Chapter 15 The Theory of Restricted Generality

CHAPTER 15

The Theory of Restricted Generality The theory which we deal with in this chapter is that called :F 2 in § 8D. Its characteristic illative primitive is the ob S, for which the associated rule is RULE

S.

SXY, XV

~

YV.

Here X, Y, and V are unrestricted obs. Thus S has the properties of an inclusion or formal implication relation; the statement ~SXY

expresses that ~

YV

holds for all obs V for which ~XV,

and so SX expresses generality over the range X; for this reason S is called restricted generality. This system was first introduced in [ACT] and [CFM],l and was further studied in [DTC] and [BVC]. It was also used as a basis for set theory by Cogan [FTS] and Bunder [STB]. The most recent developments have been those of Bunder and of Seldin [SIC]. We shall begin in § A with properties which do not require any further assumption (beyond the associated rule) concerning S. In § B we shall adjoin to the system of § A axiom schemes sufficient for the proof of a deduction theorem, and shall treat the resulting system with Gentzen techniques, obtaining in the process a proof of "Q-consistency". This latter proof will involve the definition of a set of canonical obs, or canobs.' which have a rank on which an induction can be carried out to prove the elimination theorem. In § C, we shall consider methods of formulating the conditions on canonicalness within the system and hence replacing the axiom schemes by constant axioms. Finally, § D will treat the inclusion of predicate calculus.

1. The ob E is the same as what Church called II in his [SPF. I] and [SPF. II]; but the system of these papers was afterwards shown to be inconsistent (see § OD). Because Curry had already used 'II' in another sense, he used 'E' for restricted generality. The symbol appears in that role in [RFR] and [CCT]. 2. cr. § 12B4.

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THE THEORY OF RESTRICTED GENERALITY

[15A

A. THE SYSTEM $i20 The system $i 20 is that in which we postulate a C-system with an ob 3 and its associated rule together with Rule Eq, and study the properties which follow from these assumptions with (essentially) no further axioms or adjoined rules. We study the situation first heuristically in § 1; and we discuss there the motivation behind the introduction of 3 and various notions connected with it. The remainder of the section introduces formal definitions of such notions and technical theorems concerning them. In § 2 we study the sequence of obs 3n> where 3 n is a natural generalization of 3 for n-place functions. In § 3 the subject is methods of dealing with bound variables. In § 4 we treat implication, and in § 5 similarly the universal quantifier, where both are regarded as defined in terms of 3; along with these we discuss certain related notions. In § 6, we consider an ob 3 + defined in terms of the obs P and II of §§ 4-5, and discuss the relation of this ob to 3. In § 7, we discuss the relation of 3 to F and to the system $i 12 of § 14F. Finally, in § 8, we consider a generalized functionality G. I. Informal discussion We have seen in the introduction to this chapter that 3 means a relation of inclusion or formal implication between arbitrary obs. If X and Yare interpreted as properties qJ and ljJ respectively, this relation is that which [PM] would express in the form

Of course 3XY is an ob regardless of the nature of the obs X, Y, and it may not always be feasible to regard it as meaning a proposition; but when it is asserted it entails, according to Rule 3, the relation (1)

XU

~YU

for any ob U. The motivation for introducing 3 is the fact that most generalizations - some say, indeed, all generalizations - are valid only over a restricted range; one can make this range explicit by the use of 3. In ordinary predicate calculus one has a fundamental, supposedly very restricted, range of "individuals", and what is there called "universal quantification" is really generality with respect to this restricted range. In higher order functional calculuses many different ranges enter; and a means of making them explicit is then useful. Thus 3 is the natural tool for combinatory investigation of these higher logical calculuses. As we have already noticed in § 8D, implication P and the functionality primitive F can be defined in terms of 3. We recapitulate these definitions

THE SYSTEM .'F2 0

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341

here (for the details see § 8D), thus:

P == Axy.3(Kx)(Ky), F == Axyz.3x(Byz)

= Axy.3x· By;

if we have a universal category E, then we can define universal generality by

Il == 3E. From these definitions the associated rules follow, viz.: RULE

P.

PXY, X

RULE

F.

FXYZ, XU

RULE

n.

nx,

Eu

~

~

Y. ~

Y(ZU).

XU.

When we wish to consider generalization with respect to two or more variables, one thinks at first of the analogue of the Principia's qJxy

:::> xy

tf;xy.

This cannot be represented directly by 3; one would need an analogue of 3 for functions of degree two. But if we regard x as fixed for the moment, we can regard the y as varying over a range which depends on the value of x; then we allow x to vary over a certain range.' This leads to a statement which, in the language of [PM], would be qJ1X :::>x.qJ2xy :::>y tf;xy.2

This can be expressed by 3. In fact let Xl' X 2' Y correspond to qJ1' qJ2' tf; respectively; then the relation just written becomes 3X 1 (Ax. 3(X2X)(YX)).

If we bring in the combinator from Chapter 5, (especially § 5E), this becomes 3X 1 (3X 2Y)'

In this way we can define a 3 2 , viz.

3 2 == Ax1X2y·3x1(3x2Y) which expresses a restricted generality relation for functions of two variables. Continuing in this way, we can define by induction a 3 m , expressing generality for m-variate functions, for any value of m. The formal definitions will concern us in § 2. The notation ':::> x', used as infix in [PM], is sometimes convenient as a variable-binding operation; as such it can be defined thus: 3 (2)

X :::>x Y == 3(AX.X)(Ax.Y),

1. Compare the situation in the theory of iterated integration when one integrates over a plane region; the limits of the second integration depend on the first variable. 2. If CP1 is universal this reduces to the case previously considered. 3. Cf. § 3B4, p. I 86 (bottom line). Note that 'x' here stands for any variable.

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THE THEORY OF RESTRICTED GENERALITY

[ISA

where X and Yare obs which may contain x. Generalizing this, we regard

(3) where each X k is an ob which may contain Xl> ••• , and Y may contain Xl' ••• , X m, as standing for

Xk

but not any

Xj

for} > k,

:::m~ 1" '~m11,

(4)

where (5) Alternative abbreviations for (4), with (5) understood, which we may use when confusion is not to be feared, are (~l' ... , ~m; Xl' ... , xm)·Y,

(6)

(~; X).Y.

These notations 4 have somewhat the same advantages that bound-variable notations do generally. We shall use whichever one is suitable to the context. A special case - where X k contains none of Xl' ••• , X m except X k - occurs often enough to warrant a special notation for it. We introduced such a notation in § 10El, but with F instead of::: as primitive. We shall use the same notation here for the corresponding notion, relying on the context to avoid any confusion. Thus we define (7)

(v'1X1X1' .", \flXmXm)· Y

==

0I:1X1 :::> XI .0I:2X2 :::> X2'

." •

OI:mXm :::> x m' Y 5

Note that, if we have ex.t. «11», this is the special case of (4) for which (5) holds with (8) so that (9)

(\fIX1X l' "., \flXmX m)· Y

==

(1X 1, K1X2' .", Km-1 IXm; Xl' ... , xm)Y.

In terms of the notation (7) we could define Fm so that (10)

in the sense that this is consistent with the properties of the Fm considered in § 9. These Fm suffice to formalize any reasonable sort of simple theory of 4. They must not be confused with the rather similar notations of § llB6. 5. Alternatively we could write (as, e.g., in § 10EI, p. I 369) ('
but the extra parentheses are superfluous since ('
'
is equal to ('
by Theorems 3 and 5 below.

••••

'
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THE SYSTEM

~ 20

343

types in the sense ordinarily conceived. But in terms of the notations (6) we could introduce a new kind of functionality Gm , so that (11)

in which the type taken by an argument of a function may depend on the values given to the preceding arguments. There have been indications that this may be useful." 2. The E-sequence We shall now present formal definitions of the obs Sn' and shall prove some rather technical theorems about them which are valid in $'20' As stated in § 12BI, although we are primarily interested in the case where the underlying C-system has ex.t, «11», we make certain concessions to ex.t. «P». We use lower case Greek letters for obs which appear as arguments of B, and thus mean functions; for these some property akin to the axioms [PX] of § 12B2 would be appropriate in the case. Again there is no difficulty with the ex.t. if the Bn are defined only for n > O. Consequently we define them that way; and show in a special theorem (Theorem 1) that there is, for ex.t. «11», a natural extension to the case n = O. Then in the proofs by induction we include both the cases n = 0 and n = 1 in the basic steps. The result of all this is that if we exclude the case n = 0, or include it and postulate ex.t, «11», the ~ l ' ... , ~n, 11 occurring in the discussion are arbitrary obs; we can therefore take them to be variables Xl' ••• , X n , Y and form abstractions with respect to them for use in Rule (~). Only in case we include the case n = 0 and do not wish to postulate ex.t, «11» are these Greek letters subject to a restriction. For technical reasons we reserve the letters 'x', 'y', 'z' for variables for the arguments of 3, and use 'u' instead of the 'x' of § 1. The combinator <1>: which appears here is that defined in § 5E. It has the reduction rule

«P»

(12)

When m is not given it is to be understood to be 1; when n is not given it is to be understood as 2. DEFINITION

(a)

1.

=

I-l -1 - -I-l,

(b) 6. W. A. Howard has mentioned some possible applications in connection with his work. However, although we consider a theory of types based on the Fm in Chapter 17, this chapter depends heavily on the theory of functionality as developed in Chapters 9, 10, and 14; and since we do not yet have such a theory for functionality based on the Gm , we shall not explore the use of the Gm in a theory of types in this book.

THE THEORY OF RESTRICTED GENERALITY

344

[ISA

Note that, by virtue of (12), (b) can be written 3 n+ 1

= AXoX1,,,Xny·3xo(AU.3n(X1U),,,(XnU)(Yu)).

THEOREM 1. If we replace (a) of Definition I by (13)

='0 == I,

and allow 3 1 to be defined by Definition I(b); then in ex.t, (('7))

Proof If 3 1 is defined by Definition I(b), then '31~'7 = '3~(l'30'7)·

But since

<1>1

is B (§ 5E(8)) and 3 0 is I, this gives 31~'7

=

'3~(BI'7)

= 3~'7.

Hence 3 1

= 3.

REMARK 1. An appeal to ('7) is made in the last two steps. The first of these requires a replacement of BI'7 by '7; the second an application of Axiom [133] (§ 12B2). Thus in the (([3)) case there is a restriction on '7 but not on ~. THEOREM 2. If ~ l'

... ,

~no fJ

are obs such that

(14)

and U l '

... ,

U; are further obs such that k = 1,2, ... , n;

(15)

then (16)

Proof We proceed by induction on n. If n = 0, (16) follows from (14) by Rule (I). If n = 1 the theorem follows directly by Rule 3. Suppose the theorem is true for a given n, and that (17)

k = 0, 1,2, ... ,n.

(18) From (17), Definition 1, and Rule Eq, we have

fHence by case k

=

3~0(AU. 3nC~ 1 U)(~2U) .. '(~nu)(fJu)).

°of (18) and Rule 3, f-

'3nC~lUO)(~2UO)"'(~Po)('7Uo)'

15A]

THE SYSTEM

345

$'20

Now suppose we set

Then the last relation and (18) become respectively

~ ~~U 1... Uk'

Hence by the induction hypothesis ~

rl'U 1",Un,

which, by the definition of This completes the proof. THEOREM

(19)

1'1', is the conclusion of the theorem for n + lin.

3. For all m, n = 0, 1,2, '" Sm+n~1·"~ml1t .••l1n( = Sm~1"'~m(:+1SnI11 .•. l1nO.

Proof For m = 0 this follows from (12), since ~+ 1 is I (§ 5E5 (20) and (22». For m = 1 the theorem follows by Definition l(b). For m + 11m, assume that (19) holds. Then, by Definition 1,

(20) where

By the induction hypothesis we then have U

= Au,Sm(~1u)"'(~mu)(:+1Sn<111u)"'(l1nU)«(U»

where

= n +1(<1>:+ 1Sn)111" ·'1n(· But since n+ i ' <1>:+ 1 = <1>::11, this gives us V = ::lSnI11".l1n('

Hence S~oU

=

S~O(m+ 1Sm~1' "~mV)

=

Sm+1~O~1"'~mV

(Definition 1)

= Sm+ 1~O~ 1" '~m(::11Sn'11"·'1n()·

From this and (20) we have the case m

+

11m of (19). This completes the proof.

THE THEORY OF RESTRICTED GENERALITY

346

[lSA

THEOREM 4. If ~l' ... , ~n1' I'll' ..., I'In'C U1> ... , Urn are such that

f-

Em+n~l···~ml'll· .. 1'/n"

f-

~kUl",Uk'

k = 1,2, ... , m;

then Proof This is a combination of Theorems 2 and 3. 3. Bound variables We now turn our attention to the notations involving the infix' => x' and related notions defined at the end of § 1. The conventions made at the beginning of § 2 are to apply here also. In this discussion we use the term 'constant' in the sense of an ob in which none of the variables mentioned explicitly in the same context (here U or Ul' ... , un) 7 occur. DEFINITION 2. If X and Yare combinations of constants and the variable u, then X =>u Y == E(AU.X)(AU.Y). THEOREM 5. For each k = 1,2, ... , n, let X k be a combination of constants and the variables U l ' " ' ' Uk; and let Y be a combination of constants and Ul • • • • , Un'

Let

Then (21)

Proof We proceed by induction on n. If n = 0 both sides of (21) reduce to 1'/; if n = 1 the theorem follows by Definition 2 and Theorem 1. For n > 1, let Since AU2,,,Uk,Xk AU2...Un • Y =

=

~kUl'

k > 1,

I'/U l,

we have by hp. indo

and hence

7. Thus a constant is here a relative constant, and may be a free variable or a combination containing free variables.

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THE SYSTEM

.1Jl'20

347

Therefore, by Definition 2, Xl :::>UI Z

= 3e 1(n3n-le2· .. enl'/) = 3 ne 1 • • .enl'/.

6. Let X j, Y be as in Theorem 5, and let

THEOREM

Let k

~

n, U l' Xi

for each j

q.e.d.

~

... ,

Uk be constants, and let

== [U l/Ul>'''' Uk/Uk]X j ,

n. Further, suppose that ~

Xi,

j= 1,2, ... ,k.

Then

Proof This follows from Theorems 4 and 5. 4. Implication

In § 1 (following § 8D) we saw that the ob P defined by

P == Axy.3(Kx)(Ky) satisfies the rule RULE

P.

PXY, X

~

Y,

i.e. the rule of modus ponens. Thus we can take P as the formal representative of implication. Here we shall consider some formal matters concerning this P and its relations to 3. In particular we show that we can define a sequence Pn having certain analogies to the 3 n • Since Kx and Ky are automatically Oj-obs, we have no problems in regard to the uses of (1'/); all our reasonings are valid if the ex.t. is ((/3)). We begin by defining the ordinary infix':::>' as follows: X:::> Y== PXY. This allows us to translate ordinary formulas about implication into combinatory notation and vice versa. Note that in view of Definition 2 of § 3 and the definition of P we have (22)

X:::> Y = X :::>uY,

where u is any indeterminate new to X and Y. Thus the notation is not in conflict with Definition 2 of § 3.

348

[I5A

THE THEORY OF RESTRICTED GENERALITY

To represent chain implications of the form Xl

::>

,X2

::> ••••

.X;

::>

Y,

we need an ob Pn defined as follows." 3.

DEFINITION

Po

== I,

Pn + 1 == IlXOX1",XnY,PXO(PnXl,,,XnY)· THEOREM

7.

Proof

PI

=

PI

= Ilxy. Px(ly) = Ilxy.Pxy

P.

= Ilxy. S(Kx)(Ky)

= P. THEOREM

8. For all cq, ... , an' 13,

Pna 1 ...anf3 = Sn(Kal)(K2a2).·.(Knan)(Knf3). Proof We show first that for any

~ 1, ••• , ~n' 1],

In fact, let u be a variable which is new to

~ l'

... ,

~n' 1]. Then

K(Sn~ 1" '~n1]) = Xu , Sn~ 1" '~n1]

=

Ilu. Sn(K~lu)(K~2U)... (K~nu)(K1]u)

=


From this it follows that (24)

Pcx(Sn~ 1" '~n1]) =

s., 1(Ka)(K~ 1)" .(K~n)(K1]).

PCX(Sn~I'''~n1])

=

S(Ka)(K(Sn~I'''~n1]»

by Of. of P,

=

S(Ka)(
by (23),

In fact

= Sn+ 1(Ka)(K~ d...(K~n)(K1])

by Of. 1.

We now prove the theorem by induction on n. For n = 0 the theorem is trivial. For n + lin, we have Pn+laOal· .. anf3

= Pcx O(Pnal· .. anf3)

by def.,

=

PaO(Sn
by hp, ind.,

=

s., 1(Kcxo)(K2al)" .(Kn+ l an)(Kn+ 113)

by (24), q.e.d.

8. Essentially the same as § 9E (7) and (8).

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THE SYSTEM

349

'?20

The properties stated in the following Corollary follow readily from Theorem 8 in combination with the results of § 2 and § 3. COROLLARY 8.1.

(a)

The Pn have the following properties:

Pm+nXl" ,XmYl" 'Ynz = PmXl" ,Xm(PnYl" 'Ynz),

(b)

Pn

=

AXl" ,xny,x l

::::>

,X2

::::> . . . .

.x,

::::>

y,

Pm+nXl,,,XmYl"'YnZ, Xl' ... , X m ~ PnYl···YnZ.

(c)

5. Universal quantification In § I (following § 8D) we saw that if E is present in the system then the ob IT defined by IT: BE satisfies the rule RULE

ITX, EU ~ XU,

IT.

i.e, the rule of universal instantiation. Thus we can take IT as the formal representative of the universal quantifier. Here we shall treat IT the way we treated P in § 4; we shall consider some formal matters concerning this IT and its relations to B. In particular we show that we can define a sequence ITn having certain analogous properties to the Bn • We begin by introducing the prefix '(Vx)' as follows:

(V'x)X : IT(Ax.X). Note that in view of (7) and § lOEI (4) and the definition of IT we have (25)

(V'X)X : (V'Ex)X.

To represent chains of universal quantifiers of the form

(V'Xl' V'X2' ... , V'xn)X, we need a sequence of obs ITn defined as follows: DEFINITION

4.

(a)

IT l

:

IT,

ITn+ 1 : IT . BITn • Thus, by Definition 5E2, ITn : IT[n 1•

(b)

THEOREM 9.

If we replace (a) of Definition 4 by

~~

ITo:'

and allow IT l to be defined by Definition 4(b), then in ex.t, «1]» IT l = IT. 9 9. Cf. Theorem 1 and Remark 1.

THE THEORY OF RESTRICTED GENERALITY

350

[ISA

Proof If II I is defined by Definition 4(b), then

II Ie = BII(BIIo)e = II(BIIoe) = II(Ble) = THEOREM

10. Il;

lIe.

q.e.d,

= Ax.(VEu l , ... , VEUn),XUI'''Un,

Proof We proceed by induction on n, For n = 0, the result is trivial. For n = 1, we have

III

For n

+

= SE = Ax.SEx = AX. Eu

:::J u

xu

=

Ax.(VEu).xu.

lIn, we have

Il, +I = II· BIIn = AX. II(BIInx) = Ax.Eu o

:::J uo

B(Ay.(VEu l ,

= Ax. (VEUo).(Ay. (VEu l ,

=

... ,

... ,

VEun)·yul···un)xu O

VEUn).YUI,,,Un)(XuO)

AX. (VEuo, VEUI' ... , VEun)•XUOUI' .. Un'

q.e.d.

In keeping with (25), we shall write ('lUI' ... , Vun)X == (VEu l ,

... ,

VEun)X.

