variable binders as functors

8/9/2019 Variable Binders as Functors

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VARIABLE-BINDERS AS FUNCTORS

Achille C. VarziIstituto per la Ricerca Scientifica e Tecnologica (IRST)

I-38050 Povo (Trento), Italy

(Published inPoznan Studies in the Philosophy of the Sciences and the Humanities 40 (1995), 303-19)

INTRODUCTION

The second part of Ajdukiewiczs classic paper onSyntactic Connexion is devoted to adiscussion of languages involving variable-binding operators such as quantifiers, inte-grals, and the like. Ajdukiewicz argues that such operators are not genuine functors,and consequently that those languages run afoul of the basic functor/argument schema.At most he suggests that all variable-binders might be restricted to one kind only, or tothe combination of ordinary functors with a circumflex operator (the terminology isfrom Whitehead and Russells Principia Mathematica). And that suggestion has eversince become quite popular, particularly under the impact of ChurchsLambda Cal-culus.

This fact is of no little historical interest, for there is a misleading tendency topresent lambda-equipped categorial languages as an extensionrather than an imple-mentationof Ajdukiewiczs original insights. More important, however, is thephilosophical import of this account. For apart from any considerations of esthetic ele-gance and conceptual parsimony, the claim that functional application alone is not asufficient paradigm for linguistic analysis ties in directly with a number of fundamentalissues in semantics.

In this paper I argue that this is by no means a necessary course. I argue that afairly general semantic framework can be developed where the only relevant distinc-tion is indeed the functor/argument distinction, and where the only structural operationfor generating expressions is functional application, with no need to resort to func-tional abstraction as well. I shall not prove any general results to the effect that such aframework is universally applicable. However, the overall apparatus is illustrated inconnection with some concrete examplesnotably quantificational and full categoriallanguageswhich should suffice to support my point.

In the concluding section, I then offer some remarks to illustrate the philosophicalimport of this approach. Particularly, I briefly discuss a certain conception of logic thatemerges from the proposed account: the view that logic is essentially a theory in themodel-theoretic sense, i.e. the result of selecting a certain class of models as the onlyadmissible interpretation structures (for a given language).


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LANGUAGES

As I mentioned, my starting point is essentially Ajdukiewiczs notion of a pure cat-egorial language. It is a very simple, rather abstract notion, and it is based primarily onthe idea that the expressions of any language may be divided into two kinds of syntac-tic categories1, or types: individual(orprimitive) andfunctional(orderived).

Intuitively, the former correspond to those categories of expressions whose syn-tactic status needor cannot be analyzed in terms of other categories. One of thesecould be, for instance, the category (Declarative) Sentence; another could be the cate-gory (Proper) Noun 2. We need make little effort here to select such primitives withcare. And to allow for the greatest generality, we may as well start off with an infinitenumber, bypassing the problem of specifying the intuitive status of each of them. Fordefiniteness, I shall just identify the primitive types with the natural numbers, taking0and 1 to represent the traditionally fundamental categories of sentences and nouns re-spectively.

On this basis, an infinite set of derived types is obtained by introducing some op-eration on the set of primitive types, say the operation of pair formation 3. Moregenerally, we may assume that whenever tand t'are types, primitive or derived, anew derived type may be formed, which we may identify with the ordered pairt,t'.The intuitive idea is to think of such a type as corresponding to those expressions(functors, in the traditional terminology) which produce expressions of type t' when

combined with expressions of type t. For instance, if0 and 1 are interpreted as above,then 0,0, 0,1, 1,0, and 1,1 will be the types of such functors traditionally re-ferred to as (monadic) Connectors, Subnectors, Predicators and Operators, respec-tively; 0,0,0,0, 0,1,0,1, etc. will be the types of the corresponding (mon-adic) Adverbial Modifiers; and so on4.

In general, our set of types T can then be defined as the closure of under .And the idea I wish to consider is that the expressions ofany declarative languageL can be recursively specified on the basis of some type assignment to its symbols:for each tT, the L -expressions of type twill comprise all the symbols initially as-signed to t plus all those expressions that result from applying a given structural opera-tion (usually some form of juxtaposition, but I shall leave that open) to pairs of ex-

pressions of type t',t and t'(for some t'T).

DEFINITION1. A language is an ordered triple of the form (s,g,E) satisfying thefollowing conditions:

(a) s is a one-one function with Ds :T (for some ordinal )(b) g is a one-one function with RgRs =0 (and Dg as below)(c) Eis the smallest system of sets closed under the clauses (i) for each pair

,tDs : tDEand s (,t)Et; and (ii) if t',t, t'DE, xEt',t and yEt', thentDEandg(x,y)Et.


