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FOUNDATIONS:
LOGIC,
LANGUAGE,
AND MATHEMATICS
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FOUNDATIONS:
LOGIC LANGUAGE
AND
MATHEMATICS
Edited by
HUGUES LEBLANC, ELLIOTT MENDELSON,
and
ALEX ORENSTEIN
Reprinted from
Synthese,
Vol. 60 Nos. 1 and 2
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
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TSBN 978-90-481-8406-4 ISBN 978-94-017-1592-8 (eBook)
DOI 10
.1007/978-94-017-1592-8
AlI Rights Reserved
Copyright © 1984 by Springer Science+Business Media Dordrecht
Originally published by D. Reidel Publishing Company. Dordrecht. Holland in 1984
Softcover reprint
of
the hardcover 1st edition 1984
No
part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
inc1uding photocopying, record ing
or
by any information storage and
retrieval system, without written permission from the copyright owner
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TABLE
OF CONTENTS
Series Editors' Preface 1
Preface 3
MELVIN
FITTING I
A Symmetric Approach to Axiomatizing Quan-
tifiers and Modalities 5
NICOLAS
D .
GOODMAN
I
The Knowing Mathematician 21
RAYMOND
D.
GUMB
I "Conservative" Kripke Closures 39
HENRY HIZ
I
Frege, Lesniewski, and Information Semantics on the
Resolution
of Antinomies
51
RICHARD
JEFFREY
I
De Finetti's Probabilism 73
HUGUES L E BLANC
and CHARLES
G . MORGAN I
Probability
Functions and Their Assumption Sets
-The
Binary Case 91
GILBERT HARMAN I Logic and Reasoning 107
JOHN MYHILL I Paradoxes 129
ALEX
ORENSTEIN I Referential and Nonreferential Substitutional
Quantifiers 145
WILFRIED SIEG
1
Foundations for Analysis and Proof Theory 159
RAYMOND
M.
SMULLY
AN
I
Chameleonic Languages 201
RAPHAEL STERN
I
Relational Model Systems: The Craft
of
Logic 225
J. DILLER and
A. S. TROELSTRA I
Realizability and lntuitionistic
Logic 253
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CITY
COLLEGE
STUDIES
I
N TH
E H
IST
OR
Y AN
D PH
ILO
SOP
HY
O
F SC
IEN
CE
AN
D T
ECH
NOL
OG
Y:
SER
IES
E
DIT
ORS
'
P
REF
AC
E
R
ecen t
year
s hav
e see
n the
em e
rgenc
e of
severa
l new
app
roach
es to
the history and philosophy of science and technology. For one, w hat
w
ere p
erceiv
ed b
y man
y as
separ
ate, t
houg
h per
haps
relate
d, fie
lds o
f
i
nquiry
h av
e com
e to b
e reg
arded
by m
ore a
nd m
ore sc
holar
s as a
singl
e
d
iscipl
ine w
ith d
iffer
ent a
reas
of em
phas
is. I
n thi
s dis
ciplin
e any
p
rofou
nd un
ders
tandin
g so
deep
ly int
ertwi
nes h
istory
and
philo
soph
y
t
hat i
t mig
ht
b
e
said
, to
parap
hras
e K an
t, th
at ph
iloso
phy w
itho
ut
his
tory
i
s
em p
ty an
d his
tory w
ithou
t phi
losop
hy
is
blind
.
Ano t
her co
ntem
porar
y tre
nd in
the h
istory
and p
hilos
ophy
of sc
ience
and
tech
nolog
y ha
s bee
n to
bring
toge
ther
the E
nglis
h-spe
aking
and
continental traditions in philosophy. The views of those who do analytic
philos
ophy
and
the v
iews
of t
he he
rmen
eutici
sts h
ave c
om b
ined
to
in f
luenc
e the
thin
king
of so
m e p
hiloso
pher
s in t
he E
nglish
-spea
king
wo
rld,
and o
ver
the la
st de
cade
that
influ
ence
has b
een
felt i
n the
histo
ry an
d ph
iloso
phy o
f sci
ence.
T
here
h
as als
o be
en th
e lon
g
s
tandin
g in
fluenc
e in
the
West
of c
ontin
ental
think
ers w
orki
ng on
prob
lems
in
the p
hilos
ophy
of t
echno
logy.
Thi
s syn
thesis
of
two
trad
itions
has m
ade
for a
riche
r fund
of id
eas a
nd ap
proa
ches
that m
ay
change our conception of science and technology.
S
till an
other
tren
d tha t
is
in
some
ways
a com
bina
tion o
f the
previ
ous
tw o
cons
ists o
f the
work
of tho
se ch
aract
erize
d by s
ome
as the
"Fri
ends
of D
isco
very"
and
by ot
hers
as the
brin
gers
of the
New
Fu
zzine
ss".
This a
ppro
ach t
o the
histor
y and
phil
osoph
y of s
cienc
e and
tech
nolog
y
conc
entra
tes o
n
cha
nge,
prog
ress,
and discov
ery.
It
ha
s rai
sed o
ld
episte
molo
gical
quest
ions u
nder
the g
uise
of the
prob
lem o
f rati
onali
ty
in the
scie
nces.
Altho
ugh
this a
ppro
ach h
as its
origi
ns in
the w
ork
of
Th
omas
Kuh
n in
the U
nite
d Sta
tes, a
t temp
ts to
expr
ess h
is ide
as in
explicit set-theore tical or m odel-theoretic terms are now centered
in
Germ
any.
Synth
ese
60 (1984
) 1
.
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2
S E R I
ES
E DI T OR S PR E
FA C E
The m ore
tradition
al approac
hes to the
history an
d philosop
hy of
science
and techn
ology cont
inue as we
ll, and prob
ab ly will c
ontinue as
long as
there are
skillful p
ractitioners
such as
Carl Hem p
el, Ernest
Nagel
, and t h ~ i
students.
Final
ly, there a
re still o t
her approa
ches that
address so
me of the
technic
al problem
s arising w
hen we try
to provid
e an accou
nt of belief
a
nd of rat
ional choi
ce . - T he
se include
efforts to
provide
logical
framework
s within wh
ich we ca
n make sen
se of these
notions.
This ser
ies
will
at
tempt to b
ring toge
ther work
from all o
f these
app
roaches to
the history
and philo
sophy of sc
ience and
technology
in
the belief that each has something to add to our understanding.
The
vo
lumes of t
his series h
ave emerg
ed either f
rom lectur
es given
by autho
rs while th
ey served
as honorar
y visiting p
rofessors a
t the City
C
olleg e of N
ew York o
r from c on
ferences sp
onsored by
th at instit
ution.
The
City C
ollege Pro
gram in th
e History
and Philos
ophy of Sc
ience
an
d Technolo
gy overse
es and dir
ects these
lectures a
nd confere
nces
wi
th the fina
ncial aid o
f the Ass
ociation fo
r Philosop
hy of Scie
nce,
P
sychothera
phy, and E
thics.
MA
RTIN TAMNY
RA
PHAEL STERN
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PRE
FACE
The
pape
rs in this c
ollection s
tem largely
from the
conferenc
e 'Foun
dat
ions: Log
ic, Langua
ge, and M
athematic
s ' held a t
the Grad
uate
C en
ter of the
City Unive
rsity of Ne
w York on
14
-15 N o
vember 1980
.
T h e
conferen
ce was sp
onsored by
the Phil
osophy Pr
ogram a t
the
G rad
ua te Cen
ter of th
e City U
niversity o
f New Y
ork .and
the
Associatio
n for Philo
sophy of S
cience, Psy
chotherap
y, and Eth
ics. We
w
ish to exp
ress our g
ratitude an
d appreci
ation to th
ese organi
zations
a
nd to than
k the serie
s editors, R
aphael S t
ern and M
artin T am n
y, for
their help with the conference and the preparation of this collection .