This notation then gives us, in the light of Theorem 10, (27)

The properties stated in the following Corollary follow readily from Theorem 10 in combination with the results of § 2 and § 3 and Theorem 5E2. 1 0 COROLLARY

10.1. The Il, have the following properties:

(a)

= II· BII . '" . BnII,

(b)

Il; +I

(c)

IIm+nX, EUI, ... , EUm ~ IIiXUI· .. U m ) ·

Note that in Theorems 9 and 10 and in Corollary 10.1, we have not used the fact that ~ EX holds for every ob X. Hence, the results hold for some more restricted range of quantification, i.e. they hold if E is replaced by T, where ~ TX holds for obs X in this restricted range, and II is replaced by ST. 10.

cr. tuoo § 2 Theorems 2-4.

THE SYSTEM

15AI

6. Relation to

~ 20

351

fl' 3

In [PM], the definition of the infix."> u', which is *10.02, is, in our notation, (28)

Xu

::J u

Yu

= (Vu)(X U

::J

Yu).

This does not hold in fl' 20' and Curry, in [CFM] § 8, gave the failure of this as a reason for deeming fl' 2 weaker than fl' 3. 11 However, we can obtain the effect of (28) in fl' 20 by defining the ob (29)

3+

==

hy.I1(AZ.P(XZ)(yz)).

Then if 3 is replaced by 3 + on the left side, (28) holds. Furthermore, it is easy to show that 3+ satisfies Rule 3, and hence that the results of §§ 2-3 hold if 3 is replaced by 3 +.12 On the other hand, we do not have 3 + = 3; for if x and yare distinct variables, then we have 3+ xy = I1(AZ. P(xz)(yz))

= 3E(AZ. 3(K(xz))(K(yz))), which is in normal form and distinct from 3xy. We shall consider the relation between 3 and 3+ further in § B2. 7. Relation to

fl'1

In § I (following § 8D) we saw that the ob F defined by F == hyz. 3x(Byz)

satisfies the rule RULE

F.

FXYZ, XU

~

Y(ZU).

Furthermore, as shown in § 14F, equation (F) holds. Hence, fl' 20 includes fl'120' The converse is also shown in § 14F, and hence we have that fl' 20 is equivalent to fl'120' However, there are advantages to taking 3 as primitive, for then we do not have to postulate any additional properties of =; whereas if F is primitive then we have to postulate equation (F), and the effects of adjoining this equation and what reduction relation is appropriate to it are not known. Note that if F were defined in terms of 3+ rather than 3, we would have (30)

F = hyz. (Vu)(xu

::J

y(zu)),

which was suggested as a definition of F in [FCL] and [FPF]. The F-sequence is defined as in § 14A3, and Theorem 9A3 holds for it as before. 11. Cf. [ACTI § 6. 12. Except that there are some changes on the restrictions of the arguments for ex.t. but since we needed no restrictions on the arguments of P in § 4, these changes will be relaxations of the restrictions imposed in the case of E.

«{J»,

352

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THE THEORY OF RESTRICTED GENERALITY

We conclude this section with the proofs of two theorems mentioned in § 14F1. THEOREM

11. FmIX1 ...IXmP(Bmzu) = FmIX1 ...IXm(BPz)u.13

Proof For m = 0 this is trivial. For m = 1 this is true by equation (F). For m + 11m, we proceed thus: Fm+ 1IXl ...lXm+ lP(Bm+ lZU) = FmIXl" .IXm(FIXm+ lP)(Bm(Bz)u)

by df. Bm, Th. 9A3,

= =

by hp. ind.,

FmIX l ...IXm(B(FIXm+lP)(Bz»u

FmIXl···lXm(AV.FlXm+lP(Bzv»u

= FmIXl···lXm(AV.FlXm+l(BPz)v)u

by eq. (F),

= FmIX1··.IXm(FIXm+1(BPz»u

by (0,

=

by Th. 9A3.

Fm+llXl· .. lXmlXm+l(BPz)u

This completes the proof by induction. COROLLARY

11.1. FmIXl ...lXmPI = FmIXl ...lXmlp.

Proof FmlXl' .. IXmPI = FmIXr- .. IXm(BIP)1 = FmIXl" .IXml(Bmpl)

=

FmIXl" .IXmlp.

('v'IXlX l, ... , 'v'IXmXm)X

THEOREM 12.

Proof For m S = 2". For m

= FmIX 1...IXmIY, where Y == AXl ...Xm.X. 14

= 0 this is trivial. For m = 1, this follows from the + 11m, we have AX2,,,Xm+l'X = Yx l, and hence

fact that

('v'IXlX l, ... , 'v'IXmXm, 'v'IX m+ lX m+ l)X = IX1Xl

:::>Xj

('v'IX 2X2, ... , 'v'IXm+lXm+l)X by df.,

= IXlXl

:::>x!

FmIX2...IXm+1'(Yxl)

by hp. ind.,

= FIXll(Ax l• FmIX2" .IXm+ 11(Yxl»

=

FIXll(B(FmIX2· .. lXm+11)Y)

= FIXl(BI(FmIX2 .. ·lXm+11»Y

by eq. (F),

= FIX1(FmIX2" .IXm+ 11)Y 15

by Th. 5D2, by df. of Fm'

13. Cf. [DTC] Theorem 11. 14. Cf. [DTC] Theorem 12. 15. This step is valid even in ex.t, is fJ-functional.

«fJ», because from Ax[fJF] it will follow that Fm /X2.•• /Xm+ll

THE SYSTEM .?F2 0

15Al

353

This completes the proof by induction. COROLLARY

12.1. ("lXI' ... , VXm)X = FmE... EI(A.x1 ...Xm'X),

8. The G-sequence In (11) of § 1 we noticed a certain sequence {Gn } which can represent a concept of functionality more general than that represented by F alone. We shall present here a formal definition of such a sequence {Gn } and a theorem from which it follows that it has the properties stated in § 1. We define G thus: (31)

G

=AXyZ.Sx(Syz).

Then we define the sequence {G n} as follows: DEFINITION

5.

(a)

GI

(b)

Gn+ 1

= G, = AXOX 1.. ·XnY· GXO(n+ 1GnX1,,, XnY)· = AZ.Sn(l ...(nCS[nll]z),

13. Gn(l ...(nl] Definition 5E2) by THEOREM

S[I]

(32)

where s[nl is defined (ef

= S,

Proof Since the theorem for n = 1 follows immediately from the definitions it suffices to treat the induction n + lin. This will be done as follows. By Definition 5(b),

(33)

Gn+ 1(0(1···(nl]

=

G(0(n+1Gn(1 ...(nl])

=

AZ.3(0(S(n+ 1 Gn (

=

AU·n+1 Gn(1· .. (nl]U(zu)

=

AU. Gn«(IU)"'«(nu)(I]u)(zu)

=

AU.(AW.3nC(lU)" '«(nu)(s[nl(I]u)w))(zu) by hp. ind.,

1" '(nl])Z)

by (31).

Now, by Rule (S), S(n+1 Gn(1...(nl])z

= AU,Sn«(lU)"'«(nu)(s[nl(I]u)(zu))

by (12),

by (13).

But since, by (B), (S), and (32) s[nl(I]u)(zu)

=

S(Bs[nll])zu = s[n+lll]ZU,

the preceding formula gives us S(n+1Gn(1 ...(nl])z

= Au.3n«(l U)·"«(n u)(s[n+1 ll]Zu) = <1>. + 13.( r- .. (.(s[, + 111]Z)

by (12).

THE THEORY OF RESTRICTED GENERALITY

354

[ISA

Hence, by (33), we have Gn+1~0~1'''~n17 = Az.8~0(n+18n~1"'~n(s[n+1117z»

= AZ .8n+1~0~1 ... ~n
q.e.d.

We observe that the definition of Gn in terms of G is the same as the definition of 8 n in terms of 8 in Definition I. Consequently Theorem 3 holds mutatis mutandis if we put G in place of 8 and Gn in place of 8 n•1 6 However, in place of Theorem I, we have COROLLARY

13.1. If we define Go so that

(34)

Goxy = xy,

then (b) of Definition 5 and, for ex.t. «17», also Theorem 13 hold for n = O. Proof Under the stated circumstances if we set G~

== AXy. Gx(BGoY),

G~

== AXyZ.8x(S(BG oY)z).

then by (31) we have But

S(BGoY)z = AU. BGoYu(zu) = AU. Go(Yu)(zu)

by (B),

= AU .yu(zu)

by (34),

= Syz.

Hence G~

= AXyz.8x(Syz) = G.

Thus (b) of Definition 5 holds for n = 0 even in ex.t. «P». Theorem 13 for n = 0 follows, if S[OI == I (§ 5E(18», thus:

Go17 = AZ.Go17Z

by (17),

= AZ·17Z

by (34),

= AZ .8017z

by (13).

Here we do not need «17» if Go == Z1' Instead of Theorem 4 (and its special case, Theorem 2), we have COROLLARY

13.2. If ~1'

..• , ~m'

171' ... , 17n'" X, U I ,

(35)

/- Gm+1~1·"~m'lt···17n'X,

(36)

/- ~kU1",Uk'

... ,

Um are such that

k = 1,2, ... , m,

16. Theorems 2 and 4 depend on Rule E as well as Definition 1, and so this does not apply to them.

15A]

355

THE SYSTEM :F 20

then

Proof By (35), Theorem 13, and Rule Eq, ~ Sm+n~1·"~m1J1 ...1Jn(s[m+n](x).

Hence, by Theorem 4 and (36), ~ Si1J 1U1"'Um)" .(1Jp 1"'Um)(s[m+n](xu 1'" Um)'

(38)

But

s[m+nJ(XU 1"'Um = (s[m] . Bms[n]KXU 1"' U m

= s[m](Bms[n]oxu1"' Um = ml( Bms[n]()xu r- .. Um = Bms[n](U1",Um(XU1",Um)

by Th. 5E2,

by § 5E(1l), by (12),

= s[n]«(u1'" Um)(XU 1'" Um)' Hence, by (38) and Rule Eq, ~ Sn(1J1 U 1" .U m)·· ·(1JnU 1" .um)(s[n]«(u 1" .Um)(XU 1" .U m))·

By Theorem 13, (37) follows from this by Rule Eq., q.e.d. The following corollary establishes the connection between the Gn and the Fn defined (as in § 14A3) by

F == F1 == hyz. Sx(Byz),

(39) (40)

In view of the theorems and definitions we also have (11) of § 1. COROLLARY

13.3. If

k = 1,2, ... , n,

(41)

1J == KnfJ,

then

Proof For n

=

1 we have G~1J = AZ.S~(S1Jz)

= AZ. Soc(S(KfJ)z)

by the Th., by (41),

=

AZ. Eoc(BfJz)

by (B2 ) , p. I 195,

=

FocfJ

by (39).

356

THE THEORY OF RESTRICTED GENERALITY

To complete the proof we establish n (41) for that case is ~k

(42)

==

+

[15B

lin as follows: the analogue of

Kk OCk,

k = 0, I, ... , n,

'1 == Kn+lp.

Then, by (b) of Definition 5, Gn+l~O~I' "~n'1 = G~O(n+IGn~I'''~n'1) = G~o(AU. G.c~ I u).. '(~nu)('1u))

by (12),

= GOCo(AU. Gnoc i (KOC2)" .(Kn-Iocn)(Knp)) by (42),

by hp. ind.,

= GocO(Au.Fnocl .. ·ocnP)

=

Goco(K(Fnocl· .. ocnP))

=

Foco(FnOCI" .ocnfi)

by case n= I, by (40), q.e.d.

B. DEDUCTIVE THEORY OF

$'2

In this section, we shall set up a system of axioms for a deductive theory of $'2' The objective is to formulate a set of axioms, preferably a small number of schemes, from which will follow a deduction principle which is a generalization of the deduction theorem of the ordinary logical calculus; and thus to apply Gentzen techniques to the system. Consequently, this deduction principle will play the same role here that the stratification theorem plays in the deductive theory of $'1; and, indeed, it will turn out that the axioms needed for $'2 are related to the axiom schemes for $'1' In § I we shall present the axioms, and show that from them and some very general assumptions about canonical ness the deduction theorem follows. In § 2 we shall use this deduction theorem to give a T-formulation of the system. In § 3 we shall define a set of canonical obs called "canobs" which satisfies the assumptions of § I, and in § 4 we shall use this set of obs to prove the elimination theorem for the L-formulation which is suggested by the T-formulation. In § 5 we shall consider the f"-formulation of [ACT] and [BVe] and give a simplified proof of the elimination theorem. In § 6 we extend the system to include Q. We shall not, in this section, consider formulations in which the generalizations over canonical obs are formalized within the system and the axiom schemes replaced by constant axioms. These formulations, called $'~ in [ACT], will be considered in § C.

lSBj

DEDUCTIVE THEORY OF

~2

357

1. The deduction theorem By a deduction theorem for restricted generality we mean a theorem which gives us the following. Let S be a system which is formed by adjoining new atoms and perhaps axioms to ff 2, and let S(x) be the further extension formed by adjoining a new indeterminate x to S. Let X and Y be obs of S such that by a deduction in S(x) we have Xx

rYx.

Then under certain restrictions there is a deduction in S itself of

rEX Y. The qualification "under certain restrictions" is made advisedly. For we saw in § 12B3 that if such a principle holds without restriction the system S will be inconsistent; furthermore the contradiction cannot be avoided by assuming in addition that there is an A such that XA holds in S. We study here a formulation using the idea of canonical obs as introduced in § 12B4. For the reasons stated in § 12B4 we shall carry through this theory only for the case where the canonical subjects are unrestricted. We assume that Nl-3 of § 12B4 hold, and also

r

NE.

Canl(X) & Canl(Y) ~ Cano(EXy);l

From these assumptions it follows that if the premises of an inference by Rule Eq or Rule E are in Canj, so also is the conclusion. This holds by NI for Eq; while for Rule E we have Cano(EXY)

~

Can, (Y)

by NE(rl)

~

Cano(YU)

by N3.

From this we can conclude that if the axioms and premises of a proof are in Cans, the conclusion is also. Thus we want to be sure that all axioms are in Canj. By analogy with the notation of Chapter 14 for F-obs, we shall use lower case Greek letters for canonical obs. We shall assume in each case that the degree is that which will make the asserted ob of which the canonical ob is a part a canonical ob of degree O. We shall assume that the variables explicitly mentioned in the context do not occur in any canonical obs represented by the Greek letters. 1. This assumption, like N2 of § 12B4 and NF of § 14A2, is really two assumptions; we use the same device for distinguishing the two senses as in the case of N2. In the definition of canobs in § 3, we shall, in effect, postulate only NE(lr); NE(d) will hold because the definition is inductive. But if additional assumptions are desired, then NE(rl) may not hold. One such assumption, made in [DTC], is that X = y ..... Cano(EXY), where ~ EE(WE) is postulated as an axiom. However, we shall not make such assumptions for the time being (cf. § 6 below).

358

THE THEORY OF RESTRICTED GENERALITY

[lSB

We have seen in § 14F2 that the schemes (FI), (FK), and (FS) are not sufficient for the deduction theorem in its full generality. As pointed out in [DTC] , for functions of two arguments we can only get in :F 1 theorems of the form (Xx

~ x'

f3y

~y

y(jxy),

whereas in :F 2 we may well need those of the form

ax

~x.f3xy ~

yxy.

This situation is similar to the predicament we should be in if we could evaluate double integrals over rectangles only. Nevertheless the formulation suggests axioms to choose for a more adequate formulation. Thus the scheme (FK) becomes in our present notation

=

f- ax f- ax

~x.f3y ~y

a(Kxy)

~x.f3y ~y

ax.

This suggests the axiom scheme (1)

i.e., (2)

This is the scheme (3K) of [ACT] and [DTC]. The scheme (FK) follows from it by putting in Kf3 for f3. Furthermore, it turns out that (FK) can replace (3K) in the proof of the deduction theorem. Accordingly, we prefer to postulate (FK) instead of (3K).2 Similarly, the scheme (FS) becomes

f- au ~u·f3v

~v y(xuv)'~x:au ~u f3(yu).~y.(Xz ~z

y(xz(yz)).

If we put f3u in for f3, we have

f- (Xu

~u·f3uv ~vy(xuv)'~x:au ~uf3u(yu).~y.az ~z

y(xz(yz)).

This in turn suggests the scheme (3)

f- au ~u·f3uv

~v yxuv'~x:au ~u f3u(yu).~y.az ~z

yxz(yz),

i.e., (4)

f- 3 2af3(yx)

~x.'2a(Sf3y) ~y

3a(S(yx)y).

This is the scheme chosen for (3S) in [ACT] and [DTC]. The scheme (FS) follows from it by putting in Kf3 for f3 and B2y for y. A possible alternative for (3S) would be the scheme (5)

f- au ~u·f3uv

~v yuv'~x:(Xu ~u f3u(yu).~y,(Xz ~z

2. However there are alternatives; see Remark 1 below.

yz(yz),

DEDUCTIVE THEORY OF .F2

15B)

359

i.e.,

where the x is now a dummy variable, and hence ':::> x' really means ':::>'. This scheme is sufficient for certain proofs of the deduction theorem itself. 3 It follows from (3) by the substitution of Ky for y. The converse deduction of (3) from (5) requires the substitution of yx for y and generalization with respect to x, and so is not immediately obvious; but, as will be shown in § 2, (3) follows from the deduction theorem, so if (5) is used in this proof, then (3) does indeed follow. Inasmuch as the subjects are unrestricted all the above axiom schemes are canonical of degree O. Since the axioms we have so far can contain variables, we shall also postulate the scheme: AxsCH. (3). If X is an ob, x is a variable occurring in X, and ~X

is an axiom." then, for any canonical ob IX of degree one, ~

3IX(h.X)

is also an axiom. Since we are assuming that X is canonical of degree 0 if ~ X is an axiom, then it follows that all new axioms introduced by Axsch. (3) are canonical of degree O. Note that Axsch. (3) corresponds to the assumption made in ordinary predicate calculus whose only rule is modus ponens that the axioms are closed under generalization. If the subjects are unrestricted, then we expect to have E present in the system. In § 12B2 we considered the use of Rule E in connection with this ob E, but here this extra rule is unnecessary, since it follows from a constant axiom, Axiom (E), which can be written in any of the following forms :4. AXIOM (E).

~

3E(FEE),

If E is present, we will naturally assume

NE. 3. See Remark I (below). 4. Since x is an indeterminate, such axioms can arise only by the substitution of canonical obs which contain x for the informal variables, here represented by lower case Greek letters, in the above axiom schemes or in other schemes which may be introduced into the system. If E is present in the system, then ~ Ex is an axiom, and hence, by Axsch, (3), ~ 3",E is also an axiom. Cf. [DTCl § 9, assumption (D), and Remark 2 in § 2 below. 4a. This axiom, together with (EA) for every atom A, gives us all the required properties of E in a synthetic system. In a system based on A-conversion, since Rule Eq is present, we postulate (EA) along with Axiom (E), following the suggestion of footnote 12 of § lZB2, whenever A is either an atom or a basic combinator.

THE THEORY OF RESTRICTED GENERALITY

360

[158

With these preliminaries we now define a system !F 21 to be a system formed by adjoining to a system !F 20 the schemes (FK), (ES) in the form (3) or (4), and Axsch. (E). Then !F 21 represents a class of systems, in which the subjects are unrestricted; E mayor may not be present. Following [DTC], let us define [FK] == hy. F2xyxK,5 [ES] == Axyz.E 2xy(zu) =>".Ex(Syv) =>vEx(S(ZU)V).6 From these definitions, the axiom schemes, and Rule Eq, it follows that ~

[FK](XP,

~

[ES](Xyo,

(7)

where (X and fJ are canonical obs of degree one, y a canonical ob of degree two, and 0 a canonical ob of degree three. LEMMA 1.1. It follows from the axiom schemes that ~

(EI)

E(Xa.

Proof By putting KfJ for fJ and B2 y for y in the scheme (ES), we get the scheme (FS). It can be easily shown that there is an F-deduction from (FK) and (FS) to ~

Since I as (EI).

=

F(X(X(SKK).

SKK, we can obtain from this by Rule Eq (FI), which is the same

LEMMA 1.2. E 2apy, E(X(SfJU)

~

Ea(SyU).

