Normal forms in an intensional logic for English - Deep Blue

JOYCE ~IEDMA~ A~D

DAVrD S. WA~E~

-Normal Forms in an Intensional Logic for English*

Abstract. Montague [7] translates English into a tensed intensional logic, an extension of t h e t y p e d J-calculus. W e prove t h a t each translation reduces to a form~da w i t h o u t J-applications, unique to within change of bound variable. The proof has two m a i n steps. W e first prove t h a t translations of :English phrases have t h e special p r o p e r t y t h a t arguments to functions are m o d a l l y closed. W e then show t h a t formulas in which arguments are m o d a l l y closed have a unique fully reduced A-normal form. As a corollary, translations of English phrases are contained in a simply defined p r o p e r subclass of the formulas of the intensional logic.

Introduction. I n this paper we consider 2-normal forms in an intensional ,logic m o t i v a t e d by its relevance to natural language. The system investigated is t h a t of Montague's The proper treatment of quantification in ordinary English [7], here referred to as []~TQ]. [PTQ] correlates English phrases with logical expressions. Each derivation of an English phrase is mapped, or translated, into an expression in the intensional logic, IL. IL extends the usual t y p e d 2-calculus; it includes operators for intension, extension, necessity, and tense. This logic, b u t without tense, was introduced by 1Vfontague earlier in [6] and axiomatized and investigated b y Gallin [5]. We investigate the question of when expressions in IL have a unique ~-reduced form. A counterexample shows t h a t this uniqueness does no.t hold for I L in general, ttowever, it does hold for all expressions with a simple property t h a t depends only on the form of the expression. 1Kontague's translations all have this property. These questions arose because the translations given by the rules of [PTQ] directly are long and difficult to comprehend. One reason is t h a t substitution rules in the English syntax are translated into applications of 2-functions in the logical syntax. It is natural to a t t e m p t to simplify t h e translations by eliminating all the ~-~pplieations by ~-reduction. I n [PTQ] !Kontague gives his examples in fully reduced form, although he gives no discussion of reduction. The reduced forms ~re more comprehensible to the reader and easier to interpret in a model. This increased ease also carries over if, as in our case, t h e entire process is implementecl on a computer (see [2], [3], and [4]). * This research is s u p p o r t e d in p a r t B ~ S 76-23840.

b y l~ational Science F o u n d a t i o n Gran~

J. 2'~$edman, D. S. War~en

312

We obtain the main result in two steps. The first is a theorem on the form of translations of English phrases. We show t h a t in translations of English phrases and in their reductions, the arguments to s are always modally elosed~ hence 2-applications are always contractible. The second step is a theorem on the intcnsional logic itself. I t is wellknown t h a t for the usual t y p e d ~-caleulus there is a fully reduced normal form, unique to within change of bound variable. In intensional logic, in contrast, )~-reduction does not yield a unique normal form. We show, however, t h a t if all arguments to functions are modally closed, there is a unique normal form. This combines with the result on translations to yield the main result. Example. Before proceeding to the formal development, we give an example illustrating the stages os the process to be investigated. Consider the English sentence John eats a fish. One of the derivations given for this sentence in [e~Q] is displayed in the figure. 2gonterminal nodes are labeled by syntactic rules and syntactic categories~ using TS for sentence and TE for term. The derivation uses the rule S14, which substitutes a term phrase (a fish) for a variable (bee) in a sentence form (John eats himo).

TS

I

S14

I I

I

TE

1

TS

bee

I

l

$2

$4

1 I

a

I

C2q

I

I

TE

IV

J

1

[

fish

John

$5

I

TV

I

TE

I

[

eat

bee

The word-for-word or direct translation is l~educing the outermost g-application yields

T h e subexpression [v^~ty...] can now be reduced to [2?]...]. l~edueing

313

~-novmal fo?qn-s in an inte~sional logiv for English

this ~ n e x t yields

(3x) [fish (x) A [2Q [VQ j ( Aj)] (A [eat (^ ~R [v t~] (x)) ])]. Oontinuing in this w a y leads to the reduced form

(3x)[fish(x)^ [eat(^~R[vR](x))](Aj)], in which the only ~-function lacks an argument. ~ o n t a g u e lurther transforms the formulas b y introducing extensionM constants where possible and b y replacing certain u n a r y functions b y b i n a r y predicates. F o r the lormula trader consideration t h e result is

(3 u) [fish* (u) ^find* (j , u) ]. T h e extensional forms ~re equivalent to the others only b y ~ssuming so~ne meaning postulutes and are not further considered here.

