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Philips J. Res. 46, 23-55, 1991 R 1248

AN EXTENSION OF THE MONTAGUE SEMANTICS

by PHILIPPE DELSARTE and ANDRÉ THAYSE

Philips Research Laboratory Belgium, B-1348, Louuain-la-Neuoe, Belgium

AbstractSeveral authors have pointed out some limitations inherent in Montague'sapproach to naturallanguage representation, and have attempted to remedythese limitations by working out suitable extensions of the Montagueformalism. One of these limitations originates from the fact that logicalconnectives and modal operators can only be applied to formulae and not toarbitrary logical expressions, which stands in contradiction with the commonpractice of natural language where connectives and modalities can afTectexpressions of various syntactic categories (and not only complete sentences).This paper introduces an extension of Montague's intensional logic thatprecisely addresses the problem just alluded to. The proposed extension isequipped with a well-defined 'Boolean semantics', analogous with theKeenan-Faltz semantics.Keywords: Boolean semantics, categorial grammar, intensional logic,

Montague semantics, naturallanguage representation

1. Introduetion

The formal representation of naturallanguage can be approached by meansofthe Montague semantics 1.2). The main difficulties inherent in the Montagueformalism have been pointed out by several authors 3-6). Let us give an outlineof these limitations.

The Montague semantics can deal only with isolated sentences and, exceptin some simple cases, cannot correctly interpret the anaphoras. The modalitiescan affect only sentences; they cannot affect arbitrary expressions as is normallythe case in naturallanguage. The logic used by Montague (i.e. the intensionallogic) is only able to deal with 'referentially opaque' constructions; therefore,it cannot accommodate the perception verbs, since the logic of such verbs isnot referentially opaque. (This means, for example, that .in a construction suchas 'see + infinitive', two logically equivalent infinitives may not be

Philips Journalof Research Vol. 46 No. 1 1991 23

P. Delsarte and A. Thayse

interchangeable.) In view of the limitations of Montague's semantics, severalresearchers have attempted to develop some other formalisms, more closelyadapted to the naturallanguage specificities.The formalism based on Kamp's discourse representation theory allows one

to process a set of sentences (forming a discourse), where the interpretationof each individual sentence depends on the context in which this sentence isstated. The same formalism is also able to deal correctly with anaphoricconstructions 7). The Boolean semantics introduced by Keenan and Faltzallows the modalities to affect not only the sentences but also many othertypes of expressions of naturallanguage. Furthermore, this approach enablesus to extend the notion of logical consequence from the syntactic category ofsentences to a larger family of syntactic categories 6). Finally the situationsemantics introduced by Barwise and Perry allows one to correctly deal withthe contexts that are not referentially opaque, and thus to correctly treat theperception verbs 8).

Several authors have attempted to 'translate' these different formalisms intothe language of Montague's intensional logic, and have worked out theextensions of this logic that are required for their purpose. The contributionsby Hendriks, Groenendijk and Stokhof are especially worth being mentionedin that context 4.5). A partial overview can be found in ref. 9.

This paper is concerned with a simple extension of the intensional logiclanguage, equipped with a well-defined 'Boolean semantics'. The proposedextension allows one to transfer some of the advantages of Keenan and Faltz'sBoolean formalism to Montague's formalism. In particular, the logicalconnectives and the modal operators of the extended logic can affectexpressions ofany type (not only formulae as in the classical case). This enablesus to obtain a straightforward translation for some natural languageconnectives and modalities, which can affect not only sentences but alsoexpressions of various other syntactic categories.

Section 2 is devoted to recalling the basic concepts of Montague's intensionallogic. Section 3 gives a detailed description of the proposed extension and ofits 'Boolean semantics'. Section 4 contains some illustrative examples that aimat showing how the extended intensional logic can be applied for naturallanguage representation.

Although this paper is relatively self-contained, the reader is assumed tohave 'a certain acquaintance with the Montague grammar ê) and with theelementary aspects of Boolean semantics 6.10). This paper is taken from achapter written by the authors in the book From Natural Language Processingto Logic for Expert Systems 9).

24 Philip. Journ.1 or Research Vol. 46 No. I 1991

An extension of the Montague semantics

2. Logic formalization of natural language

2.1. Categorial grammar

Each naturallanguage is endowed with a grammar, that is to say, a set ofstructures and rules by means of which all statements belonging to thelanguage, and these statements exclusively, can be produced. In order to beable to process (i.e. to 'understand') the sentences of a naturallanguage, inan automatic manner, it is necessary to formalize the description of itsgrammar. The categorial grammar formalism is one of the possibleformalizations for the syntax of a naturallanguage. The grammatical syntaxof the fragment of English language introduced by Montague is an exampleof a categorial grammar 1). In this grammar, the notion of a syntactic categoryis defined recursively as follows:

• S is a syntactic category, associated with the sentences;

• IV is a syntactic category, associated with the intransitive verbs;

• eN is a syntactic category, associated with the common nouns;

• if d and !JBare syntactic categories, then dl!JB and dll!JB are syntacticcategories.

More precisely, this is Bennett's definition of the syntactic categories 11). Itdiffers slightly from Montague's initial definition, in which the categories IV(Intransitive Verb) and eN (Common Noun) are defined in terms of the twobasic categories S (Sentence) and e (referring to some extra-linguistic 'entities'),as follows: IV = S]« and eN = Sf]«.

The Montague grammar uses some abbreviations that have becomeclassical. For example, T (Term) represents SIIV, TV (Transitive Verb)represents IVIT and DET (Determiner) represents T[Cl»,

In the categorial grammar formalism, the syntactic categories alwaysreceive a recursive definition. Each basic expression (or word ofthe vocabulary)is assigned a syntactic category. Note that the S category is assigned to nobasic expression (since a sentence cannot reduce to a single word).

The compound expressions (usually called phrases) of the language areobtained from the basic expressions by means of the syntactic rules of thecategorial grammar. Most of these syntactic rules are instances of two ruleschemata: the functional application schema (sometimes called categorialcancellation schema), and the Boolean schema.

Philips Journalof Research Vol. 46 No. 1. 1991 25

• Functional application schemaAn expression A of syntactic category .91/ f!8 or .91// &ö combines with anexpression B of syntactic category &ö to form an expression h(A, B) ofsyntactic category .91.

• Boolean schemaIf A and B are two expressions belonging to the same syntactic categoryCfJ, then 'not A', 'A and B', and 'A or B' are expressions which also belongto the syntactic category CfJ.


The Montague grammar includes syntactic rules that are defined from fiveinstances of the functional application schema:

(.91, &ö) = (S, IV), (IV, T), (T, eN), (IV, S), (S, S).

For example, the first case, i.e. .91= Sand &ö = IV, yields the following instanceof the functional application schema:

An expression of category T = S/ I V combines with an expression ofcategory IV to produce an expression of category S, i.e. a sentence.

2.2. Intensionallogic

Two types of constraints bear strong influence upon the formal logiclanguage devised by Montague: an expressiveness constraint and a translationconstraint. The first constraint requires the formallanguage to be sufficiently'expressive'; it should be able to account for the subtleties of the fragment ofnatural language that it is supposed to represent. As to the translationconstraint, it requires the syntax of the formal language to be sufficiently'close' to the syntax of the naturallanguage, in such a way that an 'automatictranslation' process can be obtained.