Proof Immediate from (6), which follows from scheme (ES). THEOREM I (Deduction theorem). Suppose that from the assumption (8)

and a set M of canonical premises in which x does not occur there is a proof in !F 2 1 that (9) 5. This replaces the definition [EK] == Axy.E2xy(BKx), which was actually used in [DTC], and reflects the fact that we are using (FK) instead of (EK). Note that [FK]xy = [EK]x(Ky). 6. This is the definition of [DTC], but Seldin in [SIC] instead adopted the definition [ESI == hyz. E2XYZ => ... Ex(Syv) => v Ex(Szv),

where u is a dummy variable. Seldin used this definition because he took (5) or (6) for (ES). 7. Since the premises and axioms are canonical, there is no loss of generality in assuming that the conclusion (9) and each step in the proof is canonical.

15B]

DEDUCTIVE THEORY OF g-2

361

Then there is a proof in fF 21 from M alone in which x does not occur that

(10)

~ E~c;.x. tI).

Proof Suppose that ~ til' ... , ~ tin' where tin == n, is the proof of (9). We shall prove by deductive induction that there is a proof from M alone in which x does not occur that

(11)

k = 1,2, ... , n.

The proof consists of five cases, where the first three are basic steps of the induction and the last two are induction steps. Case 1. tlk =

~x.

Then (11) follows by Lemma 1.1.

Case 2. The ob tlk does not contain x and ~17k is deducible from Malone. (This includes the cases where ~ tlk is in M or is an axiom in which x does not occur.) Then AX.n« = Ktlk' Hence, if u is a variable which does not occur anywhere else, it follows from ~ 17k (or ~ KtlkU) and ~ [FK](Ktlk)~ that ~ E~(B(Ktlk)(Ku».

Then (11) follows by Rule Eq. Case 3. ~ tlk is an axiom in which x occurs. Then (11) is an axiom by Axsch. (8). Case 4. ~ n« is the conclusion of an inference by Rule Eq. Suppose the premise is ~ tI i' where tlk = ni: By the induction hypothesis, there is a proof from M alone of

Then (11) follows by Rule Eq. Case 5. ~ tlk is the conclusion of an inference by Rule E. Then tlk == vU and the premises of the inference are

By the induction hypothesis, there are proofs from M alone in which not occur of ~ E~(Ax. Ellv),

By Rule Eq, ~ E~(S(Ax '1l)(h. U».

Hence, by Lemma 1.2, ~ E~(S(h. v)(h. U»,

and (11) then follows by Rule Eq.

X

does

362

THE THEORY OF RESTRICTED GENERALITY

[15B

REMARK 1. The only point in this proof in which (5) or (6) was insufficient as opposed to (35) in the form (3) or (4) is in the proof of Lemma 1.1. Hence, if (35) is replaced by (5) or (6) and if (31) is also postulated, then the theorem holds for the resulting system. Alternatively, if (6) is postulated instead of (4) and if (2) is postulated instead of (FK), then Lemma 1.3 below shows that (31) follows, and hence the theorem holds for this formulation as well. Since all of these schemes follow from the T-rules of § 2, these two alternative formulations are equivalent formulations of ff 21' LEMMA 1.3 (Bunder). The scheme (31) follows from (2) and (6).

Proof Let a be any canonical ob of degree 1. Then BKa and {3 ==

== AU. K(3ex(K(au») are canonical of degree 2. Hence, by (2) and (6),

ff-

3 2ex{3(BKex), 3 2exfJ(BKa)

:::>

.3ex(S{3y)

:::> y

3a(S(BKex)y).

Hence, by Rule 3, the definition of {3, and Rule Eq,

f- 3 2ex(Ka)(BKex) :::> y 3aCt, where y is now a dummy variable. Then (31) follows from this and

f- 3 2ex(Ka)(BKex), which is an instance of (2). COROLLARY 1.4. If x does not occur in M,

M,

e, or 1], and if in ff

21

ex f-1]X,

then, by a proof in which x does not occur, M

f-

3e1].

Proof This follows by Rule Eq since Ax.1]X

= 1].8

e

COROLLARY 1.5. Let 1> ... , em, 1] be canonical obs of degrees 1, ... , m, 0 respectively and let Xl' ... 'X mbe variables which do not occur in 1> , .. , em' Let M be a set ofpremises in which the variables Xl' ••• , X m do not occur. Suppose that from the premises in M and the additional premises

e

(12)

i = 1, .. " m,

there is a proof in ff 21 of (9). Then there is a proof in which Xl' occur of 8. For ex.t. CUJ)), we are assuming that if AXIOM [pal.

holds in the underlying C-system.

a

••• , X m

is canonical of degree n > 0, then

Bn-1Z 1a = a

do not

DEDUCTIVE THEORY OF F

lSBl

363

2

Proof By induction on m. For m = 0, the corollary is trivial. The induction step follows from the theorem. In [ACT], Theorem 1 was announced without the postulation ofAxsch. (3), which was replaced by the assumption that no axiom in which x occurs is used in the proof of (9) from (8) and M. This theorem can be proved by omitting Case 3 from the proof of Theorem 1. Moreover, if the only axioms are the schemes (3K) and (35), then it seems to be true in most cases that no axiom other than (Ex) in which x occurs is used in the proof of (9). However, the theorem cannot be iterated, for even if no axiom in which x m occurs is used in the proof of (9) from (12) and M, the application of Theorem 1 usually introduces axioms in which some of the Xi for 0 ~ i < m occur. It was to solve this problem that Curry proved the theorem in the form of Corollary 1.5 in [DTC]. However, that proof required an assumption, there called (D), which is similar to Axsch. (3) and which could easily have been used here in its singular form instead ofAxsch. (3). Hence, the main theorem of [DTC] could have been proved in the singular version and iterated as is done here. Note that the singular form of (D) follows from Axsch. (3), which is natural enough itself since it corresponds to an assumption in predicate calculus in which the only rule is modus ponens.

1.6. Let

~1> ... , ~m,

M,

be as in Corollary 1.5, and let IJ be a canonical ob of degree m in which Xl' ••• , x ; do not occur. If from the premises M and the additional premises (12) there is a proof of COROLLARY

then there is a proof in which

COROLLARY

Xl' ... , X m

Xl' ... , X m

do not occur of

1.7. The theorem and its corollaries remain true

if 3

is replaced

by 3+ throughout. Proof We show first that

In fact, suppose that Then we have, for any u, v such that

~

Eu and

K(~u)v ~ ~u ~ IJU ~

~

Ev

K(lJu)v.

Since K(~u) and K(lJu) are canonical of degree 1 by N1-N3, we have by the theorem and definition of P Eu ~

P(~u)(1Ju).

364

THE THEORY OF RESTRICTED GENERALITY

[15B

Hence, again by the theorem,

From this result we conclude that if the theorem holds for a then it holds for a+ also. The corollaries follow from the theorem if analogous definitions are made for a~ etc., q.e.d. 2. AT-formulation In order to keep things straight in this subsection, we shall understand (13)

M

f-A X

to mean that M f- X holds in the formulation of § 1, and shall call that formulation $l11' Theorem 1 and its corollaries suggest the following T-rules for $l 21:

se. axy

xz

ai:

[~x]

'1 x

yz

a~'1

where in ai the variable x does not occur in ~, '1, or in any premise except the canceled one. Accordingly, we shall define the T-formulation of $l 21' or $lIp to be the system with no axioms and with Rules ae, ai, and Eq. To indicate that f- X follows from the set M of premises in this formulation, we shall write (14)

M

f-T X.

It follows that P has the following introduction rule:

Pi:

The elimination rule is, of course, Rule P. If E is present in the system, it follows that II has the introduction rule IIi:

Where x does not occur in

'1 or in any premise. As before, its elimination rule is Rule II. Similarly it follows that F has the introduction rule Fi of § 14C3. THEOREM

2. The statement (13) is equivalent to the statement (14).

Proof Corollary 1.4 proves that (13) follows from (14). To show that (14) follows from (13), it is sufficient to show that the axioms of $l11 are theorems of $lIl' Axsch. (a) is a special case of Rule ai, so if it is applied to a theorem of $lIt, the result will also be a theorem. Axiom scheme (FK) follows from the rules Fe and Fi as in § 14C3. Thus, it remains to prove Axiom Scheme (as).

lSBj

365

DEDUCTIVE THEORY OF :7'2

Let «, 13, and y be canonical obs of degree 1, 2, and 3 respectively. Then E 2rxf3(yx) and 3rx(Sf3y) are canonical of degree O. Hence, we can proceed as follows:

1

3

1

3 2rxf3(yx)

rxz

3(f3z)(yxz)

.::.e

NP. Similarly, if E is present, II is defined as in § AS, and NE assumed, then it follows that Nil.

Cano(IlX) <=± Canl(X).

Hence, if 3+ is defined as in § A6, we have N3+.

Can o(3+XY) <=± Canl(X) & Canl(Y),

and hence, by N3, we have (15)

Can o(3 +XY) <=± Can o(3XY).

Furthermore, by Corollary 1.7 3+ satisfies Rule 3i as well as Rule Ee. From this we can argue that (16) The proof of (16 2 ) is, with x a new variable,

1 3+ ~IJ

~x _+ ----.::, e IJx

-

3~1J,

_.

":'1-

1

366

THE THEORY OF RESTRICTED GENERALITY

[ISB

whereas (16 1) was established in the proof of Corollary 1.7 (it also follows by the same argument with E and E + interchanged). This shows that although ::::+ =1= :::, the two obs :::+ and E are equivalent. This also shows why, as Schutte pointed out in [STT), set-theoretic extensionality implies the failure of ET; for although (16) holds for all ~,1], yet :::+ ~ =1= ::::~. Thus if Q is extensional, it cannot be characterized by the rules *Q and Q* of § 12C2; since ET would then require Q-consistency, and hence either :::+ ~ = :::~, which is false, or else that ~ Q(:::+ ~)(:::O fails, which contradicts the extensionality assumption. 3. Canonicalness restrictions In this article we shall define a class of obs which satisfies the canonicalness assumptions made earlier in this chapter. The obs in this class will be called graded canobs; 9 they are essentially what are called nobs in [BVe]. In making this definition, we shall assume for the reasons already stated that the subjects are unrestricted. The definition of canobs given here supposes that there is given a class of canonical simplexes. The motivation for the definition of the canonical simpelxes is the class of basic canonical simplexes of Definition 3 below, but it is not necessary to be so specific; the canonical simplexes can be taken to be any class which satisfies the following assumptions :10 (a) If X is a canonical simplex and X Y, then Y is a canonical simplex. (b) If X is a canonical simplex, x any variable, and U any ob, then [Ujx)X is a canonical simplex. (c) No canonical simplex has S as its head. (d) No canonical simplex is an Orob.11 Now, in terms of these subjects and simplexes, we shall define graded canobs, With each canob will be associated two numbers, the degree and the rank.

>-

9. The adjective 'graded' is to distinguish these obs from the "canonical obs" of [DTC], for which the notion of degree introduced below was not defined. Such obs might be called canobs without the qualification "graded". However since only graded canobs are considered in this book, the adjective will generally be omitted. 10. These are the assumptions (b)-(d) of [BVC]§ 2. Condition (b) ought to be a consequence of the definition of an indeterminate, but it does place limitations on where variables can occur in canonical simplexes; thus, with (c) it implies that no simplex has a variable as its head. Cf. the assumptions of § I2Dl about prime statements (p2). 11. This assumption was not made in [BVC], but this is an oversight. For, in [BVC], if; and Tj are nobs (canobs) of index (degree) one, then Ax.E;Tj

is a nob (canob) of index (degree) one and rank one more than rk(;) + rk(Tj), but if this assumption about simplexes is not added to those of [BVC], than this ob can also be a simplex and hence have rank zero, contradicting Theorem 4 below and Lemma 7 of [BVC]. Notice that the basic canonical simplexes of Definition 3 below do satisfy this assumption.

lSB]

DEDUCTIVE THEORY OF

367

~2

The degree and rank of a canob X will be denoted by dg(X),

rk(X),

respectively. DEFINITION I. The ob ~ is a proper canob of rank m and degree n if and only if (a) ~ is a canonical simplex and m = n = 0; (b) ~ == AX. '7 where '7 is a proper canob of rank m and degree n - I; 12 (c) ~ == 8''7 where' and '7 are proper canobs of degree I, n = 0, and rk(O + rk('7) = m - 1. DEFINITION 2. The ob ~ is a canob of rank m and degree n if and only if there is a proper canob '7 of rank m and degree n such that ~ '7. We shall now prove

>-

THEOREM 3. If Cank(X) is interpreted to mean that X is a canob of degree k, then NI-N3, and N8 hold. This theorem will follow from the following lemmas: LEMMA 3.1. If ~ is a (proper) canob of rank m and degree n, and if X is any variable and U any ob, then [U/xR is also a (proper) canob of rank m and degree n. Proof By induction on the construction of ~ according to Definitions 1 and 2. The basic step follows from assumption (b) for canonical simplexes. The induction step is straightforward.

LEMMA 3.2. If ~ is a proper canob of rank m and degree n, then there is a proper canob v of rank m and degree 0, where v is a canonical simplex if m = 0 and is of the form of clause (c) of Definition 1 if m > 0, such that ~

(17)

==

AxI ...Xn·V.

Proof By induction on the construction of ~ according to Definition 1. For the basic step of the induction, it is sufficient to note that ~ itself is such a v. For the induction step, there are the following two cases, depending on which case of Definition 1 is used: (b) ~ == AX. '7. By the induction hypothesis, n ~ 1, dg('1) = n - 1, and there is a proper canob /l of rank m and degree 0 which is of the correct form such that It follows that ~

==

AXXI",Xn-I·/l·

Let v == [Xl/X, X2/x l, ... , xn/xn-d/l. Then (17) holds. By Lemma 3.1, v is a proper canob of rank m and degree O. Moreover, it follows by the properties 12. Hence, we must have n

~

1 in this clause.

[15.8

THE THEORY OF RESTRICTED GENERALITY

368

of substitution that v is either a simplex or has the form of clause (c) of Definition 1 according to which of these alternatives holds for fl. (c) ~ == ='(1]. Then m > 0, n = 0, and ~ itself is such a v, q.e.d.

It follows from Lemma 3.2 and Definition 2 that the canobs are the same as the nobs of [BVC]. 3.3. If

~

is a canob of rank m and degree n and canob of the same rank and degree. LEMMA

~ =

Y, then Y is a

Proof By Definition 2 there is an I] such that I] is a proper canob of the same rank and degree such that ~ 1]. By Lemma 3.2 there is a proper canob v of rank m and degree such that

>-

°

I]

== AXl" .Xn· V.

It follows that

Hence by the Church-Rosser theorem there is a Z such that

>- Z, YXl",X n >- Z. v

(18)

From the second of these we have, by property

Y>- Ax

l . ..

(~),

xn.z.

This will prove the lemma if we show that Z has the form prescribed in Lemma 3.2. For m = this follows from (18) by assumption (a) about simplexes; for m > 0 it follows by induction on »rand property C-VC3 (§ llF3).

°

3.4. If ~ is a canob of rank m and degree n > 0, then for any ob U, is a canob of rank m and degree n - 1.

LEMMA ~U

Proof Suppose that ~U

~

is a proper canob. Then (17) holds, and

== (Axl""'n'v)U = Ax2",xn.[U/xtlv.

By Lemma 3.1, [U /xtlv is a proper canob of rank m and degree 0. Hence by Definition l(b) and Lemma 3.3, ~U is a proper canob of rank m and degree n - I. The general case now follows by Definition 2 and Lemma 3.3. LEMMA 3.5. The ob ~ is a canob of rank m and degree n is a canob of rank m and degree n + 1.

if and only if AX • ~

Proof Let ~ be a canob of rank m and degree n. By Definition 2 there is a proper canob I] of rank m and degree n such that ~ 1]. By Definition I(b) Ax.1] is a proper canob of rank m and degree n + 1. By property (~) for reduction,

>-

15B]

369

DEDUCTIVE THEORY OF :F 2

Hence by Definition 2, AX. ~ is also a canob of rank m and degree n + 1. This proves the only if part. Conversely suppose AX. ~ is a canob of rank m and degree n + 1. Then there is a proper canob I] of rank m and degree n + 1 such that (Ax. ~)

>- 1].

By Lemma 3.4, I]X is a canob of rank m and degree n, and hence ~ is by Lemma 3.3. This completes the proof of the lemma. LEMMA 3.6. The ob 3XY is a canob of rank m and degree 0 and Yare canobs of degree 1 and m = rk(X) + rk(Y) + 1.

if and only if X

Proof Let X and Y be canobs of degree 1. By Definition 2 there are proper canobs ~ and I] of degree 1 and the same ranks as X and Y such that X ~, Y>-I]. By Definition l(c), 3~1] is a canob of rank m and degree 0; consequently by Lemma 3.3 3XY is also. This proves the if part. Conversely let 3XY be a canob of rank m and degree O. Then there is a proper canob ( of rank m and degree 0 such that 3XY>- (. By property C-VC3 (§ llF3), ( is of the form 3~1], where X ~ and Y>- 1'/. By Definition 1, ~ and I] are proper canobs of degree 1, the sum of whose ranks is m - 1. Hence X and Yare canobs of the stated rank and degree, q.e.d.

>-

>-

Proof of Theorem 3. NI-3 are just Lemmas 3.3, 3.5, and 3.4; N3 is Lemma 3.6. THEOREM

Proof If the form

4. The rank m and, ~

if m >

0, also the degree of a canob are unique.

has two ranks and degrees then there must be an equation of

(19)

where JL and v are subject to the conditions of Lemma 3.2. Hence (20)

By the Church-Rosser Theorem these both reduce to some Z. If either JL or v has 3 as its head then so does Z, and hence neither JL nor v can be a simplex by assumption (c) about simplexes. In this case, they are both of the form 3(1], and hence so is Z, and p = O. We can now employ an induction with respect to the minimum of the two ranks (neither of which is 0). This proves the theorem. The theory of canobs as formulated so far is not complete because it is impossible to prove that the degree of a canob of rank 0 is unique. In order to be able to prove this property, it is necessary to have more information about the canonical simplexes. Accordingly, a special class of canonical simplexes,

370

[lSB

THE THEORY OF RESTRICTED GENERALITY

called basic canonical simplexes, will now be defined. If this definition for simplexes is adopted, the resulting canobs will be called basic canobsP We begin by postulating a finite or infinite sequence of canonical atoms 01 , O2 , ••• , each of which is a C-indeterminate distinct from S. An unspecified canonical atom will simply be called e. With each 0 we associate a number called dg(lJ), or the degree of e. In general, the degrees of the canonical atoms will be unspecified, but it is assumed that if 0 is to be interpreted as a propositional function of n arguments, then dg(O) = n. Thus, if E and Q are taken as canonical atoms, then dg(E)

DEFINITION 3. The ob

~

= 1,

dg(Q)

= 2.

is a basic canonical simplex if and only if ~

== OU 1 • .. Un ,

where 0 is a canonical atom of degree nand U 1 ,

... ,

U; are any obs.

THEOREM 5. If ~ is a basic canonical simplex, then (a) if ~ Y, then Y is a basic canonical simplex with the same 0 and n; (b) ifU is any ob and x any variable, then [U/x]~ is a basic canonical simplex with the same () and n; (c) the head of ~ is not S.

>-

Proof By postulate C-VC3, all reductions and substitutions in ~ must take place inside the U t» and hence, by the assumptions about canonical simplexes, (a) and (b) follow. Part (c) follows because the head of ~ is e which is, by hypothesis, distinct from S.

COROLLARY 5.1. The rank and degree of every basic canob are unique. Proof By Theorem 4, it is enough to prove that the degree of a basic canob of rank 0 is unique. In this case, both J.l and v in (20) are basic canonical simplexes, and hence they both have canonical atoms for their heads. Since Z then has the same head as both, this canonical atom must be the same in both cases, and since the degree of each canonical atom is unique, we must have p = O. 4. An L-formulation

The T-rules for S given in § 2 suggest the following operational L-rules: .3.

