a topos-theoretic approach to reference and modality

7/29/2019 A Topos-Theoretic Approach to Reference and Modality

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359N o t r e Dame Journal of Formal LogicVolume 32, Number 3, Summer 1991

A Topos-Theoret ic App roach toRe fe rence and M oda l it y *

G O N Z A L O E. REYES

Abstract This paper presents an approach to modal logic based on the no-tion of kindor interpretation of a count noun as a prerequisite for reference.It gives a mathematical formalization of this notion in the context of a lo-cally connected topos (thought of as a universe of variable sets) over atopos S (thought of as a universe of constant sets). In this context, modaloperators are intrinsically definable and the resulting formal system isdescribed in some detail.

Introduction This paper is an essay on reference and generality in naturallanguages (to borrow from the title of Geach [5]). More precisely, it is concernedwith the semantics of pronouns, proper names, and count nouns.I t is a remarkable fact of natural languages that a proper name picks up itsreference uniquely any time that it is uttered, whether or not its reference ispresent at the moment of utterance, whether or not we know the reference'swhereabouts at that moment, whether or not we are able to recognize its refer-ence, and whether or not we are referring to events that took place in the pastor may take place in the future. My main concern will be with the followingproblem: What semantical structure should be postulated for this relation (be-tween a name and its reference) to accomplish the formidable tasks just de-scribed? I shall propose an answer based on the notion of kind or sortalviewedas the semantical interpretation of count nouns.The essay is divided into two parts. In the first, "Count nouns and kinds",taken from an unpublished paper in collaboration with Marie Reyes, I state andgive arguments for a series of theseson reference and generality involving propernames, count nouns, and kinds. Although this medieval practice has long goneout of fashion, I believe it useful to understand the issues involved. Some of these*Dedicated to the memory of Isidro Suarez.Received September 27, 1989; revisedMarch 23, 1990


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360 GON ZALO E. REYEStheses have been already argued by G upta [6] and M acnam ara [13], followingthe lead of Geach [5], Bressan [3] and others. Nevertheless, the most importantof the theses for the whole development, namely the modal constancy of allkinds, is not to be found in the writings of these authors and, in fact, it con-tradicts further theses of some of these authors.The second part of this essay, "Topos semantics", is an attempt to give amathem atical formalization of the notion of kind in the context of topos theory .It is my conviction (to rephrase Montague [17]) tha t ph ilosophy, at this stage inhistory, has as its proper theoretical framework category theory (which includesset theory with "urelements") rather than set theory alone, as Montague believedin 1970. I believe that topos theory provides an adequate context to formalizethe basic notion of m odal constancy. F urtherm ore, th e very possibility of usingtopos theory imposes very strong and , I believe, fruitful constraints on the logicof reference and generality as we will show in some detail. To mention just twoof them: the logic of quantification and identity of a topos is the standard one;furthermore, under the natural assumption that kinds are exponentiable (so thathigher-order logic is interpretable), modal operators are intrinsically definablein 2i topos and need no t be introduced as a further structure. Thus this app roachdiffers from others (such as in [17]) which change the logic of quantifiers to ac-commodate the modal operators.This concludes the description of the contents of this paper. I have left outsome developments of this approach to reference and modality which have takenplace after the first draft of this paper. Connections between modal operatorsand locale theory have been studied in some detail in Reyes and Zawadowski[19]; questions of soundness and completeness of the formal systems obtainedvia this topos-theoretic approach are studied in Lavendhom me, Lucas and Reyes[11]. Finally, this approach was developed with a view to applications to cogni-tion and in close contact with the work on language learning of Macnamara andthe work of M. Reyes on the semantics of literary texts. The interested readermay consult [13], Macnamara and Reyes [14], and M. Reyes [20] for these ap-plications.

/ Count nouns and kinds The first of our theses on reference concernsproper names.(1) The denotations of prope r names are rigid.

