Theories

THE FOUNDATIONS OF MATHEMATICS and other Logical Essays By

FRANK PLUMPTON RAMSEY,M.A., Fellow

and, Director ol Stud,ies in Mathematics ol King's College, Lecturer in Mathematics i,n the Uniuersity of Cambridge

Edited bY U ,' .R.,8.'BRAITHWAITE, M.A., +.

i' '

t&4eru of'King's College,Carnbridge With a Preface by G. E. MOORE, Litt.D.,

Fellow ol Trinity

Hon. LL.D. (St. Andreus), F.B.A. College, and, Professor ol Mental Philosophy and Logic in the Uniuersity of Cambridge

1960 L I T T LEFIELD,

A DA MS

Paterson, New Jersey

& co.

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IX LAST PAPERS (1e2e) A.

TITEORTES

Let us try to describe a theory simply as a language for the facts the theory is said to explain. This need not cornmit us on the philosophical question of whether a theory is only a language,but rather if we knew what sort of language it would be if it were one at all, we might be further towards discoveringif it is one. We must try to make out account as generalas possible,but we cannot be sure that we have in fact reached the most general type of theory, since the possiblecomplicationis infinite. First, let us considerthe facts to be explained. Theseoccur in a universe of discourse which we will call the prinwry systcm, this system being composed oJ all the terms r and propositions (true or false) in the universe in question. We must suppose the primary systern in some way given to us so that we have a notation capable of expressingevery propositionin it. Of what sort must this notation be ? It might in the first case consist of na.mesof different tlpes any two or more of which conjoined together gave an atomic proposition; for instance,the niunes a, b . . . z, 'red', 'before'. But I think the systemswe try to explain are rare\r of this kind ; if for instancewe are concernedwith a series of experiences,we do not try to explain their time order (which we could not explain by anything simpler) or r Tho 'universe' of the primary system night contain 'blue or rcd ' but not ' blue ' or ' red '; i.e. we night be out to explain when a lhing wag ' blue or red'as opposedto'green or yellow', but not which it was, bluo or red. ' Blue-or-red ' would then be a term : ' blue', ' red ' nonS6ngcfor our preselt purpose.

zr3

even,assuminganorder,whetherit is a or D that comesfirst ; we take for granted that they are in order and that a comes before b, etc., and try to explain which is red, which blue, etc. a is essentiallyone beforeD,and ' a' , ' b', etc.,are not really n:unesbut descriptionsexceptin the caseof the present. We take it for granted that these descriptions describe uniquely, and insteadof ' a was red ' we have e.g. ' The 3rd oneago was red'. The syrnbolswe want are not names but numbers: the Oth (i.e. the present),1st, -1th, etc., in generalthe ath, and we can use red (z) to mean the zth is red counting forward or backward from a particular place. If the series terrainatesat say 100, we could write N(101), and generally N(n) if. m 2100, meaning' There is no ruth ' ; or else simply regarde.g.red (rz) as nonsense if nr >100, whereasif we wrote N(m) we should say red (zr) was false. I am not sure this is necessary,but it seemsto me always so in practice; i.e. the terms of our primary system have a structure, and any structure can be representedby numbers (or pairs or other combinationsof numbers). It may be possibleto go further than this, for of the terms in our primary system not merely some but even all may be best symbolizedby numbers. For instance, colours have a structure, in which any given colour may be assigned a place by three numbers, and so on. Even smells may be so treated : the presenceof the smell being denoted by 1, the absencoby 0 (or all total smell qualities may be given numbers). Of course, we &urnot make a proposition out of numbers without somelink. Moment 3 has colour 1 and smell 2 must be written X(3) : t and f(3) - 2, X and C correspondingto the general forms of colour and smell, and possibly being functionswith a limited numberof values,sothat e.g.C(3) : SS might be nonsense,since there was no 55th srnell. Whether or no this is possible,it is not so advantageous where we have relatively few terms (e.9. a few smells) to

