F.P. Ramsey - On a problem of formal logic

264 F. P. Ramsey

ON A PROBLEM OF FORMAL LOGIC

By F. P. Ramsey.

[Received 28 November, 1928.-Read 13 December, 1928.]

This paper is primarily concerned with a special case of one of theleading problems of mathematical logic, the problem of finding a regularprocedure to determine the truth or falsity of any given logical formula*.

But in the course of this investigation it is necessary to use certaintheorems on combinations which have an independent interest and aremost conveniently set out by themselves beforehand.

I.

The theorems which we actually require concern finite classes only,but we shall begin with a similar theorem about infinite classes which iseasier to prove and gives a simple example of the method of argument.

Theorem A. Let T be an infinite class, and jjl and r positive integers ;and let all those sub-classes of T which have exactly r members, or> as wemay say, let all r-combinations of the members of F be divided in anymanner into jul mutually exclusive classes C{ (i~ 1, 2, ..., jjl), so thatevery r-combination is a member of one and only one C*; then, assumingthe axiom of selections, V must contain an infinite sub-class A such thatall the r-combinations of the members of A belong to the same C-\.

Consider first the case /jl = 2. (If yu. = 1 there is nothing to prove.)The theorem is trivial when r is 1, and we prove it for all values of r byinduction. Let us assume it, therefore

, when r = p-1 and deduce itfor r = p, there being, since fx - 2, only two classes Cif namely C\ andC2.

* Called in German the Entscheidungsproblem\ see Hilbert und Acksrmann, Grwxdziigcder Theoretischen Logik, 72-81.

1928.] On a problem of formal logic. 265

It may happen that T contains a member X\ and an infinite sub-classTi, not including Xi, such that the p-combinations consisting of X\ to-gether with any p-1 members of Fx, all belong to C\. If so, Ti maysimilarly contain a member x2 and an infinite sub-class T2, not includingx2, such that all the p-combinations consisting of x2 together with p-1members of F2, belong to C\. And, again, r2 may contain an x$ and ar3 with similar properties, and so on indefinitely. We thus have twopossibilities : either we can select in this way two infinite sequences ofmembers of T (xu x2, ..., xn, ...), and of infinite sub-classes of

r (r\, F2, ..., rw, ...), in which xn is always a member of r«_i, and Tna sub-class of Tn

_i not including xn, such that all the p-combinationsconsisting of xn together with p-1 members of I

,w, belong to Ci; orelse the process of selection will fail at a certain stage, say the n-th,because Fn_i (or if n = 1, T itself) will contain no member xn and infinitesub-class rw not including xn such that all the p-combinations consistingof xn together with p-1 members of Fn belong to C\. Let us take thesepossibilities in turn.

If the process goes on for ever let A be the class (xi, x2, ..., ...).Then all these xfs are distinct, since if r>s, xr is a member of Fr_i and

so of rr_a, rr_3, and ultimately of Ts which does not contain xs.Hence A is infinite. Also all p-combinations of members of A belong toCi; for if xs is the term of such a combination with least suffix s, the other

p-1 terms of the combination belong to Ts, and so form with xs a

p-combination belonging to C\. F therefore contains an infinite sub-class A of the required kind.

Suppose, on the other hand, that the process of selecting the #,s andF,

s fails at the n-th stage, and let yi be any member of Fu_i. Then the

(p-l)-combinations of members of rn_i -(t/i) can be divided into twomutually exclusive classes C[ and according as the p-combinationsformed by adding to them t/i belong to Ci or C2, and by our theorem (A),which we are assuming true when r == p-1 (and fx = 2), Tn_i-(yi) mustcontain an infinite sub-class Aj such that all (p-l)-combinations of themembers of A! belong to the same C[; i.e. such that the p-combinationsformed by joining yY to p-1 members of A1 all belong to the same CL.

Moreover, this Ci cannot be Ci, or i/x and Aj could be taken to be xn and

r„ and our previous process of selection would not have failed at the?x-th stage. Consequently the p-combinations formed by joining yi top-1 members of Ax all belong to C2. Consider now Ax and let y2 beany of its members. By repeating the preceding argument Al - (y2)must contain an infinite sub-class A2 such that all the p-combinationsgot by joining y2 to p-1 members of A9j belong to the same Ch

266 F. P. Bamsey [Dec. 13r

And, again, this Ci cannot be Gi, or, since is a member and A2 a sub-class of Ax and so of F«_i which includes Ax, y% and A2 could have beenchosen as xn and Tn and the process of selecting these would not havefailed at the n-th stage. Now let y3 be any member of A2; then A2-(*/3>must contain an infinite sub-class As such that all ÿ-combinations con-sisting of y-A together with p-1 members of Aa, belong to the same Ciywhich, as before, cannot be G\ and must be C2. And by continuing inthis way we shall evidently find two infinite sequences yu y2, yu,and AJ} A2, ..., A,t, ... consisting respectively of members and sub-classes.of T, and such that yn is always a member of Au_i, An a sub-class of An_jnot including ynj and all the p-combinations formed by joining yn to p-1members of An belong to C2; and if we denote by A the class(yi> 1/2, ..., ynt ...) We have, by a previous argument, that all p-combina-tions of members of A belong to C2.

Hence, in either case, F contains an infinite sub-class A of the re-

quired kind, and Theorem A is proved for all values of r, provided thatp = 2. For higher values of fx we prove it by induction; supposing italready established for p. = 2 and p - v-1, we deduce it for p = v.

The r-combinations of members of T are then divided into v classes

Ci (i = 1, 2, v). We define new classes CJ for i = 1, 2, ..., v-1 by

C\ = Ci (i = 1, 2, v-2),

C.