2.2.1. Expressiveness constraint

The logic language has to be able to represent a fragment ofnaturallanguagecomprising certain modalities.In normal speech, we often talk about possibilities, about goals to be

achieved, about forecasts regarding the future, and so on. Most ofthe sentencesof natural language can be now true and now false, depending on thecircumstances, the time instant, the point of view of somebody. If we want tobe able to represent these modalities by means of a logic language, we have

26 Philips Journ.1 of Research Vol.46 No. 1 1991

An extension of the M ontague semantics

to give it the structure of a multimodal language, comprising at least thefollowing modal operators:

0: necessarily,o : possibly,[F]: always in the future,(F): sometimes in the future,[P]: always in the past,(P): sometimes in the past.

The modalities of belief or knowledge, such as 'the agent a believes (or knows)the event e', are represented in Montague's logic language by means oftwo-place predicate forms as follows:

believe(a, e): a believes that e,know(a, e): a knows that e.

Furthermore, the logic formalism has to be able to express the fact thatmost sentences of naturallanguage can be analysed according to two differentmoods: the de re mood, and the de dicta mood. Two additional operators areintroduced to that end:

-: intension operator,~:extension operator.

These operators provide us with one of the main ingredients by means ofwhich most expressions of naturallanguage can be given at least two differenttranslations into intensional logic; one of them formalizes the de re meaningand the other one formalizes the de dicta meaning.

The extension operator is the inverse of the intension operator, in the sensethat any logical expression (J, satisfies the semantic equivalence relation

(X~o:.

It should be noted that the modal operators can exclusively affect a logica]formula, while the intension operator does not have such a limitation; it çanaffect any logical expression.

Philips Journalof Research Vol. 46 No. 1 1991

2.2.2. Translation constraint

The most' essential paft of the translation constraint imposed by Montagueamounts to the existence of an isomorphism between the analysis of anexpression in naturallanguage and the analysis ofits translation in intensionallogic. This isomorphism is obtained through the introduetion of atype-theoretic syntax for intensional logic, and of a translation function thatmaps each syntactic category of natural language to a type of intensionallogic. Each natural expression A of syntactic category ~ is given a logicaltranslation, denoted by A' (an expression belonging to intensionallogic); thesyntactic category of A' is a type, denoted by f(~), where f is the translationfunction from natural syntactic categories to logical types. The essentialobservation is the following. If an expression A of category sf /~ (or d //ff8)combines with an expression B of category ff8 to form a compound expressionh(A, B), of category d, then the translation A', of type f(sf /~), combineswith the translation B', of type f(~), to form a translation h'(A', B'), of typef(d). An additional requirement bearing on the logical language is thefollowing: the function h' should be 'as close as possible' to the function h. Inparticular, if the function h simply acts as a concatenation of its argumentsA and B, then, as far as possible, the function h' should also act as aconcatenation of its arguments A' and B'.

The base symbols occurring in the definition of the logical types are t (for'truth'), e (for 'entity') and s (for 'sense'). The logical types are definedrecursively as follows:


• e and t are types;

• if a and b are types, then <a, b) is a type;

• If a is a type, then (s, a) is a type.

The translation of the fragment of natural language into the intensionallogic language is obtained by associating a logical type with each naturalsyntactic category. More precisely, a translation function f is defined; it is amapping from the sef of syntactic categories of natural language into the setof types of intensional logic. The function f is defined in a recursive manneras follows:

1. f(S) = t,2. f(IV) = f(CN) = <e, t), ' .3. f(d /~) = f(d //~) = «s, f(~), f(d) for all syntactic categories sf

and ~.

28 Philip. Journalof Research Vol. 46 No. I 1991


In fact, the translation function f given above corresponds to Bennett'sversion. In Montague's initial version, the type associated with the categoriesof intransitive verbs and of common nouns is defined as follows:

2'. f(lV) = f(CN) = «s, e), t).

To the functional application schema:

An expression of category d/~combines with an expression of category!!4 to form an expression of category d,

there corresponds a combination rule over the types:

An expression of type (b, a) combines with an expression of type b toform an expression of type a.

The isomorphism between the syntactic analysis of natural language andthe syntactic analysis of intensionallogic language results from the definitionof a correspondence between syntactic rule schemata and translation ruleschemata. Thus, to the functional application schema there corresponds atranslation schema as given below.

• Functional application schema:

(A)d/g.} + (B)g.}= {h(A, B))d'

In the particular case where the function h acts as the concatenation of itsarguments, this becomes:

• Translation schema:

(A')«s,f(g.}».J(d» + (B')f(&I) = (h'(A', B'))f(d)'

In the particular case where the function h' is the translation of theconcatenation, this becomes:

(A')«s.J(g.}».J(d» + (B')f(&I) = (A'(B'))f(d)'

(In the schemata above, each expression has an index which represents itssyntactic category or its type. The symbols.+ and = occurring in the functionalapplication schema are used to describe a syntactic rule of Montague's

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categorial grammar. The same symbols are used in the translation schema,in an informal manner.)To the Boolean schema there also corresponds a translation schema. Let r

be a dyadic connective of natural language (r = 'and', 'or'), and let r' be itstranslation in formallogic (r' = /\, v). The correspondence is the following.

• Boolean schema:

• Translation schema:

(Note that the latter writing is informal, since the symbol r does not belongto the logic language.)

2.3. The Montague semantics

In addition to the requirement of an isomorphism between the syntacticanalysis of the naturallanguage expressions and the syntactic analysis of theirformallogic translations, Montague sets the requirement of a homomorphismbetween the syntax of naturallanguage and its semantics. Thus, the semanticsof natural language has to be compositional, i.e. to comply with Frege'scompositionality principle, according to which the meaning of the whole is afunction of the meaning of the parts.If lX is an expression of intensionallogic (which may represent the translation,

lX =A', of an expression A of natural language), then

ff lXJJ.lf, w,g

represents the semantic value of lX. This semantic value is determined withrespect to a model vit, a possible world w, and an assignment function g. Themodel cé' is a fourtuple (U, W, fYt, V) having the following components:

• U is a set of individual objects; it is the domain of interpretation of theformal language.

• W is a set of possible worlds w.

• [J£ is a set of accessibility relations between the possible worlds; the differentelements of fYt are associated with the differentmodalities ofthe language.

30 Philips Journalof Research Vol.46 No. I 1991


• V is an interpretation function; it assigns elements of U to the individualconstants and it assigns sets of n-tuples of elements of U to the n-placepredicate constants, for each possible world WE W.

The assignment function g assigns elements of U to the individual variablesofthe language; more generally, for each type a, the function g assigns elementsof a well-defined set Da to the variables of type a. (Details concerning the setsD, are given below.)The semantic value of any expression IX of the formal language is obtained,

in a recursive manner, from the semantic values of the basic elements of thelanguage (the non-logical constants and the variables) and from the rules thatdefine the semantics of the logical connectives, the quantifiers, the modaloperators, and the intension and extension operators.The semantics of the basic elements is defined as follows:

• if c is a non-logical constant, then

ffcJJ.,{{,W,g = V(w, c);

• if x is a variable, thenffxJJ""{,W,g = g(x).

If IX is an element of a given type a, then the semantic value of IX, also calledthe denotation of IX, belongs to a well-determined set', which is noted Da andis referred to as the set of possible denotations of type a. The sets D; arerecursively defined by the following rules:

Dt=B={l,O};

D(s,a) = D:;' = W -T Da;

D(a,b) = Dra = D; -T Db'

(The notations yX and X -T Y both represent the set of mappings from a setX to a set Y.)The 'denotation function' D (mapping every type a to the corresponding

set of possible denotations Da), recursively defined as above, allows therequirement of a homomorphism between the syntax and the semantics to befulfilled.