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deal with. Where we have a multitude as e.g. with times, we cannot name them, and our theory will not explain a primary systemin which they have n.unes,for it will take no account of their individuality but only of their position. fn generalnothing is gainedand clarity may be lost by using numbers when the order, etc., of the numbers corresponds to nothing in the nature of the terms. If all terms were representedby numbers, the propositions of the primary systemwould all take the form of assertions about the values taken by certain one-valued numericd functions. Thesewould not be mathematicalfirnctionsin the ordinary sense; for that such a function had such-and-such a vaLuewould always be a matter of fact, not a matter of mathematics. We have spoken as if the numbers involved were always integers, and if the finitists are right this must indeed be so in the ultimate primary system,though theintegersmay, of course,take the form of rationals. This meanswe may be concernedwith pairs (tn, n) with ()r*, )*) alwaysidentical to (m,n). If, however,our primary systemis alreadya secondary systemfrom someother theory, real numbersmay well occur. So much for the primary system; now for the theoretical construction. We will beginby taking a typical form of theory, and consider later whether or not this form is the most general. Suppose the atomic propositionsof our primary system are such iu; A(n), B(m,n) . . . wherem, n, etc.,take positive or negative integral valuessubject to any restrictions,e.g.that rn B(m, nl ,n may only take the values 1, 2. Then we introduce new propositiond functions o(n), p(n), y(nt, n), etc., and by propositions of the secondarysltstem we shall mear any truth-functions of the values of.o,B, y, etc. We shall also lay down propositicnsabout thesevalues,e.g. p(z) which we shall call arioms, and whatever (n). "(n).

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propositionsof the secondarysystemcan be deducedfrom the axioms we shall call theorerns. Besidesthis we shall make a d,ictionarywhich takesthe form of a series of definitions of the functions of the primary systemA,B,C.. . .intermsofthoseofthesecondarysystem o, F, T, e.g. A(n): o(n).v .y(O, n2). By taking these 'definitions' as equivalences and addingthem to the axioms we may be able to deducepropositionsin the primary system which we shall call laws if. they are general propositions, consequences if they are singular. The totality of laws and consequences will be the eliminant when @,F, T. ., etc., are eliminated from the dictionary and axioms, and it is this totality of laws and consequences which our theory asserts to be true. We may make this clearer by an example1 ; let us interpret numbers n, ny, n2, etc., as instants of time and supposethe primary system to contain the following functions:A(n) : I seeblue at z' B(n) : I seered at r' lA(") . B1n1: I seenothing at nf. C(n) : Betweenn-7 and r I feehmy eyesopen. D(n) : Betweenz-1 and n I feel my eyesshut. E(n) : I move forward a step at n. F(n) : I move backward a step at z. and that we construct a theory in the following way: r [The example seemsfutile, therefore try-to_invent a better ; but it io fict brings but several good poiuts, which it would be diffcult otherwise to bring out. It may however nriss so"1e points wh-ich wo will consider later-. A defect in all Nicod's examples is that thoy do not give an external world in which anything happcns.-F. P. R.l

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First aawill be understoodto take only the values 1,2,3,

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I am in, if they are shut f seeno colour. The laws resulting from the theory can be expressedas follows:

0) (z).{A@)v E(n)}, {C(n)v D(")l .{E(rl v F(n)\ (2) (nr,ngl{\} n.. D(nr)\ Then we introduce o(n, m):

: At time t my e]tesare open.

Let us define 0 (nr, n7)to mean 1 2 Nc' i {n, 1v 1n, . E(v)l

And the axioms (t,m,m'l:

ns)

(2) (2) with the C's and 2'. inlslshenged

At time n I am at'place m.

F@' *) : At time * Planen is blue. y(n)

tat. C(nrl.C(n). J . (gzr) . h}

o(n,ml . o(n,,n'l . 3 . 1tr: 1n'.

(n).

(.3ml.

(n).

F@, L).

1nz. F(u)} = o (mod3) t,

o(n,m).

( n ) : p (n ,2 ).=.t\t + 1 ,2 ).

(e) t(frrJ .C(rr,\.nr1n.n)v) A(n) v B(n)

nr.f,.D(y)l

f":

(l) t(EzJ . D(nr) . nt 1 n . n ) v ) nt. :, . e(y)l lo: A(n) . B(nl.

And the dictionary A(n) : (=m) . o(n,m) . F@,n) .y(*). B(n) : (=n) . o(n, m) . p(n, m) . y(n).