Basic Definitions. F o r eonvenience~ we repeat ~ontague~s definition of meaningful expression.

Types.

The set of types is the smallest set Y such t h a t (1) e, t e :Y, (2) whenever a, b e Y, (a, b} e Y, and (3) whenever a e Y, (s, a) e :17.

Meaningful expressions. There are d e n u m e r a b l y m a n y variables and infinitely m a n y constants of each type. The set of meaningful expressions of type a is the smallest set such t h a t : (1) (2)

'(5)

E v e r y variable and constant of t y p e a is in _-M:E a. If A e ~ E a and u is a variable of t y p e b~ then ~uA a 5IE. I f d ff•E(a,b > and B @-~I:Ea~ then A(B) e ~ b. HA~ B eNEa, then A = B e N E t. If Q, R eNEa and u is a variable, then ---]Q, [Q^R]~ [OvR],

'(6) (7)

If A e :~]~a, then [^A] e lV[E. If A e~E, then I r A ] e ~ .

(3)

(4)

(3u)O, (Vu)O,

HQ, wo

B y a meaningful expression or formula of intensionM logic is u n d e r s t o o d m e m b e r of ~E a for a n y a. If u is a variable of t y p e a, then 2uB is u n d e r s t o o d as denoting t h a t function from objects of t y p e a which for a n y such object x~ takes ~s v a l u e t h e o b j e c t denoted b y B when u is u n d e r s t o o d as denoting x. The expression A (B) is as usuM u n d e r s t o o d as denoting t h e vMue oi t h e lunction denoted b y A for the functional a r g u m e n t denoted b y B. The logical s y m b o l s [~, W, H m a y be read "it is necessary t h a t " , "it will be the case t h a t " , "it has been the case t h a t " , respectively. The expression [^A] is regarded ~s denoting the intension of the expression A. The expression [VA] is well-formed only when A denotes an intension; in such ~ ease [VA] denotes the corresponding extension.

J. Friedman, D. S. Warre~

31~:

x~fodally closed formulas. We define a subclass of meaningful expressions t h a t h a v e the p r o p e r t y t h a t their syntactic form forces their denotations to be i n d e p e n d e n t of t h e point of reference at which t h e y are evaluated. The class of modally closed ( ~ c ) f o r m u l a s is the smallest clnss such t h a t : (t) (2) (3) (4) (5) (6) (7) (8) (9)

u is ~io for every variable ,u. [AA] is ~0 for e v e r y formula A. A ( B ) is ~ o whenever A and B are xo. [A ----B] is ~ o whenever A, B are ~c. 2r is ~io w h e n e v e r A is ~o. [::]Q is ~ o for e v e r y formula Q of t y p e t. -]Q is ~ o whenever Q is MO. ( ] u ) Q is ~ o ~nd (Vu)Q is ~ o whenever Q is ~o. [QAR], [ Q v R ] , [Q->R], and [Q,-~_R] ~re ~ o whenever Q, R a r e )/[o.

This definition of m o d a l l y closed is essentially t h a t of [5]. I t differs only in t h e addition of (6)-(9) for formula.s of sentence t y p e (lvm,) t h a t are i n t r o d u c e d b y GMlin only as abbreviations. (6)-(9) can easily b e p r o v e d f r o m Gallin's definitions. Formulas which contain W a n d H a r e n o t translatable into Gallin's system. Note t h a t formulas WQ and HQ are n o t m o d a l l y closed, [CA] is not m o d a l l y closed, and constants are n o t m o d a l l y closed.

Direct Translation. A direct translation of an English phrase i a translation of a derivation tree for t h e phrase into a meaningful expression of the logic b y rules T 1 - T 1 7 of [~TQ]. The relevant parts of t h e s e rules are given in t h e course of t h e proof of Theorem 1. The abbreviations used in [ ~ q ] are eliminated in the statements of t h e translation rules in order to display functional a r g u m e n t s m o r e clearly. T h e o r e m 1 shows t h a t in direct translations all functional a r g u m e n t s are variables or of t h e form [AD], hence are modally closed. Tm~OlCE~ 1. I f a function application C(B) occurs in a direct translation A, then the functional argument B is either a variable or an expressio~ of the form [AD]. Hence, B is modally closed. P~OOF:

B y induction on the construction of direct translations.