This thesis asserts that a proper name, say "Nixon" has the property of denot-ing its reference throughout all actual as well as possible situations, past, presen t,and future in which Nixon appears or may appear. A biographer of Nixon woulduse the single proper name "Nixon" to refer to the boy who grew up in Califor-nia, to the young politician who w on his first election, and to the president whowas forced to resign from office in spite of the different times and situations.He may be unsure of whether Nixon won his first election by fraudulent meansand he may discuss both the possibility that he used such means and the possi-bility that he did not to arrive at the truth of the matter. In other words, he mustconsider counterfactual situations (since clearly one of the possibilities cannotbe realized) about Nixon, i.e., the reference of "Nixon", in order to describe the


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REFEREN CE AND MOD ALITY 361actual course of events. Similarly, to explain Nixon's actions, the biographershould entertain a series of possibilities as to Nixon 's motives and evaluate themcritically. The point is once more that we are forced to consider counter factualsituations about the reference of the proper name "Nixon". The thesis of rigid-ity has been forcefully argued by Kripke [9].As an aside, let me remark that the fact that co unterfactual situations haveto be considered to describe real situations is beautifully exemplified in Classi-cal Mechanics and in the Calculus of Variations: to describe the real trajectoryof a body, we compute the Lagrangian of all its possible trajectories (most ofwhich are not physically possible!) and we take that trajectory for which the La-grangian has a minimum (or a stationary value).(2) Rigidity presupposes count nou ns.

This thesis, which is characteristic of the approach we are developing, assertsthat the only way of tracing the identity of the reference of the proper name"Nixon" throughout all real and possible situations, past or present, is by meansof a count noun such as "person" or "man". Indeed, the boy in California, theyoung politician, and the president remained one and the same person, althoughhe successively stopped being a boy, a young politician, and a president. We can-not, for instance, trace Nixon's identity through the molecules that compose hisbody, since those that constituted his body at the time of his youth were differentfrom those that constituted his body a t the time tha t he was a president. As Ar-istotle pointed out, change requires something to change. There must be a con-stancy that underlies the change of the boy into the president. The thesisguarantees such a constancy through the count nouns to the interpretation ofwhich the individual experiencing the change belongs.To m ake this thesis more precise, I shall state claims on the interpretationsof count nouns, the kinds. Kinds are the interpretations of count nouns such as"person", "dog", "atom of hydrogen". There are other nouns besides count nouns(e.g., mass nouns such as "m oney", "clay", and "oxygen"), but we associate kindswith count nouns only. Members of a kind may be individuated and counted.We can interpret expressions such as "three dogs", "two drops of water", and"three a toms of oxygen", as well as expressions of generality such as "every dog",but we cannot interpret "three oxygens" or "two moneys". I consider the notionof kind as being so basic that it cannot be analysed in terms of m ore basic no-tions. A ll that we can assert is that a kind has m embers which are individuatedand that it makes sense to say that two members are identical. Kinds are the con-stitutive dom ains which allow us to use quantifiers and the equality symbol cor-rectly. I shall not assume that equality between members of a given kind isdecidable.An important consequence of this postulate for kinds is that members of akind can be counted (at least under some limitations) and th at they are subjectto the logic of quantification. In fact, counting does not apply to heaps or con-glomerates of objects, but to kinds: the same "heap" which makes up an armymay be counted as 1 army, 6 divisions, 18 brigades, or half a million men. Thispoint was made quite forcefully by Frege and I have nothing to add.(3) Kinds are modally constan t.


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362 GONZ ALO E. REYESThis thesis is in fact a further development of the previous one. In fact, aswe saw in (2), kinds are what remains "cons tant" from situation to situation andthus kinds are precisely the required standard against which we can understand