C(n): ifut -rl . y(nl. D ( o ) : y ( n-!).i @1 . E(n): (=n) .d(n-t,m) F(n) : (=*l

-Nc' t {\ 1,

,io{n-

(f) (") :. (=m) 2 m : 0, l, or 2 : m(v,nl =, B(v) : (n + l) (vr, n) . (m + r) (rs, ,l) , vr * u, (mod 2; . J,,, ,. . A(rr,)v A(vr). 8(rJ v B(vr). fWhere2 + 1 :0

.o{n,J@)l

r, JQn)l . q\t, ml.

This theory can be said to represent me as moving iunong 3 places,'forwards' beingin the senseABCA, 'backwards' ACBA. Place,4 is alwaysblue, placeB alternately blue and red, place C blue or red accordingto a law I have not discovered. If my eyes are open I see the colour of tne place

for this purpose.]

These can then be comparedwith the axioms and dictionary, and there is no doubt that to the normal mind the axiomsand dictionary give the laws in a more manageableform. Let us now put it all into mathematicsby writing

A (rr)

I 6@ ) :

r

B(n)

u 6@): -1

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A1n7.B1n\as {(z) : e C(n)

es XQt) :

D(nl

asx@)- -1

ln

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2r9

I

(z) (n, rrr).l E x(,)i< 1. Ir-s I

I

( s)Gn ).,!^ x0 :1 : J, .6@\+0.

C(nl . D(nl as y(n) : s

s

E(n)

6 { (n ):

t

F(rrl

as $(n) :

-1

U) E n ) . E x ? ) - - 1 : ) ,.9 ( n ):s. f =6

,,

(5) (n):. (=m):,!,{?) : m (mods). f", . i\t'l + - t

E1n7F1n1 as t!(n) : o. Instead of.a(n, m) have a(z) a firnction taking values 1, 2, 3

F(n,rn) ,, F(n,*) y(n) y(n)

,,

,,

7, -l 1,0

Our axiomsare just (tl (n).a(n):1v2v3 (z) (n) . B(n,l\ : I

(s)(z). F@,z)* P@+ r, 2) (4) (n,n).p(n,m):rv-t (r) (z) . y(n) :0

vt

Of these(1) (4) (5) hardly count sincethey merely say what valuesthe functionsare capableof taking. Our definitions become.

(n 6@) : y|rl x p{n, a(n)l (ii) x(z) : y(n) y(n - r) (iii) r/(z) : Remainder mod 3 of.o(n) - o(n - l) Our laws are of coursethat {, X, rf must be such thato, B, y can be found to satisfy 1-5, i-iii. Going through the old laws we have instead of (t) 6@): - 1 v Ov l, y(n): -1 v 0 v l, {t(n): - 1 v0 v 1 [understood].

ta ut

:,!,{Ql =,!,{(r) = rn+ 1(mod3). n' = n" g r (mod2) . )n,,,r.6@')i@"1: o v - 1. So far we have only shown the genesisof.hws ; consquenccs arise when we add to the axioms a proposition involving e.g. a particular value of m,fromwhich we can deducepropositions in the primary systemnot of the form (") . . . Thesewe call the consEuences, If we take it in its mathematical form we can explain the idea of a theory as follows: Instead of saying simply what we know about the valuesof the functions with which wGare concerned,we say that they can be'conStructedin a definite way given by the dictionary out of functions satisfying certain conditionsgiven by the axioms. Such then is an example of a theory; before we go on to diScusssystematically the different features of the example and whether they occur in arry theory, let us ta^kesome questionsthat, might be asked about theorie-sand see how they would be answeredin the present case. 1. Can we say anything in the language of this theory that we could not say without it ? Obviously not; for we c€ureasily eliminate the functions of the second system and so say in the primary system all that the theory gives us.