M

lar

~Z, L; M, '1Z laJ- N, L

M, S~'7

lar N,

L,

M,

ex la, xr t1

M

X,

L

14

lar S~'7, L.

13. It is clear that the class of canobs is determined from Definitions 1 and 2 by the class of canonical simplexes. The situation is analogous to the theory of functionality in which the F-obs are determined by the F-simples. Furthermore, the basic canonical simplexes are a generalization of the basic F-simples to the case in which the system is no longer separated. 14. The lower case Greek letters in these rules indicate that the indicated obs are canobs of the appropriate degree. In this case, ; and 1J are canobs of degree one.

15B]

371

DEDUCTIVE THEORY OF :F2

Accordingly, the L-formulation of IF21, or 1F~1' will be defined to be the restricted CL-system specified as follows: (i) There are no prime statements (p2). (ii) There are no elementary rules or rules described as "additional irregular rules" in § 12Dl. (iii) The definition of canonical obs will be that of canobs in § 3. The most interesting case is the one in which only basic canobs ale used, but this assumption is not necessary. Note that the L-rules for P and II which follow from their definitions and Rules are the following:

.2.

M

la~ ~, L;

M, M, M,

M, rJ

P~rJ la~

la~

M, ~ la~", L M la~ P~", L,

N, L

N, L,

~U la~N II~ la~

M

la, x~

~x, L

M la~ II~, L.

N,

The theorems given here will apply to systems in which there are prime statements (P2), provided these are assertions of canonical simplexes, and elementary rules, provided their principal constituents are canonical simplexes, and also rules with no principal constituent; if any of these do occur, we shall assume that the corresponding axioms and rules have been added to the Aand T-formulations. THEOREM

6. If X is a constituent of a theorem, then X is a canob of degree O.

Proof By induction on the length of the proof. THEOREM

7.

ET

holds.

Proof By induction on the rank of the eliminated constituent. The proof is like that of Theorem 12C9, except that the rank is now a function of the eliminated constituent as an ob only and does not depend on the derivations in which it occurs. Stages 1 and 2 go through as in earlier proofs without change. For Stage 3, suppose that the eliminated constituent is X == 2~rJ. Then the premises for the major premise of the elimination are la~ ~U,

(21)

M

(22)

M, rJU [a} N, L,

L,

and the premise for the minor premise is

M,

~x

where x does not occur in M, L,

~,

(23)

(24)

[c, x~ rJX, L,

or rJ. By Theorem 12C6,

M, ~U la~ "U, L.

372

THE THEORY OF RESTRICTED GENERALITY

[ ISB

By the results of § 3,

+ rk(~) + rk('7) = 1 + rk(~U) + rk('7U).

rk(X) = 1

Hence, the conclusion follows from (21), (22), and (24) by two applications of the main induction hypothesis. THEOREM

8. If the L-system is taken to be singular, M la~ X +:t M ~T X.

(25)

The proof is similar to the proofs of Theorems 14E2-3, and will therefore be omitted. We cannot expect the singular and multiple L-systems to be equivalent in this system, since the singular systems have the same relation to the multiple systems that the system LA* has to LC* in [FML]. Note that Theorems 7 and 8 amount to a proof of consistency for :F 21' even if axioms and rules are added which can be taken as prime statements (p2) and elementary rules in the L-system, provided that the definitions of § 3 are taken for canonicalness. We shall come back to this point in § 6. 5.

-r-formulation

In [ACT] and [BVe] Curry proposed a system which is closely related to the singular L-system of § 4. This system, called -r 2, has two predicates of an unspecified number of arguments, forming elementary statements of the forms (26)

Can(M 1 ,

,

M m, X),

(27)

Ver(M 1,

,

M m , X).

Of these, (26) is intended to be interpreted as stating that if M l ' ... , M mare asserted, then X is canonical (of degree 0), and (27) that under these same hypotheses, X is verifiable, or true. The formula (28)

Reg(M 1 ,

... ,

M m , X)

will be used to stand for either (26) or (27), with the understanding that where 'Reg' appears several times in the same context, the same one of 'Can' or 'Ver' is intended in all the occurrences. A sequence of obs to which (26), (27), or (28) applies will be called a canonical, verifiable, or regular sequence respectively; the 0 bs M l' ... , M m will be called the antecedents, and X will be called the consequent. As with L-systems, the antecedents and consequent will be called constituents, and two constituents which are either both antecedents or both consequents will be said to be on the same side.

lSBj

DEDUCTIVE THEORY OF ff 2

373

The axioms of the system will be of the form (26) and (27) with m = O. The ob X will then be called a canonical or verifiable simplex as the case may be. It is assumed that simplexes (of both kinds) satisfy the assumptions (a)-(d) made for canonical simplexes in § 3 and, in addition, the following: (e) Every verifiable simplex is also a canonical simplex. The rules are as follows:

I.

RULE OF TAUTOLOGY.

Can(M1> ... , M m , X) Reg(M l ' II.

l.

M m , X, X).

STRUCTURAL.RULES. RULE OF PERMUTATION.

Can(M l'

... ,

M m , Y, Z)

Reg(M 1 ,

2.

.•. ,

Reg(M l'

... ,

•.• ,

M m , Y, Z,N 1 ,

... ,

M m , Y, Y, N 1 ,

M m , Z, Y, N 1 , ...

N n , X)

,Nn , X).

RULE OF CONTRACTION.

Reg(M 1 ,

,

Reg(M 1 " " M m , Y, N 1 , 3.

... ,

N n , X)

N n , X).

,

RULE OF WEAKENING.

Can(M 1 ,

... ,

M m , Y)

Reg(M 1 , III.

RULE OF EXPANSION.

Reg(M l'

••• ,

Reg(M 1 , RULES FOR

M m , N l'

... ,

••.

,N m , Y)

... ,

M m , X).

X

E.

1. If x does not occur in M 1 ,

M m , X, Y, then

••• ,

Reg(M 1 ,

2. If x does not occur in M 1 , Reg(M 1 ,

,

... ,

G1

M m , Xx, Yx)

,

Reg(M l '

M m , EXY).

M m , Y, Z, then G2

... ,

M m, 3YZ, N 1 ,

G3 ... ,

N n, X),

where G 1 == Can(M l'

•.. ,

M m , Yx, Zx),

G2 == Ver(M1>

, M m , YU),

G 3 == Reg(M l'

,

M m , ZU, N 1>

Nn> X)

N n , X).

••• ,

>-- N 1> ... , M m >-- N m'

If M 1

Reg(N 1 ,

IV.

... ,

M m , Y, N 1 ,

... ,

N n , X).

>-- Y,

then

374

THE THEORY OF RESTRICTED GENERALITY

[15B

Although this is not strictly an L-system, it has most of their important properties. Thus, theorems such as Theorems 12C6-7 can be proved for this system, and will be used without further mention. In order to make further use of this analogy, we shall extend the definitions regarding the constituents as follows: Rule I has two principal constituents and one subaltern, the indicated X in the conclusion and premise respectively; Rule II 1 has two principal constituents, Y and Z in the conclusion, and two subalterns, the replicas of them in the right premise; the replicas of them in the left premise are called the quasi-subalterns; Rule II 2 has two subalterns and one principal constituent like Rule *W; Rule II 3 has one quasi-subaltern, Y in the left premise, no subaltern, and one principal constituent, Y in the conclusion; Rule III should be regarded as broken down into parts, each of which 'expands' exactly one constituent, so that the subaltern and principal constituent can be defined in the usual way; Rule IV I has two subalterns, the Xx and Yx, and one principal constituent, SXY; Rule IV 2 has two quasi-subalterns, Yx and Zx, two subalterns, YU and ZU, and one principal constituent, SYZ. The variable x in both of Rules IV is the characteristic variable of the rule. All constituents other than those mentioned above are parametric. The semiparametric ancestors are defined using the subalterns (not quasi-subalterns) and principal constituents of Rules II and III, but not of Rule LiS Since this system is in many ways similar to the systems of relative canonicalness defined in § C5 below, we would tend to expect that given any regular sequence, any initial segment of it is a canonical sequence; in other words, if we take the first several elements of any regular sequence as premises, it should tollow that the next is canonical. This is shown to be true by the following fheorem. THEOREM

9. If

Reg(M 1 ,

(29) then for any r

~

... ,

M s) ,

s, we have

(30) Proof By deductive induction on the derivation of (29). If (29) is an axiom, then s = I and (30) follows by the assumption (e) made about simplexes. This completes the basic step of the induction. For the induction step, suppose (29) is the conclusion of a rule and that the theorem holds for the premise(s). We shall show that the theorem holds for the conclusion. There are the following cases depending on the rule. 15. This exception is essential in the proof of Theorem 1J. Without it we should not have the property that semiparametric ancestors are always on the same side. 16. Cf. [BVq Theorem 1.

DEDUCTIVE THEORY OF :F 2

lSBI

375

If the rule is I and r < s, then (30) follows immediately from the induction hypothesis, whereas if r = s, then either (29) is (30) itself (if 'Reg' is 'Can') or else we can obtain (30) from the premise of (29) by the same rule (if 'Reg' is OVer'). If the rule is II 1, let the right-most principal constituent be MI' If r > t, (30) follows by the same rule from the induction hypothesis for the right premise. If r ~ t, (30) follows from the induction hypothesis on the left premise. If the rule is II 2, let the principal constituent be M t • If r > t, (30) follows from the induction hypothesis by the same rule. If r ~ t, (30) follows directly from the induction hypothesis. If the rule is II 3, let the principal constituent be M t. If r > t, (30) follows by the same rule from the left premise and the induction hypothesis of the right premise. If r ~ t, (30) follows directly from the induction hypothesis of the left premise. If the rule is III, then (30) follows from the induction hypothesis, using the same rule again if necessary. If the rule is IV I, then (30) follows from the induction hypothesis, using the same rule again in the case that r = s. If the rule is IV 2, then let the principal constituent be M r- If r < t, then (30) follows by the induction hypothesis applied to the first premise. If r = t, then (30) follows from the first premise itself by Rule IV 1. If r > t, then (30) follows by Rule IV 2 from the first two premises and the induction hypothesis of the third, q.e.d. THEOREM 10. Any ob which appears as a constituent of a true statement of the form (26) or (27) is a canob of degree 0, where the simplexes are taken to be the canonical simplexes. 1 7

Proof Similar to Theorem 6. THEOREM

(31)

11 (ET). If Reg(M 1 ,

... ,

M m , X, N 1 ,

... ,

N n , Y)

and (32)

Ver(M 1 ,

... ,

M m , X),

then (33)

Reg(M I '

... ,

M m, N I'

... ,

N m Y).

17. Cf. [BVC] Lemma 6. This lemma contains a significant misprint: the word 'nob' appears in place of 'ob', and this gives it the apparent meaning which this theorem would have if 'canob' appeared in place of 'ob',

376

THE THEORY OF RESTRICTED GENERALITY

[lSB

Proof18 As usual, we can assume without loss of generality that no characteristic variable occurring in the proof of (31) or (32) occurs in the proof of the other or in (33). The proof is an induction on the rank of X. There are the usual three stages. Stage I. Let D be a derivation G 1,

,

Reg(L 1 , With each Gk associate a statement Reg(M 1 ,

G~

... ,

Gp of (31), and let Gk be ,

L s)'

as follows:

M m, K 1 ,

••• ,

K r) ,

where the sequence K 1 , ... , K, is formed from L 1 , ... , L, by deleting the semi. parametric ancestors of X. Let D 1 be that part of D in which there are no semiparametric ancestors of X, and let D 2 be the rest. As usual, it is sufficient to prove by induction on k that G~ is derivable for every k. There are the following cases: (IX) Gk is in D 1 • Then r = S, K, = L, for i = 1,2, ... , s. By (32) and Theorem 9, we have (34)

h

~

m.

Hence, G~ follows from (34) and Gk by repeated applications of Rule II 3. (Il) Gk is in D 2 and is an axiom. This case is impossible, since there are semiparametric ancestors of X in Gk , and since semiparametric ancestors are so defined that they are on the same side, it follows that Gk must have antecedents if it is in D 2 , and hence it cannot be an axiom. (I') Gk is in D 2 and is obtained by a rule for which all semiparametric ancestors of X are parametric. Let the premises be G i , Gj , •••• By the induction hypothesis, G;, Gj, .,. are derivable, and G~ follows from these by the same rule. (0) G k is in D 2 and is obtained by one of Rules I, II, or III for which a semiparametric ancestor of X is a principal constituent. If the rule is one of II or III, then let the (right) premise be G i • Then, by the induction hypothesis, G; is derivable. In the case of II, G~ == G;; 19 in the case of Ill, G~ follows from G; by the same rule; in either case G~ is derivable. If the rule is I, then L, == L s - 1 == K, == Z, where X)- Z. Hence, by (32) and the converse of the expansion rules (see Theorem 12C7), we have (35)

Ver(M 1 ,

••• ,

M m , Z).

18. This proof is very similar to that of Theorem 7. We give it in full here so that it can be compared with the proof of [BVCI Theorem 4. That proof is more complicated than the one given here. 19. In the case of Rule II I, it may happen that one of the principal constituents is a semiparametric ancestor of X and the other is not. This causes no problem, since one of the constituents is omitted in Gfc and Gt and hence the position of the other is the same in both.

lSBj

DEDUCTIVE THEORY OF ff 2

377

Now, the premise of the inference, Gi , is

Can(L 1 ,

••• ,

L.- 2 , Z),

and G;, which is derivable by the induction hypothesis, is Can(M 1 ,

... ,

Hence, by Theorem 9, for all h

~

M m, K 1, r -

... ,

Kr-

1,

Z).

1,

Hence, G~ follows from (35) by Rule II 3 and, if the predicate of Gk is Can, by Theorem 9. (8) Gk is in D 2 and is obtained by a rule IV for which a semiparametric ancestor of X is the principal constituent. Then the principal constituent must be an antecedent, so the rule must be IV 2. If the principal constituent is L q + 1 == SUV, then q + 1 < s; let tho premises be the following:

Gh == Can(L 1 ,

,

L q , Ux, Vx),

G, == Ver(L 1 ,

,

L q , UW),

== Reg(L 1 ,

,

L q , VW, L q + 2 ,

Gj

... ,

L,),

where x does not occur in U, V. L 1 , • • • , L q• 'Now Ux, Vx, UW, and VW are not semiparametric ancestors of X. Hence, by the induction hypothesis, G~

== Can(M 1 ,

G; == Ver(M 1 , Gj == Reg(M 1 ,

,

Mm, K 1, Mm, Kb

,

,

,

K t , UW),

,

M m , K 1,

K t , Ux, Vx),

,

K t , VW, K t + 1 ,

... ,

K r) ,

are all derivable. Since K b ... , K, is a subsequence of L 1 , ... , L q , x does not occur in it, and x does not occur in M l ' ... , M m by the assumption about characteristic variables made at the beginning of the proof. Furthermore, since all semiparametric ancestors of X are antecedents, K, == L., and since q + 1 < s, t < r. Hence, by Rule IV 2, Reg(M 1 ,

... ,

M m, K 1,

... ,

K t , SUV, K t + 1 ,

... ,

K r)

is derivable, and from this we can get G{ by the hypothesis of the stage. Since Case (8) cannot arise unless X is of positive rank, the basic step of the main induction is proved. Hence, for the rest of the proof it will be assumed that X (and hence each of its semiparametric ancestors) is of positive rank. Stage 2. Let D be a derivation G1>

Reg(L 1 ,

, ,

Gp of (32), and let Gk be

L., Z).

378

THE THEORY OF RESTRICTED GENERALITY

[lSB

Let D t be that part of D in which no semiparametric ancestors of X occur, and let D 2 be the rest of D. Since the predicate of (32) is Ver, and since if the conclusion of a multiple-premise rule is in D 2 then the right premise but no other premise is in D 2 , if Gk is in D 2 it is Ver(L t , ••• , L., Z),

>-

where Z is the only semiparametric ancestor of X, and hence X Z. With each Gk , associate a statement G~ as follows: if Gk is in D t, then G~ is Reg(M1> ... , M m , L1> ... , L., Z), where Reg is the predicate of Gk ; if Gk is in D 2 , then Reg(M t ,

... ,

G~

is

M m, L t , ... , L., N1> ... , N m Y),

where in this case Reg is the predicate of (31). Then (33) follows from G; by II 1 and II 2. As usual, it is sufficient to prove by induction on k that G~ is derivable for each k. There are the following cases: (IX) G, is in D t • Then, as in Stage 1, G~ follows from G; by 113. (13) Gk is in D 2 and is an axiom. This is impossible since every semiparametric ancestor of X has positive rank. (y) Gk is in D 2 and is derived by a rule for which the semiparametric ancestor of X is parametric. Thus, the rule must be one of II, III, or IV 2. If the rule is II 2 or III, then it is clear that G~ follows as in Stage 1 Case (y). If the rule is one of II 1, II 3, or IV 2, then only the right-most premise is in D 2 ; however, since the principal constituent must be L I for some i ~ S, the different definitions of G~ cause no difficulty in this same argument.f" (15) Gk is in D 2 and is derived by a rule I-III for which the semiparametric ancestor of X is a principal constituent. The only possible such rules (since a principal constituent is a consequent) are I and III. If the rule is III, then the premise is also in D 2 , and so the proof is similar to that of Stage 1 Case (15). If the rule is I, then L. == Z, X Z, and the premise GI , which is in Dl> is

>-

Can(L t , Since

X>- Z, we have from

... ,

L._1> Z).

(31)

(36)

By GI and Theorem 9, Can(L t ,

... ,

L t) ,

t = 1,2, ... ,s-l,

and hence, by II 3,

t = 1,2, ... , s - 1. 20. Cf. Stage 1 Case (e). Cf. also Case 6 of [BYC] on p. 185 for the case of Rule IV 2. 21. At this point in the proof, there is another error in [BYCI. For Gk in D2, GIc was defined to have the predicate Ver rather than the same predicate as (31). Thus, if the predicate of (31) is Can, this particular case (Case 1, p, 184) of [BVC] fails.

15B]

379

DEDUCTIVE THEORY OF ff 2

Thus, G~ follows from (36) by II 3. (e) Gk is in D 2 and follows by a rule IV for which the semiparametric ancestor of X is the principal constituent. Then the rule must be IV 1, and G~ follows by the hypothesis of the stage. Stage 3. In this stage, suppose that X == 3UV, that (31) is the conclusion of an inference by IV 2 with X as the principal constituent, and that (32) is the conclusion of an inference by IV 1. Then the second two premises for (31) are

(37) (38)

Ver(M t , Reg(M t ,

... ,

M m , UW),

M m , VW,N t ,

... ,

... ,Nm

Y),

and the premise for (32) is (39)

Ver(M t ,

••• ,

M m , Ux, Vx),

where x does not occur in M t , •.• , M m , U, or V. Hence, ifW is substituted for x in the derivation of (39), we have a derivation of (40)

Ver(M t ,

••• ,

M m , UW, VW).

Furthermore, rk(X) = 1 + rk(UW) + rk(VW). Hence, from (37), (38), and (40), we can get (33) by two applications of the main induction hypothesis. This system can be modified to admit elementary rules without upsetting ET, for each such rule has all subalterns and the principal constituent as antecedents and has a principal constituent of rank 0, so that these rules are excluded from Case (15) in Stage 2 by the assumption that the rank of each semiparametric ancestor of X is positive; these rules are thus fully taken care of by the Case (y) in each of the first two stages. Hence, this system can be made to incorporate all the extra rules that the L-systems of § 4 can. 6. Equality and related extensions It was mentioned at the end of § 12B2 that if we have an ob Q such that ~QXX

(41)

holds for all obs X, then we have immediately by Rule Eq (42)

X

= Y -+

~ QXY.