changes and modalities. In other words, kinds should be modally constant "bydefinition". A consequence of this thesis is that m odalities can be genuinely ap-plied to predicates only, not to kinds themselves. A member of the kind PAS-SENGER is necessarily a passenger qua passenger. On the other hand, aparticular person who happens to be a passenger need not be necessarily a pas-senger qua person . In this case, we are saying something about the predicate "tobe a passenger" of the kind PERSON, namely that it is not necessary. In the first,we are saying something about the predicate "to be a passenger" of the kind PAS-SEN GER, namely that it is necessary. It follows tha t attem pts to apply modal-ities to kinds themselves to form new "kinds" such as POSSIBLE APPLE,POSSIBLE CAR, or POSSIBLE MAN (as used by Gupta [6] to define "modalconstancy") are wrongheaded. And in fact, there are serious difficulties with anyattempt to view possible apple or possible man as kinds: neither membership inthe kind nor the relation of identity are well-defined. As regard the first: Doesa portion of jelly apple count as a possible apple? Or again, does a piece of junkmetal count as a possible car? As regards identity, Q uine [18] has already askedthe relevant question: How many possible fat men are there in the door? Onthe other hand, we may ask of a given fruit whether it is possibly an apple andthe intuition that "apples are necessarily apples " tha t Gu pta tried to express as"POSSIBLE APPLE = APPLE" could be expressed rather as "the predicate 'tobe an ap ple' of the kind FR UIT is a necessary predicate" . We cannot eliminatethe kind FRUIT in this formulation, since the statement "the predicate 'to be anapp le' of the kind INGREDIENT IN A RECIPE is a necessary predicate" shouldbe false for cooking to be possible at all!This way of considering "possibility" and "necessity" agrees with the grammarof these notions. Suppose that we find an archeological site with skeletons ofsome anthropoids. If we are asked whether some are humanoids, we could nat-urally reply that "three of these are possibly humanoids's ske letons" or "it is pos-sible that three of these skeletons are humanoid's", but we would not say "thereare three possible humanoid's skeletons". Similarly, we do not say that Mr. andMrs. X have twelve possible children, but rather that it is possible for Mr. andMrs. X to have twelve children.I shall distinguish sharply between kinds and predicates of a kind. Whereasthe first are modally constant and independent of any particular situation, thesecond are not and whether a member of a kind falls under a predicate or notwill depend on the situation envisaged. I believe that this context does better jus-tice to the dialectics of change, constancy, and modalities. As I said in the in-troduction , this thesis is basic for the whole approach to the semantics of propernames and count nouns that I develop in this essay.2 Topos semantics2.1 Constant sets vs variable sets in a topos On several occasions, Law-vere has pointed out that the dialectics of variation vs constancy (which we hinted


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R E F E R E N C E AND MODALITY 363upon in Section 1) may be given an explicit formulation in topos theory via thenot ion of a geometric morphism - S. In this case may be thought of as auniverse of variable sets, whereas S may be thought of as the universe of con-stant sets (possibly with "urelements").I recall that a geometric morphism 8 -> S is given by a couple of adjointfunctors

such that H and preserves finite limits. It is customary to call "the con-stant functor" and "the global sections functor" or "points".

Recalling the thesis that kinds are modally constant, whereas predicates ofkinds carry the burden of change, its seems natural to identify a kind with a (con-stant) set S of and a predicate of that kind S with a morphism AS - * of S,or, what amounts to the same, with a variable subset of AS. Indeed = in S isgiven by the relation of identity of the kind in question. Basic kinds such as PER-S O N , DOG will be identified with sets of "urelements".I t is natural, therefore, to define the category Q of constant sets as the fullsubcategory of where objects are of the form AS (with S E S).Remark This identification amounts to viewing kinds and their predicates asthe hyperdoctrine (S, P ) (in the sense of Lawvere [12]) where P : Sop - * Sets isthe functor defined by

P (S) = Sub (S) (5, )P (S -A T) = Sub ( ) ( A / p l > Sub (S)

where ( / ) ~ * is the pull-back along / .I t is well-known that ( / ) " 1 has both a left and a right adjoint: 3/ H(/ H v/ .We may now express modal constancy of kinds as follows:

Assumption 1 Kinds are constant sets of the geometricmorphism -> S.

2.2 The D operator In this section we show from Assumption 1 that themodal operator of necessity D is definable for predicates of kinds, essentiallyin terms of "global sections".

In fact, from the adjunction S = , H , we derive the following ad-joint maps

s r ( ) , H essentially "by restricting , to subobjects of 1".