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22O

IJ\ST

2. Can we reproduce the structure of our theory by means of explicit definitions within the primary system ? [This question is important becauseRussell, Whitehead, Nicod and Carnap all seemto supposethat we can and must do this.ll Here there are some distinctions to make. We might, for instance, argue as follows. Supposing the laws and consequencesto be true, the facts of the primary system must be suchasto allowfunctionsto bedefinedwith all the properties of those of the secondarysystem, and these give the solution of our problem. But the trouble is that the laws and concan be made true by a number of diflerent sets of sequences facts, corresponding to each of which we might have different definitions. So that our problem of finding a single set of definitions which will make the dictionary and axioms true whenever the laws and consequencesare true, is still unsolved. We can, however, at once solve it formdly, by disjoining the setsof definitionspreviouslyobtained; i.e. if the are different setsof facts satisfying the laws and consequences P1, P2, P6, and the correspondingdefinitions of a(n, m) are o(n,ml':Lr{A,

ml

B, C'',n,

Lo{A, B, C, . . ., n, ml etc. we make the definition oQt,tn\: Pt,P Lr{A,B,C .

C, (mod3)

o*

Pt)Lr{A,B,C...n,m}.

y (n ): E 7 a @)+. C ,

etc.

for a and 7.

Such a definition is formally valid and evidently fulfrIs our requirements. r Jean Nicod, Lo Glonlhic ilans le Monda Scttsibla (1924), translated in u;is Ptobtems of Gconutry atd Iniluctioa (1930): Rudoll Carnap, Dct LogischcAufbau dar Wclt (19281.

22t

What can be objected to it is complexity and arbitrariness, since LL, L, . . . can probably be chosen each in many ways. Also it explicitly.uisumes that our prinary system is finite and contains a definite number of assignable atomic propositions. Letus therefore seewhat other waysthere are ofproceeding. We might at first sight suppose that the key lay simply in the dictionary; this gives definitions of A, B, C . . . in terms of o, F, | . . . Canwe not invert it to get definitionsof o, F, y . . . in terms of A, B, C . .7 Or, in the mathematical form, can we not solve the equations for d, p, y . . . in terms of i, X, * . . ., at any rate if we add to the dictionary, as we legitimately can, those laws and axioms which merely state what values the functions are capable of taking ? When, however, we look at these eguations (i), (ii), (iii) what we find is this : If we neglect the limitations on the values of the functions they possessan integral solution provided y(n) can be found from (ii) so as always to be a factor of i?t),i.e. in generalalwaysto be * 1 or 0 and never to vanish unless {(z) vanishes. This is, of course, only true in virtue of the conditionslaid on f and X by the laws ; ass"mi"g tbese laws and the limitation on values, we get the solution a(n)= i Wf

. n,rft}.

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And for F@, m) no definite solution but e.g. the trivial one B(n, m) : $\n) (assumingy(n) :1 or 0). Here C. must be chosenso as to make y(z) alwaln 1 or 0; and the value necessar5r for this purpose dependson the facts

LAST PAPERS deduced simply from the of the primary system and cannot be nought : laws. It must in fact be one or for which X(n\ +o' (a; if there is a least poSitive or zero r or f 1' according as ;6(n) for that nis - 1 y(n) t' 0' according (D) If there is a least neg ative nf or which as X(n) for that z is * 1 or - 1' matter whether Co (c) If for no n 7@'1+ O it does not i sfl o r - 1. of C' and so of y(z)' We thus have a disjunctive definition will satisfy the limitations on Again although any value of C, one such will satisfy the the value of. o(n), probably only have to be disjunctively arioms, and this value will a6ain is not at all fixed by the equadefined. And, thirdly , F@, *\ matter in which we shail tions, and it $'ill be a complicated to say which of the many again have to distinguish cases' the axioms' p-ossiblesolutions for p(n,zr) will satisfy is neither in this case We conclude, therefore' that there of inverting the dictionary nor in general any simple way or an obviously preeminent so as to get either a unique the axioms' the reason for this solution which will also satisfy in the solution of the lying partly in difficulties of detail secondary system has a ior,r, partly in the fact that the "qrri of freedom' than the higher multiplicity, i'e' more degrees system contains three one primary. In our case the primary virtually fwe lp(n' t)' P("' 2)' valuerl functions, the secondary such an each taking 2 or 3 values' and F@,3), o@), y@)) a uniuersal characteristic increase of multipliqity is, I think ' of useful theories. alone does not suffrce' Since, therefore, the dictionary both dictionary and axioms the next hopeful method is to use in many popular discussions in a way which is referred to the meaning of a proposition of theories when it is said that we should ordinarily regard about the exterual world is what