'

-i =

Then by the theorem for p = v-1, T must contain an infinite sub-class A such that all r-combinations of the members of A belong to thesame C\. If, in this C'u i ÿ v-2, they all belong to the same Ci, which isthe result to be proved; otherwise they all belong to CI-1, i.e.

either to

Cv-i or to Cv. In this case, by the theorem for p - 2, A must contain aninfinite sub-class A, such that the r-combinations of members of A'

either all belong to C„_i or all belong to Cv; and our theorem is thusestablished.

Coming now to finite classes it will save trouble to make some con-ventions as to notation. Small letters other than x and y, whether

Italic or Greek (e.g. n, r, p, m) will always denote finite cardinals,

positive unless otherwise stated. Large Greek letters {e.g. T, A) willdenote classes, and their suffixes will indicate the number of their

members (e.g. T1M is a class with m members). The letters x and y willrepresent members of the classes I\ A, etc., and their suffixes will be

used merely to distinguish them. Lastly, the letter C will stand, asbefore, for classes of combinations, and its suffixes will not refer to the

19-28.1 On a problem of formal logic 267

number of members, but serve merely to distinguish the different classes

of combinations considered.

Corresponding to Theorem A we then have

Theorem B. Given any r, n, and /* we can find an m0 such that, ifm ÿ m0 and the r-combinations of any Tw are divided in any manner intofi mutually exclusive classes Ci (i = 1, 2, ..., jx), then must contain asub-class An such that all the r-combinations of members of An belongto the same Ci.

This is the theorem which we require in our logical investigations, andwe should at the same time like to have information as to how large momust be taken for any given r, n, and fx. This problem I do not know howto solve, and I have little doubt that the values for m0 obtained below are

far larger than is necessary.To prove the theorem we begin, as in Theorem A, by supposing that

H = 2. We then take, not Theorem B itself, but the equivalent

Theorem C. Given any r, n, and k such that n+kÿr, there is anm0 such that, if m ÿ mo and the r-combinations of any Tm are dividedinto tico mutually exclusive classes Ci and C2, then FWI must contain twomutually exclusive sub-classes An and A* such that all the combinationsformed by r members of Au+Ak which include at least one member froiiiA

n belong to the same C

That this is equivalent to Theorem B with u = 2 is evident fromthe fact that, for any given r, Theorem C, for n and fc, asserts morethan Theorem B for n, but less than Theorem B for n+k.

The proof of Theorem C must be performed by mathematical in-duction, and can conveniently be set out as a demonstration that it ispossible to define by recursion a function /(r, n, k) which will serve asm0 in the theorem.

If r = l, the theorem is evidently true with m0 equal to the greaterof 2n-1 and n+k, so that we may define

/(1, n, k) = max (2?i-1, n+k) (v ÿ 1, k ÿ 0).

For other values of r we define /(r, n, k) by recursion formulae in-volving an auxiliary function g(r, n, k). Suppose that f(r-1, n, k) hasbeen defined for a certain r-1, and all n, k such that n+k ÿ r-1, thenwe define it for r by putting

f(r, 1, k) =f(r-1, k-r+2, r-2) + l (& + 1 > r),g(r, 0, k) - max (r- 1, k),

g(r, n, k) =/|r, 1, g(r, n-1, &)} (» > 1),f(r, n, k) = /{r, n-1, g(r, n, Ar)[ (n > 1).

268 F. P. Eamsey [Dec. 13,

These formulae ca

values of r, n and

be easily seen /< k) for4-k r, and {/(r, n, k) for all values

of r greater than 1, and all positive values of n and k\ and we shallprove that Theorem C is true wWe know that this is so when r

Wq to be this /( fc).

1, and we shall therefore assume it

for all values up to r-1 and deduce it for r.When n 1, and m m0 f(r-1, k-r+2, r-2)+l, we may take

any member x of Tm to be sole member of A1 and there remain at least/(r»-1, k-r+2, r-2) members of T (x); the (r- l)-combinations ofthese members of V

VI (x) can be divided into classes C[ and C'> accord-

ing as they belong to C\ or C% when x is added to them, and, by ourtheorem for r--1, TVI (x) must contain two mutually exclusive classesA/

,

- r+ 2, Ar_ such that every combination of r-1 terms from A;._r+2+A,._2

<8ince one of its terms must come from Afc_r+ 2 iV), having only r-2members) belongs to the same C\. Taking Ak to be this Aÿ r + 2 ~}~ Ar_o

all combinations consisting of x, together wTith r-1 members of A*, be-long to the same Ci. The theorem is therefore true for r when n = 1.

For other values of n we prove it by induction, assuming it for n - land deducing it for n. Taking

m m f(r, n, ft) f\ r, n - l, g(r, n, k)\,

rm must, by em for n-l, contain and a A9 ( >', «» k) such

combination of r members of A»_i + A„/r n m, -at least one termcomes from C to C If, now,

member x and a sub

combination of x and r-1 members of Ak belongs to Cbe + (.?) and theorem If

berbe no member of \(r

,ntk) which has

, connected with it in this way. But

sub-class of ft mem

g(r, n, ft) f\r, 1, gn-l, k)},

Ag(r,n,k) must contain a member x1 and a sub-class Aÿ not including

xlf such that x1 combined with any r- 1 members of Apÿn-i,*) gives a

combination belonging to the same Cif which cannot be Clf or x1 and anyk members of Agÿn-i

.k) could have been taken as the x and At above.