STATE~CErr Of BASIS. I f C(B) occurs as part of the direct translation. of a basic phrase of English, B is a variable or is of the form [AD]. PlcooF of BASIS: The translations of basic phrases are given b y r u l e T1. W e verify t h a t the result holds in each subcase of T1.

2-normal forms i~ a~ inteq~siona~ logiv for English

315-

Tl(a).

H A is a basic expression other t h a n be, necessarily, and t h e basic terms, i.e. members of ~T~, t h e n A translates into g(A), where. g(A) is a constant a n d hence contains no function applications. Tl(b) be translates into 2P2x[V/~](^~y[Vx ---- Vy]). The only subexpression t h a t is an a r g u m e n t of a function is ^2y[Vx = Vy].

TI(e)

necessarily translates into 2p[D[Vp]], which has no function

applications.

Tl(d). The m e m b e r s of B~E, John, Mary, Bill, and ninety, translateinto j*, m*, b*, a n d n*, respectively. B y definition, A* is 2 P [ v p ] ( A A ) . The only functional a r g u m e n t is ^A, which is of the proper form. TI(e).

he~ translates into )~P[vp](x~), where the functional a r g u m e n t is a variable.

I~duetion ste19: We assume the 19ro2erty holds for all eomTonent expressions of the direct translation A, and show that it is true of A. The p r o o f is b y cases on t h e translation rules T2 t h r o u g h T17.

T2.

This gives t h e translation corresponding to the rule t h a t f o r m ~ terms from c o m m o n nouns b y adding determiners. The three possible results are: for every, )~P(Vx)[S'(x)~[vt)](x)]; for the, 22 (3y) [(Vx) [S'(x) ~ x -= y] ^ [v/~] (y)]; a n d for a, ~ P ( 3 x ) [ S ' ( x ) ^ [vp](x)], where S' is the translation of c o m m o n n o u n phrase. The only new function applications h a v e variables x and y as arguments, and b y induction all functional arguments of S' are v a riables or are of the form [^A].

T3.

This translates t h e rule t h a t adds a relative clause to a c o m m o n n o u n phrase. The result is ~ [ E ' ( x ) ^ Q ' ] , where /~' and Q' are t r a n s l a tions. As ~bove, x is a variable, and E ' and Q' have the p r o p e r t y b y induction. We note t h a t [8] (p. 261 fn.) corrects T3 to uvoid collision o f variables b y replacing x in Q' b y a new variable y. This clearly does n o t affect the property.

T~

through TIO. These give translations for the rules of h m e t i o n application. The translations are all fo t h e form E'(^B'). The only new a r g u m e n t is ^B'. Tll.

Sentence conjunction and disjunction, [Q'AR'] and [ Q ' v R ' ] , int r o d u c e no new function applications.

T12.

Verb phrase conjunction and disjunction have t h e translations 2x [C'(x) ^ E ' ( x ) ] and ~x [C'(x) v E'(x)]. The new function a p p l i c a t i o n s h a v e t h e variable x as argument.

T13.

T e r m phrase disjunction translates into ~ P [ A ' ( P ) v B ' ( / ) ) ] , where~ t h e only new a r g u m e n t is the variable/~.

J. Friedman, D. ~. Warre~

:316

s176 Quantification over sentences has t h e translation A'(^)~xQ'). The o n l y n e w a r g u m e n t is of the form [^D].

T15 and T16. Quantification over a c o m m o n noun phrase or an intransitive v e r b phrase has t h e translation 2yA'(A2x[E'(y)]). The argum e n t of 9 ' is a variable a n d t h e a r g u m e n t of _4' is of the p r o p e r form. T17. The translations of the rules of tense a n d sign are 7 A ' ( ^ 9 ' ) , WA'(AE')~ -7W_4'(AP'), HA'(^9')~ a n d 7 H A ' ( ^ 9 ' ) . The n e w function .application in each of t h e m is _4'(AE'), which has an ~rgument of t h e ~ r o p e r form.

Definitions of Reductions. The results to be p r o v e d are a b o u t t h e f o r m u l a s t h a t can be obtained from direct translations b y application .of reductions of the k-calculus. Some definitions are needed.

