I NFI NI TE CONJ UNCTI ONS A ND DI SJ UNCTI ONS
A ND THE ELI MI NATI ON O F QUANTI FI ERS
WILLIAM TODD
It is a well-k nown f ac t t hat wh e n we hav e a f init e univ ers e o f
discourse univ ers al quant if iers c a n b e replac ed w i t h c onjunc t ions
and existential quantifiers can be replaced wit h disjunctions. Furt her,
quantificational rules, such as t he rules of nat ural deduction, c an be
replaced wi t h s uc h rules as addit ion ( p v q) a n d s implif ic at ion
(13 • q p ) f r o m t he propos it ional calculus (l). This f ac t leads one t o
ask whet her quantifiers can alway s be replaced by conjunctions and
disjunctions ev en when t he universe o f discourse is inf init e a n d i nf init ely l o n g conjunctions and dis junc t ions wo u l d b e required. W e
wi l l not attempt t o give a f inal ans wer t o this question here, but we
wi l l look at some of t he problems inv olv ed and v ent ure s ome suggestions i n the matter.
Before one can make progress wit h this question it mus t be decided
whet her one is to be concerned wit h the eliminat ion of quantifiers f rom
the statements o f mat hemat ic s a n d logic o r wi t h t heir eliminat ion
f rom empiric al statements about t he wo r l d if , i n fact, i t t urns o u t
that the lat t er require an inf init e universe of discourse. We wi l l here
be concerned almos t exclusively wi t h mat hemat ic s and logic . O n l y
our conclusion t hat t here is n o general and categorical objec t ion t o
the use o f inf init e t rut h functions i n t he eliminat ion o f quant if iers
wi l l hav e relevance t o t he ot her question. Ev en wh e n we c onc ern
ourselves solely wi t h mathematics and logic, i t t urns out t hat t here
are ma n y dif f erent c ondit ions a n d circumstances wh i c h c an arise.
Much also depends on the purpose one has in setting out to eliminat e
quantifiers f rom a giv en system. This is why we can give no general
answer as t o t he legit imac y o r illegit imac y o f t he process of quantifier eliminat ion, and we wi l l instead ex amine t he v arious possible
contexts one by one.
It is ac t ually quit e c ommon f o r mat hemat ic ians and logic ians t o
take conjunctions and disjunctions, bot h f init e and inf init e, as basic
and t o def ine t he quant if iers as abbreviations f o r these t rut h f unc (
or 1i n a n y s imila r tex t.
)
A
d
i
s
c
u
s
s
i
o
n
o
289
tions. This is done by writ ers as various as Frank Ramsey and Abra
,
ham
Robinson (
junctions,
2
b u t ot hers f o u n d a l l sorts o f philos ophic al a n d logic al
positions
) ; s o on
m the
e supposed theoretical possibility of eliminat ing quantifiers
d
oi n f av or of inf init e conjunctions and disjunctions. We wi l l be
mainly
wi t h inf init e c onjunc t ions ; almos t a l l t he s ame
n
o concerned
t
a
problems
t
t arise i n the case of inf init e disjunctions and they wi l l generally
a
c have
h the same status as the corresponding conjunctions.
g In t he
r case of mathematics it is obvious f rom t he outset that there
are
e univ
a ers ally quant if ied statements wh i c h hav e a n inf init e range.
A
t s imple example would be t he statement that every nat ural number
is
i great
m erpt han —1. Th i s wo u l d be t rans lat ed as a c onjunc t ion o f
statements
o
r
t each asserting t hat a part ic ular nat ural number bas t his
property.
a
n
cHe r e t he number o f conjuncts i s denumerable, b u t ev en
this
wo
u
l
d
not be t rue i f we substitute t he corresponding statement
e
about
the
positive
real numbers. Further, it is clear that variables wit h
t
an
inf
init
e
range
are
essential t o mathematics and cannot be elimino
ated.