Hence the combinations formed by x} together with any r-1 membersof A9(r all belong to C2. But now

g(r, n-l, k) f\r, 1, g{r, n-% ft)},

md A„(r u-i n must contain an x2 and a Ag(r n-2 k), not including x2, suchcombinations formed by x2 and members


belong to the saroe Cu which must, as before, be C2, since x2 and Agÿn_otk

are both contained in Ay(r Wft) and g(r, n-2, k) k

. Continuing in thisway we can find n distinct terms xx, x2, x* and a Aÿo,/..> such that everycombination of r terms from (xl9 x2t ..., xw)+Ag(r>

0, t) belongs to C2, pro-vided that at least one term of the combination comes from (aÿ, x2, ..., xn).Since k this nroves our theorem, taking A™ to be

and A/; to be any Jc terms of Agÿto,iy

Theorem C is therefore established for all values of r, n, and k,

with trio equal to /(r, n, k). It follows that, if jjl 2, Theorem B is true

for all values of r and n with m0 equal to /(r, n-r+1, r-1), which weshall also call /i(r

, n, 2).

of fi we prove Theorem B byto

h(r, n, 2)

h(r, n, m) =

'

-fir, n-r+1, r- 1)

li\ r9 h(r, w, /x - 1), (m 2)

For, assuming the theorem for jx-1, we prove it for n by defining newclasses of combinations

Ci Cu

Ci 2 Cir-2

If /t(r, n, ju) /i{2

, rm must contain a T i)

belong either all to C C

r, 7i, jjl-1), 2}, by the theorem forhe r-combinations of whose members

In the first case there is no more to

prove; in the second we have only to apply the theorem for p-1 torh(r,Tt,n-1)

In the simplest case in which 2 the above reasoning givesm0 equal to 7i(2, n., 2), which is easily shown to be 2n(W~1),i. But for thiscase there is a simple argument which gives the much lower valuem-o = n !

, and shows that our value /i(r, n, //) is altogether excessive.For

, taking Theorem C first, we can prove by induction with regardto n that, for r 2, we may take m0 to be k . (n+1)!. (7c is here supposedgreater than or equal to 1.) For this is true when n 1, since

, ifin 2k, of the m-1 pairs obtained by combining any given member ofrm with the others, at least k must belong to the same Cit, then, for n-1, let us prove it for n.

Assuming

If m k . (n+1)! /c(n-f-l) .n\y rm must, by the theorem for n-1,contain two mutually exclusive sub-classes Au_i and A/-(w+i) such that allpairs from AH_1 + Afc(N+])< least one term of which comes from An_1?belong to the same Ci, say C\. Now consider the members of Ai:(n+\)\ in,

270 F. P. Ramsey [Dec. 13,

the first place, there may be one of these, x say, which is such that thereare k other members of which combined with x give pairsbelomnnrr to Cÿ. If so. the theorem is true

, taking" An to be A?l_i + (x):member Ai

other members of A/./„(n+i)*

Then fc-l

(« +1)

C A (u(H + l) must

combined with Xi a pair belonging to C

which combined with xx give pairs belongingist contain a A*,, any member of which givesair belonging to C2. Let x2 be any member of

Aknt then, since x2 and A./l7l are both contained in A/;(u+1), there are at most1 other members of A/-n which when combined with x2 give pairs

belonging to Ci. Hence A/.n-W contains a Awn- i) any member of

which combined with x2 gives a pair belonging to C2. Continuing in thisway we obtain Xi, x2) xn and A/., such that every pair xi9 x} andevery pair consisting of an Xi and a member of Ak belongs to Ci.

Theorem C is therefore proved.Theorem B for n then follows, with the m0 of Theorem C for n-1

and 1, i.e. with m0 equal to n!*; and it is an easy extension to show-that, if in Theorem B r 2 but =£ 2, we can take mQ to be n ! ! !, ...,

where the process of taking the factorial is performed /x-1 times.

II.

We shall be concerned with logical formulae containing variablepropositional functions, i.e. predicates or relations, which we shall denoteby Greek letters <p, x> etc. These functions have as arguments in-dividuals denoted by x, y, z, etc., and wTe shall deal with functions withany finite number of arguments, i.e. of any of the forms

${x), %(x, 7j), \[r(x, y, z), ....In addition to these variable functions we shall have the one constant

function of identityx y or (*, y).

By operating on the values of <f>, ...> and with the logical opera-tions

meaning not,V

(x)

(Ex)

1*

9 y

or,

and,

for all x.

there is an x for which,

* But this value is, 1 think, still much too high. It can easily be lowered slightly even%

when following the line of argument above, by using the fact that if k is even it is impossiblefor every member of an odd class to have exactly k - 1 others with which it forms a pair of Cufor then twice the number of these pairs would be odd ; we can thus start when k is even with& fi-k (»»+1) -1 instead of a Ajt(n + i)

1928.; On a problem of formal logic.

we can construct expressions such as

[(X, y){<j>{x, y) V x = y}]V {(Ez)x(z)}in which all the individual variables are made "apparent" by prefixes(x) or (Ex), and the only real variables left are the functions <p, x> ....Such an expression we shall call a first order formula.

If such a formula is true for all interpretations* of the functionalvariables <£, eÿC> W® Caÿ ÿ valid, and if it is true for no inter-pretations of these variables we shall call it inconsistent. If is truefor some interpretations (whether or not for all) we shall call itconsistent t.

The Entscheidungsproblem is to find a procedure for determiningwhether any given formula is valid, or, alternatively, wThether any givenformula is consistent; for these two problems are equivalent, since thenecessary and sufficient condition for a formula to be consistent is thatits contradictory should not be valid. We shall find it more convenientto take the problem in this second form as an investigation of consistency.The consistency of a formula may, of course, depend on the number ofindividuals in the universe considered, and we shall have to distinguishbetween formulae which are consistent in every universe and those whichare only consistent in universes with some particular numbers ofmembers. Whenever the universe is infinite we shall have to assume the

axiom of selections.

The problem has been solved by BehmannJ for formulae involvingonly functions of one variable, and by Bernays and Schonfinkel§ forformulae involving' only two individual apparent variables. It is solved

4

below for the further case in which, when the formula is written in»

*'normal form"

, there are any number of prefixes of generality (x) butnone of existence (Ex)\\. By

' 'normal form"H is here meant that all the

prefixes stand at the beginning, with no negatives between or in frontof them, and have scopes extending to the end of the formula.