It was also mentioned there that, although one might think of Q as being defined in various ways, it might be expedient to postulate Q as a separate atom. In a way that is the most general approach; because anything we may deduce about the new atom will apply to any corresponding defined combination, provided, of course, that the properties postulated for the former hold also for the latter.

380

THE THEORY OF RESTRICTED GENERALITY

[15B

Suppose, then, that we introduce Q as a new atom. Then it is natural, as we have already remarked in § 3, to suppose it is a canonical atom of degree 2. 22 Accordingly QXY will be a basic canonical simplex for arbitrary X and Y, and hence we have, in the notations of § 4 and § 5 respectively (43)

Cano(QXY),

Can(QXY).

Axioms which will give us (41) can be introduced into the system of § 5 by the assumption Ver(QXX), which is allowable by (432 ) , In ff 21 with unrestricted subjects we can define E == WQ,

and then use the machinery already specified for E; or we can use the axiom ~

EE(WQ)

or the T-rule

Ex ~ Qxx. In this way we have THEOREM 12. The system ff 21 with the basic canonical obs of § 3, canonical atom Q, and axioms (41) is Q-consistent.

Proof The theorem follows from the introductory discussion and Theorems 7 and 8 or Theorem 11. This result shows that we cannot derive a contradiction from the assumptions we have introduced so far, including (43). But this is an extremely weak result. For we cannot deduce any properties of Q. Thus we cannot show, in the sense of deducibility 23 that (44)

QXY,ZXr Z Y.

Possibly one could attack this situation by making a combination of the system in § 4 with that of § 12C; but that has not been done, and it is not clear that ET would hold for the resulting system.i" 22. It is not strictly necessary to do this; for we could modify the definition of basic canonical simplex so as to admit (43) only for Y"" X; cf. § C3. But the proof of Q-consistency holds with the full (43). 23. Using Q-consistency, we can indeed show that (44) is admissible. For from ~ QXY we can conclude X = Y, and then the conclusion follows by Rule Eq from the other premise. But the Q-consistency argument cannot be carried out by the rules of ofF 21. 24. If we were to introduce the axiom scheme (*)

~ Ex ::::>x.Ey ::::>y.Qxy ::::>.Cx::::> Cy,

then we have (44) under the additional hypothesis Canif.Z); and then, using (43) we can derive the symmetric and transitive properties in the forms ~ Ex ::::>x.Qxy ::::>y.Qyx, ~ Ex::::> x.Qxy ::::>y.Qyz ::::>z.Qxz.

But we no longer have a proof of Q-consistency.

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381

Now suppose we define Q, as in § 7DI, by (45) Then we cannot derive either (41) or (43). In fact since QII

= 3(h. xl)(h. xl),

this is not even canonical according to N3. Thus if we hold to N3 as stated we cannot postulate (41) without violating the condition that all asserted obs are canonical. There is, however, a possible way out. Suppose we modify N3 and clause (b) of Definition I to admit that 3XY is also canonical (of degree 0 and rank, say, 0) when X = Y. Then the obs (41) will be canonical. Furthermore, it will still be possible to show, by deductive induction, that if the axiomatic obs are canonical, so are all asserted obs. For we can divide the applications of Rule 3, the premises being canonical, into two kinds, regular and irregular: in the regular kind we have the same situation as in § I ; in the irregular kind the case of 3XY with X = y. 2 5 In either case the canonicalness of the premises entails that of the conclusion - in the irregular kind since it is equal to one premise. The theory of § 3 we have not carried through in detail, but it does not seem to offer any great difficulty. In such a system we should not have (43); but perhaps that is an advantage, because the undecidability theorem gives a motivation for not always considering QXYas a proposition. We could then adopt the axiom (as we would have to by Axsch. (3)) (46)

~

3E(W3).

However, we have not postulated this axiom here, and have not pursued the subject further. A similar remark applies if we admit obs of the form 3XE as canonical and adopt as axiom (47)

~

3E(C3E).

C. FINITE FORMULATIONS In the formulation of § B the axioms were given in the form of axiom schemes, and consequently the number of axioms was infinite; moreover we could not avoid the possibility, when we extended the system by adjoining variables, that there would be axioms in the extension which contained these adjoined variables. In this section we shall consider formulations where the axiomatic basis is finite. 25. To be sure the question of which alternative is before us, considered in isolation, is undecidable. But we can suppose as part of the hypothesis that we have a proof of the canonicalness of the premises, and from this proof the two cases can be distinguished. This hypothesi is fulfilled for the axioms.

382

THE THEORY OF RESTRICTED GENERALITY

[ISC

To accomplish this we postulate two new illative primitives, Hand Y; and we also postulate the universal category E.The ob H is to represent the canonical obs of degree 0, Y the canonical subjects. The intention is to interpret H as the category of propositions; formally this means that H represents a class of obs for which we can assume certain formal properties of propositions. In terms of these notions we can formulate conditions allied to those for canonical obs in § B. This section involves some extrapolation of § B; there are various ways of making this extrapolation, and it is by no means clear which of these ways is the most acceptable. In view of the fundamental philosophy in § HA2, it is necessary to explore more than one. The greater part of this section is devoted to an extrapolation which sticks as closely as possible to the fundamental motivation back of the developments in § B. This idea was that it may be possible to treat as propositions obs which begin with certain C-indeterminates (the canonical atoms) representing the primitive verbal functions of our usual mathematical theories, even though such combinations might be intuitively meaningless - in that case they could be regarded simply as false. From this standpoint we are interested in the case where the subjects are unrestricted;' moreover, since H is a Cvindeterminate, it is natural to suppose it is a canonical atom, so that we can assume (1)

~

H(HX)

for arbitrary X. This assumption is incompatible with certain other reasonable assumptions for H; 2 but if one does not make these assumptions it is not known that a contradiction will ensue. We begin in § I with a formulation of rules and axioms, and show that a deduction theorem, as stated in § B, follows from that formulation. In this we do not introduce all at once the full strength of the assumption made in the preceding paragraph, but allow, for example, the possibility of restricted canonical subjects. In fact the deduction theorem is deduced from six lemmas (of which one is an immediate consequence of the others), an axiom, and some further premises, all of which hold under broad conditions. We introduce the full strength at the close of § I, and then show in § 2 that the system has a Tformulation. In § 3 we treat some questions relating to the representation of equality. An L-formulation is formulated in § 4 with a cut rule; we do not have 1. Originally, in [CFM] the number of arguments was also unrestricted, the introduction of degrees being a later refinement (cf. footnote 3 in [ACT]); the addition of restricted canonical subjects is due to Seldin [SIC]. 2. On this account Curry in [CFM] p. 63, stated that from it he could "deduce a contradiction in short order". Curry actually had in mind a system of relative canonicalness (cf. § 5 and [ACT], § 6, p. 567) similar to that afterwards developed in Bunder [STB]. Bunder's derivation of the contradiction is given below in § 5.

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383

a proof of the elimination theorem for the system, and it is doubtful if a constructive proof is possible. In § 5 we consider the system of Bunder. This is an alternative extrapolation which is incompatible with (1). Unfortunately time limitations do not permit us to treat this system fully; for a detailed treatment we refer to Bunder [STB]. In § 6 we consider further some variants in the formulation, and in § 7 we discuss a finite version of the i/-formulation of § B5. The question of consistency of the systems considered in this section is left open. All we can say is that as yet no inconsistency is known. 1. The deduction theorem

We define a sequence Hn as follows DEFINITION

1.

Ho == H, Hn + i == FYH n• Because of its special importance in connection with the deduction theorem, we also define L == Hi == FYH. Our intention is to interpret (2)

to mean (3)

and (4)

~

YU

to mean (5)

Cans(U).

However, we cannot simply postulate (2) and (4) where we had (3) and (5) before; for, as we saw already in § 12B4, we cannot have (4) in case U is a variable, or contains a variable for which an arbitrary ob can be substituted, unless we have unrestricted subjects. Hence we cannot simply postulate the assumptions Nl-3, SI-2, N:=:. The present formulation of canonicalness will thus be similar to, but in some respects different from, that considered in § B. Instead of postulating a definition of canonicalness from which we can show by induction that every derivable assertion is canonical, we shall postulate outright the rule RULE

H.

x~ HX.

THE THEORY OF RESTRICTED GENERALITY

384

[ISe

Thus the system .?F~1' which will be defined in this section, will have three primitive rules, viz. Rule H, Rule 3, and Rule Eq. Along with these we postulate a finite number of axioms. From these the properties of canonicalness will be derived. Following the lead of the basic canobs of § B3, we assume that there are canonical atoms e, each of an appropriate degree. These will include H and I, each of degree 1. For each e we postulate

n = dg(e).

(Fe)

In particular we have (FH)

~

LH,

(FY)

~

LY.

Furthermore, it turns out that we need E in the system. Hence, we postulate the axioms (EA)

~

(E)

~ 3E(FEE).

if A is an atom,"

EA,

Axiom (Ex) will be the only axiom in which the variable x occurs. Finally, we have the following axioms: (3H)*

~ Lx ::::> x' H(Exy) ::::> y' 3xy ::::> 3x(BHy),

(FK)*

~

(3S)*

~ Lx::::> x'

Lx ::::>x.Ly::::>y [FK]xy,

Ev ::::>v' H(3x(Syv» ::::>y' H(3 2xyz) 3 2xyz

(3E)*

~

::::>

.3x(Syv)

::::>

::::>z.

3x(Szv),

Lx ::::>x 3xE.

The last axiom is that which would be obtained by Axsch. (3) from the only axiom in which x occurs. Before going further we state and prove the principal theorem of this section. THEOREM 1 (Deduction theorem). If (6)

M, Xx

~

Y,

where x does not occur in M or X, then there is a derivation in which x does not occur of (7)

M, LX

~

3X(AX.Y).

3. This formulation is suitable for a synthetic system; and in such a system it follows from (EA) and (E) that we have (EX) (§ I2B2), i.e., f-EX

for any ob X. In a A-formulation we have to have (EA) also when A is a basic combinator, even though these are not atoms; since we have Rule Eq, then we have (EX) (as in § 12B2).

385

FINITE FORMULATIONS

15C]

The proof requires the following lemmas: LEMMA

1.1. HX

~

L(KX).

Proof By Axioms (FT), (FH), and (FK)*, ~

FHLK.

The lemma follows by Rule F. LEMMA

1.2. X

~

L(KX).

Proof Rule H and Lemma 1.1. LEMMA

1.3. LX, LY ~ Xu ='u.Yv ='v Xu.

Proof By Axiom (FK)*, LX, LY~ Xu ='u.Yv ='vX(Kuv).

The lemma then follows by Rule Eq. LEMMA

1.4. LX, 3 2XYZ, 3X(SYV)

~

3X(SZV).

Proof By Axiom (3S)*, LX, EV, H(3X(SYV)), H(3 2XYZ), 3 2XYZ, 3X(SYV)

We clearly have

~

~

3X(SZV).

EV. Furthermore, by Rule H, 3 2XYZ ~ H(3 2XYZ), 3X(SYV) ~ H(3X(SYV)),

and so the lemma follows. LEMMA

1.5. LX

~ 3XX.

Proof We have that ~

(8)

EI

is an axiom. Hence, by Lemma 1.2, ~

(9)

L(K(EI)).

Now, putting B(K(EI)) for Y and BKX for Z in Lemma 1.4, we get (I 0)

LX, Xu ='u.E1 ='vXu, Xu ='u EI ~ 3XX.

Now, by Lemma 1.3, we have LX, L(K(EI))

~

Xu ='u.E1 ='vXu,

so by (9), (II)

LX ~ Xu ='u' EI ='v Xu.

386

THE THEORY OF RESTRICTED GENERALITY

[ISC

Also, by Lemma 1.3 and Rule E, we have L(K(EI)), LX, EI ~ Xu =>u EI,

so by (8) and (9), LX ~ Xu =>u EI.

(12)

The lemma follows by (10), (11), and (12). LEMMA

1.6. LX, EXY ~ EX(BHY).

Proof By Axiom (EH)*,

LX, H(EXY), 3XY ~ EX(BHY).

By Rule H, 3XY ~ H(3XY),

and so the lemma follows. Proof of Theorem 1. Let ~ Y1 , ••• , ~ Y n , where Y n == Y, be a derivation oJ (6). We shall prove by induction on k that there is a derivation in which x does not occur of

(13)

k=1,2, ... ,n.

The proof consists of six cases, of which the first three complete the basic step of the induction and all six cases complete the induction step. Case 1. Y k == Xx. Since x does not occur in X, AX'Yk == AX.XX = X, and (13) follows by Lemma 1.5. Case 2. M ~ Y k (this includes the case where Y k is in M and also that where Y k is an axiom) and x does not occur in Y k. Then Ax. Y k is KYk. By Lemma 1.2,

Hence, by Lemma 1.3, M, LX

~

KYku =>u 3X(KYk),

and (13) follows by Rules E and Eq. Case 3. ~ Y k is an axiom in which x occurs. Then it is Axiom (Ex), and (13) follows from Axiom (EE)*. Case 4. ~ Y k is the conclusion of an inference by Rule Eq. Then (13) follows as in Theorem B1 Case 4. Case 5. ~ Y k is the conclusion of an inference by Rule E. Then Y k == VZ, and the premises are ~

EUV,

~

oz.

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387

By the induction hypothesis, there are derivations in which x does not occur of M, LX f- SX(h.3UV),

M, LX f- SX(AX.UZ). By Rule Eq,

M, LX f- S2X(h. U)(h. V),

M, LX f- SX(S(Ax.U)(h.Z)). Hence, by Lemma 1.4,

M, LX f- SX(S(h.V)(h.Z)), from which (13) follows by Rule Eq.

Case 6. f- Yk is the conclusion of an inference by Rule H. Then Y k == HU and the premise is f- U. Hence, AX.Yk = BH(AX. U). By the induction hypothesis, M, LX f- SX(Ax. U). By Lemma 1.6, M, LX f- SX(BH(h.U)), and (13) follows by Rule Eq. q.e.d. REMARK 1. Note that the proof uses only Lemmas 1.1-1.6, together with the fact that the only axiom in which x occurs is Axiom (Ex), and the fact that Axiom (3E)* holds. Hence, the theorem holds for any system satisfying these assumptions. In particular, the addition of constant axioms does not affect the validity of the proof. COROLLARY 1.7. If M,Xxf-Yx,

where x does not occur in M, X, or Y, and

if

Mf- LX,

then Mf-SXY.

COROLLARY 1.8. If

where

Xl' ••• , X m

do not occur in M, Xl' ... , X m , Y, and

if k = 1, 2, ... , m - 1,

then

388

THE THEORY OF RESTRICTED GENERALITY

[lSC

If we take account of the definitions of P and II, then we have the following: COROLLARY 1.9 (Deduction theorem for P). If M,Xf-Y

and

Mf-HX, then Mf-PXY.

COROLLARY 1.10 (Generalization theorem for II). If

(14)

Mf-Xx,

where x does not occur in M or X, then

Mf-IIX. In order to prove Corollary 1.10, we need (i) of the following lemma. LEMMA

1.11.

(i) f- LE. (ii) HX, FHHY f- L(K(YX». (iii) L(KX), YU f- HX. Proof (i) By Axiom (2E)*, we have

LY f- 2YE. Thus, if u is any variable, we have by Axiom (FY) and Rule 2

Yu f- Eu. Hence, by Rule H,

Yu f- H(Eu). Hence, the result follows by Axiom (FY) and the theorem. (ii) We have by Rule F

HX, FHHY f- H(YX), and so the result follows by Lemma 1.1. (iii) By the definition of L and Rule F,

L(KX), YU f- H(KXU). Hence, the result follows by Rule Eq. q.e.d. We shall call (i) of this lemma (FE). Proof of Corollary 1.10. By (14) and the conventions about derivations, we have M, Ex f- Xx,

and the Corollary then follows from this and (FE) by Corollary 1.7 q.e.d.

15C]

FINITE FORMULATIONS

389

This last corollary is interesting, since Curry was originally suspicious that II was the cause of the contradictions in illative combinatory logic; see [FCL], end of § 2, and the end of [IFL]. See also § 16C2. COROLLARY U2 (Deduction theorem for .3+). If the subjects are unrestricted, then under the hypotheses of Corollary 1.7 we have M ~ .3+XY.

Proof By assumption, M,Xx

~

Yx.

Hence, by Corollary 1.9, ~

M, H(Xx)

P(Xx)(Yx).

But we also have, by assumption M~

LX.

Since the subjects are unrestricted, L == FEH, and hence this and Axiom (Ex) imply M ~ H(Xx). Thus, we have M ~ P(Xx)(Yx), and the Corollary now follows by Corollary UO and the definition of .3 +. We now turn to the general assumptions concerning canonicalness which we made in § B3. Under the interpretation Cans(U) ~ ~ YU,

(15)

we have already seen that we do not postulate (4) for obs U in which variables occur; 4 what we may postulate in this case is either all statements

(YU)

~

YU,

where U is a constant canonical subject, or else alternative constant axioms from which this can be derived. Since U is then a constant, such axioms do not upset the proof of Theorem 1. This understood, we show that the more essential assumptions of § 12B4 concerning canonicalness can be established, in some cases with the aid of additional axioms. COROLLARY 1.13. Under the interpretation (15), Nl, N2(lr), N3, and (in the case of restricted subjects) SI all hold without further axioms; if the subjects are unrestricted N2(rl) does also. 4. We may of course entertain premises of the form (4) where U contains a variable.

390

THE THEORY OF RESTRICTED GENERALITY

[ISC

Proof N1 and S1 follow by Rule Eq. N3 follows from Definition 1 and Rule F. For N2(lr), suppose that we have

I-

HkX.

Then, by the conventions on proofs, we have Yx

I-

HkX.

Hence, by the deduction theorem and Rule Eq, we have

I-

Hk + 1().x·X).

In the case of unrestricted subjects N2(rl) follows by N3 and Nl , q.e.d. For S2 we note first that if I- LZ holds, then we can show without further axioms that (16)

YU

I-

FZY(AXU).

For if X is AXU, then we have, on the hypothesis Zx

I- YU,

I- Y(Xx),

and consequently we have the conclusion of (16) by the deduction theorem. Thus a sufficient condition for S2(lr) is

I- E(FZY)Y. Taking Z to be Y, 5 this becomes

I- E(FYY)Y.

(EY)

If we postulate this, or some set of axioms which implies it, we have S2(lr). Such axioms do not upset the proof of Theorem I; and (EY) is a theorem if subjects are unrestricted. As for S2(rl) we should need (for U == ).xX) (17)

YU

I- Y(Ux),

for which we would need an axiom

(18)

I- EY(FEY)

which seems very dubious, although it is, of course, true when subjects are unrestricted. If the subjects are unrestricted, then S3 is trivially true. If the subjects are restricted, then it is false, as we saw in § 12B4. Thus we are able to establish the more essential properties of § 12B4. We turn now to consider some other properties related to canonicalness. The first of these properties is NE(lr). For this we need an additional assumption. The one which best fits NE(lr) is (FE) 5. Another possibility is Z,= E; the premise,

f- LE, follows by Lemma

1.11.

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FINITE FORMULATIONS

391

From this, it easily follows that (FP) (FIT)

However, in order to get (FF)

we would need additional assumptions. Such assumptions are the following: (FB)'

~

F2LYLB,

(Y)

~

3Y(FYY).

Both of these hold if the subjects are unrestricted. If, on the other hand, the subjects are restricted, then one of these must be postulated. Alternatively, (FF) can be postulated together with ~

(YI)

YI,

since (F3) then follows. In §§ 2-3 we shall be considering inferential formulations of ff~'t. On account of the limitations of time and space we have to pick out one particular formulation among those considered so far as the A-formulation, ffg. We choose the system with unrestricted subjects, since it is the simplest, and is most closely related to the system proved Q-consistent in § B; the other systems are left for future study. This will mean that, in ffit, Y == E and statements (YV), (SY), (FB)', and (Y) are all theorems. Furthermore, the statement we shall take for N3 will be (F3). In other words, we define the system ffit to be the system with Rules 3, H, and Eq, and Axioms (Fe), (EA) (where A is an atom, including I, K, or 5), (E), (3H)*, (FK)*, (S5)*, (3E)*, and (F3), where in all of these axioms Y == E. 2. AT-formulation

Corollary 1.7 suggests the following introduction rule for 3: where x does not occur in X, Y, or in any other premise."