Since
we
want
a
logic that can be applied t o mathematics there
t
is
n
o
hope
of
getting
along
wi t h f init e c onjunc t ions i f we wa n t t o
h
eliminat
e
t
he
quantifiers.
e
s The f irs t point to be made is that inf init e conjunctions are already
present
in most logistic systems whic h contain conjunction at all. The
e
usual
s
ort
of recursive def init ion o f a well-f ormed f ormula s imply
c
tells
us
h
o
w
t o go about constructing wffs and imposes no limit at ions
o
on
t
he
lengt
h
of t he f ormulas so constructed. A n ex ample of such a
n
def
init
ion
wo
u
l d be:
All propos it ional symbols are well-f ormed.
If A is well-f ormed t hen ( A ) is well-f ormed.
If A and B are well-f ormed then ( A )
If we- go t hrough the def init ion an inf init e number of times, we wi l l
get a f ormula
( B ) i of
s inf w
initee lengt
l l - h, and there is not hing whic h prevents
us f r o m
f oconstructing
r m e d . well-f ormed inf init e c onjunc t ions except o u r
finite lif e spans and our inabilit y to c arry out an inf init e number of
operations i n a f i n i t e t i me . Howev er, i t does n o t s eem logic allY
impossible t hat there might be some person (or angel o r s uperman)
who c ould c onjoin t wo propositions i n 1/ 2 second, a t h i r d i n 1/ 4
second, a f ourt h i n 1/8 second, and so on i n a convergent series s o
(
of 2t he s ame t it le b y t h e s ame aut hor; A b ra h a m ROBINSON, I nt roduc t ion t o
Model
)
Theory and to the Metamathematics of Algebra, publis hed by the Nort h
Holland
Publ. Co., Ams terdam.
F
r
a
290
n
k
R
A
M
S
E
Y
,
(
i
that he c ould construct an inf init e c onjunc t ion i n one second. Similarly, i t seems logic ally possible that someone might be immort al and
construct suc h a c onjunc t ion ov er an inf init e period o f t ime. This
being the case, t he fact that we do not ordinarily meet wi t h inf init e
conjunctions is a fact about the wor l d rat her t han a fact of logic.
Even t hough inf init e c onjunc t ions m a y b e c ont ained i mp l i c i t l y
in t he ordinary sorts of logistic systems, this is not really the point at
issue. We can create a system whic h wi l l contain almos t any sort of
formulas, b u t t his does n o t necessarily i mp l y t hat these f ormulas
can be int erpret ed i n s uc h a wa y t hat t hey w i l l hav e t h e des ired
meaning, even f o r t he rest of logic and mathematics. I n part ic ular,
those w h o wa n t t o us e inf init e c onjunc t ions ins t ead o f univ ers al
quantifiers wi l l have t o insist that, on t he standard int erpret at ion of
a propos it ional calculus, a n inf init e c onjunc t ion f ollows t he s ame
rules and acts i n t he same general wa y t hat f init e c onjunctions do.
We k n o w t he meaning of a f init e c onjunc t ion o f meaningf ul statements and we mus t also k now the meaning of an inf init e c o n j u n c t ion o f meaningf ul statements. I nf init e c onjunc t ions mus t n o t jus t
be well-f ormed formulas, but mus t be mat hemat ic ally meaningf ul i n
a s t ronger sense.
Since an inf init e c onjunc t ion is a f unc t ion wh i c h maps t he t rut hvalues o f t he argument s ont o a set wh i c h again c ont ains only t he
numbers 1 and 0, t his k ind of f unc t ion satisfies t he us ual def init ion,
in t e r ms o f ordered n-t uples . Howev er, t here i s n o t h i n g i n t h e
standard def init ion o f a f unc t ion t o prev ent t he n-t uples being i n finite a n d t here being a n inf init e n u mb e r o f s uc h n-tuples, or , i n
effect, t o dis allow f unc t ions wi t h an inf init e n u mb e r of arguments.
In fact, t he norm f unc t ion on Hilbert space is jus t such function. The
introduction of these conjunctions wi l l not generate any n e w paradoxes i n logic or mathematics.
While t he c ruc ial question is not t he possibility o f paradox, i t is
import ant t o see whet her t here i s a general met hod o f specifying
inf init e conjunctions. A n y f init e c onjunc t ion makes sense i n mat hematics and logic because, among ot her things, there is always a way
of specifying the f unc t ion — mak ing c lear the mapping i n question.