~-contrac~ion. A k-application is a formula of t h e form [XxA](B). 'The k-application is contractible if either (i) B is m o d a l l y closed or (if) no free occurrence of x in A lies in an intensionM c o n t e x t of A, t h a t is, w i t h i n t h e scope of a A, [2, H , or W in A. If [2xA](B) is a contractible p a r t of a formula, then its contraction is a n y result of first changing b o u n d variables in A to avoid variable collisions and then substituting B for e a c h free occurrence of x in t h e modified A. EI-contraction. An 9I-formula is a formula of the form [v [^G]]. :It is always contractible; its contraction is C. The only contractible parts of a formula are t h e contractible ~-applications and t h e E l - f o r m u l a s .

Reduction.

Let A be a w i t h contraction C. Then a B b y C in A. W e say t h a t vontr(A~ 9) t h e formula B reflexive transitive closure

formula t h a t contains a contractible p a r t B contraction of A is the result E of replacing A reduces to 9 , or red(A~ E). W e denote b y t h a t is contracted. The relation red* is t h e of red and change of b o u n d variable.

Reduced forms. A formula is in reduced form if it contains no cont r a e t i b l e parts. I t is fully reduced if it is reduced and contains no ~-applications. Translations. By a translation of an English phrase we mean its d i r e c t translation _4 b y the rules of [PTQ], or a n y formula 9 such ~hat red*(A,9), tha~ is, a n y E resulting @ore A b y reduction.

Modal closure. W e n o w p r o v e t h a t reduction of translations preserves m o d a l closure of functional arguments. W e begin with some lemmas .about properties preserved b y substitution.

~-normal forms in an intensional logic for .English

317

_Yotation. Let Sub(B, x, A) be the result of substituting B for all free occurrences of x in A. L E ~ A 1. Let B be a subexpression of A. Let A ' be the result of replaying an occurrence of B in A by an expression B'. Zet A be ~c. I f B is not ~ c or if both B and B' are x c , then A ' is ~[c. P n o o ~ : The proof is b y induction on the construction of the modally closed formula A. We follow the numbering of the definition of modally closed. We note first t h u t if A ' is B% t h e n A ' is ~c. F o r in this case A is B, A is ~c, t h a t is, B is ~c. So b y hypothesis B' is ~o, t h a t is, A ' is ~ c . This will be used in each case of the induction below. ]~ASIS:

{1) A is a variable. Then A is B and A ' is B'. So A ' is _~o by the argument above.

Induction step: (2) A is [AG]. A ' is either B' or [^C'] where G' is the result of replacing B b y B' in G a n d thus ~ c b y definition. (3) A is C(D). A ' is B' or C'(D) or C(D'). Since A is ~c, G and D are ~ c b y definition. If A ' is C'(D) t h e n C' is ~ c b y induction, so A ' is ~ c b y definition. Similarly, if A ' is C(D'). (5) A is 2uC. A ' is B' or ~uC'. :By induction C' is ~ o so b y definition A ' is ~c. The remaining arguments are similar: (4), (7), and (9) are like (3); (6) is like (2); and (8) is like (5). LEM~V~A 2. Substitution of a modally closed expression for all free occurrences of a variable in a modally closed expression yields a modally closed expression. PROOF: The proof is a finite induction on t h e n u m b e r of occurrences of the variable. A t each step we are substituting a modally closed expression for a modally closed expression, so t h e result is modally closed b y L e m m a 1. L~x 3. (Substitution preserves modal closure of arguments). I f all functional arguments of A are modally closed, and B and all of its functional arguments are modally closed, then all functional arguments of S u b ( B , x, A) are modally closed. PI~O O F :

(1) If the occurrence of x replaced b y B is not in a functional argum e n t t h e n the new functional arguments introduced b y t h e substitution are just the functional arguments of B~ which arc modally closed b y hypothesis.

J. ~u

318

D. S. Warr

(2) If x occurs in a functional argument of A, then we are substituting a modally closed expression B for a variable in a modally closed expression. Hence by Lemma 2 the result is modally closed. The o t h e r new functional arguments in the result are those of B and are modally closed b y hypothesis. One more lemma is needed before we can present Theorem 2. L E M M A 4. (Reduction preserves modal closure of functional arguments). I f all functional arguments of A are modally closed and A reducee to E, then all functional arguments of E are modally closed.