Synthese
60 (1 984) 3.
HUGUE
S
LEBL
ANC
ELLI
OTT ME
NDELSON
ALEX
ORENS
TEIN
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MEL YIN FITTING
A SYMMETRIC APPROACH TO AXIOMATIZING
QUANTIFIERS
AND
MODALITIES
1 . I N T R O D U C T I O N
We present an axiomatization of several of the basic modal logics, with
the idea of giving the two modal operators 0 and 0 equal weight as far
as possible. Then we present a parallel axiomatization of classical
quantification theory, working
our
way up through a sequence of rather
curious subsystems. It will be clear at the end that the essential
difference between quantifiers and modalities
is
amusing in a vacuous
sort of way. Finally
we sketch tableau proof systems for the various
logics we have introduced along the way. Also, the "natural" model
theory for the subsystems of quantification theory that come up
is
somewhat curious. In a sense, it amounts to a "stretching
out"
of the
Henkin-style completeness proof, severing the maximal consistent part
of the construction quite thoroughly from the part of the construction
that takes care of existential-quantifier instances.
2. B A C K G R O U N D
It is contrary to the spirit of what
we
are doing to take one modal
operator as primitive and define the other, or, one quantifier as
primitive and define the other. So, by the same token, we take as
primitive all the standard propositional connectives too. Thus,
we
have
available all of
A,
v,
~ , =>, 0 , 0,
'V,
3.
We also take as primitive a truth
constant
T and a falsehood constant
l..
For our treatment of propositional modal logic we assume
we
have a
countable list of atomic formulas, and that the set of formulas
is
built up
from them in the usual way. We will use the letters "X",
"Y''
, etc., to
denote such formulas.
For quantification theory, we assume we have a countable list of
variables and also a disjoint countable list of parameters. We will use
the letters
"x",
"y",
etc., to denote variables, and
"a",
"b",
etc., to
denote parameters. Formulas are built up in the usual way, with the
understanding that a sentence contains no free variables, though it may
Synthese
60
(1984)
5-
.19. 0039-7857/84/0601-0005 $01.50
© 1984
by D . Reidel Publishing Company
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6
M E
L V I
N F
IT T
IN
G
con
tain
pa
ram
ete
rs. W
e
fo ll
ow
th e
con
ven
tion
th
at if
cp(x
) is
a
form
ula
wit
h on
ly x
fr
ee,
then
cp(
a)
is
th
e re
sult
of
repl
acin
g a
ll fr
ee o
ccu
rre
nce
s
of
x
by occurences of
a
in
cp
W
e w
ill
use
Kr
ipk
e's
mod
el
theo
ry
for
m o
dal
log
ics
(and
an
an
alo g
for
qu
anti
fica
tion
al
theo
rie
s, t
o b
e de
scr
ibed
in
Se
ctio
n 4
). F
o r
us,
a
Kri
pke
m o
del
is
a
qua
drup
le ('
fi,
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A
X IO M A T I
Z I NG
Q U A N
T IFIERS A N D
M O D A
LITIES
7
Notice tha
t (b)-(e) in
the defin
ition of a K
ripke m o
del above
gives us
flf-
a
iff
f
If-
a
1
and flf-
a2
flf-
f
iff
flf- {31 or
f l f {32.
Inde
ed, this co
uld be use
d instead o
f (b)-(e).
W e
extend thi
s uniform
notation to
the modal
operators
(as in Fitti
ng
[1]) by
defining
the v-formu
las (neces
saries) and
1r-formula
s (possible
s)
a
nd their c
omponents
v
0
and 7To,
respectiv
ely, as foll
ows:
v
ox
~
x
v
X
~ x
7T
7To
X
~ x
N ot
ice th at (f
)-(h) in
the definitio
n of a K ri
pke m ode
l above, ta
ken
togethe
r w ith (d),
give us th
e following
equivalen
t conditio
ns:
fo
r every f E
(f')
f
If-
1r but
f
I. L
v;
for every
f
E
§ -
(x)
~
cf>(a)
The
idea is, in
a classical
first-order
model in w
hich the d
omain con
sists
of th
e set of pa
rame ters,
an
d
each
param eter
is inte rpre
ted as nam
ing
itself:
Y is true i
ff for all a ,
Y(a)
is
tr
ue;
o
is true iff
for som e a , o
(a)
is
t
rue.
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8
M E L V I N FI T T I NG
So much for background.
3 . M O D A L L O G I C S , A X I O M A T I C A L L Y
Let
us
assume that we have an axiomatization of the classical pro
positional calculus, with modus ponens as the only rule. We will build
on that in our introduction of rules and axioms pertinent to tfle modal
operators.
Modal logics are often formulated with a rule of necessitation, which
we could give
as
vo
v
But this gives a central role to the v-formulas and, for no reason other
than autocratic whim, we want to develop modal logic as far as possible
giving equal weight to both v-and 1r-formulas, giving to both neces
saries and possibles a fair share. After a certain amount of experimen
tation we hit on the following curious rule:
(M)
1To v Vo
1 T V V
(we use M for modalization). As a matter of fact, rule M is a correct rule
of inference, as the following argument shows.
Suppose
(W,
, A,
If-)
is a Kripke model, and 1r
0
v
v
0
is valid in it
(holds at every possible world). Let
r E
W; we show
flf- 1r
v
v.
Well, if
r E
, that is, if
r
is queer, then
r If-
7T; so, trivially,
r If-
1T v
v.
Otherwise,
f
E
Y
DX::::>DY
X::::>Y
OX::::>OY
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AXI OMAT I Z I NG QUANT I FI E R S
AND
M ODAL I T I E S 9
Then one may show,
in
the usual way, that replacement of proved
equivalences holds as a derived rule:
X = X'
Z = Z '
where Z' results from Z by replacing some occurrences of X in Z by
occurrences of
X' .
Here we have used = as an abbreviation for mutual
implication. (Actually a stronger version may be shown, concerning
"semisubstitutivity" of implication, in which we must take into account
the positiveness and negativeness of occurrences as well.
The
details
needn't concern us here.)
Now go back and look again at the justification we gave for rule
M.
In the argument that 7T v
v
held at the nonqueer world r we never
needed that 7To v v
0
held at every world of the model; we only needed
that it held at every world accessible from f. But to say that
7To
v v
0
holds at every world accessible from r is to say that D(
7To
v v
0
)
holds at
f. Thus, the same argument also shows the validity in all Kripke models
of the schema
(Ml)
D[
7To
v
vo]::) [ 7T
v
v].
Let
us
add
it
as an axiom schema then.
I f we do so, it is not hard to show that we have a complete axiomatic
counterpart of the Kripke model theory as given in section 2.
That is, X
is
provable in the axiomatic system just described iff
X
holds at every
world of every Kripke model.
To
show this, one may show complete
ness directly, using the now-common Lindenbaum-style construction,
or one may show this axiom system is of equal strength with one
standard in the literature, and rely on known completeness results. We
skip the arguments.
The logic axiomatically characterized thus far is called C in Seger
berg [5]. It is the smallest
regular
modal logic (see Segerberg [5]).
Now, in the model theory for C, one can have queer worlds in which
everything is possible but nothing is necessary. Also there are no special
conditions placed on the accessibility relation
rJl,
so there can be
"dead-end" worlds, worlds from which no world is accessible. In such a
world, everything is necessary, nothing
is
possible. So, our next item of
business is
to rule out such strange worlds.
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10
MEL YIN FITTING
Worlds in which nothing is necessary may be eliminated by postulat
ing that something is necessary. Let us, then, add the axiom
(M2) OT.
The model theory appropriate to this (with respect to which one can
prove completeness) is all Kripke models in which there are no queer
worlds, that is , all normal Kripke models. The logic axiomatized thus
far is the smallest normal logic, and is usually called K (see Segerberg
[5]).
