A REALIZABILITY INTERPRETATION OF THE THEORY OF SPECIES

William W. Tait

1. The main purpose of this paper is to define a realizability interpretation for

second order intuitionistic logic and to prove that every deducible sentence of this

system is realizable. As a consequence of this, we shall obtain Girard's 1970 Nor-

malizability Theorem for second order intuitionistic logic. The ideas of this pa-

per extend in a straightforward way to the intuitionistic simple type theory (logic

of finite orders); but 1 will not discuss this extension here.

2. The Theory"'& of Species.

2.1. For each of the symbols O;=i, 0, I, ••• , 1 contains infinitely many a.-variables,

denoted by

fL, .p, .p,~, •••• The i-variables are intended to range over individuals and are also denoted by

X J Y, z, Xo t ••••

The O-variables are intended to range over propositions and, for n>O, the n-vari-

abIes over n-ary species of individuals (i.e. properties of n-tuples of individuals).

The non logical constants of j are individual and function constants. An n-ary func-

tion constant (n)O) is intended to denote an n-ary operation on individuals. The

individual or i-terms are built up in the usual way from i-variables and individual

and function constants, and are denoted by

The formulae of ~are built up in the usual way from atomic formulae

n~, by means of the operations

A::JB

for o.=i, 0, I, ••• • We shall see in 2. that the remaining intuitionistic logical

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operations can be defined in terms of these. If A is a formula, n ~ 0 and Xl' ••• xn

are distinct, then

is called an n-ary predicate or n-term. We shall write

where the right hand side denotes the usual substitution of i-terms for free occur-

rences of i-variables.

n denotes the result of replacing each part X tl ••• t n of B, such that the given occur-

rence of Xn is free in B, by Tntl ••• t n• In all of these substitutions, we may have

first to rename bound variables in order to avoid confusing them with free ones. We

shall assume that this is done in some unique way.

Formal deductions in j are in the style of natural deduction (Gentzen 1934).

They are finite trees which are built up from certain initial trees by means of

rules of inference. The initial trees are the premises

n A

consisting of a formula A with an index n i? 0 over it. There are four rules of in­

ference, an introduction rule and an elimination rule for each of ~ andl(.

~ I. B

A::>B n ::> E.

YE.

A~B A B

. YXaA Ta

a A[T I xa]

Thus, in each of these rules, we take given deductions, eg of B or of A ~ B and A,

and combine them to obtain new deductions, eg of A::>B or of B, resp. We must make

a restriction on VI; but first, we must say when an occurrence of a premise in a n n

deduction is discharged: A is undischarged in A. In each of ::> E, "I and E, an

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occurrence of a premise is discharged iff it is discharged in one of the given

deductions. In~ I, an occurrence of a premise is discharged iff it is discharged n

in the given deduction of B or else the premise is A.

Restriction on ~I. XU must not occur free in any undischarged premise in the given

deduc tion of A.

A is deducible in ~(deducible in ~from BO' •••• Bn) iff there is a deduction

in l ending with A and with no undischarged premises (all of whose undischarged k

premises are of the form Bi for some i ~ n).

2.2. Prawitz 1968 showed that the logical constants.1(absurdity).-',II,Aand:![

are definable in 1, by

..,A=A:::l.L

AvB =VXO{(A ;:) X) ::l «B ::l X) ::l X»

A AB =¥XO(CA::l (B;:) X» ;:) X)

:![Z~ = 'tIXOr:vZU(B :::l X) ::l X)

With these definitions. all of the intuitionistic laws of logic are deducible inl.

Set

A == B = (A ::J B)" (B::J A)

1 E = Axy Z (Zx. Zy)

Then the theory of identity of individuals is deducible in..2 in terms of E.

Assume that.l contains the indi vidua 1 constant 0 and the unary func tion

constant I Set

Then. ~ 1a Dedekind 1888. we can derive Peano's postulates for number theory from

the premises

\{ xy( Exy V .., Exy) V x-,Eox t

Vxy(Ex'y' ::l Exy)

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- n n Let A be obtained from A by replacing each part X t l ••• t n of A by"",X tl ••• t n ,

As noted by Kreisel. the result of Gode1 1932-3 extends to show that if A 1s deduc­

ible in classical second order logic. then A- 1s deducible in~. (We are assuming

here that A is a formula of-S.> One need only note that ""A-.!! A- is derivable inl)

that each of :;) I. ::l E and VI is preserved under the trans lation A-ioA -, and that

~E translates as

When 0. = i. this is again an instance of V E.