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Let us point out that the two homomorphisms

natural syntax --+ formal syntax --+ semantics

together with the requirement of associating a (unique) logical type with eachsyntactic category of natural language are at the origin of the introduetion ofthe lambda calculus as an ingredient of intensional logic. For example, theformal translations of the noun phrases and of the determiners will be lambdaexpressions.

3. Boolean semantics of an extended intensional language

The general idea examined in this section has its origin at the contact pointbetween the theory of Montague and that of Keenan and Faltz 1.6). (Acomparison between these two theories can be found in ref. 10.) The subjectcan be introduced as follows. For certain logical types a, which will be referredto as Boolean types, the 'semantic category' D; (ofthe Montague logic) has thestructure of a complete atomic Boolean algebra; in concrete terms, D, can beidentified, in a natural manner, with the power set (i.e. the set of all subsets)of a well-defined set. This allows one to apply the logical connectives, themodal operators, and the quantifiers to the set of expressions of type a, andto endow these connectives, operators, and quantifiers with a semantics thatcan legitimately be called a Boolean semantics.The choice of this terminology, borrowed from Keenan and Faltz 6), is

motivated by the fact that the proposed semantics exactly reflects the Booleanalgebra structure of the set Da, for every Boolean type a. From a formal pointof view, the main difference between the semantics described here and thesemantics introduced by Keenan and Faltz is concerned with the ontology,i.e. with the semantic primitives. As it will be seen in what follows, one canpreserve the ontology underlying Montague's intensional logic, while.endowing this logic (after a rather immediate enrichment) with a Booleansemantics, in due form.

3.J. Structure of the set of possible denotations for a Boolean type

3.I.J. Preliminaries from intensionallogic

The types of Montague's intensional language are defined from the threeprimitive symbols e, tand s by means of recursive rules 2.3). The symbols e

32 Philips Journalof Research Vol.46 No. 1 1991


and t represent the basic types; the symbol s, which is not a type, intervenesin the formation of intensional logic from extensional logic. It will proveconvenient to give this symbol a status similar to that of a type; we will usethe term 'pseudotype'.

Let us define Ea as the set of logical expressions of type a in the intensionallanguage (extended as explained in Sec. 3.2 below), and let us define Da as theset of possible denotations (semantic values) for the expressions of type a.Thus, we can write

ITIXlI.ll,w,OED for all IXEEII a a'

for every model At, every possible world IVE W, and every assignment functiong. (In what follows, we will explicitly write only that part ofthe triple (At, IV, g)which is really useful to the intelligibility of the semantic equalities underconsideration.) The sets Da are defined recursively, as explained in Sec. 2.3,from the primitives U, Band W.

The symbol s will henceforth be referred to as the pseudotype of intensionallogic. Let us define the set Ds, associated with the pseudotype s, as follows:

Ds=W.

The definition D(s,a) = D!V can then be obtained by specializing the generaldefinition D(b,a) = D~b to the case b = s.

For each expression IXE Ea, there is a companion expression - IXE E(s,a)' whichdenotes the intension of IX. The semantic value of -IX is a well-defined functionfrom W to Da; its values r IXJJ(W) are defined by

rIXII(w) = [1Xr, for WE W.

Conversely, if IX is an intensional expression, i.e. an element of the set E(s,a)for a certain type a, then it has a companion expression -IX E Ea> semanticallycharacterized by the fact that its intension equals the extension of IX. In preciseterms, the semantic value of the expression -IX is defined as follows:

r1Xr = ffIXII(w), for WE W.

From these definitions we immediately deduce the rule of cancellation ofintension by extension, that is r- IXIl = [lXII, for all expressions IX.

Recall that from each pair (IX, fJ), consisting of expressions IXE E(b,a) andfJ E Eb' one can construct a compound expression cx(fJ) EEa, the semantic value

Philips Journalor Research Vol.46 No. I 1991 33


of which is obtained by functional composition:

ff a(f3)j = ff aj(ff {3J).

In addition, the Montague logic is equipped with the lambda calculusformalism, which plays an essential role in the translation of naturallanguageinto intensionallanguage. Recall that for each pair (x, a) consisting of a variablex, of type b, and of an expression a, of type a, one can construct an expressionÀ.x[a], of type <b, a), called a lambda expression. (It is seen that the lambdaoperator 'complexifies' the types; it plays a role which somehow is the converseof the role played by the composition operator, which 'simplifies' the types.)The semantic value of the expression À.x[a] is a well-defined function from Dbto Da; its values are given by

for r e Dç,

where g is the considered assignment function. In this writing, gr representsthe x-variant of g, defined by gr(x) = rand gr(y) = g(y) for any variable ydistinct from x. From the definitions above one can deduce the lambdaconversion rule, namely

ffÀ.x[a](f3)j=ffa~j,

where a~ represents the expression obtained from a by replacing eachoccurrence of the variable x by the expression f3 (of the same type b as x). Itis important to notice that this rule does not have full generality. The delicatecases are those where x is in the scope of a modal operator or of an intensionoperator (see ref. 2, p. 167 and p. 177). It will not be necessary for us to enterinto a general discussion of the validity of the lambda conversion rule. Wecan be content with making the required verification in the cases that are ofreal interest for us; there is no difficulty about it.

3.1.2. Boo/ean types and associated structures

A type a will be said to be a Boolean type if the rightmost symbol in theexplicit writing of a is the truth symbol t (and not, the entity symbol e). Thesimplest case is a = t. The general form of a Boolean type is

a = <bi, <b2, ••• , <bno t) ... »,

34 Philips Journalof Research Vol.46 No. 1 1991


with n ~ 0, where bi is either a type, or the pseudotype s, for i = 1, ... , n. Inmore precise terms, the Boolean types exactly are the types that are obtainedby means of the recursive construction given in Sec. 2.2.2 under the restrietionthat the initial type is t.The logical expressions of Boolean type are those which, in the final analysis,

give rise to an expression of type t, i.e. to a logical formula. (It should benoted that the translation of any expression of naturallanguage into intensionallogic always gives a logical expression of Boolean type.) It will be seen in Sec.3.2 how one can perform a semantic analysis of certain compound expressions,of Boolean type, which occur as 'meaningful pieces' of complete formulae.(These expressions belong to the extended intensional language that will beintroduced on the way.)It is easy to show that, for any Boolean type a, the set ofpossible denotations,

Da, possesses the structure of a complete atomic Boolean algebra. In moreexplicit terms, D, can be viewed as the set, represented by 2G

., consisting ofall subsets of a certain set Ga; the meet and join operations over D, can thenbe identified with the intersection and union operations over the power set 2G -.

To begin with, let us examine an example that illustrates the machinery.Consider the Boolean type

a = (s, <(s, e), t»,the denotations of which represent properties of individual concepts. (If 0: isan expression of type a and (J is an expression of type (s, e), denoting anindividual concept, then ((1. )((J) is an expression of type t, denoting a truthvalue.) In this case, the set of possible denotations D, is the following:

D, = (W ~ ((W ~ U) ~ B))

= (Buw)W = WUw x W),

where x represents the Cartesian product. This clearly reveals the propertymentioned above, i.e. D; = 2G• (identified with BG

.), where Ga = UW x W inour example.To obtain this result, we make use of the following standard identification

(with an abuse of notation):

More precisely, we identify any function jfrom Z to (Y ~ X) with a function

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P. Delsarte and-A. Thayse

from Y X Z to X by setting

f(y, z) = f(z)(y),

for all yE Y and all ZE Z. (The example above corresponds to the sets X = B,Y= UW and Z= W.)In general, for a = <bl, •.. , <bno t) ... ), i.e. for a Boolean type of 'depth'

n, we thus obtain, by a straightforward induction argument, the followingrepresentation:

Here we identify the power set 2G• with the set BG• consisting of the functionsfrom Ga to B = {l , O]. More precisely, we identify a subset of Ga with itscharacteristic function. (For the type a = t, i.e. for n = 0, we simply haveGa= {O}.)Let us finally give an elementary result which will be needed in the next

subsection. Consider a subset F of the Cartesian product Ga x Db' i.e. a relationbetween the sets Ga and Db' (The symbol b here represents either an arbitrarytype, or the pseudotype s.) By definition, F is an element of the set

D - 2G(b ••> - 2G• x Db(b,a) - - .