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asthecriterionor testof.its truth. This suggeststhat we should define propositionsin the secondarysystem by their criteria in the primary. In following this method we have first to distinguish the criterion. sufi,cientcriterion of a propositionfrom its necessary lf p is a proposition of the secondarysystem,we shall mean by its sufficientcriterion, o(P), the disjunction of all proposi tions gof theprimary systemsuchthat y' is a logicalconsequence of g together with the dictionary and axioms, and such that q is not a consequenceof the dictionary and axioms.r On the other hand, by the necessarycriterion of. p, r(!l we shall mean the conjunction of all those propositionsof the primary system which follow ftom p together with the dictionary and axioms. We can elucidatetheconnectionof.o(!) and r (p) as follows. Consider all truth-possibilities of atomic propositions in the primary systemwhich are compatiblewith the dictionary and axioms. Denote such a truth-possibility by r, the dictionary and axioms by a. Then ofy') is the disjunction of every z suchthat r ,! a is a contradiction, t (!) the disjunction of every z such that r p ais not a contradiction. If we denoteby Z the totality of laws and conseguences, i.e. the disjunction of every r here in question, then we have evidently o(P)t:zL.-t(-P),

(i)

I The laws and consequences aeed not be added, since they follow {rom the dictronary and axioms. It might be thought, however, that we should take them inslead, of. the axioms, but it is easy to see that this would merely increase the divergence between su6cient and necessary criteria and in general the difficulties of the method. The last clause could be put as that '- g must not follow from or be a lau' or cousequelce.

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t(p) :=2L.-o(-?)

(ii)

o(p)v"(-?).=.L.

(ur)

We have also olfr .f ) ::z o(f ,l . o(f )t

(rg

f.or fr. po follows from g when and only when h nd 0t both follow. Whence,or similarly, we get the dual = .r(!r)vr(Or\.

r(ftvPcl.

(v)

We also have o(p))r(t'),

(vi)

(Considerthe r's above.) o(!)v o(-?).)

.L. J. t (?)v

"

(-p),

(vii) (from iii)

and from (vi), (ii), (iii). o@).).-o(-?).L, L.-r(-f).)."@).

(viii) (ix)

Lastly we have o(f ,) v o(?s). ) . o(prv pr).

(x)

Since if g follows either from y', or from y', it follows from f tv f z; and the dual

4h .fzl

.a.r(lr)."(f).

(*i)

On the other hand, and this is a very important point, the conversesof (vi)-(xi) are not in generaltrue. Let us illustrate this by taking (x) and consideringthis ' r ' :

C(nl . =r.n--OzD(n).

225

=s.it-l.

(nl .En.Fn, i.e. that the man's eyesare only open oncewhen he seesblue. From this we car deduceo.(0,2) v a(0, 3) .'. This t ) o{o(0,2) v a(0,g)}. But we cannot deducefrom it o(0,21or c(0, 3), sinceit is equally compatible with either. Hence neither da(O, 2)) nor o{a(O,3)} is true. Hencewe do not have o {o(0, 2) v a(0, 3)} f o {c(0, 2)} v o {c(0, 3)}. It follows from this that we cannot give definitions such that, if y' is any propositionof the secondarysystem,y'wil in virtue of the definitions mea! o(p) [or alternatively r (l)),for if y'r is defuiedto mean o(0r1,f s to mean o(Pr),h v y', will mean o(y'j v o(I), which is not, in general, the sa"rreas o(hv 0). We can therefore only use o to define some of the propositions of the secondarysJstems,what we might call atamic secondary propositions, from which the meanings of the others would follow. For instance, taking our functions o, F, y we could proceed as follows : y(nl is defined 6 A(n) v B(n\, where there is.no rrifficulty as A(n) v B(n) = oty@)| = r {y(n)|. 9@, m) could be definedto mean o{B@ m)1,i.e. we should say place mwas ' blue ' at time n, only if there wereproof that it was. Otherwisewe should say it was not 'blue' ('red' in co[rmon parlance). F@, *) would then mean 6tP@, m)| not

"{F@,

Alternatively we could use z, a.nddefine

F(n,m) to be r {F@,*l}, and,p(n,n) wouldbe i lB@,all

n)1.