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[Intuitively, whereL=(s,g,E) is a language, the elements ofR s are the symbolsofL, and UR E is the class of all expressions generated from those symbols by re-peated application of the structural operation g (the requirement that s and g be one-one functions with disjoint ranges is to secure that each expression ofL can beuniquely represented as either a symbol or a compound of the form g(x,y) ). In par-ticular, an L -symbol of type tis any xR s such that x=s (,t) for some DDs(such a may be called the ordinal index ofx), while an L -expression of type tis anyelement ofEt

5 ].There is no doubt that this definition allows for some peculiarities which make

little intuitive sense. For example, clause (a) allows for the possibility that a languageL involve an arbitrary number of symbols for any type tT, but it is clear that unlesssome L -symbols are assigned derived categories, the structural operation ofL wouldbe empty, and no compound expression could be generated. Also, there is nothing inthe above definition to secure that all symbols of a given language can actually be usedto produce expressions of type 0, i.e. sentences. Such oddities, however, are irrele-vant in principle, and I shall refrain from introducing unnecessary complexities intothe account.

Indeed, the generality of Definition 1 lies precisely in the fact that it allows one tosingle out a desired language or class of languages simply by implementing clauses(a)-(c) with the appropriate additional conditions. Thus, for instance, one may want torequire that the functional expressions of a language L=(s,g,E) always yield individ-ual expressions, in the sense that wheneverEt', t is defined (for t, t'T), Et' is alsodefined, and hence so isEt. Further, among such languages one may want to pick upthose languages only whose categories of expressions always cancel to a given type t,i.e. can be used to generate expressions of type t. As I mentioned, presumably mostlanguages in use should be regarded as languages whose categories of expressionsalways cancel to 0. Overall, however, the fundamental semantic notions must be out-lined for the general case.

In this regard, it should also be noted that the above definition provides a purelycombinatorial characterization of the notion of a language. In particular, the differencebetween symbols and compound expressions is a relative one: the latter are generatedby application ofg, the former are not. This means that definition 1 does not imply

that a languages symbols must necessarily lack internal structure, but only that thisstructureif anyis irrelevant to the syntax of the language. Apart from the fact thatthis secures a certain neutrality with respect to the sort of entities languages are madeof (which was already an attractive feature of Ajdukiewiczs original account), a deri-vative advantage of this account is precisely the possibility of treating variable-bindingoperators as ordinary functors, viz. as structured symbols consisting of the operatortogether with a corresponding bound variable.

A few concrete examples will help illustrate all of this. To this end, letL=(s,g,E)


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be any language, and for all n and all t, t'T , let tn,t' be an abbreviationfor the type t0t1,,tn-1,t', where ti =tfor each i0.

To redeem some familiar notation, supposeL includes a binary connector (oftype 0,0,0). Then, for allA,BE0 , we may set:

(a) (AB) for g(g(,A),B)(b) (A) for (AA)(c) (AB) for ((AB))(d) (AB) for ((A)(B))M M

EXAMPLE 1.2. L is an elementary language iffRDs{tn,t': n0 & t,t'1}i.e. iffL comprises a set SENTL ofsentence symbols (of type 0) and a set NOUNLofnoun symbols (of type 1) along with a set CONL , n of n-ary connectors (of type0 n,0), a set SUB L , n ofn-ary subnectors (of type 0 n,1), a set PRED L , n of n-arypredicators (of type 1 n,0) and a set OPER L , n of n-ary operators (of type 1 n,1)for each n > 0. In particular, we may assume that ifL is an elementary language, thenthere exist a denumerable set VL NOUNL and various sequencesQL , n disjoint fromRs (forn >0) such thatQL , n VL= {x,y: x,yCONL , n}. In that case we may speakofVL as a set ofnoun variables, referring to a pair of the form QL , n , i , v (whereiDQL , n and vVL ) as an n-ary quantifier bindingthe variable v.That is, from apurely syntactic perspective, we may treat quantifiers as a special kind of structuredconnectors6.

Again, to introduce some familiar notation supposeL is an elementary languagewhose symbols comprise a binary quantifier ,v for each element of a set VL ofnoun variables. Then, for all vVL and allA,BE0 we may set:

(a) (AvB) for g(g(,v,A),B)

(b) (A) for (A

uA)

(c) (AB) for ((AwB))

(d) (AB) for ((A)(B))M M

(e) (vA) for ( (AvA))(f) (vA) for ((v(A)))

(where u is, as usual, the first variable foreign toA and w the first variable foreign toboth A andB).