P~OOF: We show t h a t each type of contraction preserves moda~ closure of functional arguments. (1) A-contraction consists of first a possible change of bound variab!e~ and then substitution. Change of bound variables replaces variables with other variables. Clearly this preserves modal closure of all expression~ and subexpressions. The substitution replaces a subexpression [2xC](B} by Sub(B, x, C). By Lemmas 2 and 3, all arguments t h a t are subexpressions of Sub(B, x, C) are ~rc, and by Lemma 1, any modally close4 expression t h a t contains the expression [),xC](B) t h a t has been replace4 by Sub (B, x, C) remains ~c. (2) EI-contraetion replaces a subexpression of the form IV^C] with C. The result holds by Lemma 1 and because the subexpressions of C~ are unchanged. The theorem now follows immediately. TI~EOlCE~ 2. closed.

I n translations, the arguments to functions are modall~f

PROOF: By an induction on translations. The basis is provided by Theorem 1 on direct translations. The induction stel) is b y Lemma 4.

The next theorem shows t h a t an even stronger property holds of the. arguments of translations: t h e y are either variables or of the form [^C]. Even though Theorem 2 would obviously follow from this, we h a v e deliberately c h o s e n to prove it separately, using Lemma 4. Proving it. in t h a t way makes it clear t h a t Theorem 2 would ~pp]y to any system in which the arguments of direct translations arc modally closed, a weaker constraint t h a n t h a t required for Theorem 3. LE~lvfA 5. I f all functional arguments of A are either variables or of the form [aC], and B and all of its arguments are also of these two forms~ then all arguments of Sub (B, x, A) are of these forms. P~ooF: The proof is similar to t h a t of Lemma 3. (1) If the occurrence of x replaced by A is not a functional argumenl~ then the new functional arguments introduced by the substitution are just the arguments of B~ which have the desired property b y hypothesis.

2-qwrmal forms ,in a~ intensio~a~ logio for E~Wllsh

319

(2) If x occurs in a functional argument~ t h e n there are two cases. I f the a r g u m e n t is x~ substitution preserves the property, because B ~und its arguments have the property. H the a r g u m e n t is [AC] it remains 5n t h a t form, and the other functional arguments introduced have t h e p r o p e r t y because t h e y are the arguments of B. L ~ v [ A 6. Reduction preserves the property that functional arguments are either variables or of the form E^C]. P~ooF: As for L e m m a 4. The only change is in p a r t (1), X-reduction, ~vhere L e m m a 5 is used in place of L e m m a s 2 a n d 3. THEOREM 3. I n translations~ functional arguments are either variables ~or of the form [AC]. P~ooF: B y an induction~ with the basis b y Theorem 1 and the 5nduetion step b y L e m m a 6. 2-•ormal forms. For the t y p e d 2-calculus it is well-known t h a t e v e r y expression has an irreducible X-normal form~ unique to within c h a n g e of b o u n d variable. I t is n a t u r a l to ask w h e t h e r this is also true f o r t h e intensional logic IL. One might suspect a problem for several reasons: the definition of contractible p a r t is more complicated in I L l t h e intension operator provides X-abstraction over points of reference, b u t since there is no actual variable over points of reference no change of b o u n d variable is possible; and there can be complex interactions of t h e intension operator with 2's. B y Theorem 2, in translations all functional a r g u m e n t s are modally ,closed. Thus, for application to English, we can use this modal closure a s a hypothesis and show t h a t translations have a unique fully r e d u c e d n o r m a l form. (Following our m a i n result we show t h e necessity of this hypothesis). The proof extends the proofs given in [1] and [9] for t h e t y p e d X-calCulus. modifications for I L appear in t h e definition of order of t h e con~ractible p a r t [v [AA] ] and in L e m m a 7. Order of contractible parts. We introduce a measure of the complexity o f contractible parts, based on t h e types occurring. The order of the type a, @a, is simply t h e n u m b e r of left angle-brackets occurring in t h e symbol for a. I t is i m m e d i a t e t h a t for all a and b, ~=a < @, and @a < @ . The order of a contractible part is defined b y :~=[XxA](B) : :~= a n d =~[V[AA]] ~-- @(s, a), where A is of t y p e a and B is of t y p e b. Minimal contractible part. A contractible p a r t is minimal if it conrains no proper subformulas t h a t are contractible. LEM~A 7. Let A be a formula in which all functional arguments are modally closed. I f A has a minimal contractible part B of order k, then the v.ontraetion of B contains no contractible part of order ~ or greater.

J. ~'iedma~, D. 8. Wa~'reqv

320 1Jl~OOF: I m m e d i a t e for ]YI-contraction.