We mus t n o w ask whet her t his is alway s s o i n t he case of inf init e
conjunctions.
There are i n general t wo dif f erent way s of specifying a f unc t ion.
Since a function is a set of ordered n-tuples the first element of whic h
represents t he v alue o f t he f unc t ion wh i l e t h e ot her element s r e present the values of the arguments, the most direct way of specifying
291
a function is v ia a mat rix f rom whic h these n-tuples can be read off.
In fact, t he standard wa y of specifying a t rut h f unc t ion is t o use a
truth-table whic h is jus t such a mat rix . But one can also give a rule
by wh i c h one c an det ermine t he v alue f o r t h e f unc t ion f o r a n y
given set of values f or the variables. When we do this, we are usually
specifying one f unc t ion n o t absolutely but i n t erms o f s ome ot her
already k nown f unc t ion. Thus , us ing 1 a n d 0 as t rut h values, t h e
truth function, —p, c an be giv en as 1—p. I n t he case of mos t mathematical functions i t is us ual t o giv e a rule, wh i c h mi g h t be represented by an expression such as x ' + y. Here the required mat rix
would b e inf init e a n d wo u l d n o t ev en b e denumerable s inc e t h e
function is o n t he real numbers ; henc e t he ma t r i x c annot i n f ac t
be constructed. Nevertheless, we c an be assured t hat such a mat rix
exists because t he components of it are real numbers and we k now
that i t is possible t o f o r m n-t uples ( i n t his case ordered t riads ) o f
such numbers. I t is characteristic o f s uc h f unc t ions as x ' + y t hat
there is a regular relat ions hip t hat holds between t he values of t he
variables and t he value of the function, or between the firs t member
of an n-t uple and t he ot her members. Henc e we do not really need
the mat rix to look up the v alue of the f unc t ion but can det ermine it
in accordance wi t h t he rule.
Let us n o w consider a set of inf init e n-t uples where t he non-f irs t
elements repres ent a l l v alues wh i c h mi g h t b e g i v e n t o a s et o f
variables, but where the f irs t elements bear no more t han a random
relationship to the others. Such a set of n-tuples satisfies the def init ion
of a f unc t ion jus t as muc h as any ot her set, b u t t here is n o w n o
special relat ions hip whic h holds between t he values of t he variables
and t he values whic h t he f unc t ion takes. Consequently, t here is n o
rule whic h we c an state i n a f init e way . I t t hen f ollows t hat t here
are functions whic h we cannot i n fact distinguish f rom one anot her
or specify i n any way ; i n fact, we cannot say any t hing about t hem
at all except that they exist and t hat they are not ident ic al wi t h any
of t he functions whic h we c an specify. O f course, not all f unc t ions
wit h an inf init e number of arguments are unspecifiable i n this wa y
and inf init e c onjunc t ions s eem mu c h less s t range wh e n c ompared
wit h these «inaccessible» functions.
The import ant ques t ion n o w is whet her one want s t o eliminat e
quantifiers f r o m a part ic ular system o f logic o r mat hemat ic s , o r
whet her one wants t o eliminat e t hem everywhere they occur, at least
in theory. The difference becomes apparent when we attempt t o giv e
a rule f or det ermining the t rut h value of an inf init e conjunction. We
292
might state i t as C( x
presents
t he c onjunc t ion f unc t ion and t he right -hand dots represent
l
mult
, x iplic at ion. Howev er, t here i s i mp l i c i t quant if ic at ion o v e r t h e
variables.
2
x
Similarly , w e mi g h t say i n Englis h t hat t he c onjunc t ion
n t he v alue otrue» i f and only i f a l l t he conjuncts are t rue. B u t
has
)
here
t he «all»
= again involves implic it univ ers al quantific ation. Thus
we
( r ux n i n t o dif f ic ult ies i f w e seek t o giv e a r u l e a n d eliminat e
quantifiers
i
x i n a l l languages since o u r proc edure wo u l d ult imat ely
be
2 c ircular. Howev er, t his problem disappears i f we are concerned
only
. . wi.t h a. part
. ic ular system. I n the language of the system we can
define
t he quantifiers i n t erms of inf init e conjunctions and dis junc x
tions
and,
„
) wh e n w e c ome t o int erpret t he language, w e hav e a
metalinguistic
rule f or the ev aluat ion of these t rut h functions. There
w
h
is
e then no
r harm in having irreducible quantifiers in the metalanguage.