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EXAMPLE 1.3. L is a full categorial language iffRDs =Ti.e. iffL com-prises a non-empty set S L , tof symbols for every type tT. In particular, we may as-sume that ifL is a full categorial language, then for each type tT there exist a se-quenceA L , tdisjoint from R s and a denumerable set VL , tS L , tso that, for everyt'T, {x,y,t': x,y,t'S L , t',t,t'}= A L , tVL , t{t'}. In that case we may speak ofeach VL , t(tT) as a set ofvariables of type t, referring to a triple of the form A L , t,i,v,t' (t,t'T, iDA L , t, vVL , t) as an abstractor of type t',t,t' binding the vari-able v.

To illustrate, suppose now L is a full categorial language with an abstractor

,v, t' for each variable v and each type t'T; then the usual lambda-combinatorialnotation can easily be introducedfor instance:

(a) (vA) for g(,v, t', A)(b) I t for (vv)(c) K t,t' for (v(vv))(d) S t,t',t" for (v(v (vg(g(v,v),g(v ,v)))))

(provided of course vU{VL , t: tT}, t, t', t"T, AEt', and v , v , v, v are fixedvariables of type t, t', t,t' and t,t', t' respectively ).

MODELS

We can now turn to the notion of a model (of a language L ). The general idea is sim-ply that a model must act as a sort of semantic lexicon: it must determine what sort ofthings may be assigned to the basic components of the language as their contensivecounterparts, and it must do so within the limits set by the relevant type distinctions.Thus, in general, a model for a given language L of the form (s,g,E) will be a struc-tureM of the form (d,h,I), with d, h andIa triple of functions satisfying essentiallythe same conditions ass , g andErespectively: to each category of expressionsEtREthere will correspond some non-empty domain of interpretationItRI; to each sym-bol s (,t)Et there will correspond a denotationd(,t)It ; and to the structural op-eration g(URE)3 there will correspond an operationh(UR I)3 subject to the sametype restrictions, i.e. such thatxIt',t andyIt'always imply h(x,y)It

7.

DEFINITION 2. A modelfor a language (s,g,E) is a structure (d,h,I) satisfyingthe following conditions:

(a) d is a function on the domain ofs(b) h is an operation distinct from d(c) I is a system of sets closed under the clauses (i) for each pair ,tDs :

tDI and d(,t)It ; and (ii) if t',t, t'DI, xIt',t and yIt', then tDI andh(x,y)It.


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I think this is more or less the notion of a model that Ajdukiewicz had in mind, atleast relative to the general notion of a language introduced above. Of course this is allquite abstract, and the intuitive appeal of the account depends on the actual make up ofthe functions d, h, I. For example, if the sentence type 0 were in DE, then I0 mightbe construed as a set of truth-values, or perhaps as a set of propositions. And if1or1,0 were inDE, one might think ofI1 as a set of individuals and I1,0 as a setof properties. However, in general I believe semantics ought to remain neutral withrespect to the philosophical question of what kind of entities should be taken as inter-pretations of what expressions, particularly where commitment is not necessary (for

the same reason, in the definition of a language the notions of sentence, noun,predicator, etc. are not defined independently, but only in relation to the auxiliarynotion of a type).

In fact, what I want to stress here is that the generality of this approach lies pre-cisely in the fact it allows one to single out a desired model or class of models simplyby implementing clauses (a)-(c) with the appropriate additional conditions, just as withthe notion of a language. For instance, the definition allows the domains of interpreta-tion associated with the basic categories of expressions of the given languageL to bearbitrary sets, and hence to vary from model to model. That in practice one is ofteninterested only in the study of models which agree in assigning the same, fixed do-main of interpretation to certain basic categories ofL (say, only models M such thatI0if definedis a fixed set of truth-values) is another matter, and may be accountedfor by singling out that class of models whose elements satisfy the desired conditions.And of course the same criterion may be used to account for the fact that some sym-bols are often assumed to have a fixed meaning (i.e. denotation in the case of indi-vidual symbols, and denotation plus relative conditions on the structural operation inthe case of functional symbols): in general, such assumptions can be regarded as de-termining a certain selection of models as the only admissible ones, and different as-sumptions will correspond to different selections. Thus, as I take it, to say e.g. that alanguage L has a classical conjunction connective is to say that the only models tobe considered are among those models where I0 is, say, the set 2={0,1}, and where denotes that function : 22 2 such that (x)(y)=xy for all x,y2; to say that Lhas a standard multiplication operator is to say that the only models to be con-

sidered are among those models where I1 is, say, the set , and where denotesthat function : such that (x)(y)=x.y for allx,y; and so on.