Next, worlds in which nothing is possible may be eliminated
by
postulating that something
is
possible. We take as an axiom
(M3) T.
The model theory appropriate to this is all normal Kripke models in
which every world has some world accessible to it.
The
logic
is
generally called D. (Again, see Segerberg [5]).
Finally, we might want to restrict our attention to models in which
each (normal) world
is accessible to itself (in which the accessibility
relation is reflexive). To do this one adds either (or both) of
(M4)
v ~
vo
7To
~ 7T
The logic thus characterized
is
T.
Note that T T is an instance of the second of these schemas, and
since
T is
a tautology, T follows by modus ponens. Thus with M4
added, M3 becomes redundant.
REMARKS.
One
goes from C to K by adding
OT
as an axiom. It
is
of
some interest to consider a kind of halfway point, where instead of
adding
OT as an
assumed truth,
we take it as a
hypothesis.
That is, form
the set
S
of formulas
X
such that OT ~ X
is
provable
in
C. This set
S
is,
itself, a (rather strangely defined) modal logic, intermediate between C
and K. I t is closed under modus ponens, but not under rule M. Rather, it
is closed under the weaker rule.
0( 7To
v
vo)
0(
7TV
v)
'
which
is
equivalent to Becker's rule. It
is,
in fact, the logic axiomatized
in Lemmon [4]
as
P2, but without his axiom
O X ~ X
(our M4).
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AXI OMAT I Z I NG QU ANT I FI ER S
AND M ODAL I T I E S 11
If
we
strengthen things a bit, by considering those X such that
DT
:::>X
is provable in C plus axiom M4, one gets the Lewis system S2 .
(Again, see Segerberg [5], chapter four.)
One can also play similar games with axiom M3 (0 T) to produce
interesting logics. We know very little about them.
4. QUANT I FI E D L OGI C S , AXI OM AT I C AL L Y
There is an obvious analogy (of sorts) between the modal operators and
quantifiers. What we do in this section is to parallel the development of
section 3, substituting quantifiers for the modal operators to see how far
the analogy extends when things are done the way we did. The idea
is
simple: ('v'x) may behave likeD, (3x) like 0, y like
v
and o ike ?T. This
may be so;
we
will see.
The language now is first order. Once again we assume a pro
positional-logic base with modus ponens as the sole rule.
First, the analog of rule M
is
rule (Q)
o(a)
v y(a)
OV )
And, as a matter of fact, this is a correct rule of inference in classical
first-order logic. This argument is left to the reader.
Using rule
Q,
one may show analogs of the results listed in section 3
based on rule
M.
Thus, one may derive the usual inter definability of the
quantifiers (again, as mutual implication), and one may show that
replacement of proved equivalences holds as a derived rule.
Now, if you actually thought through the "justification" of rule Q,
almost certainly you also showed the validity, in all first-order models,
of the schema
(Ql) ('v'x)[o(x) v y(x)] :::> [o v y].
So let us add it as an axiom schema. We might, by analogy, call the
resulting logic QC. I t thus has rule
Q
and schema Ql.
The following
is
a reasonable question: What
is
an adequate model
theory for QC,
one with respect to which completeness can-be shown?
Well, the following rather curious one
will
do. We simply translate the
corresponding modal model theory, making suitable adjustmer:ts to
take care of the fact that quantified sentences have many instances, but
modalized formulas have single components. We have chosen, for
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12
MEL YIN FI T T I NG
simplicity, to leave out any mention of the notion of an
interpretation
in
a model. A more elaborate treatment would have to include it, but the
following is enough for our purposes.
A model
is
a quintuple
(Cfi, « >, < P,
< fl,
If-)
where: (1) CfJ is a nonempty
set (of possible worlds); (2) « ~ Cfi; (3) < Pis a mapping from members of
CfJ to nonempty sets of parameters; (4) < fl
is
a relation on Cfi; and (5) If-
is
a relation between members of CfJ and sentences such that
for every f E
Cfi,
conditions (a)-( e) as in section 1;
for every f E
« >,
(f) flf-(3x)cp(x) but fi,V-(Vx)cp(x);
for every f E Cfi- « >
(g) flf-(Vx)cp(x) iff for every dE CfJ such that f< /ld, and for
every a E < } (d), d If- cp(a)
(h) f If-: (3x)cp(x) iff for some dE CfJ such that f< fld, and for some
a
E
< P(d),
fl.
If-
cp(a).
Now, a sentence
X is
a theorem of the logic
QC
iff
X
holds at every
world of every such model. We leave the correctness half to the reader.
Note that
if
cp(a) is provable in QC;so
is
the parameter variant cp(b).
This
will
be of use in proving correctness. And we briefly sketch the
completeness half in the next section.
We note that we could restrict models so that, for each world f,
< P(r)
is a singleton. I t makes no difference. Now
we
can continue with
our
development, paralleling that of section 3.
In the quantifier models above there can be
"queer"
worlds (mem
bers of cp) in which everything exists, and hence no universal sentences
hold. Such anomalies can be eliminated by adding the postulate
(02) (Vx)T.
Doing so gives us a logic we may call QK. An adequate model theory
for it
is
one that consists of all models of the sort described above, but
with« always empty, that
is
no "queer" worlds.
Next, there may still be worlds from which no world is accessible. In
such a world, every universal sentence holds, but no existential. They
behave rather like empty-domain models of first-order logic. They may
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A X
IO M ATIZI
N G Q U
NTIFIE
R S
AN
D M O D A L
ITIES
1
3
be ru led out
by add
ing the ax
iom
(03)
(3x)T.
Let
us call th
e resulting
logic QD.
Its m od e
l theory
is
tha
t of QK
with
the addi t
ional requ
irement that
for ever
y w orld r
there m us
t
be
so m e
w
orld L
l
su c
h tha t
f ~ L l .
Finally (?)
we may
add an ana l
og to M 4 ,
namely
(Q 4
) y => y(a)
or
«5(a) > «
5
(
or both).
A s m ight
be
ex pec ted,
Q3
then b ecom e s redun dan t .
We can
call the
logic thus
axiom atize
d
QT.
Its
m o d el the
ory is that
of Q
D
w
ith the
restr ict ion
tha t the a
ccessibilit
y rela t ion ~
b e reflex
ive.
We
have reac
hed the
end
of
o
ur
parallel
developm
ent (obviou
sly,
since we
have matc
hed every
th ing we d
id in sect io
n 3).
But
we do n
ot
yet
have the u
sual first-
order logic
.
O
ne
d
oesn't
w ant
classical fi
rst
order m
odels with
lots of po
ssible w o r l d ~
in them
: a classic
al model
should be
a one-w
orld m odel
.
N ow
, to only co
nside r one
-wor ld Kr i
p k e m odal
m o d els is
to trivialize
m odal l
ogic; it re
nders the
m o d al ope
ra to rs usel
ess. N eces
sary tru th
b
e c o m es th
e sam e th
ing as tru
th.
T
hat i
s, A => DA
is valid
in all
o n
e -world m
odels. Of c
ourse , this
is no t desi
rable. M od
al operato
rs are
suppo
sed to do
s o m ething
; they are
supposed t
o have an
effect; the
y
ought n
ot be
vacuou
s .
W ell, it i
s precisely
at this poi
n t tha t m o d
al operat
ors
and
qua
ntifiers
diverge.
Quan tifiers
can
be va
cuous.
L
et cp be
a se
ntence (he
nce with
no free va
riables). Then
( Vx)cp o
ught to
mean no t
h ing
mor
e
than cp
itself. So
our
final quantificational axiom
schema
is, simply,
(Q 5)
cp =>
( Vx)c
p,
w here
cp is a sente
nce.
When thi
s is ad d ed ,
convent io
nal c lassica
l first-orde
r logic is
the
result.
5.