This
s it uat ion ma y ex plain t he f ac t t hat philosophers of t en t ak e a
e
dimmer
v i e w of inf init e conjunctions and quant if ier reduc t ion t han
i
mathematicians;
the philosophers tend t o t hink of the more ambitious
(
C
project.
As stated i n the beginning, t he most ambit ious project whic h
»
we
r w i l l c ons ider is t he eliminat ion o f quant if iers f r o m systems o f
mathematics
and logic. There wi l l always be an English metalanguage
e
(if
- not an England) i n whic h quant if iers wi l l remain.
At t his point questions as to t he legit imac y of inf init e conjunctions
seem t o be closely relat ed t o t he question o f whet her inf init e c onjunctions are meaningf ul. Howev er, philos ophic al c rit eria o f meaningfulness w i l l not help us muc h here. Giv en a statement i n hand
one c an apply one o r t he ot her o f t he ma n y possible theories and
decide whet her t he statement has meaning, o r decide what k ind o f
meaning i t has. But logic and mathematics constitute a v ery special
context since t he primit iv es of t he systems are not even supposed t o
have meaning in themselves but are only supposed to be interpretable.
Further, one of the main problems about inf init e conjunctions, as we
wi l l soon see, is t hat it is not obvious h o w t o construct t hem i n t he
first plac e. This i s a problem wh i c h nev er occurs t o philos ophers
who promulgat e t heories o f meaning. Thus i t does n o t s eem v ery
rewarding to ask whet her an inf init e c onjunc t ion can be meaningf ul.
We hav e t o ask instead whet her it wi l l serve its int ended purpose.
We c an n o w state f our dif f erent demands wh i c h s omeone mi g h t
make o f inf init e conjunctions before admit t ing t heir legitimacy. We
wi l l see h o w f ar these demands c an be satisfied f irs t and lat er see
how import ant each demand is t o persons of various points of v iew.
These demands are as f ollow:
293
1) That there be a rule f or det ermining t he v alue of t he f unc t ion
given the values of the arguments.
2) That there be a rule f or the construction of t he conjunction.
3) That there be a rule f or constructing a mat rix f or such a f unc tion.
4) That t here b e r u l e s f o r prov ing conjunctions o f t his s ort as
theorems i n a system and also t hat there be rules f or deriv ing
non-theorems.
We hav e already discussed t he f irs t demand i n s ome det ail and
have seen that it can be satisfied. We n o w pass to the second whic h
might be t hought to be t he most basic of all.
The univ ers ally quant if ied variables of a system have some range
whic h is specified i n t he int erpret at ion of t he system, but this does
not necessarily t ell us how t o construct t he conjunction. I f the quantifiers of the language range ov er the nat ural numbers, f or instance,
there i s n o problem i n reconstructing such a statement as t he one
whic h asserts that all t he nat ural numbers are greater t han or equal
to O. We c an t ak e t he nat ural numbers i n t h e i r nat ural order, let
the f irs t c onjunc t say t hat 0 is great er t han o r equal t o 0, let t he
second s ay t he s ame t h i n g about 1, a n d s o on. I f , howev er, t h e
original v ariables ranged ov er t he real numbers , t here i s n o s uc h
obvious princ iple t o f ollow.