* To avoid confusion we call a constant function substituted for a variable <p, not a valuebut an interpretation of <p ; the values of <p (x, y, z) are got by substituting constant individualsfor x, 7/, and z.

| German erfilllbar.| H. Behmann, "Beitrage zur Algebra der Logik und zum Entscheidungsproblem",

Math. Annalen, 86 (1922), 163-229.§ P. Bernays und M. SchSnfinkel, "Zum Entscheidungsproblem der mathematischen

Xiogik Matli. Annalen, 99 (1928), 342-372. These authors do not, however, include identityin the formulae they consider.

|| Later we extend our solution to the case in which there are also prefixes of existenceprovided that these all precede all the prefixes of generality.

ÿ1 Hilbert uad Ackermann, op. cit., 63-4.

272 P. P. Ramsey

4

[Dec. 13,.

The formulae to be considered are thus of the form

(#1, Xty ..., X<nj F{<f>y iff) i *ÿ1j *ÿ2j ...> Xn)>

where the matrix F

<f>, x, etc., and = for arguments drawn from xu ÿ2, ..., %*-This type of formula is interesting as being the general type of an

axiom system consisting entirely of "general laws

,,*. The axioms for

order, betweenness, and cyclic order are all of this nature, and we arethus attempting a general theory of the consistency of axiom systems ofa common, if very simple, type.

If identity does not occur in F the problem is trivial, since in thiscase whether the formula is consistent or not can be shown to be in-

a function of values of the functions

dependent of the number of individuals the universe, and we have

only the easy task of testing it for a universe with one member onlyf.But when we introduce identity the question becomes much more

difficult, for although it is still obvious that if the formula is consistentin a universe U it must be consistent in any universe with fewer membersthan U, yet it may easily be consistent in the smaller universe but not

the larger I

(xv xa) [u;l z2 V | tiixj. 0 (*2)} ]

is consistent in a universe with only one member but not in any other.We begin our investigation by expressing F in a special form. F is

a truth-function of the values of <f>. x> "ft, ..., and for arguments drawnfrom X\t %2, ..., %n- If <f> is a function of r variables there will be iir\

values of <f> which can occur in F, and F will be a truth-function of *

2ri

values of <j>, x, and , which we shall call atomic propositions.With regard to these 2ny atomic propositions there are 22 nr possibilitiesof truth and falsity which we shall call alternatives, each alternative beinga conjunction of propositions which are either atomic propositionsor their contradictions. In constructing the alternatives all the 2;i

atomic propositions are to be used whether or not they occur in F. Fcan then be expressed as a disjunction of some of these alternatives,

namely those with which it is compatible. It is well known that such an

* C. H. Langford, *, Analytic completeness of postulate sets", Proc. London Math. Soc.(2), 25 (1926), 115-6.

f Bernays und Schonfinkel, op. cit359. We disregard altogether universes with m>members.

X Here and elsewhere numbers are given not because they are relevant to the argument,but to enable the reader to check that he has in mind the same class of entities as the author.

1928/ On a problem of formal logic 273

mann call tlie i i

)le; indeed, it is the dual of what Hilbert and Acker-ausgezeichnete konjunktive Normalform

"*, and is

Wittgenstein The only exception is whenF is a tradictory truth-function, in which case our formulacertainly not consistent.

F having been thus expressed as a disjunction of alternatives (in ourspecial sense of the word), our next task is to show that some of thesealternatives may be able to be removed without affecting the con-

formula be

remov

consider the alternatives that remain

In the first place an alternative may violate the laws of identity bycontaining parts of any of the following forms :-

Xi =£

x Xj . Xj =£ Xi

X Xj . Xj Xj- . Xi X); (*=£./> J =£ k, kÿi),

or by containing Xitradictory ~ <p for sets osubstituted for xj [e.ÿ. xx

ling Xi = Xj (i and values of a function <j> and its con-<f> for sets of arguments which become the same when Xi is

Xq . <p{Xi, X%i X§) . xv X8)] .Any alternative which violates these laws must always be false and can

evidently be discarded without affecting the consistency of the formula.The remaining alternatives can then be classified according to the numberof xf

s they make to be different, which may be anything from 1 up to n.Suppose that for a given alternative this number is v, then we can

derive from it what we will call the corresponding y alternative by thefollowing process :-:

For xly wherever it occurs in the given alternative, write yi; next, ifin the alternative x2 #i, for x2 write yi again, if not for write y2. Ingeneral, if x-t is in the given alternative identical with any Xj with j lessthan i, write for x

t the y previously written for Xj; otherwise write forXi, ?//.+], where k is the number of y,s already introduced. The expressionwhich results contains v t/,s all different instead of n x,

s, some of which

are identical, and we shall call it the y alternative corresponding to thegiven x alternative.

* Op. cit., 16.t We write x zfzy ioi ~ (x ~ij)

ser. 2. vol. 30. no. 1726.

F. P. Ramsey Dec. IB

Thus to the alternative

<f>(Xi) <j> (x2). <f> (x3) <p(x±). xl *3 . X2 Xÿ.XiÿX

corresponds the y alternative

$(2/2) * Ui ÿ 2/2

We call two y alternatives similar if they contain the same numberof 1/ swe ca

and can be derived from one another by permuting those y1

s, and

1 two x alternatives equivalent if they correspond to similar (oridentical) y alternatives.

Thus

0(«i) - (ÿ2) * 0 (*a) <f>(Xt) m X . X xi.xiÿ= x2, (a)

is equivalent to

. ÿ(*ÿ4) . "4~ «ÿ2 * *ÿ2 - *ÿ3 - *ÿ4»

since they correspond to the similar y alternatives

<t>(y 1) . ÿ 0(y2) . =£ &

- ÿ(yi) . ÿ(1/2)

derivable from one another by interchanging t/i and t/2j although (a)and (j3) are not so derivable by permuting the x

i

s.