C
O M PL E T E
N ESS, HO
W PR O V E
D
In
a sense, t
h e p ro o f o
f the co m p
leteness of
the quant
ifier system
QC
w ith r
espec t to t
h e m odel
theory pr
esented in th
e p revious
sect ion is
a
stret
ching out
of th e usu
al
H
enkin
co m p leten
ess p ro o f f
or first-ord
er
logic
.
Let
us
sketch w hat
we mean
b
y this.
R ecall tha
t th e u
sual Henkin
argume
nt runs as
follows.
Take a
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AXI OM AT I Z I NG QUANT I FI E R S A N D M ODAL I T I E S 15
model to
F,
since ~ F E r
F ·
Notice that in the proof just sketched, the maximal consistent
construction gives the worlds, while the existential instantiation moves
things from one world to another. This
is
what we meant by "stretching
out" the Henkin construction.
Adding axioms (02)-(04) modifies the construction in obvious ways.
We leave this to the reader.
Now we can ask, What
is
the effect on this construction of also
imposing that final axiom schema (05), cp:::;)
(Vx)cp?
Very simply, it
makes things cumulative. Notice that, in our model, if fq[A, then
moving from
r
to
A,
in effect, drops one quantifier from each sentence.
But cp:::;) (Vx)cp allows us to add one quantifier,
so-the
effect
is
neutralized. Briefly, if we assume (05), then if fll- cp and
fq[A,
then
A II- cp.
For, if fll-
cp,
since also fll- cp:::;)
(Vx)cp,
we must have fii-(Vx)cp.
And since fq[A, then
A II-
some-instance-of-cp. But since the quantifier
was vacuous, this m e a n s ~
II-
cp. Now, that things are cumulative if
(05)
is
imposed means the limit (=chain-union) part of Henkin's proof can
be carried out. And thus a conventional classical model results.
This sketch must suffice. Details, though slightly devious, are far
from devastating.
1. S E M ANT I C T AB L E AUX
We show how the tableau system of Smullyan [7] for propositional logic
may be extended to handle the logics discussed in section 4. We begin
with a brief sketch of the system that suffices for propositional logic.
First, proofs are in tree form (written branching downward). There
are two branch-extension rules:
a
(If
a
occurs on a branch,
a
1
and
a
2
may be added to the end of the
branch.
I f
f3 occurs on a branch, the end of the branch may be split, and
{3
1
added to the end of one fork, {3
2
to the end of the other.)
A branch
is
called closed
if
it contains A and ~ A for some formula
A, or if it contains _1_, or if it o n t a i n s ~
T.
A tree is called closed if every
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16
M EL
VIN FIT TING
branch is
closed. A
closed tre
e with ~
X at the ori
gin
is
, by d
efinition, a
proof
of X.
W e begin
by addin
g to the ab
ove a tab
leau rule to
give the
modal
l
ogic C . In
words, th
e rule is as
follows. If,
on a b
ranch, ther
e are
v-for
mulas, an
d there is
a 7T-for
mula, then t
hat 7T-f
ormula may
be
repla
ced by 1r
0
,
all the v-f
ormulas by
the corre
sponding
v
0
-formula
s,
and a
ll oth er fo
rmulas del
eted.
W
e schemat
ize this as
follows. Fi
rst, if S is a
set of form
ulas, defin
e
S#
=
{
Vo
J v
E S}.
Then
the rule
is
s,
7T
S#,
7To
(provided S#-
4>),
where this
is to be int
erpreted as
follows: if
S U {
1
r} is
the set of f
ormulas
on a bran
ch, it may
be replac
ed
by S
#
U {
7
To} (pro
vided
S
# is
not
empty).
EXAMPLE.
W e show
OX
::
J
~ o
~ X
is
provable using this rule.
The p
roof begin
s
by
puttin
g
~
O X ::J
~ 0 ~ X
) at the ori
gin, then
two a
-rule applicati
ons produ
ce the follo
wing one-
branch tre
e:
~
O X
::J
~ O
~ X)
ox
~ ~
o ~
x
o ~ x .
Now take
S
to consist of the first three formulas, and
7T
to be 0
~ X .
The
n
S#
= {X}
, which is
n
ot em pty,
so the rule
says the se
t of formul
as
on th
e branch m
ay be rep
laced by S#
U {1r
0
}, n
amely
X
~ x
and this is
closed
RE
MARK. Beca
use of the
way trees
are written
, an occu
rrence of a
for
mula may
be com m
on to seve
ral branch
es, b u t
we may wis
h to
m
odify (or d
elete) it u
sing the ab
ove rule o
n only o n
e branch.
T h e n ,
s
imply, first
add new
occurrence
s of the fo
rmula at th
e ends of
all the
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AXI OM AT I Z I NG
QUANT I FI E R S
AND M ODAL I T I E S
17
branches that are not to be modified, then use the above rule on the
branch to be worked on.
Now the other modal logics can be dealt with easily.
The logic K was axiomatized by adding OT. In effect this says there
are always v-formulas available, hence S# can always be considered to
be nonempty. And, in fact, the appropriate tableau system
forK is
the
one above without the provision that S# be nonempty.
The logic D had
T
as an axiom. In effect, this says there
is
always a
7T-formula around, so an explicit occurrence of 7T need not be present to
apply the tableau rule. Properly speaking, a tableau system for D results
from that of K by adding the additional rule:
s
S#
Finally the logic
Thad
as an axiom schema v => v
0
• Well, simply add the
tableau rule
JJ
JJo
(it can easily be shown that the D-rule above is redundant).
For quantifiers
we
proceed analogously of course. Thus, for a
parameter a, and a set S of first-order sentences, let
S(a) = {
y(a) I
y E
S}.
Then a tableau system for QC
is
the Smullyan propositional system
plus the rule (to be read in a similar fashion to the one for C above)
S, 5
S(a), o(a)
provided
(1) a
is
new to the branch;
(2)
S
(a) :f= c/>.
For QK, drop the requirement that S(a) be nonempty.
For
QD
, add the rule
s
S(a)
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1
8
M
E L V
IN F I
T T IN
G
p
rovid
ed a
is ne
w to
the b
ranch
.
Fo
r Q
T, a
dd th
e rule
Y
-y(a)
for
any p
aram
eter a.
(
This
m ake
s the
QD
rule r
edund
ant).
And
fin
ally,
for f i
rst-or
der lo
gic p
roper
we w
ant th
e "cu
m ula
tiven
ess '
tha
t
cp
::::>
(
Vx)cp
bro
ught;
tha t
is, as
we go
on
in a t
ablea
u con
struc
tion ,
sente
nces s
hould
nev
er be
delet
ed. W
ell, w
e cou
ld re
place
the QK
ru
le
by
S
, o
S , S(a
),
o(a)
pro
vided
a is
new
to th
e bra
nch.
T
he onl
y other
qua
ntifie
r ru l
e now
is
Y
-y(a)
It i
s no
t har
d to s
ee th
at the
se ar
e equ
ivale
nt to
the si
m ple
r set
0
o
(a)
a n
ew
Y
-
y(a)'
an
d we
have
exac
tly th
e firs
t-orde
r sys
tem o
f Sm
ullyan
[7].
B IB L IO G R A PH Y
[1
] Fitti
ng, M
.: 1973
, 'Mod
el Exi
ste nce
T heore
m s fo
r M oda
l and
ln tuitio
nistic
Logics
',
Jo
urnal
of
Symb
olic L
ogic
3
8
6
13-627
.
[2]
Kripk
e , S.:
1963,
'S em a
ntical
Analys
is o f
M odal
Logic
I, No
rmal P
ropos
it ional
Cal
culi', Zeit
schrift
fur
math
ematische
Logik u
nd
G
rundla
gen d
er M a t
he m a t
ik
9,
67-9
6 .
[3] K
ripke,
S.: 19
65, 'S
em ant
ic al A
na lysis
of M
odal L
ogic
II , N o
n-norm
al M o
dal
Pro
positi
onal C
alculi '
, The The
ory
o
f
M
od
e l
s, J.