In cons tructing a c onjunc t ion, o r a sequence o f a n y k ind, t h e
import ant t hing is t o k now where t o start and, at any point , where
to proceed next. I t hink that this is what we mean by k nowing h o w
to do something. Thus it seems reasonable t o say that we k now h o w
to construct an inf init e c onjunc t ion i f we are t old what the f irs t c onjunct is and hav e a rule f o r t he selection o f t he nex t c onjunc t at
any giv en point . Thus t he import anc e o f well-ordering i s obv ious
here. I f t he set o f conjuncts c an be well-ordered t hen ev ery subset
wi l l have a f irs t element. By t ak ing the set itself as a subset we are
assured t hat it wi l l hav e a f irs t element and t his gives us o u r f irs t
conjunct. We t hen consider the remaining conjuncts and, since t hey
constitute a subset, we t ak e t he f irs t element o f t hat subset as o u r
next conjunct, a n d so on. We are t hen i n a v ery good pos it ion as
f ar as eliminat ing quant if iers wh e n t h e v ariables range o v e r t h e
natural numbers , t he integers, o r t he rat ional numbers . Since these
sets c an a l l be well-ordered and t here wi l l be a c onjunc t f o r each
number, we c an c laim t hat t here is a met hod f o r constructing t he
required inf init e c onjunc t ion. W e c an als o, o f course, dec ide i m mediately whet her a n y giv en statement i s inc luded i n t he l i s t o f
conjuncts.
294
If we are quant if y ing ov er the real numbers t he situation is more
difficult. W e k now of no wa y of well-ordering t hem and i t c an be
proven t hat t hey c an be well-ordered only i f we assume t he choice
axiom or an equiv alent principle. As long as we d o not k n o w h o w
to well-order t he real numbers we do n o t k n o w h o w t o order t he
conjuncts i n a n inf init e c onjunc t ion c orres ponding t o a s t at ement
about the real numbers. Hence we do not k now how to construct such
a conjunction and still would not k now how to even if we could perf orm an inf init e number of operations i n a f init e t ime. O n t he ot her
hand, it is not as if we were in ignorance about the contents of such a
conjunction. A statement wi l l be one o f t he conjuncts i f and o n l y
if it predicates some specified property, F, o f a real number. Henc e
we can giv e a c rit erion f or members hip i n t he c onjunc t ion but not
a c rit erion f or constructing t he c onjunc t ion.
The nex t demand concerns t he construction of matrices. This demand is lik ely t o be made b y someone wh o considers t hat giv ing
a rule only specifies one function i n terms of some ot her function. I t
then appears t hat ult imat ely s ome f unc t ions mus t be s pec if ied b y
matrix. One mus t decide whet her c onjunc t ion is t o be one of these
when one decides whet her o r not t o mak e t he demand. A c ruc ial
fact here is t hat a t rut h t able has 2" rows where n is t he number
of arguments. Henc e n o inf init e c onjunc t ion w i l l ev er hav e a denumerable t rut h table, and, according t o our previous argument, we
wi l l k now how t o construct such a mat rix only i f we k now h o w t o,
well-order t he set. This is not guaranteed ev en i f t he choice ax iom
is assumed. Thus t he t hird demand is largely unsatisfiable.
Let us n o w look t o the last demand. Since a c onjunc t ion is a non
thesis i f any o f it s conjuncts are non-theses, i t f ollows t hat i t wi l l
be practically possible t o f ind counter-instances t o a proposed thesis
of this f orm; one needs only t o f ind one conjunct whic h is not prov able. Howev er, i t wo u l d be mo r e dif f ic ult t o prov e a thesis o f t his
f orm as a t heorem i f we d o not jus t t reat a n inf init e c onjunc t ion
as a n alt ernat iv e not at ion f or univ ers al quant if ic at ion a n d us e t he
usual quant if ic at ion rules . I f one takes t he v i e w t hat inf init e c onjunctions are really basic and t hat univ ers ally quant if ied expressions
are just abbreviations f or them, one has t o ask whet her, on t he one
hand, i t is sufficient t o prov e t hat one arbit rarily chosen c onjunc t
is t rue and inf er t hat the rest are, or, o n t he ot her hand, t o prov e
that each c onjunc t is t rue. Muc h depends here o n whet her we are
t hink ing of logic or mathematics. I t is t radit ional i n mathematics t o
select an arbit rary member o f a set and t hen generalize t he result,
295
as i n Euclidean geometry. Even i n modem mathematics it is customary t o use induc t ion t o prov e s omet hing about a n inf init e n u mb e r
of numbers i n a f init e wa y by generaliz ing a propert y of an arbit rarily chosen number i n t he as s umption o f t he general case. I n a
logistic system, howev er, t his us e o f inf init e c onjunc t ions has t he
effect of reducing the predicate calculus to the propositional calculus.