Ifwe identify F with a function from Db to 2G., then wecan write the equality

F(r) = {qE Ga:(q, r)EF}, for each rEDb•

From this we deduce the following two Boolean identities:

F*(r) = (F(r»*,

(n Fi)(r) =n Fi(r),iel iel

where (F;}iel is any family of subsets F, of Ga x Db' (In the first identity, F*represents the complement of F with respect to Ga x Db' while (F(r»* representsthe complement of F(r) with respect to Ga.) In view of de Morgan's law, thereexists a dual version of the second Boolean identity, in which the intersectionoperator is replaced by the union operator.

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3.2. Enrichment of Montaque's logic

3.2.1. Logical connectives

The first enrichment idea (the simplest one) consists in allowing the logicalconnectives (---;, r., v, ::::>, ==) to act on the logical expressions of any Booleantype a, and not only on the formulae (which are the expressions of type t) asin Montague's logic. If IX, IX! and IX2 are expressions belonging to the set Ea,then the expressions ---; IX, IX! /\ IX2, IX! V IX2, IX! ::::> IX2 and IX! == IX2 are acceptedas well formed and belonging to the same set Ea·

For example, if IX! and IX2 represent two proper names, such as 'John' and'Mary', then IX! /\ IX2 will be an acceptable representation of the noun phrase'John and Mary'. Similarly, if IX! and IX2 represent two intransitive verbs, suchas 'sing' and 'dance', then IX! /\ IX2will be a suitable representation of the verbphrase 'sing and dance'. (These naive examples aim only at intuitivelyexplaining the main motivation of the approach. It will be seen in Sec. 4 howthe enrichments proposed here can actually be used in the framework ofnaturallanguage processing.)The semantics of the logical connectives thus extended (to each class Ea

with a Boolean type a) is defined by means of the following equalities:

i---; IXJ = iIXr = o,\iIXJ,iIX! /\ IX2J = iIXd niIX2J,iIX! v IX2J = iIX!J u iIX2J,iIX! ::::> IX2J = iIX!r uiIX2J,iIX! == IX2J = iIX!J IJJ iIX2J)*,

where the symbols *, n, u and E9 represent respectively the set-theoreticcomplementation, intersection, union, and symmetric difference over the powerset Da = 2Ga formed with the subsets of Ga. In the classical case, a = t, thisexactly corresponds to the usual definition of the semantics of the logicalconnectives.The main usefulness of the proposed approach lies in the fact that it allows

one to perform some 'partial analyses' of a formula involving connectives. Itis interesting to examine the question of verifying whether different possibleanalyses produce the same result. For example, it seems to be desirable thatthe expression '(Mary and Lucy) sing' always receives the same semantic valueas the expression '(Mary sings) and (Lucy sings)'. Similarly, it seems to bedesirable that 'Mary (sings and dances)' is semantically equivalent to '(Mary

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sings) and (Mary dances)'. As explained in Sec. 4 below, the equivalences inquestion are actually satisfied. An important difference between these twoexamples should, however, be emphasized. In the first case, the equivalenceis preserved if the proper names 'Mary' and 'Lucy' are replaced by arbitrarynoun phrases. In the second case, if the proper name 'Mary' is replaced by anoun phrase such as 'a woman', then the equivalence ceases to be satisfied.These informal comments aim at pointing out that the results obtained

through the proposed approach often agree with the intuitive requirements.In fact, from a semantic point of view, our approach is analogous with theKeenan and Faltz approach. However, there are some noticeable differences.Consider, for example, the two expressions 'loves (John and Paul)' and '(lovesJohn) and (loves Paul)', analysed as intransitive verb phrases (whose subjectcan be a proper name, e.g. 'Mary'). Keenan and Faltz's approach allows oneto consider these expressions as semantically equivalent, while the approachintroduced here does not allow it. (This is due to the specific structure of thetranslation of transitive verbs, such as 'love', in Montague's theory.)In what follows, we will closely examine the above-mentioned questions of

semantic equivalence (in the framework of our extended intensional logic).Let IX, 1X1 and 1X2 be expressions of type (b, a), and let {J be an expression oftype b. Assuming that a is Boolean, we can prove (in full generality), thefollowing semantic equivalences:

P. Delsarte and A. Thayse .

(-, IX)({J) ~ -,(IX({J)),

(lXI 1\ 1(2)({J) ~ 1X1({J) 1\ 1X2({J).

(There are similar results for the other connectives, and they require noadditional proof.) The equivalence relation ~ used here is defined as follows.For two logical expressions 1'1 and 1'2 of the same type, writing 1'1 ~ 1'2means that the semantic values {1'IIJ and {1'2IJ are equal, for any model, anypossible world, and any assignment function.The semantic equivalences written above can be deduced immediately from

the definitions, via the Boolean identities given at the end of Sec. 3.1. Moreprecisely, the first identity, applied to the function F = {IXIJ and to the pointr = {{JIJ, yields the equivalence involving the logical negation; the secondidentity, applied to the functions F,= {IX;J/ for i = 1, 2, and to the point r = {{JIJ,yields the equivalence involving the logical conjunction.A very similar argument allows us to obtain analogous results about the

extension operator (instead of the composition operator). If IX, 1X1 and 1X2 areexpressions of type (s, a), where a is Boolean, then we have the semantic

38 Philips Journul of Research Vol.46 No. 1 1991


equivalences

-(-,(X) ~-[(X),

((XI-/\ (X2)~ (etIl/\ C (X2)·

It is also useful to see how the lambda operator (the 'converse' of thecomposition operator) interacts with the extended connectives. Let usmention the following semantic equivalences:

AX[ -,(X] ~ -.(h[(X]),

AX[(XI /\ (X2] ~ AX[etl] /\ ..1.x[(X2]'

where et, (Xl and (X2 are expressions of type a (Boolean) and x is a variable oftype b (arbitrary). The proof is obtained, from the definition of the semanticsof the lambda operator and of the extended logical connectives, by asimple application of the Boolean identities. Let us give analogous resultsabout the intension operator. If (X,(Xl and et2 are expressions of type a (assumedto be Boolean), then we have the semantic equivalences

~(-,(X) ~-.( (X),

~ ((Xl /\ (X2)~ (etIl/\ ((X2)·

Let us now go back to the question of the 'distributivity of compositionwith respect to conjunction (or disjunction)'. In analyses like 'Mary (singsand dances)', for which the connectives occur on the right, the semanticequivalences given above cannot be applied as such; the desired results areonly obtainable through the commutation effect of the lambda operator. Thisquestion deserves to be examined in detail. In Montague's translation system,a proper name, e.g. 'Mary', is represented by a logical expression [J of thefollowing form:

[J =he x(m)],

where m is a constant of type (s, e), denoting an individual concept, andwhere x is a variable of type b = (s, «s, e), t», denoting a property ofindividual concepts. By definition, this expression [J, of type (b, t), denotes aset of properties of individual concepts.If etl and (X2 are two constants, of type «s, e), t), which denote sets of

individual concepts, and which are the formal translations ofintransitive verbs

Philips Journalof Research Vol. 46 No. I 1991 39

p((CXI 1\ CX2)) ~ CXI (m) 1\ cx2(m)

~ p(CXI) 1\ p(cx2).


such as 'sing' and 'dance', then the expression p((cxI 1\ cx2)), of type t, can beconsidered as a translation of the sentence 'Mary (sings and dances)' in theextended intensionallanguage. By successively applying the lambda conversionrule, the rule of cancellation of intension by extension, and the distributivityof composition with respect to conjunction, we obtain the semantic equivalence

As a result, the analysis 'Mary (sings and dances)' is equivalent to the analysis'(Mary sings) and (Mary dances)'.