W.
Addiso
n, L.
H enki
n and
A .
Tarsk
i (eds.}
, N orth
-H olla
nd Pu
blishin
g Co.,
A mste
rdam,
pp. 2 0
6 -2 2 0 .
[4) Lemm on, E.: 1957, 'N ew Foundati ons for Lew is M odal Systems',
Journal
of
Symbolic
Logic
22
176-
186 .
[
5] Seg
erberg
, K .: 1
971, An E
ssay i
n Clas
sical M
odal L
ogic,
U p
psala.
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A
X
IO
MA
T
IZI
NG
Q
U
AN
T
IFI
ER
S
AN
D
M
O D
A
L I
T IE
S
1
9
[
6]
Sm
ully
an,
R
.: 19
63,
A
Un
ify
ing
Pri
nci
ple
in Q
ua
nti f
ic a
tio n
T h
eo
ry
, P
roc
eed
ing
s of
th
e N
atio
na
l A
cad
em
y of
Sc
ien
ces.
[7) Smullyan,
R.:
1968,
First
Order
Logic,
Springer-V erlag , Berlin .
D e
p t.
of
M a
the
m at
ics
L
ehm
an
C o
lle
ge
B
ro
nx, N
Y
1
046
8
U
.S
.A .
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NICOLAS
D. G O O D M A N
THE
KNOWING MATHEMATIC
IAN
1.
Mathematics is at the beginning of a new foundational crisis. Twenty
years ago there was a firm consensus that mathematics
is
set theory and
that set theory
is
Zermelo-Fraenkel set theory (ZF). That consensus
is
breaking down.
It
is
breaking down for two quite different reasons. One
of these
is
a turning away from the excesses of the tendency toward
abstraction in post-war mathematics. Many mathematicians feel that
the power of the method of abstraction and generalization has, for the
time being, exhausted itself. We have done about as much as can be
done now by these means, and it is time to return once more to hard
work on particular examples. (For this point of view see Mac Lane [16,
pp. 37-38].) Another reason for this turning back from abstraction is
the economic fact that society is less prepared now than it was twenty
years ago to support abstract intellectual activity pursued for its own
sake. Those who support research are asking more searching questions
than formerly about the utility of the results that can reasonably be
expected from projects proposed. Mathematicians, moreover, are in
creasingly obliged to seek employment not in departments emphasizing
pure mathematics, but in departments of computer science or statistics,
or even in industry. Thus there
is
a heightened interest in applied and
applicable mathematics and an increased tendency to reject as abstract
nonsense what
our
teachers considered an intellectually satisfying level
of generality. The set-theoretic account of the foundations of mathe
matics, however,
is
inextricably linked with just this tendency to
abstraction for its own sake. Mathematics, on that account, is about
abstract structures which, at best, may happen to be isomorphic to
structures found in the physical world, but which are themselves most
definitely not in the physical world. Thus as mathematicians turn away
from pure abstraction, they also become increasingly dissatisfied with
the doctrine that mathematics is set theory ·and nothing else.
There is also another reason for the breakdown of what we may call
the set-theoretic consensus on the foundations of mathematics. That
is
Synthese
60
(1984) 21-38. 0039-7857/84/0601-0021 $01.80
©
1984 by D. Reidel Publishing Company
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2
2
N IC
O L AS
D. G O
O D M A N
the b
reakdow n
of the con
sensus with
in set theo
ry. Th
e
w
ork of G od
el
and
Cohen and
Solovay a
nd the rest
has shown
that Zerm
elo-Fraen
kel
is a
n astonishi
ngly weak t
heory, whi
ch settles f
ew of the is
sues of cen
tral
con
cern to the
set theori
st. Not on
ly does it n
ot settle t
he continu
um
problem
, but it also
does not
settle the S
ouslin hyp
othesis, th
e Kurepa
hyp
othesis, th
e structur
e of the
analytic h
ierarchy,
or the ga
p 2
conjectu
re. A con
temporary
set theori
st, faced
with a dee
p-look ing
problem
, asks first
n ot for a p
roof or a co
unterexam
ple, but fo
r a model.
H e doe
s not exp
ect to prov
e or refut
e a co njec
ture, but
to prove it
indepe
ndent. The
num ber
of indepen
dent set-th
eoretic ax
ioms grow
s
alarmingly. More and m ore exotic large cardinals are invented and
studie
d, though
none of th
em can be
proved to
exist. M o
re and m o
re
comp
lex combi
natorial pr
inciples ar
e extracte
d from the
st ructure
of
G odel's c
onstructib
le universe
or from th
e techniqu
e of some
intricate
forcing a r
g u m ent an
d are then
shown to h
old in som e
models bu
t not in
others. I f
mathemati
cs is set t
heory, wh
ich set the
ory is it?
No one
knows ho
w to choo
se am ong
these man
y conflicti
ng princip
les. This
situation
, moreove
r, has gone
beyond t
he point w
here it is o
f interest
on
ly to logi
cians. A lg
ebraists in
terested in
the struc
ture of in
finite
abelian g
roups must
watch the
ir set theor
y (see [10]
and [24]
). Analysts
inte
rested in th
e structur
e of topolog
ical algeb
ras must do
the same
(see
[27
]
).
Increas
ingly it se
ems that e
very m athe
matic ian w
hose inter
ests
are at a
ll abstract
is
going t
o be faced
with the p
roblem of
choosing
w
hich set-th
eoretic ax
ioms to wo
rk with. B
ut it also
seems clea
r that
th
ere is no w
ay within
the presen
t fram ewo
rk to distin
guish whi
ch of
these a
lternative
set theorie
s is the tr
ue one - if
that que
stion even
m akes
sense any
longer. Ea
ch m athe
matic ian m
ust rely o
n his own
,
increasingly bewildered, intuition
or
taste. Evidently it
is
time to try to
f
ind a new
framework
.
2.
I suggest
that both
of the ab
ove difficu
lties with
the set-th
eoretical
fou
ndational
consensus
arise from
the sam
e source
- namely,
its
str
ongly redu
ctionistic
tendency.
Most math
ematical o
bjects, as
they
o
riginally pr
esent them
selves to u
s, are not s
ets. A natu
ral number
is
not
a transi
tive set line
arly o rder
ed by the m
embership
relation.
An ordered
pair is
not a doub
leton of a s
ingleton an
d a double
ton. A fun
ction is no
t
a
set of o r
dered pairs
. A real .n
u m b er is n
ot an equ
ivalence c
lass of
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24
N IC
O L
A S D .
G
O O
D M
A N
di
sjo
int f
rom
th
e un
ive
rse
of d
isco
urs
e of
the
res
t of
sci
enc
e. In
th
inki
ng
abo
u t
pur
e m
athe
m a
tics
w
e en
d up
no
lo
nge
r th
inki
ng
abo
ut a
pa
rt o
f
scie
nce
, bu
t ra
the
r ab
out
a b
eau
ti fu
l, au
ste
re s
ubs
titu
te fo
r sc
ien
ce.
Thi
s
dev
elo
pm
ent
has
me
ant
a
prog
res
sive
im
po
veri
shm
ent
of
the
ma
the
m
at
icia
n 's
in tu
itio
n.