F o r 2-contraction we n o t e t h a t if [2xO] (D) is minimal, t h e n its contraction can contain (or be) a contractible p a r t in only t w o w a y s : (i) D is [byE] of t y p e ( f , e> = d a n d x occurs in C in a subformnla x ( ~ ) . The order of the resulting contractible p a r t [hyE](~) is @ < f , e> = @d~ < ~ 4 ( d , c} = k.

(ii) D is [AE] where ( s , e} = d a n d x occurs in C in a subformnla [Vx]. The order of the resulting contractible p a r t IV [AE]] is @ ( s , e} = @d < @ ( d , c} = k. D can contain no h-applications, since b y t h e hypothesis t h e y w o u l d be contractible and hence D w o u l d n o t be minimal. I~E~AIr Lelnma 7 does n o t require its hypothesis, b u t w i t h o u t it t h e proof is more difficult. The proof requires t h a t the minimal contractible p a r t B cannot contain an uncontractible A-application t h a t becomes contractible when B is reduced. This follows from two observ a t i o n s : (i) If D is n o t m o d a l l y closed, neither is Sub(B, x, D)~ a n 4 (ii) If C contains y in an intensional context, so does Sub(B, x, C)r p r o v i d e d x is n o t y.

LE~M_A 8. /Let A be a formula in which all functional arguments are modally closed. I f A reduees to D by contraction of B to C, all new contractible parts of D are contained in C. PI~ooF: There are no n e w A-applications n o t contained in C. U n d e r t h e hypothesis, all old ).-applications are ahoeady contractible. ~L~At~K. shows.

Lemlna 8 needs the hypothesis, as the following e x a m p l e

[huAu](VAv) has only one contractible p a r t [rAy], which is of order L The h-application is n o t contractible because [vA~] is n o t Inodally closed a n d [Au] is an intensional context. After contraction of the arguinent, t h e forinula is [huAu](v), which is a contractible p a r t of order @(e, (s, e}} = 2, where b o t h u and v are of t y p e e. ImM~A 9. _Let A be a formula in which all functional arguments are modally closed. I f D results from A by contraction of a minimal contractible ~art of order k~ then any new contractible parts of D are of order less than k. Pl~ooF:

B y Lelnmas 7 and 8.

L e t h z* be t h e set of ~ll finite tuples of n a t u r a l nuinbers, ordered b y t h e relation > as follows: (x~, ..., x~} > {Yl, ..., Y~} iff (a) u > m or (b) n = m and there exists k such t h a t for all i ( 0 < i < k ) , x~_ l = y m _ , and x._ k < y , ~ _ k . (This is sometimes called "reverse iexicographic order".)

A-q~ormal forms i~ aq~ inte~sional logic for English

321

L I ~ A 10. I f XI~ X 2 , . . . is a sequence of elements of N* such that X~ < Xi+l, then this sequence is finite. As Pietrzykowski obsexves~ the lemma is easily proved by a double. induction on the length of X~ and on the value of its rightmost components.

Definition of L. We define a mapping Z el formulas into ~ * : for any formula A~ Z ( A ) = (il, ..., ik} , where ij is the number of contractible parts of order j in A. L E p t A 11. Let A be a formula in which all functional argument~ are modally closed. I f D results from A by contraction of a minimal contraetible part B, then Z ( A ) > Z(D). P~oo~: L e t Z ( A ) ---- ( i ~ ..., ik, ...~i~} a n d l e t ~ B = k (1 ~< k~< m)~ Comparing D with A, D has one fewer contractible part of order k, possibly some new contractible parts of order less than k~ but no new contractible parts of higher order~ by L e m m a 9. ]~enc% L(D) : ( j ~ . . . , j k _ ~ i k - - l ~ . ik+~, ..., i~}. By the definition of the ordering relation, Z (A) < L (D). L E p t A 12. Let A (1} be a formula in Which all functional argumenta are modally closed. Then there is a fully reduced formula A (n} such tha~ red*(A (1} , A (n}). ])~ooF: Let A(1}~ A ( 2 } ~ ... be a sequence el formulas such that red(A(i}~A(i§ ( i ~ 1 ) , and e o n t r ( A ( i } ~ A ( i + l } ) is minimal. I f A (i} is not fully reduced, then it has at least one minimal contractible part, so there exists an A ( i + 1 } such that red(A (i}, A ( i § But b y Lemma 117 L ( A ( i } ) > L ( A ( i § for all i ~ 1. Hence by Lemma 10, the sequence ~ L ( A ( 1 } ) , . L ( A ( 2 } ) , ... must be finite. Thus there exists a fully reduced formula A ( n } and red*(A(1}, A ( n } ) . Co~o~L~Y. Zet A be a formula in which all functional argumenta are modally closed. Then reduction by contraction of minimal parts yields a fully reduced formula B such that red*(A~ B).