In this regard, there is yet an important feature of our definition that marks a de-parture from (or rather, a generalization of) the standard Ajdukiewiczian approach:there is in fact no requirement that every domain of the formIt',t be a Fregean set offunctions :It' It , hence no requirement that a models structural operation alwaysyield the value h(x,y)=x(y) forxIt',t and yIt'. Again, this is in the spirit of ab-stractness and philosophical neutrality. But we shall see in a moment that this gener-


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ality is particularly important for the purposes stated in the Introduction. For surely,we cannot treat quantifiers as connectors andinterpret all connectors as functions fromtruth-values to truth-values.

In fact, as I see it this general attitude with respect to the notion of a model has farreaching consequences, for it also seems to permit a uniform treatment of what aresometimes regarded as different types of semantical modeling. For example, the se-mantical analysis of so-called intensional languages is usually supposed to induce aconceptual departure from that of simpler case studies, for in that case the intendedmeanings of a languages symbols seem to depend on various factors which are dis-

regarded in the simpler accounts. Yet, within the present framework such cases can bedealt with like any other: to account for the relevant factors one only has to refer to theappropriate classes of models, requiring e.g. that the domains of interpretation be notjust sets of entities (truth-values fort=0, individuals fort=1, etc.), but sets of func-tions ranging over those entities and taking as arguments indices from an appropriateset of intensional features. More precisely, for any language we can single out theclass of its indexicalmodels as consisting of all those models M=(d,h,I) such that,for some setA of possible features and some system U=Ut: tDE of universesof entities,It =UtA for each tDE. And among such models, we can take the onesthat best fit our purposes. (The nature ofA is in fact liable to further qualifications de-pending on ones specific needs. For instance, to account for a certain standardmeaning of such modal connectors as necessarily, possibly, etc. one may construeA as a set of possible worlds; but if the language is also supposed to contain in-dexical locutions such as demonstratives, personal pronouns, etc., one might preferthinking ofA as a set of ordered pairs consisting of a possible world and a contextof use; and in connection with tensed languages one would perhaps go even further,requiringA to be a set of ordered triples consisting of a possible world, a contextof use, and a moment of time. All of this, as I see it, is just a matter of case-by-casespecifications.)8

It is of course understood that ifM is an indexical model relative to a given set offeaturesA, the values ofh should in general not depend onA unless the arguments dotoo. That is, one should have h(x,y)(ai )= h(x,y)(aj ) wheneverx(ai )=x(aj ) and y(ai)=y(aj) (ai, ajA). Also, such values should behave coherently with respect to the en-

tities that make up the system of universes U=Ut : tDE. Thus, in general oneshould require that h(xi,y)(a)= h(xj,y)(a) ifxi(a)=xj(a), and h(x,yi)(a)=h(x,yj)(a) ifyi(a)=yj(a). Indexical models satisfying these additional conditions may be referred to asmodels rootedon the couple (U, A). As we now shall see, it is precisely models ofthis sort that can be employed to provide an interpretation of languages whose vo-cabulary includes variable-binders.

For concrete illustration, we can refer directly to the examples considered earlier.


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EXAMPLE 2.1. If L is a sentential language, then M is a bivalent sententialmodelforL iffI0 =2. In particular, suppose L contains only a binary connector asin Example 1.1: then we may say thatM is astandardsentential model forL iffM in-terprets as thejoint-denial connector:

(*) ifs (,t)=, then h(h(d(,t),x),y)=1(xy) for everyx,yI0 .

EXAMPLE 2.2. If L is an elementary language with a denumerable set VL ofnoun variables, thenM is a bivalent elementary modelforL iffM is rooted on a cou-ple (Ut: tDE,A) such thatU0 =2 andA=U1

VL.

Thus, to account for the fact that some L -expressions may involve variable sym-bols of type 1, we may just requireM to be an indexical model relative to the set offeaturesU1

VL, i.e. expressions may be interpreted as functions of the relevant value

assignments a:VL U1 . In particular, suppose L includes a binary quantifier ,vfor each variable vVL , as in Example 1.2: then we may say that M is a standardelementary model for L iff the following additional conditions hold for every,tDs , every vVL and everyx,yI0 :

(a) ifs (,t)VL , then d(, t)(a)=d(, t)(b) for all a,bA(b) ifs (,t)=v, then d(,t)(a)=a(v) for all aA(c) if s (,t)=,v, then h(h(d(,t),x),y)(a)=I{1(x(a(v/u)) y(a(v/u))): uU1 }

for all aA

(i.e., iffM interprets ,v as the joint denial quantifierbinding the correspondingvariable v, every constant symbol denoting constantfunctions of the right sort).

EXAMPLE 2.3. IfL is a full categorial language with a denumerable setVL , t ofvariables for each type tT, then M is a bivalent categorial modelfor L iffM isrooted on a couple (Ut:tT, A) such thatU0 =2 andA=Ut

VL,t: tT.