M an
y m
ath
em
atic
ian
s w
ho
kno
w s
om
e c
om p
lex
an
alys
is d
o
not
kno
w,
or
do
not
th
in k
of,
the
co
nn
ecti
on
bet
w ee
n
a
nal
ytic
fun
cti
ons
and
th
e flow
o
f a
flu
id. W
he
n I
wa
s ta
ugh
t co
m p
lex
an
alys
is in
g
rad
uate
sc
hoo
l, I
wa
s no
t ta
ugh
t t
hat
con
nec
tion
. H
en
ce
w
hen
th
ink
ing
abo
u t
ana
ly ti
c fu
nct
ions
th
ese
ma
the
m at
icia
ns
can
no t
rel
y o
n th
eir
com
m o
nse
nse
in
sig h
t in
to t
he
beh
avi
or o
f w
ate r
. A
gai
n,
many mathem aticians who know some of the theory of rings and ideals
do
no
t k
now
th
e a
lgor
ithm
s w
or
ked
out
by
K
ron
eck
er a
nd
oth
ers
to
co
m p
ute
suc
h id
eals
. W
hen
I w
as
taug
h t
rin g
the
ory
in
grad
uat
e sc
hoo
l,
I
was
not
tau
gh
t tho
se
alg o
rith
ms
. F o
r th
ese
ma
the
m at
icia
ns a
n i
deal
i
s
not
a c
om p
uta
tion
al
obje
ct
but
mer
ely
an
abs
trac
t se
t sa
tisfy
ing
cer
tain
clo
sure
co
ndit
ion
s. T
hus
the
y fi
nd i
t dif
ficu
lt e
ven
to
con
side
r no
ntr
iv ia
l
exa
m p
les
of t
he
abs
trac
t th
eor
y th
ey
hav
e le
arn
ed.
A n
y in
tu it
ion
the
y
ma
y h
ave
mu
st b
e p
urel
y fo
rm
al.
T
he
se t
-the
ore
tic
red
ucti
onis
ts h
av
e ex
pla
ined
aw
ay
the
ob
ject
s
we
w
ere
try
in g
to s
tu dy
, a
nd n
ow
tho
se o
bje
cts
are
no
long
er
ther
e to
gu
ide
u
s. P
erh
aps
i
t is
tim
e
to b
ring
so
me
of t
hem
ba
ck.
3.
T h
rou
g h
ou t
mo
st o
f th
e tw
ent
ieth
cen
tur
y ph
ysi
cis t
s ha
ve
held
th
a t it
is
no
t p
ossi
ble
ade
qua
tely
to
de
scri
be
the
phy
sica
l w
orld
w
itho
ut t
aki
ng
in
to a
cco
un t
the
ob
serv
er w
ho
col
lect
s th
e d
ata
that
t he
ph
ysic
al t
heo
ry
is
in tended to predict and explain. B oth the theory of relativity and
qu
an
tum
m
ech
anic
s a
re i
n v
ery
la
rge
par
t th
eor
ies
of
the
re
lati
on
bet
w ee
n th
e o
bse
rve
r an
d th
e p
hys
ical
rea
lity
tha
t he
ob
serv
es.
N ei
the
r
the
ory
, ho
we
ver,
de
nies
th
e re
alit
y o
r th
e o
bjec
tivi
ty
of t
he
exte
rna
l
rea
li ty
th
at is
b
ein
g o
bse
rved
. A
lth
oug
h
bo t
h t
heo
ries
in
volv
e
a
f
ar-r
each
ing
ad
m ix
tur
e o
f ep
iste
m o
log
y in
th
eir
bas
ic d
esc
ript
ion
of
p
hys
ical
rea
li ty
, ne
ithe
r th
eor
y c
an l
egit
im a
tely
be
acc
use
d o
f id
eali
sm.
T
he
ob
serv
er
and
th
e
exte
rna
l p
hys
ical
re
ality
w
ith
wh
ich
he
is
c
on
fron
ted
ar
e b
o th
ir
redu
cib
ly
pres
upp
ose
d.
In
thi
s se
nse
th ese
th
eori
es a
re
fund
am
ent
ally
dua
list
ic .
T w
enti
eth
-cen
tur
y m
ath
em a
tici
ans
, on
th
e o
th er
ha
nd,
hav
e s
oug
ht
to
ma
inta
in
the
mo
nis
tic
cha
rac
te r
of
thei
r di
scip
lin
e.
Th
e cla
ssic
al
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26
NICOLAS
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G O O D M A N
extensive or meaningful mathematics. Nevertheless, the constructive
tradition has the very great merit of having emphasized and studied the
epistemic aspects of mathematics. Those aspects have been largely
ignored by classical mathematicians.
The debate between the classical and intuitionistic positions on the
foundations of mathematics can thus be viewed
as
a disagreement
between two opposed, mutually exclusive, monisms. Each denies that
the other's reality is of any fundamental significance.
It
is like a debate
on the foundations of physics between a strict Newtonian who denies
that observability is of any fundamental consequence and a strict
phenomenalist who denies that our sense experiences refer to any
knowable reality outside of ourselves. Obviously such a debate is
unlikely to be fruitful. In physics the fruitful step was the step to a
dualistic view which emphasized both the role of the observer and the
role of the reality being observed. It is the thesis of the present essay
that a similar dualistic view on the foundations of mathematics is both
possible and desirable.
4 .
The observer in quantum mechanics or in the theory of relativity is a
very highly idealized physicist. He has no subjective bias. He does not
forget anything. He has no deficiencies of experimental technique. His
attention never wavers. He never sleeps. In our theory, then, we may
expect that the knowing mathematician will be similarly idealized. In
particular, we
will
assume that his powers of concentration are poten
tially infinite.
There
is to be no finite bound on the complexity of the
computations he can carry out
or
on the length of the proofs he can
construct. As I have argued in another place (see [9]), this is a very
considerable idealization of what is the case for human mathematicians
or even for the human race viewed collectively as a single mathemati
cian. Thus the knowing mathematician of our theory is himself to be
conceived as a mathematical abstraction. His introduction into the
theory is a step not toward a constructivistic impoverishment of
mathematics, motivated by doubt about the metaphysical underpin
nings of set theory, but rather toward an enrichment of classical
mathematics by the introduction of a new and more extensive vocabu
lary.
The
point
is
not to consider only those objects which can be
known, but rather to consider what aspects of arbitrary objects are
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T H E KNOWI NG MATHEMATIC IAN
27
knowable
in
principle.
Thus, although our knowing mathematician
is
to be thought of as
only potentially infinite, he
is
not to be thought of as somehow situated
in
the physical world. All he does
is
mathematics. He has no properties
that do not follow from his being an idealized mathematician. In this
respect he resembles the idealized physicist of quantum mechanics
or
of
the theory of relativity.
Although our goal is to reverse the set-theoretic reductionism of the
last few decades, it
is
clear that we cannot do that in one fell swoop. It
makes no sense to write down a theory having as its primitives all the
notions which, in an unanalyzed form, have
ever
played a role in
mathematical practice. Such a theory would be ugly and, presumably, a
conservative extension of its set-theoretic fragment. It would give no
new insight. The point is not to reintroduce old notions for the sake of
not explaining them away.
The
point, rather,
is
to try to rebuild our
mathematical intuition by gradually enriching it with notions and
principles that are not known to be reducible to set theory.
Thus
I
suggest that our first draft of such a theory should be a set theory, but a
set theory enriched with intensional epistemic notions.
The
first theory
of the sort I have
in
mind was based on arithmetic and
is
to be found
in
Shapiro [23].
Then
Myhill in [17] proposed a theory based on set theory
but
in
which the arithmetic part was still, so to speak, singled
out
in the
very syntax of the theory. Finally, in [8], I proposed a theory that
is
strictly set theoretic. A similarly motivated but independent and
formally very different theory can be found in the recent work of
Lifschitz (see [14] and [15]).
The theory I will discuss here is the theory of my [8]. I t is a modal set
theory, with membership as its only nonlogical primitive, and with
Lewis's S4 as its underlying logic. The modal operator of S4
is
to be
read epistemically. Thus
O
means that is knowable.
5.
In one respect, at least, our theory will resemble quantum mechanics
more than it does the theory of relativity.