We can also obtain this conclusion, in a more direct manner, by examiningthe semantics of Montague's formal representation of the proper names. Forthe expression p defined from the constant m as above, the semantic valueffpr can be identified with the principal filter generated by the atom {(ffmp, w)},in the lattice Db = 2Gb

, with Gb = UW X W. In other words, we have the equality

ffpr= {re: UW x W:(ffmp, W)EI'}.

The equivalence p((cxI 1\ cx2)) ~ P(cxd 1\ p(cx2) immediately follows from thisproperty. Note that the filter structure for the formal semantics of the propernames also occurs in the Keenan-Faltz approach 9).

Remark: The semantic equivalence p((r:x.l 1\ cx2)) ~ p(cxl) 1\ p(cx2) is valid forany Boolean type a, under the following general assumption. The expressionp is a lambda expression of the form ÀXC x(y)], where x is a variable of typeb = (s, <c, a», and y is an expression, of type c (arbitrary), containing nofree occurrence of x; and CXiis an expression of type (c, a), for i = 1,2.

3.2.2. Modaloperators

By a similar method, we can extend Montague's intensionallogic for whatregards the modalities. Let us first recall some classical notions belonging tothis area. Consider a pair [M], <M) consisting of a universal modal operator[M] and of the corresponding ('dual') existential modal operator <M),together with an associated accessibility relation R, defined over a set W ofpossible worlds. In the framework of natural language modelling, oneespecially needs the following three (universal) modal operators:

1. The necessity operator, D, characterized by wRw' for all wand w' in W;

Philip. Journni or Research Vol.46 No. I 1991



2. The operator 'always in the future', [F], characterized by (co, ,)R(co', r") ifand only if, < r';

3. The operator 'always in the past', CP], characterized by (co, ,)R(cv', r") ifand only if r < r.

In this context, the set W is assumed to have the structure of a Cartesianproduct, W = Q x T; the elements cvofQ are interpreted as the possible worlds(in a restricted sense), while the elements, of T are interpreted as the timeinstants. The symbol < represents a total order relation over the set T.

In the proposed extension, the modal operators [M] and (M) can act onthe logical expressions of any Boolean type. (The classical theory restrictsitself to the type t.) More precisely, if a is an expression of type a, assumedto be Boolean, then we consider [M]a and (M)a as well-formed expressionsof type a. For example, if a represents a common noun, such as 'president',then the expression (P)a could represent the phrase 'former president'. In thenext section, it will be seen how such an extension of the modal operatorscan actually be used in the formal translation of natural language intoMontague's intensionallanguage.Let us mention a particularly interesting application, concerning the tense

modifications of the verbs. If a is a constant, of type «s, e), t), representingan intransitive verb, like 'sing', then the modal expression (F)a can be usedto represent the future form 'will sing'. As explained at the end of this section,such a representation is satisfactory in that the analysis '(Mary) ((in the future)sings)' is equivalent to the analysis '(in the future) (Mary sings)'.For any expression a (of Boolean type), the semantic values of the modal

expressions [M]a and (M)a are defined as follows:

!f[M]ar·w= n t-r-:w'eR(w)

!f(M)ar·w= U t-r-:w'eR(w)

where R(w) is the subset of W containing the possible worlds that are accessiblefrom w, that is

R(w) = {w' E W :wRw'}.

These definitions are natural extensions of the classical concepts (concerningthe type t) to the case of an arbitrary Boolean type.


Let us now examine (in a similar manner as in the preceding subsection)the question of semantic equivalence for the compound expressions that involvemodalities. We will see that, under a 'suitable assumption, the followingequivalence is satisfied:

([M]er:)(P) ~ [M](er:(P)),

with er:EE(b,a> and PE Eb' for a given Boolean type a. (Here, [M] representsany universal modal operator, The case of the existential modal operator (M)is exactly the same, owing to the duality property (M)er: ~-,[M](-,er:),resultingfrom de Morgan's law.) For the reason explained below, the semanticequivalence written above is not satisfied in full generality. We will be led toassume that b is an intensional type, i.e. a type of the form b = (s, c) for acertain type c. Under this assumption (which is quite natural in this context),the equivalence we are interested in is always satisfied, for any modal operator,whatever the expressions er:and P,

By use of the second Boolean identity given in Sec. 3.1 we see that thedesired semantic equivalence amounts to the equality

IV'eR(IV) IV'eR(IV)

Assume that b is an intensional type, which means that the expression Pdenotes an intension. In this case, the semantic value of P does not dependon the choice of the possible world (see the first part of Sec. 3.1). As aconsequence, the desired equality is obviously satisfied.The equivalence that we have just established does not directly apply to

the above-mentioned question about tense modifications of the verbs. Let usnow examine how to treat this question. Consider an expression P such that

P ~ ÀXCx(y)],

where y is an expression of type c, assumed to be intensional, and where x isa variable of type b = <s, <c, a», for a certain Boolean type a. Assume thaty contains no free occurrence of x. Let er:be an expression of type <c, a). Then,for any modal universal operator [M], we have the semantic equivalences

p([M]a)) ~ ([M]a)(y)

~ [M](a(y))

~ [M](p(er:)).

42 Philip. Journalor Research Vol.46 No. 1 1991


Let us finally mention the 'commutativity' between the lambda operatorand any modal operator. If x is a variable (of an arbitrary type), and if a isan expression of Boolean type, then we have the semantic equivalence

Äx[[M]a] ~ [M](Äx[a]).

3.2.3. Quantifiers

A similar approach can be used for the quantifiers vv and 3y (whatever thetype of the variable y). Let a be any Boolean type. If a is an expression oftype a, then we consider Vya and 3ya as two well-formed expressions of typea (in the extended intensional logic that we are introducing). This allows usto quantify some expressions that are not formulae. Before giving the precisesemantics of quantifiers in the general case, let us examine an example statedin a 'natural language' style. (It should be pointed out, however, that theextension introduced in this third paragraph does not seem to have directapplications in the framework of the translation of natural language intointensional logic language, as this translation is conceived by Montague andhis continuators.)Consider a two-place predicate, A(tl, t2), where the arguments ti and t2

are of type Cl and C2, respectively. The symbol A can be viewed as a constantof type (C2, (Cl' t», via the equivalence

Assume that the constant A represents a transitive verb, e.g. 'admire', andthat the predicate A(tl, t2) represents a sentence such as 'tl admires t2'. If theconstant i, of type Cl' represents the proper name 'John', then the formulaVxA(j, x) can be paraphrased by 'John admires everybody'. The extensionsuggested here gives a meaning to the quantified expression 'IY2A(Yz), of type(Cl' t), and ensures the semantic equivalence

In our example, we thus allow the intransitive verb phrase 'admire everybody'to combine directly with the subject 'John' so as to produce the consideredsentence.