Thus, again, since the expressions ofL may involve variable symbols of varioustypes, we require here that M be an indexical model based on the appropriate set offeatures: sequences of value assignments at:VL , tUt for each tT. In particular,suppose L includes an abstractor,v, t' for each vU{VL , t: tT} and each t'T ,as in Example 1.3: then we may say thatM is astandardcategorial model forL iff the

following additional conditions hold for every ,tDs andeveryxIt':(a) ifs (,t)U{VL , t: tT}, then d(,t)(a)=d(,t)(b) for all a,bA(b) ifs (,t)=v, then d(,t)(a)=a(v)(v) for all aA(c) ifs (,t)=,v, t' , then h(h(d(,t),x),y)(a)=x(a((v)/a(v)(v/y(a)))) for all yI(v)

and all aA, where (v) is the type ofv

(i.e. iffM interprets each ,v,t' as thefunctional abstractorof type t',t,t' bind-ing the corresponding variable v, every constant symbol denoting constantfunctionsof the right sort).


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VALUATIONS

At this point we just need to build a bridge between languages and models in the obvi-ous way, via the notion of a valuation. This is straightforward, for it suffices to notethat every language is homomorphic to any of its models. That is, ifL =(s,g,E) is alanguage and M =(d,h,I) a corresponding model, then there exists a unique map: URE UR Isuch that (i) (s (, t))=d(, t) for each ,tDs , and (ii) (g(y,z)=h((y),(z)) for each y,zDg 9. This means that M is sure to provide allthe infor-mation that is needed in order to assign a value to each expression ofL : d assigns adenotation to each basic expression, andh tells us how to compute the denotation of acompound expressiongiven the denotation of its component parts. Since all this infor-mation is perfectly reflected in the homomorphism that relatesL and M , such a mapcall itVis the natural candidate for the role of a valuation ofL on M: it yields thevalueV(x)=d(,t) forx=s (,t), and the valueV(x)=h(V(y),V(z))forx=g(y,z).

DEFINITION 3. The valuation of a language L on a corresponding model M isthe unique homomorphismV betweenL and M.

Just for the sake of completeness, one may refer to our previous examples toverify that V behaves normally when certain standard conditions are satisfied.

EXAMPLE 3.1. IfL is a sentential language with a binary connector (as in Ex-ample 1.1) and M a standard sentential model forL (as in Example 2.1), then theusual conditions hold for allA,BE0 :

(a) V(AB)=1 iff V(A)=0 and V(B)=0(b) V(A)=1 iff V(A)=0(c) V(AB)=1 iff V(A)=1 orV(B)=1(d) V(AB)=1 iff V(A)=1 and V(B)=1M M

EXAMPLE 3.2. IfL is an elementary language with a binary quantifier ,v foreach variable vVL (as in Example 1.2) and M is a standard elementary model forLrooted on a pair (Ut: tDE,U1

VL) (as in Example 2.2) then the following usual con-

ditions hold for allA,BE0 , all vVL , and all aU1VL :

(a) V(AvB)(a)=1 iff V(A)(a(v/u))=0 and V(B)(a(v/u))=0 for all uU1

(b) V(A)(a)=1 iff V(A)(a)=0(c) V(AB)(a)=1 iff V(A)(a)=1 orV(B)(a)=1(d) V(AB)(a)=1 iff V(A)(a)=1 and V(B)(a)=1M M

(e) V(vA)(a)=1 iff V(A)(a(v/u))=1for some uU1(f) V(vA)(a)=1 iff V(A)(a(v/u))=1for every uU1


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EXAMPLE 3.3. IfL is a full categorial language with a denumerable set VL , t ofvariables for each tT and an abstractor ,v,t' for each variable v and each type t'(as in Example 1.3) and M is standard categorial model forL rooted on a pair (Ut:tT, Ut

VL,t: tT) (as in Example 2.3), then the following conditions hold for all

t, t', t"T, all vU{VL , t:tT}, allAEt', allxEt, allyEt,t',t" , and allzEt,t' :

(a) V(vA(v))=V(A)(b) V(I t(x))=V(x)(c) V(K t,t'(x,A))=V(x)(d) V(S t,t',t"(y,z,x))=V(y(x,z(x)))

10.

DISCUSSION

With Definition 3 our semantic framework is complete. It is in fact a semantic frame-work, not a semantic theorya framework wherein each of a variety of semantic theo-ries falls rather naturally: as the particular form of the language and the specific con-ditions on the relevant models are varied, the theory varies.