[Xx] Yx

Si*: LX

SXY 6. It is well, even at the risk of some repetition (cf. § 12Cl) to be completely explicit about what this rule means. We are reasoning in some extension S' of our fundamental system S (which is a C-system with extra assumptions concerning the illative primitives); this extension is formed by adjoining to S certain indeterminates, say Yl, ... , Yn, axioms I- En, ... , I- EYn, and a set M, perhaps void, of other axioms. The left premise is that LX is assertible in S', which can be expressed thus MI-LX,

392

THE THEORY OF RESTRICTED GENERALITY

[ISC

§"rp

Thus, the T-formulation of or §"i[, will be the system whose axioms are (EA), where A is a constant atom (including I, K, or S), together with Axiom (FE), and whose rules are Eq, 2 (or, to use its other name, :Be), 2i*, H, and the following: RULE E. EX, EY f- E(XY),

f-

RULE He. EX!, ... , EXn RULE H2. LX, LY,

f-

H(eXt ...x

n) ,

where n = dg(e),

H(2XY).

We now derive some simple consequences of this formulation. In the first place we note that F, P, and II all satisfy their normal elimination rules and the following introduction rules:

where x does not occur in X, Y, Z, or in any other premise.

[Xx] Y(Zx)

Fi*: LX

FXYZ

[X] Y

Pi*: HX PXY

IIi:

where x does not occur in X or in any other premise.

Xx

IIX

Next we derive a number of special results. We can obtain the axiom (Fe), viz. (Fe) thus: by Rule He we have

EXt, .•. , EXn

f-

H(Ox!...x n ) ·

Then we obtain (Fe) by applying Fi* n times, the premise f- LE being valid by (FE). As a special case we have in the case 0 == H, the rule (1), viz.

Ex f- H(Hx); the. EYI not being necessarily mentioned in M because, according to the meaning of E (§ 12B2),they are necessary for the adjunction of the indeterminates. As for the right premise, it has to be understood, not only that x does not occur in X, Y, or M, but that x is new for S', so that the reasoning has reference to a system S'(x) in which we have not only the axioms for S' holding, but f- Ex also; the premise is that if we adjoin f- Xx as new axiom to S'(x) then f- Yx follows. Thus we can express this premise as M,Xxf-Yx.

If there is no premise Xx, we understand X is E. The conclusion is that f- EXY holds in the original S', i.e, Mf-EXY.

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393

from this we have the axiom ~ LH.

(FH)

Again, as in Lemma 1.1, we have Hx ~ L(Kx),

(19)

thus:

1 Ey

Hx H(Kxy)

Fi*E~ 1

L(Kx).

Next we derive the property (cf. discussion of (FF) in § 1): Lx, Ly ~ L(Fxy).

(20)

thus:

Hp Ly

Hp Lx

As a corollary we have, by induction on n,

For n = 0, this was established above. The induction follows since Hn + 1 == == FEHm and we have ~ LE and ~ LH n • Finally we establish the following inferential principle: Suppose X does not contain x, but Y may, then LX ~ L(Ax. XY).

(21)

The proof is as follows

1 LX

--Df FEHX

Ex

- E EY

------Fe H(XY) Eq H((Ax . XY)x) F. L(Ax.XY).

1* -

1

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THE THEORY OF RESTRICTED GENERALITY

[ISC

This will be used below for the special case X == H. We are now in a position to prove the equivalence of the A- and T-formulations of .?F~1' Corollary 1.7 proves that Rule 2.i* is admissible in .?Fit, and the proof that the remaining rules are admissible in that formulation is either trivial or else makes use of the corresponding axioms. This proves the right-left part of the following theorem: THEOREM 2.

Mir.X +% M r X. T

The proof of the other half of this theorem consists of showing that the axioms of the A-formulation are theorems of the T-formulation. To show this, we proceed as follows: AXIOM (Fe). This was established above. AXIOM (EA). This is an axiom of the T-formulation in each case. AXIOM (E).

1

1.

Ex

Ey E

E(xy) Fi* _ 2 FEEx ___-_,__ 2.i* - I 2.E(FEE).

AXIOM (2.H)*. We divide the proof into two stages. In the first stage we show that Lx, H(2.xy) f- Y,

where

Y == 2.xy

::>

2.x(BHy).

The first stage is:

1

1. 2.xy

4 3 H(2.xy)

Lx

Y. In the second stage we complete the proof using (21) (for X 2.i*, thus:

4 LH L(.icy. H(2.xy» (21) LL

.3

Lx H(2.xy) ---Y-_-.*-'- 3stage 1

=--'----,---'-------.=.1 H(2.xy) ::>y Y

Lx

:;J". H(2.xy)

::>y

Y.

-

2.i* - 4

== H) and

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AXIOM

(FK)*. This is straightforward, thus:

1.

1

3 xu yv Ly x(Kuv) Eq --=-----,...,.------ Fi* - 1

4 Lx _----=----=--,-F~y:-x_(K_u_) Fi* Fx(Fyx)K.

_2

We can then complete the proof by two applications of 8i*. In the same way we can prove the more general scheme (8K)*

1.

4

1 yuv

xu

3 FELy L(yu)

Lx

LL Here

r L(FEL) follows by (FH

AXIOM

2) .

(8S)*. This is the axiom Lx

::> x.

Ev ::> •• H(8x(Syv»

::>)'.

H(8 2xyz)

where Y

==

8 2xyz ::>.8x(Syv)

::>

8x(Szv).

We show first that Lx, H(8x(Syv», H(8 2xyz)

thus:

rY,

::> z

Y,

396

THE THEORY OF RESTRICTED GENERALITY

IISC

We then continue the proof using (21) with H for X as in the proof of (3H)*, but condensed somewhat, thus: 7 6 S 4Lx, Ev, H(3x(Syv», H(3 2xyz)

y Stage 1 - - - - (21), 3i* - 4 3 2xyz =>z Y (21), 3i* _ 5 Z,

Z == H(3x(Syv» =>y' H(3 2xyz) =>zY.

where Finally we have

71

6

Lx

Ev

- - - as above Z - - - - 3i* - 6 LL Ev =>v Z '::'1 -'* - 7 ----q.e.d. Lx :» x' Ev => v Z,

LE

AXIOM

1

(3E)*.

1. Lx

LL

Ax

xu

Eu Eu

- - - - 3i*-1 3xE

- - - - - 3i* - 2 Lx => x 3xE.

AXIOM

(F3).

LL

1.

1

Lx

Ly

--~(H;::')

H(3x )

-'*

Y --------'--.::.1

~

-

LL FLH(3x) - - - - - - 3i* - 2 FL(FLH)3.

1

This completes the proof of Theorem 2. The extra axioms mentioned after Corollary 1.13 are not relevant because we have unrestricted subjects. We now derive some properties relating to the ob 3+. By definition we have 3 + = hy. 3E(AU. 3(K(xu»(K(yu»).

We then have Lx, 3xy ~ 3+ xy.

(22) For we have Lx

LE

3xy

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Next we have Lx, E+xy ~ Exy.

(23)

For we have for any ob A E+xy

~~-Df

EE(Au. E(K(xu))(K(yu)))

Eu xu zse - - - Eq E(K(xu))(K(yu)) K(xu)A _ - - - - - - - - - - - - - - - - .::.e K(yu)A

--'-- Eq

Lx_ _ _ _ _ _ _-'-y_u Ei* _ 1 Exy.

As corollaries of (22) and (23) we have Lx ~ L + x,

(24)

where L

L + x ~ Lx,

=h.EE(BHx),

Since ~ LE, we have (24 1 ) , as corollary of (22), and (24 2 ) as corollary of (23). Next we note that the rule which is the analogue for E + of Ee, viz. E+ XY, XU

(25)

~ YU,

follows easily; the proof in fact is (with A arbitrary): E+XY

EU

--------Ee E(K(XU))(K(YU))

K(YU)A YU

XU K(XU)A

:q

.::.e

Eq

Likewise the similar analogue of Ei* holds, viz., (26)

In fact this follows at once by (24 2 ) , Ei*, and (22). Now let the superscript d ' denote systematically the result of replacing .E everywhere by E+. Then we have: COROLLARY

2.1. For all M, X,

(27) The converse holds if the deduction on the right is obtained solely by using the + -analogues of the rules for E. Proof We show that the + -analogues of all axioms and rules of .?FiT hold. Then (27) holds by deductive induction for the T-formulation; for the Aformulation it then follows hv Theorem 2.

398

THE THEORY OF RESTRICTED GENERALITY

usc

For all axioms which do not contain 2 the +-transform is the same as the original and hence is valid. This applies to the axioms f- EA, where A is not S. For the rules which do not contain S explicitly, namely Eq, E, H, He the transformed rule is a specialization of the original rule. Likewise (2.e)+ is a specialization of (25), and (2.i) + is a specialization of (26). This disposes of all cases except Axioms (E2.) and (FE) and Rule H2. For Axiom (E2.), we have f- E2.+ by Rule E, Axiom (E2.), and the other axioms (EA). Likewise Axiom (FE)+ follows from (FE) by (24 1 ) , Finally we establish Rule (H2.) + thus

L+x 1 (24 2 ) Lx Eu - - - - - Fe H(xu) (19) L(K(xu))

L+y -

Ly

(24 2 )

1 Eu

H(yu) L(K(yu))

Fe (19) H2.

H(2(K(xu))(K(yu))) H(l1>Pxyu) Eq Fi* - 1 LE L(l1>Pxy) H2. H(2.E(l1>Pxy))

---:-:--.,---'-- Eq H(2.+xy).

Thus we have (27) by deductive induction. By a similar deductive induction we have the converse under the restrictions stated. We leave open the question of whether the converse holds without these restrictions. 3. Formulation of equality

It is a simple matter to adjoin to the ~~ 1 formulated as above an atom Q such that for all obs X (28)

f-QXX

holds. In the A-formulation this can be accomplished by adjoining the axiom 7 (29)

f- 2E(WQ);

in the T-formulation by adjoining the rule

(30)

Ex f- Qxx,

from which (29) would follow as in the deduction of (Fe) from the rule He. As we noted in § B6, this entails Rule Q, i.e.

(31) 7. This is Ax. [QE] of § 7Dl.

x =Y

-+ f-

QXY.

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399

We also noted there that this can be done without assuming that Q is a basic canonical ob, which in our present system would give us 8 (32) This may be important; because we now have no proof of Q-consistency, and from a certain interpretation there are objections to (32) in view of the undecidability theorem. However, that interpretation is quite different from our present one, and the analogy with § B6 suggests that (32) may not lead to contradiction." Nevertheless there are some objections to doing this with Q defined, as in § 7DI, by Q == 'P3(CI).

(33)

To be sure one can postulate the reflexiveness for this Q just as we could for the atomic Q, and it seems unlikely that a contradiction would result from such an assumption. But this reflexiveness requires that we have (34)

~

zX

::::>z

zX

for all X. In the system of § B this is not canonical, and so is unprovable, even for a particular X, say X == I, - unless, indeed, we extend the system in ways suggested at the end of § B6. For the specialization of z to 1it gives ~

PXX

and

~ 3+XX;

the first of these runs counter to our intuition that P means a connective between propositions. Moreover, in the case of such a Q, the relation (32) seems especially counterintuitive. A possible alternative is to restrict the z in (34) to the case where ~ Lz holds. For the time being let Q'

== AXy. Lz ::::>z .zx

::::>

zy.

Then the reflexiveness of Q' can be proved for any given X thus

1-

1

Lz zX --Fe zX H(zX) - - - - - - - - Pi - I

LL zX::::> zX ------::---- 3i* - 2 Q'XX.

8. This is the special case Y == E of (FQ) as defined below in § 5. 9. The contradiction derived in Bunder [STBj § 2.6 involves several other assumptions of a dubious nature.

400

THE THEORY OF RESTRICTED GENERALITY

[ISC

Moreover if Q is an atom for which (28) is postulated, then if (32) holds we have Q'XY ~ QXY

(35) thus:

H Q

Q'XY

EX

-2 - - - - Fe L(QX) ~

- - - - - Of Lz ;:) z .zX ;:) zy

--------.::.e QXX;:) QXY

QXY.

The converse implication will follow from the axiom (36) which seems a quite reasonable one. Thus if we have (32) and (36), Q and Q' are equivalent. Suppose, then we adopt Q' as definition of Q, i.e. Q:; Q' :; Axy.Lz ;:)z.zx;:) zy.

(37)

Then Q will satisfy the following rules: Qe*:

QXY

ZX

LZ

[Lz, zX] zy

Qi*:

ZY

QXY.

where z does not occur in X, Y, or any other premise

In fact Qe* follows by Ee and Pe; Qi* thus (cf. the above derivation of Q' XX):

1

1Lz EX ---Fe H(zX)

zX zy

Pi* - I zX ;:) zY ----- Ei* - 2 QXY.

Conversely, these rules, considered as T-rules for Q as a new atom, determine that Q shall be equivalent to Q'; in fact Qi* gives (35), and Qe* gives the converse. There is no involvement of (32); but if (32) holds we also have (36). Thus Qe* and Qi* are a weaker way of saying that Q is to be Q'. Although the rules Qe* and Qi* are not precisely the rules of § 12CI, they are sufficient for the proofs of the usual properties of equality, namely (I), (K), (5),1° (0), (0-), (r). (11), and (v). For example, using RULE

HP. HX, HY

~

H(PXY),

10. The properties (I), (K), (S) are direct consequences of (e) by Rule Eq.

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401

FINITE FORMULATIONS

which follows easily from Rule HE, we get the following proof of (rr}:

4

LE QXY

1

4

4

Lz Eu Lz EX - - - Fe Fe H(zu) H(zX) -HP H(P(zu)(zX» Fi* _ 1 L(Au. P(zu)(zX»

EX

Lz

.~--

Fe

1.

H(zX) zX Pi* _ 2 P(zX)(zX) (Au. P(zu)(zX»X QEq e*

.3 (Au. P(zu)(zX»Y Eq zY P(zY)(zX) - - - - - - - - - Pe zX - - Qi* - 3 - 4 QYX

The remaining properties of equality follow in a similar manner. However, without (32) we cannot universally quantify these properties. The best we can do for (a), for example, is to prove: (38)

thus:

1

1.

(a) L(Qx) Qyx Ei* _ 1 Qxy :;:'y Qyx ~.

_ LL (21) L(h. L(Qx» -

~xy

L(Qx) :;:'x.Qxy:;:,y Qyx

~l* -

2

In a similar fashion we can establish, for example (39)

~

L(Qy) :;:, s: H(Qxy) :;:, x ' Qxy :;:,.Qyz :;:, z Qxz,

(40)

~

L(Qx) :;:'x.Qxy :;:,y.Ez :;:'z.Q(zx)(zy).

If we have (32), however, the quantification of the properties of equality is easy. There is then no difficulty about such as the following: (41)

~Ex:;:'x.QxY:;:'y.Qyx,

(42)

~

Ex :;:'x.Qxy :;:,y.Qyz:;:,z Qxz,

(43)

~

Ex :;:'x.Qxy :;:,y.Ez :;:'z.Q(zx)(zy).

On the other hand there is one property of equality, viz. (44)

X, QXY~ Y,

which is the analogue of Rule Eq, that we do not obtain from the definition of Q as Q', even with (32), unless we have Q-consistency. If we want this

402

THE THEORY OF RESTRICTED GENERALITY

[ISC

property we must postulate it, say by an axiom (45)

f- Hx ::Jx.Qxy ::Jy'Pxy.

Let us return for a moment to the case where Q is defined by (33) and satisfies (28). In such a case (44) is obvious. We also have Qe* and Qi* without the extra premise f- Lz; 11 moreover the proofs of (0), (r), (Il), etc. as unquantified inferential rules, are much simpler, and so are those of (38)-(43). Now we have pointed out that such a system has anomalous properties. Yet as of now we know of no actual contradiction, even if (32) is adjoined. Until such a contradiction is forthcoming, our philosophy (cf. § llA2) requires that the system should not be totally abandoned. It may well be that these anomalous properties are harmless. We may now summarize this discussion as follows. We have quite a variety of possibilities for a Q with (28). For none of these do we yet have a proof of Q-consistency; neither do we have an actual contradiction. The systems however, seem to involve various degrees of risk. The most conservative is that where we postulate Q as a new atom with just (28). This gives us Rule Q; but without Q-consistency we have no "properties of equality" unless we add further postulates. The adjunction of (32) does not help in that connection. Intermediate in risk is the case where we postulate that Q is Q'; which is equivalent to postulating Qe* and Qi* with the premise Lz as above stated; in that case we have the properties of equality as unquantified inferential rules, and quantified in a weakened form as (38)-(40). If we adjoin (32) we have (41)-(43), but we still do not have (44). Finally the riskiest case is that where we define Q by (33). Then we have (44); the proofs of the other properties of equality are much as in the previous case, but are easier. The last system, particularly with (32), has anomalous properties which make one suspicious; but whether it, or in fact any of these systems, actually leads to contradiction is an open question. 4. An L-formulation The T-formulation of § 2 suggests the following L-formulation, or ff~t: let the system be the unrestricted CL-system whose prime statements (p2) are the axioms of the T-formulation, i.e.,

EX* FE*

laf- EX, laf- LE,

where X is any ob

1ta

,

and whose rules are, besides the structural and expansion rules, the following: 11. In the case of Qi* this follows simply by substituting Qx for z in the derivation which is the premise of the rule. Note that these are precisely the rules Qe and Qi of § 12CI. l la. This is to avoid postulating the nonelementary rule E*. With E* we could of course restrict X to be an atom (or basic combinator). Cf. § 12C2.

lSCj

FINITE FORMULATIONS

xu, L

M lar

403

M, YU lar N

M, SXY lar N, L. M, Xx

la, xr Yx, L

M lar LX, L

M lar SXY, L.

MlarX,L M lar HX, L.

HO. If n = dg(O),

M lar EX lo L; ... ;M lar EX., L M lar H(OX 1 ••• X.), L.

M lar LX, L

M lar LY, L

M Iar H(SXY), L.

Actually, this system is not quite a CL-system, since prime statement FE· fails to satisfy the assumptions about prime statements. Note that the L-rules for P and II which follow from their definitions and 12 the above rules .S. are and .P of § 16C2 and, in addition,

.n.

P.

M, X lar Y, L

M lar HX, L

M lar PXY, L.

A similar situation holds with respect to rules for F. We have no proof of ET for this system; and the considerations of Godel's second theorem make it seem doubtful if a finitary one is possible. We do, however, have the following: THEOREM

3. The statement

Mlar N

(46)

is derivable

if and only if M, SH(WP) lar N

(47)

is strictly derivable." Proof For the if part, note that the following is a valid strict derivation: 14

Hx, x [u, xr x Hx [u, xr Hx - - - - - - - - - - - P.

Eu [o, ur Eu

H H •

Hx lu, xr Pxx Eu [c, ur H(Hu) lar LE F. Exp. Hx lo, xr WPx lar LH ~ - - - - - - - - - - - - - - - - - .:::.. lar SH(WP)

From this and (47), (46) follows by Rule Cut. 12. Note that for Th. the second premise required by E. follows from FE·. 13. Cf. Schutte [SSPI Theorem II (7.1). See § 12Dl for the terminology. 14. The rules P. and F. used here follow from E. in much the same way that Pi· and Fi· follow from Ei·.