In a similar manner, we can analyse the converse sentence 'everybodyadmires John' as a combination of the expression 'everybody admires' (of asomewhat unusual appearance) with the proper name 'John'. To that end we



introduce a two-place predicate constant .4, of type (~l' (C2' t», with thefollowing equivalence:

By use of the quantified expression Vy1À(Yd, of type (C2' t), we obtain thesemantic equivalence

The left-hand side can be paraphrased by the following natural languagereading: '(everybody admires) (John)'.Let us go back to the general theory. For a variable Y of type c (arbitrary)

and an expression a of type a (Boolean), the semantic values of the expressionsVya and 3ya are defined in the following manner:

{VyajO = n t-r.«»;

{3yajO = U {aIr,reDe

where g is the considered assignment function (which assigns values to thevariables), and gr represents the y-variant of g specified by g(y) = 1'. This is anatural extension of the classical definition (which corresponds to the casea = t).We are mainly interested in the commutativity between the quantifiers and

the composition operator. More precisely, we wish to have the semanticequivalence

(Vya)(f3) ~ Vy(a(f3)),

for a E E(b.a) and f3EEb' where a is a Boolean type and b is an arbitrary type.(The existential quantifier can be treated in the same way, in view of theduality property 3ya~-,vy(-,a).)The commutativity in question is easily seen to be satisfied under the

following restrictive assumption: the expression f3contains no free occurrenceof the quantified variable y. The argument is basically the same as in the twopreceding paragraphs (about connectives and modalities); it also uses thesecond Boolean identity given in Sec. 3.1. The equivalence we are interested

44 Philip, Journal of-Research Vol.46 No. I 1991


in can be written as follows:

reDe reDe

Under our assumption concerning {3, this equality is obviously satisfied, sincewe have ff {3JO' = ff {3JO for every value r belonging to the set o;

Analogously, the extension operator can be shown to commute with thequantifiers. If IX is an expression of type (s, a), then we have the semanticequivalence

- (\fylX) ~ 'v'y( IX).

Let us finally mention the commutativity properties concerning the lambdaoperator and the intension operator. For any expression IX of Boolean type,we have the following equivalences:

h['v'ylX] ~ 'v'y(..1.x[IX]),

- (\fylX) ~ 'v'Y(IX).

4. Application to natural language representation

In the preceding section we introduced three ideas of enrichment ofMontague's intensional logic; they are concerned with the way of using thelogical connectives, the modal operators, and the quantifiers. This sectionaims at demonstrating the effect of the first two ideas upon the formalizationof natural language. In particular, we will compare the representation of afragment of naturallanguage by means of the extended intensionallogic withthe representation based on the classical intensional logic. To make thiscomparison we will follow the same pattern as in the preceding section; wewill successively examine the use of the logical connectives and of the modaloperators in the formalization of natural language.

4.1. Logical connectives

In natural language, the connectives 'not', 'and', 'or', 'if· .. then', 'if andonly if' are informal translations of the logical connectives=-, /\, v, ::::>, ==.Now, in classical intensionallogic, these connectives can only affect expressionsof type t, i.e. formulae. Remembering that the sentences of naturallanguage


46 Philip. Journal of Research Vol.46 No. I 1991


are precisely those expressions which are translated into logical formulae, wecome to the conclusion that using the logical connectives as the translationsof the natural connectives allows us only to translate logical combinations ofsentences (and of no other kind of natural expressions) in the logic language.In more precise terms, if we use the symbol => to represent the translationoperation (from natural language to formal language), we can write thefollowing schema:

(sentence)! => (formula)!

(sentence), => (formula),

(sentence)! and (sentence), => (formula)! 1\ (formula),

(sentence)! or (sentence), => (formula)! v (formulal,

Natural language constructions not only use logical combinations ofsentences, but also logical combinations of expressions belonging to variousother syntactic categories. Let us for example mention the combinations 'Johnand Paul', 'work and talk', 'rich and happy'. It would be interesting to beable to translate all these logical combinations of natural expressions by meansof a translation schema that would be identical (or nearly identical) with thesimple schema indicated above for the translation of sentences. Now, for thereason that we mentioned, we cannot use this schema by merely replacing'sentence' by 'natural expression' and 'formula' by 'logical expression'.However, Montague's intensional logic allows one to represent logicalcombinations of certain natural expressions, but this requires a different wayof translating the natural connectives 'and', 'or' into logic. According to theMontague formulation, the logic translation ofthe natural connectives dependson the syntactic category of the expressions involved in the combinations. Inthe initial form of Montague's grammar, such a translation is defined for thesyntactic categories of terms (T) and of intransitive verbs (J V). The definitionwas recently extended by Hendriks, first to the syntactic category of transitiveverbs (T V) and then to any syntactic category 4). The principle of this methodis given hereunder.

As a translation of a natural conjunction or disjunction, Hendriks proposesa certain lambda expression, the form of which is determined by the type ofthe logical expression it acts on. Assume that the connectives are used tocombine natural expressions whose logical translations are expressions of type<a!, <a2, ... , <am t) ... » (which we simply write as <ä, t»).Let xai be a variable of type ai> for i = 1, ... , n. The translation of the


conjunction and of the disjunction, involving natural expressions A and Bwhose logical translations A' and B' are of type (d, t), is defined as follows:

(A and B) => Axa.[· .. [ÄxaJA'(xa,) (xaJ A B'(xa,) (XaJJJ J(A or B) => Äxa,[' .. [ÄxaJA'(xa,) (xaJ v B'(xa,) (Xan)]J J

In this kind of translation, the expressions A' and B', of type (d, t), aretemporarily transformed into expressions A'(xa,) ... (xaJ and B'(xa,) ... (xa.),of type t; since these expressions are formulae, it is possible to apply the logicalconnectives A and v to them so as to produce some new formulae. Then,the formulae thus obtained are transformed, by means of lambda operators,into expressions of type (d, t) which are the intensionallogic translations ofthe phrases 'A and B' and 'A or B'. In summary, the Hendriks method amountsto first transforming arbitrary logical expressions (of Boolean type) intoformulae, which can be combined by means of the logical connectives, andthen transforming the resulting formula into a logical expression of the initialtype. This is seen to require introducing a rather complex mechanism for thetranslation ofthe naturallanguage connectives into formallogic. Furthermore,the form of the translation depends on the syntactic category of the expressionsaffected by the connectives.

The language of the extended intensional 'logic, introduced in Sec. 3.2,often allows us to preserve the simplest translation for the natural languageconnectives: 'and' and 'or' are directly translated into the logical connectives A

and v, whatever the syntactic category of the expressions involved in thecombination. This way of solving the problem is near to the method devisedby Keenan and Faltz in the framework of a formal language which is strictlypropositional (and, therefore, significantly less expressive than Montague'sintensionallanguage).

Let us illustrate and substantiate the statements above by way of threeexamples. The first of them aims at demonstrating the translation of acombination of terms (proper names or noun phrases) by means of naturalconnectives. The formalization of this example within the intensionallanguageis based on the semantic equivalence

established in Sec. 3.2. When applied to our example, this equivalence allowsus to combine logically two proper names, and then to combine functionallythe result with an intransitive verb so as to produce a sentence. We will analyse

Philip. Journalor Research Vol.46 No. I 1991 ,47

• Example 1: John and Mary talk

Translation of the vocabulary words:

John =>À.pCP(j)]Mary => ÀQC Q(m)]talk => talk'


this example (and the other ones) by means of the Montague-Hendrikstranslation schema, and by means of the translation schema associated withthe extended intensionallogic.We will use the symbol => to represent the meta-expression 'is translated

into'. When it is necessary to distinguish between the rules of the strictintensional logic and those of the extended intensional logic, we use thetranslation symbol :b. in the first case (translation based on Montague's rules)and the translation symbol J. in the second case (translation based on therules proper to the extended intensionallanguage).