We now see that we can discard any alternative contained in Funless F also contains all the alternatives equivalent to it; e.g. if Fcontains (a) but not i a) may be discarded from it. For omittingalternatives clearly cannot make the formula consistent if it was not sobefore; and we can easily prove that, if it was consistent before, omittingthese alternatives cannot make it inconsistent.

For suppose that the formula is consistent, i.e. that for some particularinterpretation of <p, x> ...> F is true for every set of x,s, and let pbe an alternative contained in F an alternative equivalent to p but notcontained in F. Then for every set of x

,

s one and only one alternativein F will (on this interpretation of <p, ...) be the true one. and thisalternative can never be p. For if it were p, the corresponding y alterna-tive would be true for some set of t/,s, and the similar y alternativecorresponding to would be true for a set of t/,s got by permutinlast set. Giving the x,s suitable values in terms of the y q would then

* We take one function of one variable only for simplicity; also to save space we omitexpressions which may be taken for granted, such as xx = xlt xx zfz x4.

1928.; Ox A PROBLEM OF FORMAL LOGIC. 275

be true for a certain set of x,s and F would be false for these x,s con-

trary to hypothesis. Hence p is never the true alternative and may beomitted without affecting the consistency of the formula.

When we have discarded all these alternatives from F, the remainder

will fall into sets each of which is the complete set of all alternativesequivalent to a given alternative. To such a set of x alternatives will

correspond a complete set of similar y alternatives, and the disjunction ofsuch a complete set of similar y alternatives (i.e. of all permutations of

form A. form containing v y:s we

shall denote by an Italic capital with suffix v, e.g. Av, Bv.The force of our formula can now be represented b}'

conjunction, which we shall call P.the following

For every yv

For every distinct yu y2,

Ax or Bx or

A a or Bo or

t t I .

4

For every distinct yu y2, ..., yv, Av or Bv or. .

For every distinct yv y.2, .yH> A n or Bn or ..

(P)

where Avi B„+, etc., are the forms corresponding to the x alternatives

still remaining in F. If for any v there are no such forms, i.e. if noalternatives with v different #,

s remain in F, our formula implies thatthere are no such things as v distinct individuals, and so cannot beconsistent in a world of v or more members.

We have now to define what is meant by saying that one form is in-volved in another. Consider a form Av and take one of the y alternativescontained in it. This y alternative is a conjunction of the values of<f>, an<ÿ their negatives for arguments drawn from(We may leave out the values of identity and difference, si]for granted that y1s are always different.) If fx «< v we cfor granted that y1s are always different.) If fx «< v we can select fi ofthese y,s in any way and leave out from the alternative all the terms init which contain anv of the v-a iy,s not selected. We have left anit which contain any of the v-fi y"s not selected. We have left analternative in fx y

,

s which we can renumber yu 1/2, ..., I/M, the form

Ep to which this new alternative belongs we shall describe as beinginvolved in the Av with which we started. Starting with one particulary alternative in A,, we shall get a large number of different Eÿs by

* Cf. Langford, op. cit116-120.f The notation is partially misleading, since A„ has no closer relation to Athan to Bÿ

T 2


ng the fx t/'s which we select to preserve; and fromwhichever y alternative in A

., we start, the E 's which we find to be

olved in Av w

For example,

be the same

{0(2/1. 1/d ÿ 9%1> Vi) . yi) .

Vi ~ 2/1) . yd ÿ yd ÿ fÿy2. yi>)is a form A2 which involves the two Ei,s

<p(yi> Vi)

<p(yi> Vi)-

It is clear that if for some distinct set of v «?/,s a form Av is true, thenevery form involved in Av will be true for some distinct set of fx t/

"

s

contained in the v.

We are now in a position to settle the consistency or inconsistencyof our formula when A7, the number of individuals in the universe, is less

number of re,s in our formula In fact, if N n,

it is necessary and sufficient for the consistency of the formula that Pshould contain a form AN together with all the forms EM involved in it

for every jjl less than N.This condition is evidently necessary, since the A7 individuals in the

this A

in :

must be for

of w,

s. and so contained in P if P is to be true for

any 0, x> ...; an<ÿ forms involved in this AN must be true fordifferent selections of \f s, and so contained in P if P is to be true forthis <f>, x, ÿ.

Conversely, suppose that P contains a form AN together with all formsinvolved in then, calling the N individuals in the universeVi> 2/2, yk, we can define functions 0, x> Vs ... to make any assignedy alternative in A N true; for any permutation of these N y's anotheralternative in AN will be the true one, and for any subset of y

f

s some

y alternative in a form involved in Ay. Since all these y alternatives areby hypothesis contained in P, P will be true for these <f>, and ourformula consistent.

When, however, N n the problem is not so simple, although itclearly depends on the A w s in P such that all forms involved in them arealso contained in P. These An's w.e may call completely contained in P,and if there are no such AnS a similar argument to that used whenN n will show that the formula is inconsistent. But the converse

argument, that if there is an An completely contained in P the formulamust be consistent, no longer holds good; and to proceed further we haveto introduce a new conception, the conception of a form being serial.


In view of this fact we shall find it more convenient to take P in

its new form, and denote the new set of functions by <p0, \o, Y,o, ....Suppose, then, that <p0 is a function of r variables; there are

n(n - 1)... (w -./.+1)

values of <£0 with r different arguments drawn from yi, y2, ..., yn andevery y alternative must contain each of these values or its contradictory.r! of these values will have as arguments permutations of t/i, y2, . yr-

Any other set of r y*s can be arranged in the order of their suffixes asy*i> yy*r,

sx < s2< s3 ... < sr, and it may happen that a givenalternative contains the values of <£0 for those and only those permuta-tions of ySl1 ySi, yH which correspond (in the obvious way) to thepermutations of yi, i/3, ..., yr for which it (the alternative) contains thevalues of $0; e.g. if the alternative contains <f>o(yi, #2, yr) and</>o(yr, yr-1, 2/1), but for every other permutation of y1% t/2, ..., yr con-

tains the corresponding value of <£0, then it may happen that thealternative contains <p0(y,„ y,„ y,) and <po0j.-r,

»/«,.„ y,,), but forevery other permutation of ?/<v ?/v y,r contains the correspondingvalue of ÿ <pQ.