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In this case we should say m was 'blue' whenever there was no proof that it was not; this could, however, have been achieved by means of o if we had defined p(n, m) to be m), and 9'@, *) tobe o{p'(n,m)},i.e. applied o to p - F'(n, instead of B. In general it is clear that z always gives what could be got from applying o to the contradictory, and we may confine our

they stand the definitions look circular, but are not when interpretedin this way. For instanceoto(n,1)) is Z, i.e. laws (1)-(5)togetherwith

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attention to o. It makes, however, a real difference whether we define F or F by means of o, especially in connection with place 3. For we have no law as to the values of p(n, B), nor any way of deducing one except when a(a,3) is true and.r{(z) or B(z) is true. If we define F(n, 3) to be o{p(n. 3)}, we shall say that 3 is never blue except whcn we observe it to be; if we define F@,5) tobe o{p(n,3)} we shall say it is always blue except when we observe it not to be. Coming now to o(n, m) we could define o( n, l) :

o { o (n , 7 )} ,

a( n, 2) :

o { a (n , l ) v a (n ,2 )\ . a { a (n , 7 )1 ,

d( n, 3) :

o { a (n , l ) v a (n ,2 )} ;

and we should for any n have one and only one of. a(n, ll, o(n,2'), o(n,3) true: whereas if we simply put o(n, m):

o{a(n, tn)y,

this would not follow- since o{a(n, r)}, o{a(n, z)1, o{a(n, B)l could quite well all be false.

[E.s.if(n).A(n).Bt"ll Of course in all these definitions we must suppose o{a(n, n)l etc., replaced by what on calculation we find them to be. As

(1nr, n zl . 2(nr, n) . 2(nr, n) . h * n, (mod2) . Bn, . Bn, . v.

(anr,nr,nr).2(!tr,n).2(nr,n).2(nr,n).nr*nr(mod z). An, . Anr. Bni.

v.

(1nb na).l(nr, n) .2(!rz,n) . Bnt , Bn,

Such then seemto be the definitions to which we are led by the popula.rphrasethat the meaningof a statementin the secondsystemis given by its criterion in the first' Are they such as we require ? What we want is that, usingthesedefinitions,theaxiomsand dictionary should be true wheneverthe theory is applicable, are true ; i.e. that i.e. wheneverthe laws and consequences interpreted by means of these definitions, the axioms and dictionary should follow from the laws 3nd consequences. It is easyto seethat they do not sofollow. Ta.kefor instance the last axiom on p. 216:

(o): F(n,2). = . B@{ r, 2), which meansaccordingto our definitions (n) : o{p(n,2)} . = . d{P(n + r, 2)}, which is plainly false, since if, as is perfectly possible,the man has neveropenedhis eyesat place2, both o {P (tt,2)} and * 1, 2)} will be false. "{F@ [The definition by r is no better, sincez {F (n, 2)} and (n + 1,2)) would both be true.l " $ This line of argumentis, however,exposedto an objection of the following sort: If we adopt thesedefinitionsit is true that the axiomswill not follow from the lawsand consequences, but it is not really necessarythat they should. For the

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laws and consequencescannot represent the whole empirical (i.e. primary system)basisof the theory. It is, for instance, compatible with the laws and consequences that the man shouldneverhavehad his eyesopenat place2; but how could he then have ever formulated this theory with the peculiar law of alternation which he ascribesto place 2 ? What we want in order to construct our theory by means of explicit definitions is not that the axioms should follow from the laws and consequences alone,but from them togetherwith certain propositions existential of the primary system representing experiencesthe man must have had in order to be able with any show of reasonto formulate the theory. Reasonablethough this objection is in the present case, it can be seen by taking a slightly more complicated theory to provide us with no general solution of the difficulty; that is to say, such propositions as could in this way be added to the laws and consequenceswould not always provide a sufrcient basis for the axioms. For instance, suppose the theory provided for a whole system of places identified by the movementsequences necessaryto get from one to another, and it was found and embodiedin the theory that the colour of each place followed a complicated cycle, the same for each place,but that the placesdiffered from one another as to the phaseof this cycleaccordingto no ascertainablelaw. Clearly such a theory could be reasonably formed by a man who had not had his eyesopen at eachplace,and had no grounds for thinking that he everwould open his eyesat atl the places or evenvisit them at all. Supposethen tn is a placehe never goesto, and that F@, *) is a function of the secondsystem, signifying that m is blue at r; then unless he knows the phaseof m, we cannever have o{p(n, m)},but if e.g.the cycle gives a blue colour once in six, we must have from an axiom v Bg, m). We have, therefore, FQ, mlv p(t, n) v the sarne difrculty as before. iust

