERSITY OF CALIFORNIA, LOS ANGELES
DEPARTMENT OF PHILOSOPHY
LOS ANGELES
,
CALIFORNIA
90024
November 22, I968
Dear Professor Prior
Thank
you for your letter and especially the information about Fine
and Bull. I had earlier thought
about
some of the matters
in my letter to you (in
particular S5 with quantifiers), and reading your article
stimulated
me to prepare
the enclosed abstracts, which I submitted for the
January A.S.L. meetings. But
now
it appears that these results are
not new. I hope that Fine and Bull will send me copies of
their papers
(you
did not send me their addresses or I would write directly to them). [Marginal note reads: Both have
written since this letter sent to the typist (Ed)]
I
wonder if Fine has axiom sets with strong separation p
roperty
I describe in S5 with Multiple
Possibility.
I formulated my axiom in a slightly more complex way than
necessary in
the (vain) hope of
achieving strong separability.
The
theory known to be decidable is what Church calls singulary 2nd
order
logic with identity,
not
necessarily unifor
reading my letter to you, I see that I expressed
theory is decidable, hence the
which
are expressible in the
sublanguage is also decidable. It is clear from the proof
of decidability for the
full
theory that
for quantification
theory
(Tautologies,
Distribution, Vacuous Quantification, Specification) for both individual a
nd m
onadic
predicate
variables, plus the usual identity axioms (
x
x
=
x
,
Leibniz law). Since identity is definable with
predicate
variables (
"
$
for
F
(
F
"
6
F
$
)),
the identity
formulas and axioms are superfluous. Thus since
F
[
Fx
v
G
(
Gx
6
x
(
Fx
6
Gx
))] is valid, it must
be provable just from the usual quantification axioms.
And
so it is. The
F
in question is, intuitively, singleton
x
({
x
}), so take for the formula
F
"
,
H
(
Hx
6
Hx
).
Obviously,
/
H
(
Hx
6
Hx
) and
/
G
(
Gx
6
y
(
H
(
Hx
6
Hy
)
Gy
)), thus
/
F
[
Fx
v
G
(
Gx
6
y
(
Fy
6
Gy
))], hence
/
F
[
Fx
v
G
(
Gx
6
x
(
Fx
6
Gx
responding axiom
in S5
ntifiers was independent.
-2-
It
seems to
validity, so possibly I have still not answered your question about the subtheory with the axiom (A)
(A)
F
[
Fx
v
G
(
Gx
6
y
(
Fx
6
Gx
)].
Are
you thinking of singulary
s
econd order logic sans identity and with only one individual
variable
(say
) ? The formulas of this last theory (call it
V) are completely isomorphic
to those of S5
quant
fact the converse of the 2 of S5 with Quantifiable Propositional Variables will
translate
any formula (in the language of V into
a formula in the language of S5Q.
Thus my completeness
proof
for S5Q tells us that we can axiomatize V with the usual axioms pl
proof for my axiom (8) tells us that (A) is not derivable from the usual axioms so
long as the
derivation
stays within the language of V.
If you still have reprints of Egocentric Logic, I would like one, and other reprints also.
Should
a vi
siting posi
or at a nearby institution for a logician become available I will
certainly mention Fine. (or is a student position e.g. teaching assistant appropriate?)
Sincerely,
David Kaplan
2
DK:bs
P.S. there has been a 3 week delay in typing this letter for which I apologize.
following formulae appear to have been written on the flap
of the envelope. Its not
obviously Kaplans, and could be Priors since it is partly, though not entirely, in Polish notation.]
(
*
)(
(
) (nec.{
*
. {
(
. ~nec.}
(
(
p
)(
q
) (nec.
Np
.
Nq
. ~nec.
Nq
)
(
*
)(
(
) (nec.
F
*
F
(
F
(
A
pN
*
p
N
(
p
N
A
p
(
p
[Transcribed
by M.J.
Cresswell, from a version obtained by OCR from Kaplans typewritten letter.
The
Virtual Lab does not list any letters from Prior to Kaplan.]