If this happens, no matter how the set of r y*s, ySv ySi, ySr is

chosen from y 1, y2, y.„, then we say that the alternative is serial in<po

*, and if an alternative is serial in every function of the new set we

shall call it serial simply.Consider, for example, the following alternative, in which we may

imagine <f>0 and to be derived from one "old" function <f> by thedefinitions

</>o(y» yd =

ÿo(yd = <p(yi, yd,

<Po(y\> 2/2) . ÿ 2/1) . 2/3) . ÿ Vi)' 0oO/2> yÿ . ÿ 0o(ys. 2/2)'

ÿ0(2/1) . ÿ ÿ0(2/2). ÿo(ys)'

This is serial in <f>0, since we always have 0o(2/v but notin yp-Q, since we sometimes have \Js0(y s),Jbut sometimes - Henceit is not a serial alternative.

We call a form serial when it contains at least one serial alternative,and can now state< our chief result as follows.

* Thus, if fp0 is a function of n variables, all alternatives are serial in </>ÿ.


Theorem.-There is a finite number m, depending on n, the numberof functions <p, ..., and the numbers of their arguments, such thatthe necessary and sufficient condition for our formula to be consistentin a universe with m or more members is that there should be a serial

jorm An completely contained in P. For consistency in a universe offewer than m members this condition is sufficient but not necessary.

We shall first prove that, whatever be the number N of individuals inthe universe, the condition is sufficient for the consistency of the formula.If N ÿ n, this is a consequence of a previous result, since* if An is com-pletely contained in P, so is any AN involved in An.

If N>n, we suppose the universe ordered in a series by a relationR

. (If N is infinite this requires the Axiom of Selections.) Let q be anyserial alternative contained in An. If <j>o is a function of r arguments, q

will contain the values of either $0 or ~ 0o (but not both) for every per-mutation of yi, y2, ..., yr. Any such permutation can be writtenypJ, ypn, yPr where pi, p2, ..., pr are 1, 2, ..., r rearranged. We makea list of all those permutations (pi, p2} ..., pr) for which q contains thevalues of <p0, and call this list 2. We now give <£0 the constant inter-pretation that z2, ..., zr) is to be true if and only if the order ofthe terms Z\, z2, ..., zr in the series R is given by one of the permuta-tions (pi, p2, pr) contained in 2, in the sense that, for each i, Zi is the

ÿ-th of zij z2, ..., 2r as they are ordered by R.Let us suppose now that yi, y2, ..., yn are numbered in the order

in which they occur in R, i.e. that in the R series yi is the first of them,

y2 the second, and so on. Then we shall see that, if <p0 is given theconstant interpretation defined above, all the values of <p0 and ÿ <po in qwill be true. Indeed, for values whose arguments are obtained by per-muting yi, y2, ..., yr this follows at once from the way in which <p0 hasbeen defined. For <pQ(y<rV y*) is true if and only if the orderof y<TV y*2, y<rr in the R series is given by a permutation (pl9 p2, pr)contained in 2. But the order in the series of yav yvv y<Tr is in

fact given (on our present hypothesis that the order of the t/'s isVn Vn ...> yi) by (<t15 0*2, ..., o*r), which is contained in 2 if and only if

...J V*) CQntained in q. Hence values of <f>o for argumentsconsisting of the first r y

i

s are true when they are contained in q andfalse otherwise, i.e. when the corresponding values of ÿ<p0 are containedin q.

For sets of arguments not confined to the first r y>s our result followsfrom the fact that q is serial,' i.e. that if sx < s2 < ... < sr, so that

ySl, ySi, y$r &re in the order given by the R series, q contains the


values of <p0 for just those permutations of y$l, //,,,, ySr which corre-spond to the permutations of 1/1, y2i ..., yr for which it contains thevalues of <f>o, i.e. by the definition of <£0 and the preceding argument, forjust those permutations of y,v yÿ ..., ySy which make <j>o true.

Hence all the values of <p0 and - <j>o in q are true when yv y2, ..., yn

are in the order given by the R series.If, then, we define analogous constant interpretations for xo, "ÿo, etc-,

and combine these with our interpretation of <p0, the whole of q will betrue provided that t/i, t/2, ..., yn are in the order given by the R series,and if 1/1, y2, ..., yn are in any other order the true alternative will beobtained from q by suitably permuting the y1 s, i.e. will be an alternativesimilar to q and contained in the same form An. Hence An is true for

any set of distinct ylt t/2, yn. Moreover, for any set of distinctt/i, 1/2, ..., yv (v<n) the true form will be one involved in A1h and sinceAn and all forms involved in it are contained in P, P will be true for

these interpretations of <p0, xo< V'o, and our formula must be con-sistent.

Having thus proved our condition for consistency sufficient in anyuniverse, we have nowT to prove it necessary in any infinite or sufficientlylarge finite universe, and for this we have to use the Theorem B provedin the first part of the paper.

Our line of argument is as follows : we have to show that, whatever<p0, xo> ÿ0, . .. we take P will be false unless it completely contains aserial An. F°r this it is enough to show that, given any <p0t xo, Vro, ...,there must be a set of n y1s for which the true form is serial*, or, sincea serial form is one which contains a serial alternative, that there mustbe a set of values of y\, y2, yn ÿr which the true alternative is serial.