The whole point of the theory
of relativity is to relate the observations of different observers. In the
mathematical case this does not seem to make much sense. Troelstra in
[26] considers two mathematicians, one of whom communicates num
bers to the other without giving him complete information about how
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28 N IC O L
A S D. G
O O D M AN
those num
bers a re o
btained. The
o th er m
athem atic
ian, then,
is able to
make certa
in predict
ions abou t
the future
behavior
of the first
mathe
matic
ian. This
entire p ic
ture, how
ever, seem
s to m e
more clos
ely
relat
ed to what
happens in
empirical
science th
an to anyth
ing that g
oes
on in
mathemati
cs. M athem
aticians e
xchange c
omplete i
nformation
.
T hey
do no t gen
erally hid
e all or
part of their
algorithms
from eac
h
other. M
oreover,
in contrast
to the situa
tion in phy
sics, I do n
ot see that
it make
s sense t
o suppose
that one
mathem a
tician som
ehow has
preferen
tial access
to some p
art of math
ematical r
eality. On
this basis,
th
en, it seem
s to m e ad
equa te to c
onsider th
e case o f o
nly on
e
kn
ower.
T his
is
analogous to the situation in quantum m echanics, w here
one
does not
usually co
nsider the
effect of
having m o
re than
one experi
m enter.
Thus
w h
en, in
our theory, we
write Ocp,
we take th
is to m ean
that the
f
ormula
cp is know ab
le to our id
ealized m a
them aticia
n . W e say
"know
able" ra th
er than "know
n" bec
ause of th
e logic of S
4. Let us r
ecall the
rules of
that system.
In additi
on to th
e usual rule
s of the c
lassical
p
redica te ca
lculus, we
have the
follow ing m
odal rules
:
1.
Ocp--7cp.
2
. 0cp--?0
0cp.
3.
0 1\ 0
p --7 1/1)
--7
01/J.
4.
From
1-
cp
infer Ocp.
If we tak
e 0 to m e
an 'know n
', then the
third of the
se rules as
serts tha t
whenever
our math
ematician
can make
an infere
nce, he ha
s already
done so.
Even
an idealized mathem atician presumably does not follow
out every p
ossible ch
ain of infe
rence. Thus
we must
ta
ke 0 to
m ean
'
know able ' .
On this re
ading, the
first of ou
r rules asse
rts tha t eve
rything
k
nowable
is true. T
he second a
sserts that
anything
knowable
can be
k
nown to
be knowabl
e.
O
ne m ig
ht argue f
or this by n
oting
that if < > is
ever kn
own, it wi
ll then
be
k
nown t o b
e know n, a
nd hence k
nown to b
e
knowab
le. The t
hird rule a
sserts th a
t if both
a conditio
nal and it
s
an teced
ent ar
e
kn
owable, the
n the cons
equent is k
nowable. F
inally, the
fourth ru
le derives
from our co
nfidence i
n our
very
system o f a
xioms. If
we
actually pr
ove a claim
, then w
e know th a
t claim to
be true, an
d so
tha
t claim
is knowable.
t seems
to have b
een G ode l
in [7] w ho
first obser
ved that S
4 can be
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31
function such that . . . . In order to see that there really are mathemati
cal notions independent of set theory, let us consider a familiar example
that is actually rather closely connected with
our
idea of mathematical
knowability in principle. I am thinking of Church's thesis.
The standard account is that there was a vague and unanalyzed
informal notion of computability in principle, and that
Church, Turing
,
Kleene, and Post offered alternative analyses of this notion in set
theoretic terms that turned out to be equivalent. As usual, let us refer to
a function satisfying the formal definition as recursive. Church's thesis
is
then the vague and premathematical claim that recursiveness coincides
with computability in principle. It has even been urged that the
replacement of computability in principle by recursiveness
is
analogous
to the replacement of the informal eighteenth-century notion of con
tinuity by the formal nineteenth-century
concept
defined using epsilons
and deltas (see Shapiro [22]
).
Note, however, that the two analyses
function quite differently
in
practice.
The
informal notion of continuity
is
used only heuristically to motivate the epsilon-delta definition.
Once
the epsilon-delta definition has been given, no further reference is made
to the informal notion. The practicing analyst who uses the notion of
continuity thinks in terms of the epsilon-delta definition, or in terms of
some other definition equivalent to that definition on the real line.
Every assertion about continuity is explicitly justified by means of the
epsilon-delta definition. The contemporary recursive-function theorist,
on the
other
hand, uses the informal notion of computability constantly.
He thinks in terms of that notion, rather than in terms of one of the
standard formal definitions. Proofs in the theory of recursive functions
usually no longer refer to the formal definitions
at
all.
If
it becomes
relevant to relate the theory to
one
of the formal definitions, this
connection
is
established by a global appeal to
Church's
thesis.
The
analyst says, "This function
is
continuous because I have shown that it
satisfies the epsilon-delta definition."
The
recursion theorist says, "This
function
is
recursive because I have shown how to
compute
it."
The
role of the formal, set-theoretic definition
in
the two cases could not be
more different.
It might be suggested that we are still so close to the time when the
definition of recursiveness was first given that there has not been time
for the formal notion to drive
out
the informal notion it is intended to
replace.
The
historical evidence, however, points in the opposite
direction.
The
early papers in the theory of recursive functions were
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NICOLAS
D.
G O O D M A N
recursively enumerable set can exhaust the truths of any nontrivial
mathematical theory. Thus, at least as far as we could know, the
problem of intensionality does arise.
9.
When we do set-theoretic mathematics, we reason about sets, not about
descriptions of sets. Nevertheless, when we reason epistemically about
particular sets, asking ourselves questions such as whether we could
actually construct a set with a certain property, we are necessarily
dealing not with the sets themselves, but only with defining criteria. As
a mathematical abstraction, at least, it is
not difficult to construct a
language in which every set has a defining criterion. For, form a
"language" in the logician's sense having a name for every set. Then
the set
A is
defined by the criterion, 'x belongs to A'
.
Of course, in this
language, the set A
will
also have many other defining criteria, some of
which may not be knowably equivalent to this one. Although no human
being could learn this language, we often talk as though we were using a
finite fragment of it. A mathematician who has proved the existence of
a set with a certain property but does not know any actual criterion for
membership
in
such a set will not hesitate to introduce a name for a
particular such set. For example, he may write as follows:
"Thus
we see
that there exists a regular ultrafilter. Let D be such an ultrafilter. We
know that
D
has such and such property. Hence . . . . " As an illustration
of this procedure, let L be a language of the sort described.
Every set has a defining criterion expressible in
L.
These criteria can
be thought of as representatives , of the sets. For all mathematical
purposes, these representatives will do the work of the sets. Member
ship
in
a set
is
just satisfaction of the corresponding membership
criteria. The usual axiom of extensionality, which asserts that two sets
with the same members are identical, tells us that two defining criteria
which are satisfied by the same objects are representatives of the same
set. Thus, in the spirit of the usual set-theoretic reductionism, we may
think of a set as an equivalence class of these defining criteria under the
relation of being satisfied by the same objects. (Actually, as this
construction
is
carried out in my paper [8], it is technically somewhat
more complex. The problem is that some criteria will not be exten
sional. Thus sets should be thought of as equivalence classes of
hereditarily extensional criteria. But these details need not concern us
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K N O W I N G
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35
here.)
The relation of extensional identity is not the only interesting
equivalence
relation
on the
defining criter ia expressible in
our language
L. A
somewhat stronger
relation
is
the relation which holds
between
two of these criteria when it is knowable that they apply to exactly the
same
objects.
Let me
refer to this relation as
modal
extensional identity.
Criteria
that
are modally extensionally identical
can
be thought of as
representatives
of the
same set-theoretic property or
attribute.
More
formally, by a
property
let us
mean
an equivalence class of defining
criteria
under
the relation of modal extensional identity.
Then
we
can
interpret
our
modal
set theory
as being
about
properties.