Now I believe this is rather close to what Ajdukiewicz had in mind. What is at-tractive in the conception of semantics originated with his work is precisely that itleads to a general framework fixed in advance, rather than a theory to be worked outanew in each case: we only have to specify a type assignment for the languages sym-

bols (eventually along with a suitable structural operation) and then specify which,among the indefinitely many structures that give a homomorphic interpretation of thelanguage, are to count as admissible models. The account outlined here is a bit moreabstract than Ajdukiewiczs, hence more general. But it is precisely this trade-off be-tween abstractness and generality that shows the enormous potentiality of Ajdukie-wiczs original approach to semantics. I have on purpose refrained from consideringextending the basic notion of a categorial language, as suggested in more recent litera-ture 11 , to emphasize this aspect.

I would now like to conclude with a couple of remarks on the resulting overallpicture, both of which tie in with this general perspective.

The first remark concerns specifically the fact that also languages involving vari-

able-binding operators seem to fit in naturally with the proposed account. I dontknow whether this could by itself justify a claim of generality12. Certainly the fact thatwe can analyze a language with quantifiers by the simple functor/argument schema isnot unattractive, but that is only one way of putting it: there are several alternativesavailable that do the job as well. What I think is more interesting is the fact that also alanguage with, say, a lambda operator can be treated in the same fashion. There is arather widespread belief that within a purely categorial framework one cannot go veryfar in the analysis of complex linguistic structures, though much can be done with the


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additional help of the functional abstraction operator: arguably there is a strong con-nection between abstraction and certain types of syntactic/semantic transformationsthat allow one to come near to the surface of most natural language sentences(probably as close as we need for most purposes) 13 . From this point of view, then,the present account becomes fairly attractive and justifies a different, more positiveattitude towards categorial investigations.

The second remark is also related to the above, but it bears on a more general is-sue in the philosophy of semantics. Very often, the necessity of using variables andvariable-binding operators is taken to have important consequences with respect to the

role of semantic theorizing. In particular, it is often taken to imply that certain crucialaspects of our language lie outside, or perhaps above, the realm of semantics: in-sofar as such symbols mustreceive a fixed interpretation, there must be a correspond-ing set oflogicalprinciples that we need keep in mind when we come to spell out thesemantic framework we want to use. This is rather typical, for instance, in the cus-tomary model-theoretic way of presenting first-order logic, where the meaning ofbound variables and quantifiers is characterized only indirectly through the recursivedefinition of the truth-value of a sentence. And the same applies to lambda-equippedcategorial languages, where the meaning of the lambda operator is usually fixedduring a recursive definition of the value of an expression. To a certain extent this wayof proceeding reflects neither less nor more than a natural need for short cuts: if weare not going to consider other ways of interpreting those symbols, there is no need todo otherwise. But the question is howand somehow even whetherone could dootherwise.The question is whether it is possible in principle to do semantics withoutalso doing logic.

As I see it, there has never been much clarity on this point 14 . Nevertheless mysuggestion is that within the approach outlined here the question seems to admit a sim-ple answer. Insofar as quantifiers, functional abstractors, and the like can be treated asordinary functors, one has good grounds to regard these symbols as being on a parwith all the othershence good grounds to treat logics as true theories in the usual se-mantic sense. To treat something as a logical word is to select a certain class ofmodels as the only logically admissible ones, just as to treat something as, say, amathematical symbol is to select a certain class of models as the only mathe-

matically admissible ones. Earlier I exemplified this point with reference to the usualinterpretation of the conjunction connective and the multiplication operator. Now thesuggestion is that a similar argument could be given for any other logical word, wit-ness the examples given in the previous sections. Classical sentential logic can be ob-tained by selecting a sentential language along with the class of its standard bivalentmodels; classical elementary logic can be obtained by selecting an elementary languagealong with the class of its standard bivalent models; the classical typed lambda-calcu-lus can be obtained by selecting a full categorial language along with the class of all of


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its standard bivalent models; and so on. Including, as I mentioned, logics usually clas-sified as intensional (e.g. modal or tense logics).

Further generalizations can also be considered. For instance, as it is the semanticframework outlined here embeds the requirement that every model provide a homo-morphic interpretation of the corresponding language, which reflects the standard as-sumption that a model must be made of well-defined, sharp-cut entities, neatly linkedto one another and to the languages expressions in a univocal way. Philosophicallyas well as for practical reasonsone could find this a serious limitation in the scope ofa semantic theory, particularly in the spirit of the above considerations. There is no a

priori semantic reason to rule out the possibility that (our representation of) what wetalk about may involve gaps and/or gluts of various sorts. Indeed, since there is nogeneral linguistic criterion for incompleteness, there is no general way that incomplete-nesses can be ruled out without weeding out a variety of unproblematic cases as well.And since there is no general decision procedure for inconsistency, there is no generaland effective way that inconsistencies can be ruled out without rendering a great dealof perfectly innocent thinking impossible. Without going into any details here, it istherefore worth remarking that the approach described above can rather easily be gen-eralized so as to cover such cases as well. This would of course make things a bitmore complicated. For there is no homomorphism between a language and an in-complete model, while there can be more than one between a language and an incon-sistent model. Still, the point remains that as long as we can define a general notion ofa valuation linking arbitrary languages and models thereof, be it or not a pure ho-momorphic valuation15 , a general semantic framework is available relative to which avariety of cases can be treated in a uniform fashion. And I think this is definitely in thespirit (methodologically, if not philosophically) of Ajdukiewiczs original insights.