Let us suppose that among our functions <p0, Xo> "ÿo, ... there are a\functions of one variable, 02 of two variables, ..., and au of n variables,

and let us order the universe by a serial relation R.The N individuals in the universe are divided by the a-i functions of

one variable into 2"J classes according to which of these functions theymake true or false, and if N ÿ 2®1A1 we can find ki individuals whichall belong to the same class, i.e. agree as to which of the functionsthey make true and which false, where ki is a positive integer to beassigned later. Let us call this set of individuals I\v

Now consider any two distinct members of I\p Z\ and z2 say, and

let Z\ precede z2 in the R series. Then in regard to any of the a2 func-tions of two variables, cpo say, there are four possibilities. We may either

* For then P can only be true for <f>0, xo, fa, ... by completely containing this true serial form.


have

(1) ÿoÿi) ÿ2) . ÿ0(ÿ2» ÿl))

or (2) <f>0(zl9 z.J . - <Pq(#2> ÿ1),

or (3) - 0O(ÿ, z2). ÿ0(ÿ2, ÿ

or (4) - (j>Q{zly z2). - 0O(*2, zj.

<po thus divides the combinations two at a time of the members of I,

*-

,

into four distinct classes according to which of these four possibilitiesis realised when the combination is taken as Z\, z2 in the order in whichits terms occur in the R series; and the whole set of a2 functions of twovariables divide the combinations two at a time of the members of I\

,

into 4°a classes, the combinations in each class agreeing in the possibilitythey realise with respect to each of the a2 functions. Hence, byTheorem B, if ki == h(2, k2i 4°*), r;>1 must contain a sub-class I\2 of k2members such that all the pairs out of I\2 agree in the possibilities theyrealise with respect to each of the a2 functions of two variables.

We continue to reason in the same way according to the followinggeneral form :-

Consider any r distinct members of I\r_x; suppose that in the R

series they have the order z1} z2, ..., 2r. Then with respect to anyfunction of r variables there are 2rJ possibilities in regard to zi} z2y ..., zrtand the ar functions of r variables divide the combinations r at a time

of the members of I\V1 into 2rJ°r classes. By Theorem B, ifkr_x - h(r, kri 2',J£1r)*, IVj must contain a sub-class l\r of kr memberssuch that all the combinations r at a time of the members of Fjcr agree

in the possibilities they realise with respect to each of the ar functionsof r variables.

We proceed in this way until we reach TjCit lt all combinations n-1 ata time of whose members agree in the possibilities they realise withrespect to each of the an_x functions of n-1 variables. We then deter-mine that &w_i shall equal n, which fixes kn_2 as //.(« -1, n, 2(T1"1);""-1)and so on back to kx, every fcr_i being determined from kr.

If, then, Nÿ2a1ki, the universe must contain a class Tkn l or Tn

(since &n_ 1 = n) of n members which is contained in Tkr for every r,r = 1, 2, n-1. Let its n members be, in the order given them by R,Ui, 1/2, ..., !/«. Then for every r less than n, y\, y2, ..., yH are contained inTk

r and all r combinations of them agree in the possibilities they realise

* If ar = 0 we interpret h(r, hr} 1) as kr and identify r*r l and r*r

.


with respect to each function of r variables. Let ySi, ySs, ..., ySr

(sL < s2 < ... < sr) be such a combination, and xo a function of r variables.Then ?/v ySi, ..., y,r are in the order given them by R, and so areyl9 y2j yr; consequently the fact that these two combinations agreein the possibilities which they realise with respect to xo means that xo istrue for the same permutations of y$v ySv y,r as it is of ?/1? y.2. ..., yr.

The true alternative for yi, y2, ..., yn is therefore serial in xo, andsimilarly it is serial in every other function of any number r of variables* ;it is therefore a serial alternative.

Our condition is, therefore, shown to be necessary in any universeof at least 2f"A-1 members where hi is given by

frn-l = -w,

fir-1 = h(r, kr, 2r'*r) if ar =£ 01> (r = 7i- 1, n - 2, 2).

= kr if ar = OJ

For universes lying between n and 2tJ kx we have not found a necessaryand sufficient condition for the consistency of the formula, but it isevidently possible to determine by trial wThether any given formula isconsistent in anv such universe.

III.

We will now consider what our result becomes when our formula

(iCj, 2/2* ...» ÿn) X' , ...> » *ÿ2» ...>

contains in addition to identity only one function <p of two variables.In this case we have two functions </>0, Vro given by

<f>o(y» yk) = <p(yu yk) (i =£ A),

Xo(2/i) = 2/i)»

so that a± = 1, a2 = 1, ar = 0 when r > 2. Consequently

Jc2 = ]c3= ... = = n and = 7*(2, n, 4);

but the argument at the end of I shows that we may take insteadhi = n\ ! !, and our necessary and sufficient condition for consistencyapplies to any universe with at least 2. n! ! ! individuals.

In this simple case we can present our condition in a more strikingform as follows.

* We have shown this when r < n; we may also have r - n, but then there is nothing toprove since in a function of n variables every alternative is serial.


It is necessary and sufficient for the consistency of the formula thatit should be true when </> is replaced by at least one of the followingtypes of function :-

(1) The universal function x = x . y = y.

(2) The null function x =£ x . y ÿ y.

(3) Identity x - y.

(4) Difference x =£ y.

(5) A serial function ordering the whole universe in a series, i.e. satisfying

(a) (x) - <f>(x, x),

(W (*, y)\.

X = y V {4>b, ?/). ÿ <p(y> .*)[V i<P(y> *). ÿ 2/)}]v

(c) (x, y, *){ y) V - <f>(y, z) V </>(x, z)).

(6) A function ordering the whole universe in a series, but also holdingbetween every term and itself, i.e. satisfying

(a') (x)<p{x,x)

and (b) and (c) as in (5).