Definition: Let / A / denote the equivalence class defined by change of bound variables on A. L ] ~ r A 13 (Uniqueness). s A (1} be a formula in which all functional arguments are modally closed. I f red* (A 2) such t h a t red(B (i}, B (i + l } ) holds for 1 ~ i ~< m - - 1 . (The reduction sequence here might include some non-minimal contractions). W e introduce the deiinition of t h e image(s) oi an expression E u n d e r reduction. L e t red(B~ G) a n d let E be a subexpression oi B. If contr(B, C) is equal to E, t h e n there are no images of E in G. If E does n o t overlap vontr(B, C), t h e n the image of E in C is j u s t t h e corresponding occurrence of E in C. If contr(B, C) is a proper subexpression of E, t h e n t h e image of E in G is E', where red(E, E') b y contr(B, C). If E is a proper subex19ression oi eontr(B, C), t h e n there are three eases. If eontr(B, C) is iv AD] ' t h e n the image of E is t h e corresponding occurrence of E in C. If contr (B, G) is [2xD](/~), E occurs in D, and red(D, D') b y contr(B, C)~ t h e n t h e image .of E is t h e corresponding expression Sub(B, x, E') in D', where [E/ = /E']. I I c o n t r ( B , C) is [2xD] (1~) a n d E occurs in 2w, there m a y be zero or more images of E in C. T h e y are the occurrences of E in t h e copies of ~ t h a t a r e introduced for t h e free x in D (after change of b o u n d variable in D). If red(B, G) a n d red(G, D) a n d E is a subexprcssion oi B~ t h e n t h e images of E in D are t h e images of t h e images of E in C. W e note t h a t t h e image ,of a contractible p a r t is always a contractible part. N o w let E = v o n t r ( A ( k } , A ( k - k l } ) a n d let E ( 1 , 1 } - - - - E and for 1 < i < m let E ( i , 1}, . . . , E ( i , hi} b e t h e s u b p a r t s of B ( i } t h a t are t h e images of the E ( i , j} u n d e r t h e r e p l a c e m e n t induced b y t h e reduction •I B (i --1} to B (i}. (Note t h a t E might b e duplicated, or might disappear, ;so t h a t n t m a y be 0). There m u s t exist B ( p } (1 < p ~ m) such t h a t no image E ( p } oi E is present, for otherwise B w o u l d h a v e a contractible part. lqow define t h e seqnence B ' (1}, . . . , B ' ( m - - l } us follows: for 1 ~< i < 10, let B ' (i} b e the result of replacing each E ( i ~ j } in B ( i } b y its contraction. F o r 2 ~< i < m, let B ' ( i } = B ( i § I t c a n b e seen t h a t red*(B'(i}, B ' ( i § f o r 1 ~ i ~ m--2 and t h a t /B'(m--1}/ -~ /B(m}]. There are t w o cases in showing t h a t red*(B'(p--1}~ B'(p}). Y~ contr(B(p--1}, B(p}) is an i m a g e of E, t h e n B ' ( p - - 1 } is B(p}. If contr(B(p--1},B(p}) is n o t itself an image of E, t h e n it m u s t b e a 2-contraction [2xC](D) where the final images of E occur in D and where x does n o t occur free in C. Then red(B'(p--1}, B(p}) b y t h e contraction of [2xG](D') where red*(D, D') b y contracting all images of E in D. 3s since A (k + 1 } is B ' (1} we h a v e t h a t red*(A (Ir +1}, B' (2}). These t w o results i m p l y t h a t r e d ( A ( k + l } , B) holds, which contradicts t h e definition of /r and completes t h e proof.

I f all functional arguments of A are modally closed, there is a fully reduced formula B such that red*(A,B) and B is unique to within change of bound variable. TI~Ol~E~ 4.

Pgoor: ~niqueness.

L e m m a s 7 t h r o u g h 12 p r o v e existence; L e m m a 13 proves

2-~ormal forms in an intensionat logic for English

323

The main result n o w follows.

T ~ o ~ E ~ 5. Translagons of English l~hrases have a fully reduced )~-normal form, unique to within change of bound variable. P~ooF:

B y Theorem 2 and Theorem 4.