But, when 0. = n. it is

However. we do have the application

VXnA- r n -

A-[Tn-, xn]

VXnA- ,.,rn -

n-A-C,T 'Xn]

~ Tn n - n-of V E. from which we obtain A[ 'X] using .. ,T t l ••• t n lii

Thus. in this sense. classical second order logic is

result applies also to simple type theory.

n-T t1 ••• t n •

embedded in j. This

2.3. It will be convenient and suggestive to introduce another notation for natural

deductions in~. Accordingly, we introduce the notion of an A-term for each formula

A, by induction.

means that a is an A-term. Let VO' VI'

XO' etc. will denote arbitrary Vn"

a 1= A

be an infinite list of symbols. X, Y. Z,

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xA is called an A-variable.

c F= A => B, a F A => (ca) 1= B

The free occurrences of variables in xA are xA itself and the free occurrences of

variables yU in A. The free occurrences in ~b and Ax~ are those in b other than

occurrences of xA and xa , resp. The free occurrences in (ca) and (eTa) are those in

c and those in a and Ta , resp. We must impose a

Restriction on Axa • Ax~ is a VX~ term only when Xa is not free in A whenever vA is free in b.

With each natural deduction D of A we associate an A-term Inl as follows:

i~\ = .J' n

I v:a.! Axali I

A::JB I) (Ii => Bllll) B

Note that discharged and undischarged premises transform into bound and free vari-

245

ables, resp. Hence, the restriction on ~I transforms into the restriction on Axa •

Thus, D ... 'D/ is a bi unique correspondence between natural deductions of A in ...g and A-terms. In view of this, we may use the language of terms from now on, rather

than of natural deductions.

2.4. In formulating Gentzen's 1934 Hauptsatz for the system of natural deduction,

Prawitz 1965 introduced conversion rules for deductions which, in the language of

terms, are just Church's 1932 lambda conversion:

where the substitutions on the right for free occurrences of variables are defined

in the usual way. From now on, we shall not distinguish between formulae or terms

which can be obtained from one another by renaming bound variables.

a » b

will mean that b is obtained by replacing a part c of a by d, where c=t d. If there

is no b such that a,» b, then a is called normal. A reduction is a sequence

ao' a1' ••• (finite or infinite) such that ai,»ai + 1. aO is itself a reduction of

length 1.

a b

means that there is a reduction beginning with a and ending with b. a is normal-

izable iff there is a normal b with a ~ b. a is well-founded (strongly normalizable

in Prawitz 1970) iff all reductions beginning with a are finite. If a is well-

founded, it is clearly normalizable.

Well Foundedness Theorem for j. If a 1= A, then a is well-founded.

Girard 1970, Martin-Lof 1970 and Prawitz 1970 all proved that every A-term is

normalizable. (All of the proofs are based on an idea of Girard.) The arguments

easily modify~however, to prove well-foundedness. We shall obtain the Well-Founded­

ness Theorem below as a consequence of the Realizability Theorem (whose proof is

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again not much more than a reformulation of Girard's idea).

The isomorphism between natural deduction and the lambda calculus with type

structure, utilized in 2.3 - 4, was first noted in Curry-Feys 1958 for the case of

positive imp1icationa1 logic.

3. The Lambda Calculus and Realizability. 'e.

3.1. The terms of the lambda calculus/are built up from the variables Vn and the

constant K by means of the operations

Axb (ab)

of lambda abstraction and evaluation, resp. The notions of a free occurrence of a

variable, and of substitution for free variables are defined as usual. The rule of

lambda conversion is

(AXb)a

The relations a»b and a~b and the notions of normality, normalizability and ..ell­

foundedness are defined just as before in 2.4. Note that, unlike,3, l: has non nor-

malizable terms, eg s = «AX(XX» (Ax(XX»); and also has normalizable non well-

founded terms, eg «AXK)s). We shall write

ab = (ab), abc = «ab)c), abed = «abc)d), etc.

M

will denote the set of closed well-founded terms ofl. If a6M and a~ b, thenbEM

a:;:b

means that for some c, a ~ c and b ~ c. The Church- Rosser Theorem (1936) asserts

that = is an equivalence relation on the terms oft:.

We shall need the follOwing result, which 'He prove in 4.

~ If b[a/x]Cl ••• cn and a are in M, then so is (Axb)acI ••• cn (n ~O).

3.2. Let Ii denote the set of all closed i-terms of~.

A proposition ~ L is a species (i.e. property) R of elements of M such that

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i) If aE R and a ~ b, then bei R.

ii) If b[a/x]CI ... c n _ R, n~ ° and ae M, then (AXb)acl ... cnCi R.

If Ka1 ... a n is in M (i.e. a1' •••• an are in M), n;" 0, then Kal ... anE R.

An n-ary propositional function over t, n ~ 0, is a species R of n + I-tuples

i E ri • •••• tn' a with t l •••• , tn E I and aG M such that for all t l •••• , tn

the species

Rt1 ••• tn={a:<t1 , •••• tn' a)CRl

is a proposition over r (i,e. satisfies i) - ii1».

We extend J to the systemTby adding an n-ary relation constant for each

n-ary propositional function over r and a proposition constant for each proposition

overt. If pn is such a constant. then pn is the proposition or propositional func-

tion it denotes (n ~ 0).