If, therefore, our theory is to be constructed by explicit definitions,thesecannot be simple defnitions by means of o (or z), but must be more complicated. For instance, in regard to place 2 in our original examplewe can define

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2)}, BP,2laso{P(o, F@,2) as o{p(0,2)) if.n is even, -

o{p(o,2)l if' n is odd.

I.e. if we do not know which phaseit is, we assumeit to be a certain one, including that 'assumption' in our definition. E.g. by saylng the phaseis blue-even,red-odd, we mean that we have reasonto think it is; by sayingthe phaseis blue-odd, red-even, we mqul not that we have reason to think it is but merely that we have no reasonto think the contrary. But in general the definitions will have to be very complicated ; we shall have, in order te verify that they are complete, to go through all the casesthat satisfy the laws and conseguences(together with any other propositions of the primary system we think right to assume)and seethat in each casethe definitions satisfy the axioms, so that in the end we shall come to something very like the general disjunctive definitions with which we started this discussion (p. 220). At best we shall have disjunctions with fewer terms and more coherenceand unity in their construction; how much will dependon the particular case. We could see straight ofi that (in a finite scheme)such definitionswere alwayspossible,and by meansof o and z we have reachedno real simplification. 3. We have seenthat we canalwaysreproducethe structure of our theory by means of explicit definitions. Our next question is 'Is t};lis necessaryfor the legitimate use of the theory ? '

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To this the answerseemsclear that it cannot be necessary, or a theory would be no use at all. Rather than give all these definitions it would be simpler to leave the facts, laws and consequencesin the language of the primary system. Also the arbitrarinessof the definitions makesit impossiblefor them to be adequateto the theory as somethingin processof growth. For instance, our theory does not give any law for the colour of place 3; we should, therefore, in embodying our theory in explicit definition, define place B to be red unlessit was observedto be blue (or elseoiceaersa). Further observation might now lead us to add to our theory a new axiom about the colour of place 3 giving say a cycle which it followed; this would appear simply as an addition to the axioms, the other axiomsand the dictionary being unaltered. But if our theory had been constructedby expticit definitions, this new axiom would not be true unless we changed the definitions, for it would depend on quite a different assignrnent of colours to phce 3 at times when it was unobserved from our old one (which always made it red at such times), or indeed trom any old one, except exactly that prescribedby our new axiom, which we should never have hit on to use in our definitions unlesswe knew the new axiom already. That is to say, if we proceedby explicit definition we cannot add to our theory without changingthe definitions, and so the meaningof the whole. [But though the use of explicit definitions cannot be necessary, it is, I think, instructiveto consider(aswe havedone) how such definitions could be constructed, and upon what the possibility ol Srving them simply depends. Indeed I think this is essential to a complete understanding of the subject.l

Clearly in such a theory judgment is involved, and the judgments in question could be given by the laws and the theory being simply a languagein which consequences, they are clothed, and which we can use without working out the laws and consequences. The bestway to write our theory seemsto be this (3c, B, 7) : dictionary. axioms. The dictionary beingin the form of eguivalences. Here it is evident that o, B, y are to be taken purely extensionally. Their extensionsmay be filled with intensions or not, but this is irrelevant to what can be deducedin the primary system. Any additions to the theory, whether in the form of new axioms or particular assertionslike c(0, 3), are to be made within the scopeof the original o, F, T. They are not, therefore, strictly propositions by themselvesjust as the difierent sentencesin a story beginning'Once upon a time' have not complete mearringsand so are not propositions by themselves. This makes both a theoretical and a practical difierence :

a. Taking it then that explicit definitions are not necessary, how are we to explain the functioning of our theory without them ?