Schema of Montague's intensionallogic:

(A and B); A, BET :b.Àx[A'(x) 1\ B'(x)](John and Mary) :b.ÀX[ÀpC P(j)J(x) 1\ ÀQC Q (m)] (x)](John and Mary talk):b. (Àx[À.pC P(j)J(x) 1\ ÀQC Q(m)] (x)]) (talk')

~ ÀpC P(j)] (talk') 1\ À.QC Q(m)](talk')~ talk'(j) 1\ talk'(m)

Schema of the extended intensionallogic:2

(A and B) => A' 1\ B'2 ~ ~(John and Mary) => ÀP[ P(j)] 1\ ÀQ[ Q(m)J)

2 ~ ~-(John and Mary talk) => (ÀP[ P(j)] 1\ ).Q[ Q(m)J)( talk')~ ÀpCP(j)] (talk') 1\ ÀQC Q(m)](talk')~ talk'(j) 1\ talk'(m)

The second and third examples are concerned with the formalization oflogical combinations of verb phrases. In the framework of the extendedintensionallogic, we can make use, in this case, of the semantic equivalence

also established in Sec. 3.2. As in the first example, we give first theMontague-Hendriks version and then the extended intensionallogic version.

48 Philip. Journalof Research Vol.46 No. I 1991


• Example 2: John talks and works

Translation of the vocabulary words:

work => work'

Schema of Montague's intensional logic:

(A and B); A,BEIV J.Àx[A'(x) 1\ B'(x)]1(talk and work) => Àx[talk'(x) 1\ work'(x)]1 - -(John talks and works) => ÀP[ P(j)]( Àx[talk'(x) 1\ work'(x)])~ Àx[talk'(x) 1\ work'(x)](j)~ talk'(j) 1\ work'(j)

Schema of the extended intensionallogic:2(A and B) => A' 1\ B'2(talk and work) => talk' 1\ work'2 - -(John talks and works) => ÀP[ P(j)]( (talk' 1\ work'))

~ (talk' 1\ work')(j)~ talk'(j) 1\ work'(j)

• Example 3: John examines and solves a problem

Translation of the vocabulary words and of the compound expressions:

examine => exm'solve => solv'a =>ÀZ[ÀY[3x(Z(x)l\- Y(x))]](prob')problem => prob'(a problem) => ÀZ[ÀY[3x(Z(x) 1\ - Y(x))]](prob')

~ ÀY[3x(prob'(x) 1\ - Y(x))]

Schema of Montague's intensionallogic:1(A and B); A, BE TV =>Àv[Àw[A'(v)(w) 1\ B'(v)(w)]]1=> ÀV[Àw[exm'(v)(w) 1\ solv'(v)(w)]](examine and solve)

(examine and solvea problem) 1

=> ÀV[Àw[exm'(v)(w) 1\ solv'(v)(w)]](ÀY[3x(prob'(x) 1\ - Y(x))])~ Àw[exm'(ÀY[3x(prob'(x) 1\ - Y(x))])(w)1\ solv'(ÀY[3x(prob'(x) 1\ - Y(x))])(w)]


(John examines andsolves a problem)

1 ~ A

= ÀP[ P(j)]( ÀW[exm'(ÀY[3x(prob'(x)" ~ Y(X))])(W)"solv'(ÀY[3x(prob'(x)" ~ Y(X))])(W)])~ Àw[exm'(ÀY[3x(prob'(x)" ~ Y(X))(W)"solv'(ÀY[3x(prob'(x)" ~ Y(X))])(W)])~ exm'(ÀY[3x(prob'(x) ,,~ Y(X))])(j)"solv'(ÀY[3x(prob'(x)" ~ Y(X))])(j)


Schema of the extended intensionallogic:2(A and B) =A' "B'

(examine and solve) J. exm' " solv'(examine and solvea problem) J. (exm' " solv')(ÀY[3x(prob'(x),,"- Y(x))])(John examines and 2 ~ A

solves a problem) = ÀP[ P(j)] ( (exm' " solv')(ÀY[3x(prob'(x)" ~ Y(x))]))~ (exm' " solv')(ÀY[3x(prob'(x)" ~ Y(x))])(j)~ exm'(ÀY[3x(solv'(x) ,,~ Y(x))])(j)"solv'(ÀY[3x(prob'(x)" ~ Y(x))])(j)

The case of negation in naturallanguage is the subject of another interestingapplication. In the framework of the extended intensional language, thenegation connective (-,) can act on any logical expression of Boolean type(not only on a formula as in the classical Montague language). Consider, forexample, the negated forms of the verbs. The sentence 'John does not talk'can be analysed syntactically as '(John) ((does not) (talk))', which promptsus to translate the phrase 'does not' into the connective -, and to apply thisdirectly to the translation of the verb 'talk'. The detailed formalization canbe presented as follows.

• Example 4: John does not talk

Johndoes nottalk

=ÀpCP(j)]=-,= talk'

(does not talk) =-, talk'(John does not talk) = ÀpC P(j)]((-, talk'))

~ (---; talk')(j)~ ---; talk' (j)




4.2. Modaloperators

In usual speech, we often talk about possibilities, hypothetical events, goalsto be achieved, forecasts regarding the future, and so on. In naturallanguage,the modes 'necessary', 'possible' and 'permitted' are expressed throughauxiliary verbs such as 'must' and 'may'. The temporal modalities are expressedby means of the conjugation tenses, such as future, perfect, imperfect andfuture perfect. The possibility, the necessity, the future, the past, the belief,the knowledge are modalities that can affect the sentences of naturallanguage(and, more generally, the expressions of natural language) so as to modifytheir meaning. The sentences 'John works', 'John must work', 'John will work','John will have to work' have different meanings, since their verb 'work' isaffected, in four different ways, by the modalities of the necessity ('must', 'haveto') and of the future ('will').

The researchers working in the field of classical logic (propositional logicand first-order predicate logic) observed the inability of this formalism torepresent the sentences and the expressions ofnaturallanguage that are affectedby modalities. The study of this limitation of the classicallogic formalism ledto the development of some other logic systems, which are generally referredto as non-classical logics. Modal logics belong to this family; their nameoriginates from the fact that these logic systems comprise modal operatorsthat can affect the formulae so as to modify their interpretation. For example,in statements such as 'It is possible that F', 'It will probably be true in thefuture that F', 'John thinks that F', 'It is often true that F', the expressionswritten in front of the logical formula F are formalized by means of modaloperators. The truth value of such statements depends not only on the truthvalue of the given formula F; it depends also on the time instant when thisformula is stated (temporallogics), on the person who thinks or believes thatF (logics of belief), or on the necessity, possibility or randomness of an event(logics of possibility). In the framework of the extended intensional logicintroduced in Sec. 3, the modalities can affect not only a formula F, but alsoany well-formed expression Cl. (of Boolean type).

Let us now examine the question ofthe translation ofnatural modalities intothe extended intensional logic. A modality of natural language is treated asan element of the syntactic category (&r& if it combines with an element ofthe syntactic category (& to produce a new element of the same syntacticcategory rJ.

To satisfy the compositionality principle in the interpretation of the phrase'Mod A' (where A represents a natural expression and Mod represents a naturalmodality) we have to apply the translation of Mod to a logical expressionwhich denotes the intension of the translation A' of A (we cannot apply it to


A' itself). The translation of Mod is expressed in terms of the modal operator[M] ('universal' case) or of its dual <M) ('existential' case). In particular, thenotation D and o is used for the necessity and possibility operators.