NOTES

1Ajdukiewicz (1935) spoke ofsemantic categories, following the terminology of Husserls Un-tersuchungen (1900-1901) and Lesniewskis Grundzge (1929). Presumably this is because he wasalready thinking of their semantic role, but I find that a bit misleading. In the following I shallmainly speak of types.

2These are indeed the two main primitive categories Ajdukiewicz had in mind, though he did notrule out the possibility of different choices. Some authors, for instance Lewis (1970), have insisted ona third one, Common Noun, and others have considered dispensing with Proper Noun in favour of

Noun Phrase, or even Verb Phrase.3I shall use standard set-theoretic notation and terminology: note only that if is a function, I

write D and R for the domain and the range of , respectively, while (x/y) denotes the func-tion exactly like except that its value atx is y.

4Since we define derived types by means of a binary operation, we can of course speak of


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monadic functors only. However, it is a well-known fact that this implies no loss of generality: forwhenevern > 1 and t1, , tn, tn+1 are types, the derived type t1,t2,,tn,tn+1may be used torepresent the category of those n-adic functors that combine with n-tuples of expressions of type t1,t2, , tn (in this order) to produce expressions of type tn+1 . In this sense, as long as the second co-ordinate of a derived type is allowed to be a derived type itself, our relying on imposes no signi-ficant restriction on the class of possible functors. (To illustrate, the English word and is a dyadicfunctor that makes a sentence out of two sentences; but it can equally be regarded as a monadic functorthat, when applied to a sentence, produces an expression that behaves again as a monadic functor: afunctor that makes a sentence out of a sentence). The idea goes back to Schnfinkel (1924) and reflectsthe set-theoretic isomorphismAB1B2 Bn+1 (((AB1)B2))Bn+1.

5Although we speak of the type of a symboland consequently of an expressionit could be

argued that this is only an apparent limitation: it is not difficult to imagine words belonging to morethan one syntactic category, but we can always deal with such cases by treating them as distinct sym-

bols with a common surface realization. (For instance, the word light in English may seem tofunction as either a noun or an adjective, but we can also deal with this peculiarity the way commondictionaries do: by distinguishing between a word light1 and a word light2, to be classified into twodistinct syntactic categories). Apparently the point was made by Ajdukiewicz himself, though severalauthors have later proposed alternative accounts.

6The requirement that QL , n VL= {x,y: x,yCONL , n} is of course but one way of secur-ing syntactic precision. Note also that in a similar fashion one could for instance treat descriptors assuitable structured subnectors: given sequences DL , n disjoint from Rs so that DL , n VL = {x,y:x,ySUB L , n}, one could speak of a pair of the form DL , n , i , v as a n-ary descriptor binding thevariable v. In the present context, I shall not go into languages with descriptors to avoid certain well-known complications in their semantics.

7The requirements that each domain of interpretation Itbe non-empty , that dbe a total functionon Ds , and that h be a total operation on U{It',t It' : t',t',tDI} reflect a major assumption ofstandard semantics, namely that each model must provide a complete and consistentinterpretation ofthe language. It is of course possible to give up such requirements so as to admit incomplete and/orinconsistent structures as models bona fide. For instance, allowing d to be a partial function wouldcorrespond to a more liberal attitude with respect to the possibility of non-denoting symbols; allow-ing it to be a relation would leave room for symbols with more than one denotations; etc. Althoughthis is in the spirit of greater generality, in the present context such a departure from standard seman-tics would take us too far. A fuller account may be found in a forthcoming work entitled UniversalSemantics .

8I think this is a rather natural exploitation of some basic ideas of Montague Grammars, par-ticularly of the treatment in Montague (1970). Note of course that one could also take the notion of anindexical model as fundamental, regarding all other models as models indexed by a features singleton.

9That is unique is easily seen, provided RgRs =0. Obviously (x) is defined ifx is a sym-bol. And if(y) and (z) are both defined, then (g(y,z))=h((y),(z)) is also defined, sinceh is a func-tion.