This means t hat the only rule of inference we would have would be
modus ponens or something similar, and that only the t radit ional rules
of the propositional calculus would be deriv able f rom it. Hence there
would be n o wa y o f generaliz ing f r o m a n arbit rarily chosen c onjunct, and to prove such a thesis we would have t o prov e each conjunct. Since we f eel t hat theses o f t his f o r m are prov able i n logic ,
we wo u l d t hen be c ommit t ed t o proofs of inf init e lengt h.
The idea of proofs of inf init e lengt h may be upsetting if one thinks
that there ought to be an effective (finite) met hod of deciding whet her
any given wf f is a t heorem or not. Howev er, one c ould under some
circumstances a d mi t proof s o f inf init e lengt h a n d s t ill satisfy t his
demand. Th e system mi g h t be s uc h t hat one c ould i n t he met alanguage characterize t h e f o r m o f a n inf init e c onjunc t ion, perhaps
using met alinguis t ic quant if iers , wi t h o u t s t at ing t h e c onjunc t ion.
One might then have an effective decision procedure, again stated i n
the metalanguage, whic h would t ell one whet her any giv en wf f is a
theorem, or, in ot her words, whet her a proof, eit her f init e o r inf init e,
exists whic h has any specified last line. Howev er, t his sort of system
would be unusual, and one would i n any case no longer be able t o
determine in a f init e and mechanical way whet her any given sequence of f ormulas constitutes a proof.
It is now t ime to summarize our results and draw conclusions. The
first is that if one is just considering a part ic ular mathematical system
and is not t ry ing t o reduce it to logic, t here is n o part ic ular problem
in t ak ing inf init e c onjunc t ions a n d dis junc t ions as p r i mi t i v e a n d
introducing quant if iers a s def ined t erms . Ev en t hough w e c annot
construct a mat rix f o r these t rut h f unc t ions a n d c annot giv e rules
for doing so, we can, giv en an appropriate metalanguage wi t h quantifiers, express t he f unc t ion arit hmet ic ally , and i n these respects i nfinite conjunctions and disjunctions wi l l be lik e mos t ot her mat hematical f unc t ions . Statements inv olv ing thes e expressions w i l l b e
unusual i n t hat we cannot i n practice writ e t hem out i n t heir unabbreviated f orm, and, i f t hey are not denumerable w e c annot giv e
rules f o r so doing. Howev er, we do hav e a perfectly adequate wa y
of abbrev iat ing t hem, a n d t here are ot her cases where t he objec t
296
language is t oo cumbersome t o deal wi t h unless we hav e abbrev iations o f s ome sort. Some systems, f o r instance e v e n hav e a n i n finite number of axioms. Most important, we can deal wit h statements
of this sort in that we can prove t hem to be non-theorems by counterexample and prove t hem t o be theorems using the methods discussed
above. Wh i l e we are i n a s light ly bet t er position wh e n t he inf init e
conjunctions are denumerable, ev en when they are not, t here seems
to be no decisive objection t o reduc ing quantifiers ult imat ely t o conjunctions and disjunctions i n a t y pic al mat hemat ic al system.
The s it uat ion i s s t ill largely unc hanged ev en wh e n w e c ons ider
a system of logic prov ided t hat we are not t ry ing t o collapse a l l o f
logic a n d mathematics int o one large system. Th e ma i n dif f erenc e
here is t hat we are lik ely t o hav e v ery limit ed rules o f inf erenc e
whic h wi l l not a l l o w us t o prov e t heorems inv olv ing inf init e c onjunctions o r dis junc t ions i n t h e objec t language. Instead, w e w i l l
have t o prov e met alinguis t ic ally t hat such inf init e proof s d o ex is t
in t he object language. Again, t his need not be a f at al difficulty, and
there may even be an effective decision procedure f or theoremhood.