Types (l)-(4) include only one function each; in regard to types (5)and (6) it is immaterial what function of the type we take, since if onesatisfies the formula so, we shall see, do all the others*.

We have to prove this new form of our condition by showing that Pwill completely contain a serial A n if and only if it is satisfied by func-tions of at least one of our six types. Now an alternative in n yf s isserial in xo if it contains

either (i) Xo&i) . Xo (to) . . . . Xo (to) Qr> fQr 8hQrt> nXo(to)>r

01- (ii) ~ XoQ/i) . . . . ~ Xo(y«) » » n ~ Xo(yr),r

but not otherwise, and it will be serial in <f>0 if it contains

either (a) II </>0(yr, ys). <p0{ys, yr),r < s

or (b) II <p0(yr, y,). ~ </>„(//„ y,),r < s

or (c) II ~ 4>0{yr, y,) -fain*, ijr),r <s

or (d) II ~ 4>o(yr,y*). ~ <pQiy„ yr) ÿr < s

* A result previously obtained for type (5) by Langford, op. cit


There are thus altogether eight alternatives serial in both <f>0 and xo gotby combining either of (i), (ii) with any of (a), (b), (c), (d); but theseeight serial alternatives only give rise to six serial forms, since thealternatives (i) (b) and (i) (c) can be obtained from one another byreversing the order of the y

,

s and so belong to the same form, and so dothe alternatives (ii) (b) and (ii) (c).

It is also easy to see that any formula completely containing one ofthese six serial forms will be satisfied by all functions of one of the sixtypes according to the scheme

Form (i) (a) (i) (b and c) (i) (d) (ii) (a) (ii) (b and c) (ii) (cl)

Type of function 1 6 3 4 5 2

raid that conversely a formula satisfied by a function of one of the sixtypes must completely contain the corresponding form. For instance,a function of type 6 will satisfy the alternative (i) (b) when pi, y2, ..., ynare in their order in the series determined by the function, and whent/i, 1/2, ..., yn are iQ any other order the function will satisfy an alterna-tive of the same form.

In the language of the theory of postulate systems we can interpretour universe as a class K, and conclude that a postulate system on a base(K, R) consisting only of general laws involving at most n elementswill be compatible with K having as many as 2. n! ! ! members if andonly if it can be satisfied by an R of one of our six types.

IV.

Let us, in conclusion, briefly indicate how to extend our method inorder to determine the consistency or inconsistency of formulae of themore general type

(EzXi *2> 2, iCgj ..., ÿn}F{(j>, \fr, , Zfy ...> "ÿJi)

which have in normal form both kinds of prefix, but satisfy the condi-tion that all the prefixes of existence precede all those of generality.

As before, we can suppose F represented as a disjunction of alterna-tives and discard those which violate the laws of identity. Those left

4

we can group according to the values of identity and difference for argu-ments drawn entirely from the z

f

s. Such a set of values of identityand difference we can denote by Hi (=, zlf zs, ..., zm), and F can be putin the form

(H,. Ft) V(ff2. Fo) V (Hz. Fa) V

On a problem of formal logic-

and the whole formula is equivalent to a disjunction of formulae.

...> ...>

. 0®1» . . . j *ÿw) ÿ > ÿ 1 > ., Xn) }

V (Ezu Zo, ) %ni) (- , %if . . .j

. (;ÿi> *ÿ2? . . .» ÿ2ÿ* , #1, . .., Znii t T ÿ -. > .<- n/ i

V etc.

Since if anv one of these formulae

of its terms must

be consistent, it is enough for us to show how to determinethem, sav the first In this H

VI

of identity and difference for every pairWe renumber ÿ11 ÿ2) . . .) Z the same suffix for every

Hi. and our formula becomes

(-

£/<? 2, Z%, ..., Zp} ipÿij «Eoj . . .» *£ft) l(ÿS X' *, * * - , ÿ1> ÿ2» . . . > *ÿ1> *ÿ2> . . . > Xfi) j (0

in which it is understood that two 2,s with different suffixes are alwaysdifferent.

Now supposing the universe to have at least /x+n members, we con-sider the different possibilities in regard to the :r,s being identical withthe z,s, and rewrite our formula

(Ezv Zfy ...) *ÿ2> *.*> Xfi)

n»=1, «j = ], /ut

X =£ *3 G(.<Pi X» . . .' Zÿf %fXf X}, ...> 2-7l) )

J'

in which means<<

. . . ) iEfl)

if, then " and

= n y. , ÿ2' ÿn)»

the product being taken for

(9a

Xl9 Zl9 z9, ..., 2

ÿ2> ...>

0H ÿ1> ÿ2, * 9 * 9 ÿ/M

and in G any term xi = ÿ is replaced by a falsehood (e.g. x% =£ £i) notinvolving any z.

Next we modify G by introducing new functions. In G occur valuesof, e.g. <f>, with arguments some of which are 2,s and some x,s; from

286 On a problem of "formal logic.

these we define functions of the x,s only by simply regarding the zfs asconstants, and call these new functions <j>of xo> .... Values of <j>,which include no #'

s among their arguments, we replace by constantpropositions p, q, .... The only values of identity in G are of the formXi = Xj and these we leave alone. Suppose that by this process G turnsinto

X* * ...» 00, Xo» ,f*i]?* ...> » ...> *ÿh)*

Then the consistency of formula (i) in a universe of N individuals isevidently equivalent to the consistency in a universe of N-fi individualsof the formula

(x-ii Xfy ...» xn)L(</>, j ...> *Poj Xo» ...» P* ...» . . ..» .£»).

But this is a formula of the type previously dealt with, except for thevariable propositions p, q, ..., which are easily eliminated by consideringthe different cases of their truth and falsity, the formula being consistentif it is consistent in one such case.

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F.P. Ramsey - On a problem of formal logic