In this way, interpreting
our
theory as being a
theory
of
properties
rather
than a theory of sets, the problems of interpretation we men
tioned
above
disappear. For, two properties
that can
be known to apply
to the same objects will have exactly the same properties expressible in
the language of
our
theory.
Thus
if we adopt a modal axiom of
extensionality asserting that
properties
that can be known to apply to
the same things are identical, then we will have full substitutivity of
identity.
10.
Let me now suggest that the
above
construction of properties from sets
is exactly backwards. The historical development, at least, went in the
other
direction. Sets first appeared in mathematics in the form of
properties expressed in some language - say mathematical
German.
That is to say, the defining criteria are historically the primary objects .
These
properties
were studied by impredicative
and
nonconstructive
methods, so that it was clear very early that mathematical German was
not
adequate
to formulate all possible such
properties
(for what
amounts to this point , see Borel [3, pp. 109-11 0]).
Thus
sets were
introduced
as
mathematical
abstractions to support a notion of
property
that
had
come detached
from the idea of expressibility in any particular,
or even
any possible, language. The
problem
then
arose about
what
criterion of identity
one
should use for these strange new mathematical
objects
. In
the
positivistic intellectual climate of the turn of the
century,
it was not difficult to arrive at a consensus that extensional identity was
the only possible criterion. As a
matter
of fact, however,
some
sort of
intensional identity may be more suitable for pre-set-theoretic mathe-
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36
NICOLAS D . G O O D M A N
matical practice. Thus I suggest that we should go back to the situation
in, say, 1880, when it was not yet clear what it should mean to say
that
two sets are identical. In
that
situation it makes sense
to
suggest
that
two sets are identical just in case it can be known that they have exactly
the same elements. Given
that
we
have
an adequate logic of knowabil
ity, which we do , this suggestion may be more fruitful than the generally
accepted one. To avoid confusion today, however, we should probably
not refer to the resulting objects as sets, but as properties.
1 1 .
Once
it
is
clear
that the
theory we want to write down
is
a theory of
properties in
the above
sense, it is not difficult to write down the axioms
by imitating ZFC (that is,
Zermelo-Fraenkel
set theory with the axiom
of choice).
For
the details, I refer thti
reader
to [8]. The resulting theory
is as rich as set theory. ZFC is faithfully interpretable in it. As a matter
of fact, the theory is richer than set theory. For example,
in
it we can
express that we have actually constructed an object, rather than merely
proved
it to exist. Most of the basic
constructive
notions are easily
expressible. Of course, the underlying metaphysical position is not at all
intuitionistic,
but
Platonistic.
What remains is to try to develop set theory and analysis in this new
framework, trying to exploit its additional expressive power. As we do
so, we should be able to develop sufficient intuition for the epistemic
component
of the new framework that we
can
begin to think informally
about
the knowing mathematician without having to rely
on
any
formalization, including ours.
Then
we should be led to ask new
questions and to find new
phenomena
that will enrich classical
mathe
matics.
It
seems rational to hope that the resulting sharpening of
our
set-theoretic intuition will
either
lead to new insight into the apparently
unanswerable questions of set theory, or else will enable us to see that
those questions are not really very important or very central after all.
What is more important is that by working with intensional notions
directly connected with
our
actual mathematical experience - by
considering mathematical knowledge, computability, and construction
as ingredients of our universe of discourse rather than as merely
psychological aspects of
our
work - we may restore
some
of the
concrete mathematical intuition that we
have
lost. In this way we may
gradually break down the prestige of
set-theoretic
abstraction and
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3
8
NI C OL A
S D
.
G O O D
M A N
[1
8] Quine, W
. V.: 1976,
'Three Grades
of Mod al Invo
lvem ent' , in
The Ways
of Paradox
and Other
Essays, rev. e
d. , Harvard
U.P. , Cambri
dge, Mass., p
p. 158-176.
[19] Randolph ,
J.
F.: 1968,
Basic Real and Abstract Analysis
,
Academic Press, New
York.
[20] Roge
rs, H.: 1967
, Theory of
Recursive
Functions an
d Effective C
omputability
,
McGraw-Hil
l, New York .
[21] Ro
senlicht, M.:
1968, Intr
oduction
to
A
nalysis, Scott,
Foresman, G
lenview, III .
[22] S
hapiro, S.: 'O
n Church 's T
hesis ' , Typesc
ript.
[23] Shapi
ro, S.: 1984, '
Epistemic
r i t ~ m e t i c
and
Intuitionistic
Ari thmetic ',
to ap pear in
S. Shapiro (e
d .), Intensio
nal Mathema
tics, North-Ho
lland Pub. C o
., New York.
[2
4] Shelah,
S. : 1979,
'On Well Orderin
g and More o
n Whitehead
's Problem', A
bstract
79T-E47,
Notices Amer
. Math. Soc
., 26, A-442 .
[2 5] Shoenfield , J. R.: 1972,
Degrees
of
Unsolvability,
N orth-Holland , Amsterdam .
[26] T
roelstra,
A .
S
.: 1968, '
The The ory of C h
oice Sequenc
es', in B. Va n
R ootselaar a
nd
J. F
.
Staal (eds.),
Logic , Met
hodology and
Philosophy
of Science I l
l
N
orth-Hollan d
,
A m st
erdam , pp. 20
1 - 223 .
[27]
Williamson,
J
.
H.: 1979,
Review of
Topological
Algebras by E .
Beckenstein
, L.
Na
rici, and
C. Suffel, Bull
. Am
er
. Mat
h. Soc. (New S
eries) 1
,
237-
244.
Dept. of M
athematics
Suny-Buff
alo
A m h
erst, NY 142
60
U .S.A.
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R AYM OND D. GUMB
"CONSERVATIVE"
KRIPKE
CLOSURES*
0. I N T R O D U C T I O N
Computable Kripke closures are properties of relations which have
closures in, roughly speaking, the sense of the transitive closure. They
were introduced in [8] to generalize Kripke-style tableaux con
structions and were studied from a model-theoretic perspective by
Weaver and Gumb [19].
In section 1 of this paper, we review the properties of computable
Kripke closures. In section 2, we state four additional laws that can be
imposed on computable Kripke closures and state properties of the
closures determined by these laws. In a sense, all four of our laws
require closures to be "conservative". However, regarding later sec
tions, it
is
more revealing to classify two of the laws as being
commutative and two as being conservative.
In the remaining sections, we sketch applications of these laws
in
modal logics having a Kripke-style relational semantics: simplifying
Kripke-style tableaux constructions (section 3), proving the Craig
Interpolation Lemma (section 4), establishing Henkin-style complete
ness proofs (section 5), and providing a somewhat plausible prob
abilistic semantics (section 6). At least in modal logic, our laws carve
out natural classes of properties of binary relations.
1. C OM P UT AB L E KR I P KE C L OS UR E S
The presentation of the computable Kripke closures
in
this section
is
much the same as that
in
[8, section 5]. A model-theoretic study of the
first-order Kripke closures using a different notation can be found
in
[19].
A (binary) relational system
is
a pair i i =(a-, r), where a-
is
a nonvoid
set and
r
sa-Xu
is
a (binary) relation on
a-.
Let
BR
be the class of
relational systems. Understand Pr
s;
BR to be a property of relations if Pr
is
closed under isomorphisms. Let Pr be a property of relations, and let
Synthese
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(1984) 39-49. 0039-7857/84/0601-0039 $01.10
© 1984
by
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Reidel Publishing Company
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4
0
R A
YM O
N D D
. G
UM B
i
i = (u,
r
) an
d ii+
= (u+,
r+)
be re
lation
al sy
stems
. W e
say t
hat r+
is a
Pr-re
lation
(on
u+) if
i
i
+
E Pr
. W e
call
r
+
the
Pr
-closu
re
of r (on
u +
) and
writ
er+=
p
ru •
(r) if u s
u +,
r s r+
, a
nd r+
is th