More dif f ic ult ies appear wh e n we c ome t o constructing a unif ied
system o f logic and mat hemat ic s (inc luding at least arit hmet ic and
set theory). Historically, i t has been primarily the logicists wh o have
want ed to do this b u t the program need not be limit ed t o those wh o
have want ed t o f ound mat hemat ic s i n logic. One mi g h t als o wa n t
to bring t he f ormalis t plogram t o it s logic al conclusion a n d model
as ma n y systems as possible i n a giv en system, wh i c h mi g h t t urn
out t o be a propos it ional calculus, a system o f arithmetic, o r s ome
other k i n d o f system. W e need not ask wh a t s ort o f system is t o
have all the others modelled in it in order to see what happens when
we eliminat e quant if iers i n it . W e are as s uming t hat we s t ill hav e
the quantifiers, «all» a n d «some» i n Englis h s o t hat w e c an s t ill
express t h e r u l e f o r t h e ev aluat ion o f inf init e c onjunc t ions i n a
metalanguage and in a completely non-symbolic way. There does not
seem t o be any great loss here: t he r u l e is s t ill t he s ame whet her
we can state it in arithmetic, i n logical symbolism, or jus t in English.
If one takes t he logic is t rat her t han t he f ormalis t point o f v iew,
one is not unif y ing systems just f or purposes o f elegance or t o bring
about greater unif ormit y i n mathematics, but ult imat ely i n order t o
insure t he reliabilit y o f mathematics. Th e propos it ional c alc ulus i s
(
dations
3
of mathematics are i n c hapter I I o f this book.
)
J
.
M
.
K
E
Y
N
E
S
,
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the system wit h the most int uit iv e axioms and such logicists as K e y nes would have been very muc h attracted by the t hought of reducing
the rest of logic t o t he propositional calculus v ia t he eliminat ion of
quantifiers, a n d t h e n r e d u c i n g mat hemat ic s t o t h i s c ondens ed
logic (
3 cannot actually produce the required proofs in the object language
we
) . t he propos it ional calculus, b u t c an only prov e t hat s uc h proof s
of
H o and do that only in the metalanguage. I n order to produce these
exist,
w e
metalinguistic
proofs we mus t be us ing all sorts of princ iples o f inv e
ference
t hat go f a r bey ond t he rules o f t he propos it ional calculus.
r ,
Even
i f t he metatheorems are s elf -eliminat ing i n t hat t hey giv e us
m
directions f o r c arry ing out t he proof s i n t he objec t language, s t ill
uthat mus t be prov en i n some way . Fr o m this point of v i e w not hing
ccan ever replace the lac k of real f init e proofs i n t he object language
hthat one can look at and check. I f one's point of v iew is less logicist
oand one is jus t t ry ing t o unif y systems and eliminat e quant if ic at ion
fat the same t ime, t his is muc h less troublesome. Howev er, even here
tone may f ind t hat more t han one had init ially supposed has grav it at hed f r o m the objec t language t o t he metalanguage and t hat mos t o f
ethe real mat hemat ic al and logic al argument remains in a completely
aunforrnalized o r s emi-f ormaliz ed state i n v arious ,metalanguages.
t Of course, t here wi l l again be n o rule f o r t he construction o f a
tmat rix corresponding t o a n inf init e c onjunc t ion unles s we pres uprpose t he ax iom o f choice and discover a wa y o f well-ordering t he
arows o f a t rut h t able of t his sort. This f ac t continues t o constitute
cno dif f ic ult y f or the formalist, even one wh o want s t o unif y systems
tto t his ex tent a n d t o eliminat e quantifiers. H e is probably wi l l i n g
ito admit t he ax iom of choice i n t he f irs t place, and i n t he second
oplace, h e wo u l d be unlik ely t o mak e a demand o f t rut h f unc t ions
nin logic whic h he does not mak e of other functions in ot her branches
iof mathematics. Thirdly , and most important, i f the f ormalis t models
severything else i n logic , t he primit iv es o f logic w i l l r e ma i n u n i n lterpreted. Th i s means t hat a n inf init e c onjunc t ion need n o t be a
ofunction a t all unless one chooses t o int erpret i t t hat way . Ev en i f
sone does so int erpret logic, t he specification of t he functions is part
tof t he interpretation of t he system and not part of t he system itself.
iThe logicist, on the other hand, regards the primit iv es of the propos if tional calculus as hav ing a set meaning, o r he at least regarde t he
i customary interpretation of the system as being of prime importance.