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(a) When we ask for the meaning of e.g. o(0, 3) it can only be given whenwe know to what stock of 'propositions' of.thef,rst anil secondsystemso(0, 3) is to be added. Then the meaning is the difference in the first system between (=o, F, y) : stock. o(0,3), and (3c, p, 7) . stock. (Weinclude propositions of the primary system in our slock although thesedo not contain o, F, y.) This account makes c(0, 3) mean something like what we calledabover{c(0,3)}, but it is really the differencebetween r{o(0,3) f stock} and r (stock). (D)In practice,if we ask ourselvesthe question " Is o(0, 3) true ? ", we have to adopt an attitude rather different from that which we shouldadopt to a genuineproposition. For we do not add o(0, 3) to our stock wheneverwe think

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we could truthfully do so, i.e. whenever we suppose (Ao,9,7): stock . a(0, B) to be true. (ao, F, y):stock. d(0, 3) might also be true. We have to tlink what else we might be going to add to our stock, or hoping to add, and considerwhether o(0, 3) would [s ssrfain to suit any further additions better than d(0, g). E.S. in our little theory either p(n,S) or p(n,3) could always be added to any stock whictr includesd(n,3, .v .41n1 . B("1. But we do not add either, becausewe hope from the observed instances to find a law and then to fill in the unobserved ones according to that law, not at random beforehand. So far, however,as reasoningis concerned,thaf the values of these functions are not complete propositions makes no difference, provided we interpret aU logical combination astaking placewithin the scopeof a singleprefix (3 c, F , y) ; e.g,

p(", 3l . FQt,s, must be (lF) z F@, B) . p(n, s),

not(3fl 9@,s).(rF)F@,q. For we call reason about the characters in a story just as well as if they were really identifed, provided we don,t take part of what we say as about one story, pa^rtabout another. We can say, therefore, that the incompletenessof the 'propositions' of the secondarysystem affects ow disfutes but not our rcasoning. 5. This mention of 'disputes' leads us to the important questionof the relationsbetweentheories. What do we mean by speaking of equivalent or contradictory theories ? or by saying that one theogyis containedin another,etc. ? In a theory we must distinguish two elements: (1) What it asserts: its meaning or content. (2) Its symbolic form. Two theories are called quiaalent if they have the same

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content, contradictoryif they have contradictory contents, compatibleif their contents are compatible, and theory z{ is said to be containedin theorv B iI A's content is contained in B's content. If two theoriesare equivalent, there may be more or less resemblancebetween their symbolic forms. This kind of resemblanceis difrcult if not impossibleto define precisely. It might be thought possible to define a definite degree of resemblanceby the possibility of defining the functions of B in terms of thoseof ,4, or conversely; but this is of no value without some restriction on the complexity of the definitions. If we allow definitionsof any degreeof complexity, then, at least in the finite case,this relation becomessimply equivalence. For each set of functions can be defined in terms of the primary system and therefore of those of thc other secondarysystemaia the dictionary. Two theoriesmay be compatiblewithout being equivalent, i.e. a set of facts might be found which agreedwith both, and another set too which agreedwith one but not with the other. The adherentsof two such theories could quite well dispute,although neither affirmed anything the other denied. For a dispute it is not necessarythat one disputant should asserty', the other p. It is enough that one should assert something which the other refrains from asserting. E.g. one says 'If it rains, Cambridgewill win', the other 'Even if it rains,they will lose'. Now, taken as material implications (aswe must on this view of science),theseare not incompatible, since if it does not rain both are true. Yet each can show grounds for his own belief and absenceof grounds for his rival's. People sometimes ask whether a 'proposition ' of the secondary system has any ineaning. We can interpret this as the questionwhethera theory in which this proposition wasdeniedwould be equivalentto onein which it wasaffirrned.

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This dependsof courseon what elsethe theory is supposedto contain ; for instance,in our examplep(n, g) is meaningless coupled with d(n, B) v y(n). But not so coupled it is not meaningless,since it would then exclude mv seeingred under certain circumstances,whereas Ft", Sl woulcl exclude my seeingblueunderttresecircumstances.It is possiblethat these circumstancesshould arise, and therefore that the theories are not equivalent. In realistic language we say it could be observed,or rather might be observed (since . could, implies a dependenceon our will, which is frequently the case but irrel,annl), but not that it will be