404

THE THEORY OF RESTRICTED GENERALITY

[ISC

The only if part will be proved by an induction on the proof of (46). If (46) is prime, then (47) follows from it by *K, which can be applied whatever the range since the introduced constituent is a constant. This completes the basic step. For the induction step, if (46) is the conclusion of an inference by any rule except Cut, then (47) follows from the induction hypothesis by the same rule (with the added constant parameter on the left). None of these steps introduces any uses of Rule Cut. If (46) is the conclusion of an inference by Rule Cut, let X be the eliminated constituent, let T == 3H(WP), and let N == Y, L. Then the following shows that (47) follows by a strict derivation from the induction hypotheses: M, T, X

If-

Y, L

M, T

If- X,

L

M, T, PXX If- Y, L -Exp M,T,WPXIf-Y,L

*P

M, Tlf- X, L - H* M,TIf-HX,L ~ - - - - - - - - - - - - - - - - *.::. M, T, 3H(WP) If- Y, L - - - - - - - - - *W M, Tlf- Y,L.

5. Relative canonicalness Ever since [ACT] we have felt a need for a theory in which the canonicalness of a subordinate part of an ob might depend on parts which, so to speak, are senior to it. For instance, we might suppose that, X being a proposition, PXY is a proposition not only when Y is also a proposition, but when from the hypothesis that X is true it follows that Y is a proposition. This might be expressed in a rule such as (48)

HX, PX(HY)

f-

H(PXy).14b

A system with (48) or a similar property will be said to be a system of relative canonicalness. It was originally assumed that all finite formulations would be such systems.P The system of Bunder [STB] is such a system. He suggested that we follow the lead of the three-valued logic which Kleene [IMM] proposed in connection with recursive functions, and adopt the following tables for P and H

IrfNl T T

F T

HX

N T

T T

T N N

N

These tables are given a constructive interpretation in the following manner. 14b. It is easy to show that (48) entails (FP). 15. See [ACT] § 6.

15C]

FINITE FORMULATIONS

405

If, at a given stage of our knowledge, we have established ~ X, then the value T is assigned to X; if we have established that X is false (in some sense), then F is assigned to X; if neither of these has occurred, then X has the value N. Thus the tables are a three-valued logic only in a modified sense; there are really only two values T and F; the "value" N indicates that neither of these values has yet been assigned, but we do not know what may happen later. So interpreted these tables motivate an approach to his system of relative canonicalness. It is clear that (48) agrees with these tables. On the other hand (1) does not agree with them; in fact if X has the value N, HkX has the value N for all k. Indeed, in the presence of (48), Rule H, and the properties of absolute implication 16 over the range H, (1) leads to a contradiction. This was mentioned in [CFM] and [ACT], but the proof there mentioned was lost (except for the case k = 2); the proof was rediscovered by Bunder.!? We give Bunder's proof here in full. Suppose that for some k we have, for all X, ~

(49)

Hk + 1 X.

Let Y be an arbitrary ob, and define Go Gn+1

=AX.X

Y,

:::>

=Ax.Hn+1 x

:::>

Gnx.

We prove first, by induction on n, that (50)

For the basic step, we have by (48) HY, Hx ~ H(Gox).

For the induction step, assume (50); then by (48), HY, H(Hn+1x)

~

H(Hn+l x

:::>

Gnx),

so HY, Hn+2x ~ H(Gn+ IX),

which completes the induction step and hence the proof of (50). Now let X

=YG

k,

where Y is a fixed-point combinator; 18 then X

=

GkX.

16. I.e. the system HA of [FMLj. These are the properties deducible from Pe and Pi only, with all X, Y, etc. restricted to the range H. Cf. § D2 below. 17. See his [STBj § 2.4. See also his [PIC], which appeared after the text was written. 18. See § llF7.

THE THEORY OF RESTRICTED GENERALITY

406

[ISC

But by (50), HY, Hk+IX ~ H(GkX),

so, by Rule Eq, HY, Hk+IX ~ HX.

Hence, by the assumption (49) and Rule H, we have HY ~ HiX,

(51)

all i

~

1.

all i

~

O.

all j

~

1;

From this result and (50) it follows that HY~ ~ 1,

Now, for all j

H(GjX),

it follows by definition of Gj that HY, X::> GjX ~ X ::>.HjX

::>

Gj-IX;

and so by the properties of absolute implication, we have HY, X::> GjX ~ HjX ::>.X::> Gj_IX.

Thus, by (51), HY, X::> GjX ~ X::> Gj_IX,

hence, by iterating this j times, we have HY, X

Now since X

=

::>

GjX ~ X

::>

GoX,

all j ~ 1.

GkX and HY ~ HX, we have HY ~ X

::>

GkX,

and hence, by the definition of Go, HY

Now, since HY

~

(52)

~

X ::>.X::> Y.

HX, we have by the properties of absolute implication HY~

HX ::>.X::> Y,

i.e., HY ~ GIX.

But we also have HY ~ H 2X,

so by the usual properties of implication and Rule H, HY/- H2X::> GIX,

i.e., HY~

G2X.

lSCj

FINITE FORMULATIONS

407

Similarly, we can prove

i.e., HY~

X.

HY~

Y.

Hence, by (52), Since Y is arbitrary, we also have

Hence, by our assumption (49),

Similarly, we can get ~ Hk - 1Y, ... , ~ H Y and thus ~

Y.

Since Y is arbitrary, this is a contradiction. Hence if we have a system of relative canonicalness we must give up (1). Thus Bunder cannot consistently postulate ~ LH

(FH)

with L defined as for unrestricted subjects. Instead he introduces an ob Y,19 corresponding to that in § 1 except that S2 and S3 are not assumed, defines the ob L in terms of it just as we did there, and postulates (FH) for the L so defined. This Y is to be understood as meaning a category sufficiently broad to include all the notions of set theory, yet not so broad as to lead to contradiction in combination with a deduction theorem. Since most systems of set theory are based on an applied first order predicate calculus and we can take the universal domain of that calculus as such a Y, this seems a not unreasonable hope. As we shall see it is going a little further than necessary to assume that (FH) holds; but that too is not unreasonable. Bunder then derives the deduction theorem from a set of axioms which is essentially only a variant of that used in § 1. In fact he postulates the axioms for E (with E defined as WQ), (31)*, (FH) as above formulated, (3E)*, modified forms of (3K)* and (3S)* as follows (53)

~ Ey :::>y' Lx :::>x.xu :::>u·yuv :::>v xu,

(54)

~

Ev :::>v.Lx :::>x.3x(Syv) :::>y.32xyz:::>z 3x(Szv),

19. He calls it A.

408

THE THEORY OF RESTRICTED GENERALITY

[ISC

and an axiom ~

(H)

EIH

(concerning which we shall have more to say in § 6), from which he obtains Rule H. Lemmas 1.1 to 1.6, and hence the deduction theorem, follow. The axiom (FH) is really not necessary for this theory: it is used only to derive Lemma 1.1, which would follow from the axiom ~

(55)

EH(BLK); 20

and Lemma 1.2, which is really all that is necessary, would follow if we replaced (H) by ~

(56)

EI(BLK),

provided category Y was not void, and then we could get Lemma 1.1, but not (55), by the deduction theorem and (FY) (which Bunder assumes later anyway). However without (FH) we could not use the deduction theorem to deduce theorems of the propositional calculus in closed form; we could, for example, have HX, HY~ X ::J.Y::J X,21

but not ~ Hx ::Jx.Hy ::Jy'x ::J.y::J X.

To this system Bunder now adjoins the axiom (57)

~ Lx

::J x'

FxHy

::J

s : H(Exy)

as an axiom to replace (FE)·. This axiom is motivated by (48), and may be shown to include (48). He also adds (FY) and some other axioms which we shall not discuss here.P The system as thus formulated has the drawback that the axioms used to derive the deduction theorem are stronger than the deduction theorem itself. Thus we are unable to derive in it (FL)

~

LL,

20. If we were to take L as primitive and define H as BLK, we should want an axiom ~ E(FYH)L. However, Bunder writes that in his [PPCl, which we have not yet seen, he shows that this is not needed. 21. This is obtained by the deduction theorem. By arguing direct from (53) we can get rid of the premise ~ H Y. 22. These axioms are used for special purposes in connection with the predicate calculus. But two of them have some interest in the present connection. The first of these, his Axiom 11, is ~ Ex ::J x' H(Exy) ::J y.XU ::J u H(yu). Cf. Axiom (EH)*. The other, his Axiom 5', is a strengthening of Axiom (EE)* to ~ EX::J x.IIY::J y.Exy. The latter axiom, however, is scarcely used; it is merely mentioned as a possibility which fits the truth table.

15C]

409

FINITE FORMULATIONS

and thus axioms with LX::Jx in front cannot be obtained by the deduction theorem. However the matter goes further than this. The example in a preceding footnote is connected with the fact that we cannot derive (53) from the deduction theorem even if we postulate (FL).22a In this particular case we can get around that by using (FK)* in place of (53); but we have a similar and more serious difficulty with (54). In fact replacing (54) by (25)* does not help, because the derivation of (25)* from the T-formulation in Theorem 2 depended on (21), which does not hold if we put Y in the place of E. Thus we are not able to frame a T-formulation from which the axioms of the A-formulation can be derived. One suggestion in this connection is to adjoin additional axioms such as those suggested in the discussion following Corollary 1.13. The effect of doing this has not been investigated. But if these methods give us (FF) then in combination with (FH) and (FY) we have (FL). From this we are able to derive (FQ) for the Q defined by (37), as follows

1

3

Lz Yx - - - Fe H(zx)

1

1-

Lz

Yy

H(zy)

Fe HP

H(P(zx)(zy)) Fi* _ 1 LL FLH(AZ. P(zx)(zy)) LY H(Qx ) (57), Eq ------------'y'- Fi* - 2 FYH(Qx) Fi* _ 3 F2YYHQ. LL

LY

However, although we do not know of any inconsistency in the system so constituted, the above contradiction makes it look suspicious. Besides the new axioms look unnatural from the standpoint of the above interpretation of Bunder's Y. The Bunder system has many features of naturalness. The questions we have raised concerning it are among the more important open questions concerning ff~. One important advantage is that it is counterintuitive, in view of the undecidability theorem, to assume f- FEHX except for trivial X. 22a. Bunder, in private correspondence, has pointed out that (53) can be obtained from the deduction theorem, (F L), and his Axiom 5' (see preceding footnote). But this Axiom 5' does not appear to follow from the deduction theorem, and we still have the problem of deriving (54).

410

THE THEORY OF RESTRICTED GENERALITY

I[ISC

Bunder's system was adapted for the study of set theory from the standpoint where the set-theoretic X e Y is represented in combinatory logic by YX. But there is another way of representing set theory, viz. where we take the set-theoretic s, and possibly other set-theoretic notions, as canonical primitives. In such a case we might base it on the system of §§ 1-4. This approach has not been explored. We had hoped to include a treatment of it in this volume as Chapter 18, but found ourselves unable to get to it. 6. Modifications and extensions

In § 1, we remarked that the axioms we have used are not the only ones from which the deduction theorem can be proved, but that any set of axioms will do provided that the only axiom in which a variable x occurs is (Ex) and that Lemmas 1.1-1.6 hold. Some alternative axioms were considered in § 5. In this subsection, we shall consider some further alternatives. The principal result of substituting these alternatives for the axioms of § 1 is that the Aformulation is no longer included in the T-formulation, and hence any consistency proof obtained by using the inferential formulations will not apply. We met this same situation in § 5. We shall consider modifications under three headings: an axiom to replace Rule H, Seldin's formulation (in his [SIC», and additional axioms. a. An axiom to replace Rule H. It is trivial to show that if ~

(H)

SIH

is postulated, then Rule H is admissible. This would allow us a formulation with only two rules, viz. Eq and S. However, since we do not have (or want) ~

L1,

(H) does not follow in the T-formulation. This is the reason we did not use it in § 1. Note that if we postulate Axiom (H) instead of Rule H, then Case 6 of the

proof of Theorem 1 becomes redundant. Since this case is the only one in which Lemma 1.6 is used, this lemma is not needed for the deduction theorem 1 is modified by replacing Rule H by Axiom (H), in this case. Hence, if then Axiom (SH)* can be omitted. Thus the effect of Axiom (H) so far as the deduction theorem is concerned is the same as Rule H and Axiom (SH)*. An analogous remark applies to

g-;

(56).

b. Seldin's formulation. In his [SIC] § 4Cl, Seldin proposed a system for 23 in which Axiom (H) is postulated instead of Rule H, Axiom (S5)* is

g-i

23. Seldin called his system same name proposed in § 1.

JF:

1,

but this should not be confused with the system of the

lSC]

FINITE FORMULATIONS

411

replaced by ~ Lx :::>x' H2y :::>y. H2 z :::>z [ES]xyz,24

(58)

and in addition there are the following axioms: (EL I )

25

~ Ex :::>x.Ez :::>z.H(E 2xyz):::>y H 2y,

(EL 2 )

~ Ex :::> x' Ey :::> v: H(E 2xyz) :::> z H 2 z,

(EI)*

~

Lx

:::> x Exx.

Of these axioms, (EI)* is redundant by Lemma 1.5. In addition, despite Seldin's claim on p. 114 of [SIC], it is not true that all the axioms follow from his T-formulation, for the remark of § a above applies to his system. However, a formulation of his system which does follow from his T-formulation can be obtained by omitting Axiom (H), postulating Rule H, and adjoining the following axioms: (59)

~

(EL)

~ Ex :::> x' H(Exy) :::> y Ly.

Lx :::>x' Ly :::>y.Exy

:::>

Ex(BHy),

Axiom (59) is a substitute for Axiom (EH)*. Axiom (EL) is a partial converse for Axiom (FE); furthermore, in this axiom and in (ELI) and (EL 2), the first 'Ex' can be replaced by' Lx' without affecting the proof of the deduction theorem. c. Additional axioms. There are other axioms which have been considered from time to time in illative combinatory logic and do not follow from the T-formulation. Here, we shall list some of the most important ones. The axioms of Bunder were discussed in § 5. In [UQC], Curry introduced a set of axioms for Il which are equivalent to the following: (n!)

~

(ilK)

~

FII1 IK,

(TIS)

~

FIl 2Il 2S,26

Fill,

These axioms can be adjoined to any system with unrestricted subjects, and no contradiction arising from this is known. Rosenbloom, in [EML],27 proposed two axioms which amount to (FQ). We have shown that this follows in a certain system of relative canonicalness but not in other systems. However, no known contradiction follows from postulating this in systems considered so far. 24. Here we suppose the form of [ES] in § B1 footnote 6 is used. 25. Seldin called this and the next axiom (EHl) and (EH2) respectively, but these names would be confusing here. 26. The axioms actually introduced were for B, C, W, and K, since these were taken as the basic combinators in Curry's early work. In [UQc] they were stated without F, which had not yet been introduced, but they appear with F in [CFM]. 27. See pp. 126-127. He actually proposed them for a system ~:.

THE THEORY OF RESTRICTED GENERALITY

412

[ISC

7. r-formulations If we are to have for want to have Can(M 1 ,

(60)

~i

a r-formulation such as that of § B5, we shall M m , X) ~ Ver(M 1 ,

... ,

.•• ,

M m , HX),

at least from the point of view of interpretation. If this is taken together with Rule IV 1, we have Ver(Xx, H(Yx))

-+

Ver(H(3XY)),

or, in an equivalent form, Ver(FXHY)

(61)

-+

Ver(H(3XY)).

Since this implies an analogue of (48), we have a system of relative canonicalness. Thus, by the result of § 5, we cannot have (FH) holding in this system. Hence, the formulation will have to be different from the formulations used so far in this section. Curry, in [ACT] § 7, suggested that H be an operational constant and that a pair of operational rules be postulated for it. This system requires some modifications in the rules I-IV of § B5, and hence all its rules will be given in full. The following convention is used in stating the rules: if X is the consequent of a regular sequence, then

where U 1, ••. , u, are the variables which occur in X and do not occur in any antecedent of the sequence. The rules are as follows: I. RULE OF TAUTOLOGY.

Can(M l , ... , M m , {Uju}X) Reg(M l , •.. , M m , X, X). II. STRUCTURAL RULES. 1. RULE OF PERMUTATION.

Can(M l ,

... ,

M m , Y, {Uju}Z)

Reg(M l ,

2.

... ,

Reg(Ml> , M m , Z, Y, N l , M m , Y, Z, N l , , N n , X).

RULE OF CONTRACTION.

Reg(Ml> Reg(M l ,

, M m , Y, Y, N l , .•• , N n, X) , M m , Y, N l , ... ,Nn , X).

... ,

N n , X)

15C)

3.

FINITE FORMULATIONS

413

RULE OF WEAKENING.

Can(M, ... , M m , {Uju}Y) Reg(M 1 ,

III.

... ,

M m , Y, N 1 ,

If M[

RULE OF EXPANSION.

, M m , N 1,

Reg(Mj,

>-- N 1 ,

... ,

,

... ,

N m X)

N m X).

>-- Nm X >- Y,

Mm

then

Reg(N 1 , · .. , N m, Y) Reg(M 1 , ... , M m , X). IV.

RULES FOR

=:.

1. If x does not occur in M 1 ,

M m , X, Y, then

... ,

, M m , Xx, Yx)

Reg(Mj,

, M m , =:XY).

Reg(Mj, 2. If x does not occur in M l'

... ,

M m , Y, Z, then

G1

G2

Reg(M 1 ,

G3

M m, =:YZ, N 1 ,

... ,

... ,Nn ,

X),

where

V.

RULES FOR

G1

== Can(M 1 , ... , M m , Yx, {Uju}Zx),

G3

== Reg(M 1 ,

••• ,

M m,ZW,N 1 , · · . , Nn,X).

H.

Can(M 1 ,

1.

,

Reg(M 1 ,

,

2.

M m , X)

M m , HX). [Can(M 1,

Can(M 1 ,

... ,

M m , {Uju}X)

Reg(M l '

... ,

Reg(Nj,

M m , HX, N l '

... ,

,

M m , X)]

, N n , Y)

N n , Y).

If Q is defined by (37), and if the subjects are unrestricted, then its rules are the following, where, in both rules, z is a variable which does not occur in M 1 , ... , M m, X, or Y: Ql

Q2

Reg(M 1 ,

... ,

Reg(M 1 ,

G1

G2

Reg(M I,

... ,

M m , FEHz, zX, zY) ... ,

M m , QXY).

G3

G4

M m, QXY, N 1 ,

... ,

Gs N n , Z).

414

THE THEORY OF RESTRICTED GENERALITY

[15D

where G 1 == Can(M 1 ,

.,',

M m , FEHz, {Uju}zX, {UjU}ZY),28

G2 == Can(M 1 ,

••• ,

M m , WX, {Uju}WY),

G3 == Ver(M 1 ,

,

M m, FEHW),

G4 == Ver(M 1 ,

,

M m , WX),

Gs == Reg(M l'

, .. ,

M m , WY, N 1> ••• , N m Z).

In this system if Can(EX) and Ver(EX) hold for all obs X, then Can(QXy) holds for arbitrary obs X and Y. The proof of this is in Fig. 1. But there are some deficiencies in this system. The analogue of Theorem 12C7 goes through all right, but it is not clear that this is true for Theorem 12C6 or for Theorem B9. For example, in Fig. 1 there is a proof of Can(H(zY), zY),

but if V =f. E, it is not at all clear that Can(H(UY), UY)

is provable. And furthermore, it is not clear that Can(H(zY» is provable. And this is not all. For if we introduce extra rules to take care of these problems, we still have Can(Xx, Yx)

--+

Can(3XY),

and from this and ET and Ver(H(QXY», we can obtain a contradiction, as shown in Bunder [STB] pp. 22-23. For these reasons, this system is unsatisfactory, and we shall not consider it further. D. THE PREDICATE CALCULUS So far in this chapter, we have considered only formulas involving implication, a universal quantifier, and a restricted universal quantifier. We have not considered negation, conjunction, alternation, or any kind of existential quantifier. In this section we shall consider these other connectives and 28. This is a departure from the general convention; it means that the Uj are variables which do not occur in the M, or in FEHz, and that if one of the Uj occurs in both X and Y, then the same Vi is substituted for it in both cases.