Let p be a variable of intensional type, i.e. of type <s, a) for a suitable typea. The translation rule alluded to above is the following:

Natural expression Formal expressionuniversal modality =Àp[[M]-p]existential modality =Àp[ <M) - p]

Associated type<<s, a), a)<<s, a), a)

Note that these translations generalize the translation given by Montague inthe special case where the type a is t and where the modality [M] is thenecessity D:

Necessarily =Àp[D-p] <(s, t), t)

It is then possible to combine functionally a modality with an expressionoe,of type a, prefixed by the intension operator; this functional combinationis easily seen to satisfy the following semantic equivalences:

}.p[[M] - p]( IX) ~ [M]o:,

}.p[<M) - p](oe) ~ <M)o:.

As a first example, we examine the sentence 'John will talk'. Here, we havea temporal modality, represented by <F), the meaning of which is 'at leastonce in the future'. Since the modality is applied to the intransitive verb 'talk'(syntactic category IV), the type of the corresponding variable p is(s, <<s, e), t» .• Example 5: John will talk

John =).QCQU)]= ).p[<F) - p]= talk'= ).p[<F) - p](talk')

~ <F)talk'(John will talk) = ).QCQ(j)]n <F)talk'))

~ «F)(talk'))(j)~ <F)(talk'(j))

will. talk(will talk)

52 Philips Journalof Research Vol.46 No. I 1991


This example demonstrates the usefulness of one of the generalizationsproposed in the framework of the extended intensional logic. Roughlyspeaking, the fact that a modal operator can affect any logical expression(here, the translation of an intransitive verb) allows us to 'conjugate' the logictranslation of a verb. According to Montague's classical analysis, prior to anytranslation, a sentence in the future tense such as 'John will talk' first has tobe transformed into the statement 'In the future John talks'. In the translationof this version, the modal operator (F) acts on the logical formula thatrepresents the sentence 'John talks'. If we use the extended intensionallogic,then we can make the formal representation of the future modality act directlyon the translation of the verb. Note that the last expression occurring in theexample above, i.e. the expression (F)(talk'(j)), is obtained from the precedingexpression via the semantic equivalence

(M)et.)({3) ~ (M)(et.({3))

established in Sec. 3.2. This property ensures the equivalence between theformulae (F)(talk'))(j) and (F)(talk'(j)), which are the respectivetranslations of the two formulations '(In the future) (John talks)' and 'John((in the future) (talks))' of our sentence 'John will talk'. For the result to becorrect, it is necessary that the constant j be of the type (s, e) (intensional)and not of the type e; thus, we have to use the Montague version of thetranslation function and not the Bennett version.The schema that we have used to analyse the preceding example can also

be applied to any other modality acting on a verb. For instance, the sentence'John must talk' can be translated as follows:

• Example 6: John must talk

must => À.p[O-p](John must talk) => (O(talk'))(j)

~ O(talk'(j))

In the last two examples above, the auxiliary verbs 'will' and 'must' modifythe meaning of the verb ('talk') that follows them in the two sentences; theseauxiliary verbs act as modifiers of the meaning. In fact, the modifiers can actnot only on verbs, but also on expressions belonging to various other syntacticcategories. The adjectives, acting on common nouns, are typical examples ofmodifiers. It is worth emphasizing a distinction between adjectives such as'former' or 'alleged' (in the noun phrases 'the former president' or 'an allegedproof', for instance), which deeply alter the meaning of the noun they refer

Philips Journal of Research Vol.46 No. 1 1991 53


to, and adjectives such as 'big' or 'nice' (in the noun phrases 'a big house' or'a nice picture', for instance), which express a mere quality of the noun. Theformalism ofthe extended intensionallanguage, which admits modalities actingon logical expressions of any type, enables us to translate the modifiers in asyncategorematic manner. The example below is concerned with the analysisof a sentence in which the adjective 'difficult' modifies the common noun'problem' that follows it.

• Example 7: John will examine a difficult problem

difficult => ..1.q[<D) ~q](difficult problem) => ..1.q[<D) ~q](prob')

~ <D)prob'(a difficult problem) => ..1.Z[..1.Y[3x(Z(x) 1\~ Y(x))]]((<D)prob'))

~ ..1.Y[3x«D)prob'(x) 1\~ Y(x»]~ ..1.Y[a] (abbreviation)

(examinea difficult problem) => exm'(..1.Y[a])(will examinea difficult problem) =>..1.p[(F)~p](exm'(..1.Y[a]»

~ <F)exm'(..1.Y[a])(John will examinea difficult problem) => ..1.QCQ(j)]( (F)exm'(..1.Y[aJ»)

~ «F)exm'(..1.Y[a]))(j)~ (F)(exm'(..1.Y[a])(j»

Finally, let us give the formal analysis of a sentence that contains both anegation connective and a modality .

• Example 8: John will not talk

willnottalk(not talk)(will not talk)

=>..1.QCQ(j)]=> ..1.p[<F)~p]

John

=>-.

=> talk'-. talk'=> ..1.p[(F) ~p]((-. talk'»

~ (F)(-. talk')(John will not talk) => ..1.QCQ(j)]( (F)(--, talk'))

~ «F)(-. talk'))(j)~ <F)(-. talk'(j))



REFERENCES

I) R. Montague, Formal Philosophy, Yale University Press, New Haven, CT, 1974.2) D. Dowty, R. Wall and S. Peters, Introduetion to Montague Semantics, Reidel,

Dordrecht, 1989.3) A. Thayse (Ed.), From Modal Logic to Deductive Databases, Wiley, New York, 1989.4) H. Hendriks, Flexible Montague grammar, First European Summer School on Natural

Language Processing, Logic and Knowledge Representation, Groningen, 1989.S) 1. Groenendijk and M. Stokhof, Dynamic Montague grammar, Proc. 2nd Symp. on

Logic and Language, Hajduszboszlo, Hungary, 1990.b) E. Keenan and L. Faltz, Boolean Semantics for Natural Language, Reidel, Dordrecht,

1985.7) J. K amp, A theory of truth and semantic representation, in Groenendijk et al. (Eds), Truth,

Interpretation and Information, Foris Publications, Dordrecht, 1984.8) J. Barwise and J. Perry, Situations and Attitudes, MIT Press, Cambridge, MA, 1983.9) A. Thayse (Ed.), From Natural Language Processing to Logic for Expert Systems, Wiley,

New York, 1991.10) P. Dupont and A. Thayse, Montague's semantics and Boolean semantics for natural

language representation, Philips J. Res., 45, 41 (1990).11) M. Bennett, A variation and extension of a Montague fragment of English, in B. Partee

(Ed.), Montague grammar, pp. 119-163, Academic Press, New York, 1976.

Authors

P. Delsarte: Ir. degree (electrical engineering and applied mathematics), University of Louvain, 1965 and1966; Dr. degree (applied sciences), University ofLouvain, 1973;Philips Research Laboratory Brussels, 1966- .His research interests include algebraic coding theory, combinatorial mathematics, the theory and applicationsof orthogonal polynomials and Toeplitz matrices, and formal semantics for natural language.

A. Thayse: Ir. degree (electrical engineering and applied mathematics), University of Louvain, 1964 and 1965;Dr. degree (applied sciences), Ecole Polytechnique Fédérale de Lausanne, Switzerland, 1980; Philips ResearchLaboratory Brussels, 1967- ; Visiting Professor at the Ecole Polytechnique Fédérale de Lausanne, 1983;Visiting Professor at the University ofLouvain, 1986-1990. His research interests include logicand its applicationto natural language representation, speech recognition and artiûcial intelligence.

Philip. Journalof Research Vol.46 No. I 1991 55

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