Remarks on I L . An example shows t h a t t h e unique normal form result does n o t e x t e n d to I L in general. Consider t h e formula =

where x a n d y are variables of t y p e a, e is a constant of t y p e a, a n d u is a variable of t y p e ( a , (s, a ) ) . B o t h X-applications are contractible. Contracting t h e Ax yields [~y[Ay] = [u(c)J](e), which cannot b e f u r t h e r c o n t r a c t e d because e is n o t m o d a l l y closed and y occurs in the intensional c o n t e x t [^y]. Contracting the ~y first instead yields [2x[^x] ---- [u(x)J](v), which cannot be further contracted because e is n o t m o d a l l y closed and x occurs in the intensional context [^x]. B o t h of these formulas are therefore r e d u c e d forms and t h e y are n o t the same. The example depends on t h e particular definition chosen for 2-contraction. E a c h of t h e reduced forms obtained is eqniv~lent to [2x [^x]] (e) -----u(c), which is in some sense further reduced. If we were to redefine A-contraction to get this result~ uniqueness might be provable. Some combination a n d modification of the axiom schemata ASd.X t h r o u g h ASr and AS6 of [5] (pp. 19-20) could be used in this way. Conclusions. Our results are a first step t o w a r d t h e characterization of t h e subset of formulas of intensional logic t h a t are obtained as translations of English sentences. These formulas h a v e unique fully reduced normal forms, in contrast to the general case for t h e intensional logic. The main theorems h a v e immediate application in c o m p u t e r processing of t h e expressions o b t a i n e d as translations of English phrases fro G t h e [PTQ] fragment. The unique normal form can b e o b t a i n e d b y contractions of minimal parts, and this process will always terminate. The resulting form is easier to comprehend t h a n t h e direct translation and is an appropriate form for display or for evaluation.

References [1] P . B . _&xD~ws, l~esolution in type theory, Journal of S y m b o l i c Logic 36 (1971), pp. 414-432. [2] J. FRI~])_~AN, D. ~OR~N, and D. WARREN, An ~te~'pretation system fo~ Mo~tag~e g~'ammar, A m e r i c a n Jou~'nal o[ C o m p u t a t i o n a l Linquistics, microfiche 74 (1978), pp. 23-96.

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J. ~iedman, D. S. Wa~'~'e~,

[3] J. F~I]~D~AN, D. I~ORAN, and D. W~a~REN, Evalq~ating English sentences iq~ a logivaZ model, presented to the 7th International Conference on Computational Linquistics, ~niversity of Bergen, l~orway, August 14-18, 1978; Report ~-15, Computer and Communication Sciences Department, TJniversity of ~ichigan, Ann Arbor, l~Iichigan, 1978, mimeographed. [4] J. FRIV.D~A~ and D.S. W~a~Rv.N, A parsing method for Montague grammars, Linguistics a n d Philosophy 2 (1978), pp. 347-372. [5] D. GALLX~, Intensional a n d Higher.order M o d a l Logic, l\rorth-I~olland Publishing Company, Amsterdam, 1975. [6] 1~. iYIONTAGUv., U~iversal g~ammav, Theoria 36 (1970), pp. 373-378; reprinted in [8], pp. 222-246. [7] R. t~ONTAGUV., The p~ope~ treatment of q~aq~tifieation i~ o~dinary English, (PTQ), in: J. Hintikka, J. ~r and P. Suppes (eds.) A p p r o a c h e s to Natural Language, D. Reidel, I)ordrecht, 1978, lop. 221- 242; reprinted in [8], pp. 247- 270. [8] R. MONTAGVv., F o r m a l Philosophy: Selected Papers o[ R i c h a r d Montague, edited and ~dth an introduction by Richmond Thomason, "Yale ~niversity Press, I~ew ttaven, 1974. [9] T. PI:~TlaZYKOWSKI,A complete mechaq~izatioq~ of second-order" type theory, Journal o f She Association ]or C o m p u t i n g Machine~'y 20 (1973), pp. 333-365.

D ~ P A ~ E N ~ O~ CO~rV~Ea AND ~O~MUNICATION SCIENCES THe. ~NIY~I~HITY OF ~ICHIGAN ANN ARBOR, ~V~ICHIG~N 48109

l~eeeived J~ly 27, 1979

St~dla ~og~ca X X X I X , 2/3