Can(Ex, H(Ex))

--~----':'-Vl

Can(Ex) ----I Can(Ex, Ex)

Can(QXY)

Fig. 1

Can(FEHz, zX, zY) Q 1

Can(EX) - - - - - - - - - II 3 Can(FEHz, EX)

Can(Eu) ----I Can(Eu, Eu) -----Vl Can(Eu, H(Eu)) IV 1 Can(FEHE)

... til

VI

~ .-

~

~ o

(")

l"l

~

(")

~....

~

"'tl

.Q

[15D

THE THEORY OF RESTRICTED GENERALITY

416

quantifiers, and we shall show that any of the systems of predicate calculus considered in [FML] Chapter 7; i.e., any of the systems LA *, LM*, LJ*, LD*, LC*, LE*, or LK*, can be set up within the framework we have considered. In § I we set up these systems in an extension of the system ff 21 of § B. In § 2 we do the same for the system ff~1 of § C.

1. Predicate calculus in

ff 2 1

In order to introduce negation, conjunction, alternation, and an existential quantifier into the system ff 21' we adjoin to the system the C-indeterminates 0, A, V, and X. The C-indeterminate 0 is intended to represent a standard false proposition, 1 so that negation can be defined by

r ==

(1)

CPO.

Then, in a system ff~l to which 0 has been adjoined and I' is defined by (1), if we assume that 0 is a canob of degree 0, we have the following L-rules for I":

-r-

M, ~ la~ 0, L

M la~ ~, L M, 0 la~ N, L ----------M, r~ la~ N, L,

M la~ r~, L.

These rules correspond to Rules *N* of [FML] § 681. By analogy, for the other connectives we want the following rules: *A*

M, 'llat- N M, A~'l la~ N,

lat- N M, A~'l la~ N, M, ~

M

-~--,---,._._-

M, V~'l la~ N, *X*

M, nx la,

la, xt- N M, X~'llat- N,

M,

~x

M, X~'l

x~

N

lat- N,

M la~ n, L

M

lat-

~V,

M

L lat-

lat- n, L -M_ __ ...

..

_-,,~.

M la~ V~'l, L,

M

lat- 'lV, L

X~'l, L.

If we now define the unrestricted existential quantifier by 1: rules are M,~xla,xt-N

L,

M la~ A~'l,

lat- ~,L M lat- V~'l, L,

M,'llat-N ----

M,~la~N

M la~ ~,L

== XE, then its

M lat- ~V,L M lat- 1:~, L. -~-'---~'--

M,

1:~

lat- N,

In these rules, it is assumed that the definition of canobs is suitably modified to include the new connectives. In addition, we may want to consider the 1. Bunder in [8TB] uses '0' for the empty set, so that a standard false proposition would be represented by 'OX' where X is any canonical subject. Thus, if K were a canonical subject in this system, OK would be a false proposition.

15D]

THE PREDICATE CALCULUS

417

following rules:

Px

M, P~11

la~ N

M, ~

la~ N

Mla~N,

rx

M,nla~N

M,~la~N

Mla~N,

OJ

M la~ 0, L M la~~, L.

And finally, if there are counteraxioms 01' 02' ... , then we may also want the rule M la~ 0;, L M la~

0, L.

These last four rules correspond respectively to the Rules Px, Nx, Fj, and F* of [FML]. Since the seven systems LX* referred to in the introduction are formed by taking various combinations of these rules, it is natural to suppose that these systems can be set up in ff 21 by taking appropriate combinations of these rules and modifying the definition of canobs accordingly. This can, indeed, be done, and the rest of this subsection will be devoted to doing this in a precise way. The ordinary predicate calculus involves quantification over a "fundamental domain" which is usually thought of as rather a circumscribed totality. Let us call this domain a. Then universal quantification in predicate calculus corresponds to 3a, and existential quantification to Xa. But in terms of 3 we can express situations more general than this; in particular we can treat many-sorted predicate calculuses of a rather general kind. Although the most general situation of this sort belongs under Chapter 17, yet we shall first treat a situation which covers some of the generalizations. Let LX*(6) be anyone of these systems over an underlying formal system 6 as defined in [FML] § 7A5. Ignoring the 6 for the moment, we define a family of associated systems ff~~ as follows. (A) We adjoin to ff~l a sufficient supply of new canonical atoms of various degrees, and also of new atoms which are not canonical. 2 In systems which require it, one of these atoms is 0; it is a canonical atom of degree O. (B) We adjoin the new C-indeterminates A, V, and X. The canobs of the system are the basic canobs as defined in § B3, modified by adjoining to Defi2. The different systems of the family will differ in these adjunctions. We may suppose that we have at our disposal infinitely many indeterminates of each of the various kinds; then the supply will certainly be sufficient.

THE THEORY OF RESTRICTED GENERALITY

418

[15D

nition BI the following clauses: (d) e == A(I] where ( and I] are proper canobs of degree 0, n = 0, and rk(O + rk(I]) = m - 1; (e) == V(I] where ( and I] are proper canobs of degree 0, n = 0, and rk(O + rk(I]) = m - 1; (f) e== X(I] where ( and I] are proper canobs of degree I, n = 0, and rk(O + rk(I]) = m - 1. (C) The prime statements of ~~~ are

e

(pI)

X

laf-

X,

where X is any basic canonical simplex; in addition there may be prime statements of the form

laf-E,

(p2)

where E is a specified basic canonical simplex. (D) The system ~~~ is singular or multiple accordingly as LX* is singular or multiple." The rules are the structural rules; the expansion rules; the operational rules *A*, *V*, *3*, and *X*; possibly certain specified elementary rules of the form (2)

M

laf- E 1 , L; ... ;M laf- Em, L M laf- Eo, L,

where the E, are basic canonical simplexes; and those of the irregular rules Px, Tx, 0*, and OJ whose corresponding rules Px, Nx, F*, and Fj respectively are rules of LX*(6). THEOREM 1. If ~~~ is defined as in (A)-(D) above, then Theorems B3-4 and Lemmas B3.I-B3.6 hold for ~~~, provided that in Lemma B3.2 the phrase "the form of clause (c)" is replaced by "one of the forms of clauses (c)-(f)".

The proof of Theorem I is very similar to the proofs of the theorems and lemmas to which it refers. THEOREM

2. ET holds for ~~~.

Proof Similar to the proof of Theorem B7. The additional operational rules do not affect Stages I and 2, and the prime statements (p2) and elementary rules are taken care of by that proof. The rule 0* is an elementary rule, and so does not affect these stages, and rules Px and Tx do not affect them because they have no principal constituent. The rule OJ can occur in Stage 2 Case (15), and it is treated there as follows: let the premise of the in3. For the present we are not allowing mixed systems. We do not yet know whether the theorems proved below all hold for mixed systems. (Note the complications in the proof of ET for such systems in [FML).)

lSDj

THE PREDICATE CALCULUS

419

ference be A i' which is

x, lal 0, Vi' L k , where Vi consists of the constituents of Vk which are parametric in the inference. By the secondary induction hypothesis, there is a derivation of A;, which is M,

x, lak~ 0, N, Lk.4

Let Y be a constituent of N. Then by OJ, we have M,

x, lak~ Y, N, t.;

and A~ follows from this by W. and perhaps C•. Thus, it remains to prove Stage 3. There are four cases, depending on the head of the eliminated constituent. If the head is E, we have the case of Theorem B7. If the head of the eliminated constituent is A, the major and minor premises reof the elimination theorem are conclusions of inferences by rules spectively. In this case, the eliminated constituent has the form A~I'/ and the premise for the inference by .A is one of

.A.

(3)

M,

~ la~

M,

N,

I'/Ia~

N.

Similarly, the premises for the inference by A. are (4)

M,

la~~, L,

Now, if n = 1 or 2 according to whether (3 1 ) or (32 ) is the premise of .A, then the conclusion of ET follows from (3n ) and (4n ) by the main induction hypothesis." If the head of the eliminated constituent is V, we have the case dual to that of A. If the eliminated constituent is X~I'/, then the major and minor premises for ET are conclusions of inferences by .X•. The premise for the inference by .X is one or the other of M,

~x

[u, x~ N,

whereas for the inference by (5)

M

M, nx [n, x~ N;

x. we need

la~ ~U, L,

both of

M

la~ I'/U, L.

From the former, substituting U for x we have (6)

M, ~U la~ N,

or

M, I'/U la~ L.

From that one of (6) which we have, and the corresponding one of (5), we have the desired result by the hypothesis of the primary induction. We now further specialize g-~f; the specialized system we call g-~~(6). 4. If the system is singular, then N is not present here, and the Y of N referred to below is just Y itself, and so the application of W. and C. is vacuous. 5. We are using the fact that rk(M'/)) = rk(;) + rk('/)) + 1.

420

THE THEORY OF RESTRICTED GENERALITY

[15D

We first establish a correspondence between the primitives of 6 and the new atoms adjoined under (A). In this it is not necessary to use different symbols for the original primitives and their correspondents; here it is permissible to identify them in the sense that we name them by the same symbols in our Alanguage. Let a be a new canonical atom of degree I; in interpretation this will stand for the fundamental domain of the predicate calculus. Then the correspondence can be extended to all the compound notions of LX·. The manner of doing this is probably clear; but to be perhaps excessively explicit we list below on the left certain notations of [FML], and on the right the corresponding notations in the present treatment:

e,

el

0h(t 1> ••• , tm )

Wk t 1" .t mk

CPk(t 1, ••• , tn )

CPk t 1" .. tnk

E1

Ei

A::JB

PAB

AAB

AAB

AvB

VAB

-,A

rA

F

0

(Yx)A(x)

3ae

(e == Ax.A(x))

(3x)A(x)

Xae

(e == Ax.A(x)).

This understood, we specify §~~(6) as follows. The prime statements (p2) shall be those of the form la~ E,

where E is an axiom of 6, 6 together with all statements I~

«e.;

where the e, are the atomic terms of 6. The elementary rules are those of the form (2), where either E i and Eo are elementary propositions of 6 such that

is an auxiliary statement of LX·, or else for some terms t i,

••• ,

tn and an n-

6. If the axioms of 6 are given in terms of axiom schemes, we can have such a statement where a is the set of term variables functioning as schematic variables in the scheme. This is a possibility not explicitly considered in LX·, an axiom scheme being there merely a set of axioms, and the introduced axioms were the statements of type (p2). This is evidently a minor variation. An analogous remark applies in some later cases.

15D]

THE PREDICATE CALCULUS

421

place operator w the rule is

M la~ at;

i = 1,2, ... , n

M la~ a(wtj ...tn ) ·

It then follows that, if t is a term of t(a), then la~

(7)

at.

THEOREM 3. Let LX* be singular or multiple (but not mixed), and let its counteraxioms, if any, be elementary,' Let

(8)

Mla~N

in LX*.

a.

Let K consist of ax for all variables x in

Then in ~~~(6).

M,Kla~N

(9)

Proof We use deductive induction on the proof of (8) in LX*. The basic step is clear; for if (8) is a prime statement of LX*, then it is a prime statement of ~~~(6), and from this we have (9) by *K. It remains only to consider the induction step. Suppose, then that (8) is obtained by a rule R. In all cases except where R is a quantification rule, there is a rule R' in ~~~(6) which is, so to speak, a literal translation of R, and has the same range in premises and conclusion." In such a case the theorem holds for the premise(s), and (9) follows by R' from the translated premise(s). Thus only the quantification rules require special consideration. If R is Il-, the premise is M [u, b~ A(b), L. Let K be determined from the conclusion as in the theorem, and let

~

be

AX. (A(x)). Then by the hypothesis of the induction M, K, ax lu, x~ (A(x)), L. Hence, by Exp»,

M, K, ax lu, xr

~x,

L.

From this we have by 8*

M, K

la~ 8a~,

L,

which is (9). 7. Our proof holds only under these limitations, but we do not claim they are essential for the truth of the theorem. 8. In those cases where something is added in the conclusion (e.g. *K*, V*, and, except in the Ketonen form, *A) this property is a consequence of [FML] Theorem 7BZ; but an analogous property holds for SWrlx(EI) by Theorem 12C5.

[150

THE THEORY OF RESTRICTED GENERALITY

422

If R is .IT the premise is

M, A(t)

la~

N.

By the induction hypothesis M, (A(t)), K

Hence by Exp», with

~

(10)

la~

N.

as in the case of Il«,

M,

~t, K la~ N.

On the other hand by induction on the structure of t we have K la~ at;

(11)

hence by.K M, K

la~

at.

From this and (to) we have (9) by .3. If R is ~. the premise is M la~ ACt), L.

Hence we have, with

~

as before,

M, K,

(12)

M, K

la~

(A(t)), L

la~ ~t,

by hp, ind., by Exp•.

L

On the other hand we have, by (11) and .K,

M, K

la~

at, L.

From this and (12) we have (9) by X•. Finally, if R is .~ the premise is M, A(b) [u, b~ N.

Let

~

be as before. Then we argue as follows: M, (A(x)), K, ax [u, x~ N M, K, ax,

M, K,

~x

[u, x~ N

Xa~ la~

N

by hp, ind., by .Exp, .C, by.X.

The conclusion is (9). This completes the proof of Theorem 3. 2. Predicate calculus in ~i 1 The introduction of negation, conjunction, alternation, and an existential quantifier into the system ~i I is similar to the introduction of these connectives

THE PREDICATE CALCULUS

15Dl

423

and quantifiers into ff 21 in § 1. As we did there, we define negation in terms of the standard false proposition. In this case, the rules for negation become the following:

-r-

°

M la~ X, L M, la~ N, L M, rx la~ N, L,

M, X

la~

0, L M

la~

M

la~

HX, L

rx, L.

However, for the operational rules for A, V and X it is not immediately clear what if any, extra premises involving Hand L should be added to these rules. For the present, we shall not require any such premises, so that the rules will be the rules .A. and .V. of § 16C2 and .X.

M, Xx, Yx

la, x~ N

Mla~XU,L

M,XXYla~N,

Mla~YU,L

M la~ XXY, L.

This procedure seems reasonable in view of the consistency result proved in Theorem 16C2. However, future research may lead us to modify this definition and introduce some of these extra premises into these rules. With these additional rules, we then proceed to define ffi~x much as we did in § 1. However, in (A), the only assumption we made there about the canonicalness of the new atoms is that 0, if it is needed, is a canonical atom of degree O. Here we postulate instead ~ HO. Furthermore, in (B), instead of (d)-(f), we take the following rules: M la~ HX, L

M la~ HY, L

M la~ H(AXY), L.

M la~ HX,L M

la~

Mla~LX,L

M la~ HY,L

H(VXY), L. Mla~LY,L

M la~ H(XXY), L.

Finally, in (C) and (D), we drop all reference to canonicalness restrictions in connection with the prime statements and elementary rules, since the system is unrestricted, so that (pi) is a prime statement for any X, and the appropriate restrictions on the prime statements (p2) and the elementary rules are just those for any CL-system. Since we have not yet found a proof of ET for ffit, we certainly do not have a proof of it for ffi~x. However, we can prove a theorem which corresponds to Theorem 3. In order to prove this theorem, we first define ffi~X(6) for a formal system (6) very much as we defined ff~~(6) in § 1. As there, we assume that the primitives of 6 are adjoined as atoms under (A). However, we shall not assume that any of these are canonical atoms. Furthermore, the only prime statements

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THE THEORY OF RESTRICTED GENERALITY

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(p2) shall be those of the form la~ E,

where E is an axiom of e. Furthermore, the only elementary rules we shall add will be those of the form (2) where all the E, and Eo are elementary propositions of e such that

E 1 , E2 ,

... ,

Em

~o

E

is an auxiliary statement of LX*. For each elementary statement (8), we define the sequence of grammatical conditions for (8) to be the sequence consisting of the following: (i) Lx, (ii) «e, for each atomic term e, of (e) which appears in (8), (iii) Fmrx..• rxHl/J (with m «'s) for each primitive predicate l/J of e of degree m which appears in (8) (including the case m = 0), (iv) Fmrx•. •tu» (with m + I a's) for each primitive operator w of degree m in e which appears in (8), and (v) rxx for each variable x in a. THEOREM 4. Let LX* be as in Theorem 3, and let (8) hold in LX*(e). Let K be the sequence of grammatical conditions for (8). Then (9) holds in ffi~X(e).9

Proof The proof is similar to that of Theorem 3. First, however, we note the following easy lemmas: LEMMA 4.1. If t is a term oft(a) which appears in (8), and theorem, then (13)

if K

is as in the

K la~ «t.

Proof It is easy to show by induction on the structure of t that K ~F «t,

and (13) follows. LEMMA 4.2. If A is a proposition of $(a) which appears in (8), and as in the theorem, then (14)

if K

is

K la~ HA.

Proof If A is an atomic formula, then it is easy to show that K ~F HA

in a way similar to Lemma 4.1. This proves (14) if A is an atomic formula. Now proceed by induction on the structure of the formula A using Rules H3*, HA*, HV*, and HX*, and, where necessary, the rule in ffit corresponding to F* in ff 21 (the second premise of this rule will be valid since Lrx is in K). We now proceed with the proof of the theorem. It is, as is that of Theorem 3, 9. This is essentially Theorem 25 of Chapter 4 of Bunder [STB].

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THE PREDICATE CALCULUS

425

an induction on the proof of (8). The basic step of that induction goes through as in Theorem 3. For the induction step, the only difference from the proof of Theorem 3 will be for those rules in LX* whose corresponding rules in ~~f require an extra premise involving Lor H. Since the only such rule is S*, we need only consider the case in which (8) is the conclusion of an inference by P* or Il«. If the rule is P*, then the premise is

M, A where N == A

:::>

la~

B,L,

B, L. By the induction hypothesis, M,K,Ala~B,L

holds in ~i~X(®). Furthermore, A is a proposition of S~(a), so by Lemma 4.2, (14) holds. Hence, (9) follows by P*. If the rule is Il-, then the proof is as in Theorem 3, except that the presence of Lex in K is now needed to justify the second premise of S*. This completes the proof of Theorem 4. COROLLARY 4.3. Theorem 4 still holds if the definition of ~ifx is modified to include additional premises involving H in the algebraic rules or L in the quantification rules.

Proof The sequence K will always justify these extra premises. In his [STB], Bunder suggests the following definitions for A, V, and X :10

A == Axy.Hz :::>z.(x :::>.y:::> z):::> z, V == Axy.Hz :::>z:(x:::> z) :::>.(y:::> z)::::> z, X == Axy.Hz :::>z.(xu ::::>u'Yu:::> z):::> z.

However, we cannot use them in ~ifx as we have defined it because we cannot derive HA*, HV*, and HX*. These last three rules can be derived if the system is a system of relative canonicalness. But then we cannot derive any of the operational rules for these connectives and quantifiers unless we have either that H is a canonical atom of degree 1 (so that we have lo~ LH), or else a special deduction theorem involving H. In Bunder's system, H is taken as a canonical atom of degree 1, and, to avoid the resulting paradox, E is replaced by a more restricted 1 with the property that lo~

10. Bunder called this

.~'

Sexl.

and had no quantifier corresponding to our use of that symbol.

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THE THEORY OF RESTRICTED GENERALITY

Under these assumptions, Bunder was able to obtain all the properties he wanted for conjunction, alternation, and existential quantification. However, some of his results depend on the fact that his system includes axioms which cannot be derived from the T-formulation. Ifwe work within the L-formulation, we can derive Rules A... and V"" but none of the others as they stand. Instead of X"" we can derive the reasonable rule M la~ XU,L

M la~ YU,L

M la~ LX,L

M

la~

LY, L

M la~ XXY, L

However, we have not been able to obtain any rules for these connectives and quantifier on the left that do not involve a premise of the form M la~ HZ, L

where M, Z, and L consist only of parametric constituents. Hence, we do not yet feel that we can use these definitions as an alternative to adjoining A, V, and X as C-indeterminates.