10 For readability, I use here a notation of the form (w(w1, , wn)) as an abbreviation forg((g(g(w, w1),w2),),wn).

11 The first substantial refinements or extensions of the pure categorial paradigm go back tothe work of Bar-Hillel (1960) and Geach (1970). Some indications of the current trends may be foundin Bach (1984), Oehrle, Bach and Wheeler, eds. (1988) and Buszkowski, Marciszewski and van Ben-them, eds. (1988).


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12 Certain aspects of the proposed account might still sound unpalatable. For instance, as PeterSimons has pointed out to me, there is no way one can single out an elementary language wherevacuous quantification is forbidden. However, I am inclined to regard such limitations as symp-tomatic of a more general problem concerning the generative power of Definition 1, typical of virtu-ally every categorial grammar and independent of how one feels about variable-binding through func-tors. For instance, there is no way one can single out a sentential or elementary language whereuseless conjunctions such asAA are forbidden. From this point of view, the corresponding generalsolution would simply involve redefining the structural operation g as required: rather than a totalfunction on U{Et',t Et': t',t',tDE}, it should be allowed to be a partial function (of some sort).Thus, for instance, an elementary language where g(,v,A) is defined only if the noun variable v is aconstituent symbol of the sentenceA would be a language with no vacuous quantification.

13 The work done in the tradition of Cresswell (1973) is most indicative of this approach.14 The question is not whether one can live without any logic, as it were. For of course, one

could always say that semantics is set theory, and thatsurely embeds quite a bit of logic.15 In my Universal Semantics, I propose a general solution based on an extension of Van

Fraassens notion of a supervaluation. An alternative approach has been proposed in Muskens (1989).

REFERENCES

Ajdukiewicz K. (1935), Die syntaktische Konnexitt, Studia Philosophica 1, 1-27 (Eng. trans. byH. Weber: Syntactic Connexion, in S. McCall (ed.),Polish Logic 19201939, Oxford: Claren-don Press, 1967, pp. 207-231).

Bach E. (1984), Some Generalizations of Categorial Grammars, in F. Landman and F.Veltman (eds.)Varieties of Formal Semantics, Dordrecht/Cinnaminson: Foris, pp. 1-23.

Bar-Hillel Y. (1960), On Categorial and Phrase-Structure Grammars,Bulletin of the Research Coun-cil of Israel3, 1-16.

Buszkowski W., Marciszewski W., van Benthem J., eds. (1988), Categorial Grammar, Amster-dam/Philadelphia: John Benjamin.

Church A. (1940), A Formulation of the Simple Theory of Types, The Journal of Symbolic Logic5, 55-68.

Cresswell M. J. (1973),Logics and Languages, London: Methuen.Geach P. T. (1970), A Program for Syntax, Synthese 22, 3-17.Husserl E. (1900-1901), Logische Untersuchungen, Halle: Max Niemeyer (2nd ed. 1913/21; Eng.

trans. by J. N. Findlay, Logical Investigations, London: Routledge and Kegan Paul, 1970).Lesniewski S. (1929), Grundzge eines neuen Systems der Grundlagen der Mathematik, Fundamenta

Mathematicae 14, 1-81 (Eng. trans. by M. P. ONeil: Fundamentals of a New System of theFoundations of Mathematics, in S. Lesniewski, Collected Works, ed. by S. J. Surma, J. T.Srzednicki, D. I. Barnett, and V. F. Rickey, Dordrecht/Boston/London: Kluwer/Nijhoff, 1992,Vol. 1, pp. 129-173).

Lewis D. K. (1970), General Semantics, Synthese 22, 18-67.Montague R. (1970), Universal Grammar, Theoria 36, 373-398.Muskens, R. (1989), Meaning and Partiality, Ph.D. Thesis, Katholieke Universiteit Brabant.Oehrle R., Bach E., Wheeler D., eds. (1988), Categorial Grammars and Natural Language, Dor-

drecht/Boston: Reidel.


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Schnfinkel M. (1924), ber die Bausteine der mathematischen Logik, Mathematische Annalen 92,305-316 (Eng. trans. by S. Bauer-Mengelberg, On the Building Blocks of Mathematical Logic,in J. van Heijenoort (ed.), From Frege to Gdel: A Sourcebook in Mathematical Logic, 1879-1931, Cambridge, Mass.: Harvard University Press, 1967, pp. 355-366.

Van Fraassen B. C. (1966), Singular Terms, Truth-Value Gaps, and Free Logic, The Journal of Phi-losophy 63, 481-95.

Whitehead A. N, Russell B. A. W. (1910-1913),Principia Mathematica , Cambridge: Cambridge Uni-versity Press.

variable binders as functors

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