t Further, i t i s characteristic o f Ramsay's k i n d o f logic is m t o plac e
t great emphasis o n t he concept o f a t rut h-t able t aut ology f o r philou
r298
n
s
o
u
t
t
h
a
t
sophical reasons mos t of whic h we cannot discuss here. One reason,
however, is that if the ax ioms of the system can be seen to be t rut htable tautologies t hey are c ert ain i n a v ery s t rong sense, a n d t hus
of c ruc ial import anc e t o t he logicist. I f t he system has as ax ioms
conjunctions of inf init e lengt h, we c annot be assured t hat t hey are
tautologies i n t he same wa y that we should i f they were f init e; t he
situation is made worse if they are not even denumerable. Of course,
we might s t ill convince ourselves t hat the ax ioms and theorems are
all tautologies, but we wo u l d hav e t o do t his i n t he Englis h met alanguage in an int uit iv e way, and one c ould not really call this giv ing
a proof. Anot her dif f ic ult y is t hat t he Ramsayan logicist is lik ely t o
t hink t hat t rut h functions i n part ic ular mus t be specified by mat rix
rather t han by rule because t hey c ome t o have a v ery special role
that ot her functions do not have. But we hav e seen t hat we do not
even k now how t o go about constructing these matrices.
On t he ot her hand, ev en i f t he system does c ont ain inf init e c onjunctions and disjunctions, these need not necessarily appear i n t he
axioms. I n part ic ular, we k n o w t hat t he rules o f quant if ic at ion c an
be reduced t o t he propos it ional rules of addit ion and s implif ic at ion
so it might be possible t o do wit hout any expressions i n t he ax ioms
that wo u l d ordinarily inv olv e quantification. I n t hat case t he Ramsayan logicist's s it uat ion wo u l d b e c ons iderably improv ed because
we c ould t hen ac t ually construct t r u t h t ables f o r t he ax ioms . O f
course, if there are theorems of inf init e length, as there mus t be, then
there wi l l s omewhere hav e t o be proofs o f inf init e lengt h wi t h t he
difficulties t hat we ment ioned above. Thus t he ax ioms may n o w be
certain i n a v ery strong sense, but we wi l l nev er be quit e so c ert ain
that t he theorems f o l l o w f r o m t he axioms. Muc h of t he mot iv at ion
f or this k ind of program lies i n uneasiness about the us ual methods
of proof in mathematics and even i n logic. Hence the attempt t o f ind
principles, s uc h as t rut h-t able tautologies, wh i c h are impos s ible t o
doubt (alt hough s ome hav e been doubted), a n d t he consequent reduction of a l l ot her principles t o t hem. I t is here t hat t he necessity
of hav ing inf init e c onjunc t ions a n d dis junc t ions i n t h e ax ioms o r
inf init e proof s intrudes. Thes e d o not bot her t he ordinary wo r k i n g
mathematician o r logic ian, w h o nev er has t o d e a l wi t h t h e m i n
practice, but they lac k t he k ind of certainty whic h f init e t rut h-t able
tautologies hav e and t heir presence vitiates t his k i n d o f program.
These difficulties a p p l y only if one is t ry ing t o make mathematics
and logic more reliable by perf orming this sort of reduction. I f one
is not wo r r i e d about arit hmet ic tottering, as Frege was , b u t is as -
299
serting t hat t he whole consists of tautologies jus t as a n at t empt t o
define t h e nat ure o f mat hemat ic s a n d logic , w e hav e a n ent irely
different case. An inf init e tautology wit h a truth-table we do not k now
how t o construct is s t ill a tautology, a n d t he presence o f inf init e
conjunctions a n d disjunctions does not i n a n y wa y cast doubt o n
this thesis about the nature of mathematics what ev er other difficulties
it ma y have. I n fact, i t is as lik ely t hat t he f ollower o f Ramsay is
saying something about the nature of mathematics as i t is t hat he is
trying to provide mathematics wit h a more secure foundation. Hence,
our c onc lus ion about t h e legit imac y o f inf init e c onjunc t ions a n d
disjunctions must be that it all depends on what one proposes t o use
th
e
m
Wi l l i a m TODD
University of Cinc innat i
f
o
r
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