In denotes the set of all pn in~

For each sentence (i.e. closed formula) A OfZ we define the species A

of terms of C by induc tion on the number of occurrences of ::J or"" in A:

Proposition 1. For each sentence A Of~ i is a proposition over~. n This is clear if A is atomic, i.e. = P tl, •• t n , Assume that A and Bare

propositions over rand c f. A::J B. KC i and so cK C 'B. Hence, cK E M and so cC M.

Then for all aE i, ca~ da and so daE 'B. Hence, dE. A::JB, Let

- d -and dE. M. For all aE A, c[ /X]el ... enaE B and so

(Axc)del ... ena E 'B by ii). Hence ().xc)de1 ... e n£ ~. Let Kal, .. an e M. Then

for all a E i, Ka l , .. ana is in M and so in 'B, Hence, Kal .. ,anE~, Thus, A::J B

is a proposition over~. ----- a

A similar argument proves that'V' XCLA is a proposi tion over r' if A[P / XCL] is

for each pa e la,

248

I":-' Let T be a closed n-term of), n >, O. We define the n-ary propositional

function T over 1: (a proposition if n = 0) by

n -Thus, there is a relation constant Pr in I with Pr = T. By induction on A, it

easily follows that

(1)

From (1) and the definition of~XUA, we obtain

(2) c e 'f/XUA => cK € A[T/XU]

for each closed a-term r of~

When a e A, we say that a realizes A. This is closely related to Kleene's

1945 recurSive realizability interpretation, except that, instead of coding functions

by their Godel numbers, we use the corresponding term of~.

3.3. It will be convenient to make the purely notational restriction on A-terms of

'&that a variable occurs in one of them with at most one superscript. I.e. if xA and XB occur in c F C, then A = B; if XU and XB occur in c, then U = 13, and not both

xA and XU occur in c.

With each A-term a we associate a term a of r as follows:

x

ca c a

cK

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L _nl an Let aF" A and let Yi ' ••• , Yn include -­form.p. If Ti is a closed ai-term of../, set

all the free variables in a of the

... ,

fun Then AO is a sentence. Let zBl Z 1 J ••• , m

••• , En are sentences ofT. Let biG ii'

is called an instance of a for AO'

be all the free variables in aO' Then Bl ,

Then AO is called an instance of A and

Realizability Theorem. If a j:: A, AO is an instance of A and a1 an instance of a for

AO' then ale Ao '

We shall prove this in the next section. As an immediate consequence, we

obtain the Well-Foundednes5 Theorem for~. First, let a ~ A and let a (and so A) be

closed. Then ~ is an instance of a for A. Hence, a is well-founded, by the Realiz-

ability Theorem. But, an infinite reduction a» b »c» ••• would yield an infinite

reduc Uon ~» b »c » ••• ; and so, a is well-founded. Let a i=' A and let a be open

If Xal am Bl yBn now, 1 ' ••• , Xm ' Yl , ••• , n are its free variables, then

lxal 1

But, this term is closed, and hence, well-founded. But, that implies that a is well-

founded.

4. Proof of the Realizability Theorem.

4,1. First, we shall prove the Lemma.

Proof. Let

250

be a reduction. We must show that it is finite. Note that b 1s well-founded, since

an infinite reduction of b would yield an infinite reduction of b[a/x]c1 ••• c n• Like­

wise, each c i is well-founded. If each di is of the form (AXbl)alci"'c~ where

b >/ b ' • a >/ a l and c t #' ci' the reduction is finite because b, a, cl' ••• , c n are

well-founded. If some of the di are not of this form, then for some j, dj = bl[a'/x]

ci'''c~ where b~ b', a;;ya t , and ci~ci' But

reduction dj ~ dj+l~ ••• must be finite, since

then b[a/x ] c1 ••• c ~ d., and so the n J

b[3/x]c 1 ••• c n is well-founded.

4.2. We prove the Realizability Theorem by induction on a.

~. a = x"-. Then by definition, a 1 G AD.

~.

~. a = AXCb, where b 1= B and so A = C ::> B. Let c E CO. Then bl[C Ix] is an

instance of b for BO; and so by the induction hypothesis, b1[C /x] e Bo' Hence, by

clause 11) in the definition of a proposition over 'C, alc G BO. Thus, a 1 E: AO.

~. a = AXc" where b I:: B and so A ="'X~. Let P e I U• Then bI[K/X] is an

instance of b for BO (since by the restriction on AXU, XU is not free in any C when

yC is free in b). So, blCK/x]e BO by the induction hypothesis; and so

aIKE: Bo' I.e. ale Ao'

~. a= bc, where b 1= B::>A and c I:: B. By the induction hypothesis, bIG B~O

and c 1 E- BO' SO a l = b1c1E.' AO

Case 5. a= bT, where b I=Vx~ and A = B[T! XU]. bIt: WX~)o and so,

~.

University of Chicago

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