In more detail, given p9 we define (p) as follows:( classifies subobjects).P ^ - X

Applying we obtain the equivalences


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364 GONZALO E. REYES P > >AX

( classifies subobjects)AX > X^TW ( H >We remark that may be defined, alternatively, as the transpose of the clas-sifying map r : -> of 1 A 1>- .Similarly, y(K) is defined for a given K as follows:

(H )AX (Q classifies subobjects).K^AXWe apply and form the pull-backT ( K ) >>TAXI I -

N ( K ) > Xwhere Id - ^ is the unit of the adjunction H . The lower horizontal mapin the pull- back diagram is classified by X > .

An easy computation gives that H . Furthermore, since preserves finitelimits

(T) = T and (p/\q) = (p) (q).We may now define the modal operator

D = : ( ) ^ ( ) .The following can be proved easily (or invoke the fact that D is a lexcotriple)

1. D < I d2 . D2= D3. UT =T4. (K 1AK2) = K1A K2.From here we can define an operator s on predicates of AS as follows:if S - is an arbitrary predicate of S, we let S s > be the trans-pose of S ot ( ) > ( ), where t ( ) : S - ( ) is the transpose of .

Example 1 = Sets Sets7 = ,

where / is a set. In this case, (5) = (S)iGI and (( i)/ e/) = * i

F r o m these functors we derive = 2 2 7 = ( )


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REFERENCE AND MODALITY 365which are easily checked to be

\l iip=Tl ifp = { T if K = I

We shall write (p) ={iei\p},y( ) = \\viewe )l

where || . . . | stands for "truth- value of . . .".The action of D son predicates of AS may be described simplyasi\\- s [s] iff Vjelj\\- [s]

for alls G 5.Example2

= Sets Sets**" = 8,where = (P, / 5 ( C ) . For proofs of these assertionsand further details, seethe Appendix.


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366 GONZALO E. REYES2.3 Locally connected topos In several examples of a topos definedover S, 8 - S, which include the case that -> S is given as the category ofpresheaves over a category in S, the functor : S -> has a left adjoint r0 H .When this is so, r0 is called the "connected components functor" for reasons tobe discussed later. I shall be especially interested in cases that r0 satisfies someFrobenius type conditions.T he following theorems are due to Barr and Pare [1]:Theorem2.3.1 Let &- +$bea geometricmorphism. Thefollowingare equiv-alent(a) A is a cartesianclosed functor(b) has a left adjoint 7r0 satisfying the Frobenius reciprocitycondition

ro(5 x E) = S x r0E.Theorem2.3.2 Let 8>- ^Sbea geometricmorphism. Thefollowingare equiv-alent(a) A is a locally cartesianclosed functor(b) has a left adjoint satisfying the generalized Frobenius condition:

(ASxE\ cM AT ) S **oE.Remarks(1) The first condition of Theorem 2.3.1 may be reformulated as " pre-serves exponentials", whereas the first condition of Theorem 2.3.2 may be refor-mulated as " preserves the operations / s".

(2) The second condition of Theorem 2.3.2 is equivalent to the statement that7o is S- indexed, which according to [1] is the right generalization of the notionof a locally connected topos over sets due to G rothendieck. The reason for thisterminology comes from the particular case ShA") - Sets, where A" is a topo-logical space. In this case, all the conditions of both theorems are equivalent toth e statement that X is locally connected. Furthermore, by identifying Sh(^Owith the category of etale spaces over X, ro turns out to be the functor whichsends an etale space over X into the set of its connected components.

I now state a second assumption on kinds.Assumption 2 Thegeometricmorphism & - S is locally connected.

This assumption may be stated in the equivalent form (cf. Theorem 2.3.2): has a left adjoint r0 H satisfying the generalized Frobenius condition.A consequence of this assumption is that kinds (= constant sets, by Assump-tion 1) are exponentiable. Another consequence is the fact that the possibility op-era tor 0 is definable for predicates of kinds.2.4 MAO operators In this section we show how to define a couple of op-erators (0,D ) on predicates of kinds from Assumptions 1 and 2. Whereas D wasdefined in terms of global sections, 0 will be defined in terms of connected com-ponen ts .


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R E F E R E N C E AN D M OD ALITY 367We start from

s r ( )an d we define a left adjoint H as follows:

( H )( classifies subobjects)Js -X

We apply r0 and take the image factorization o( K ) r o ( * )

I I "P(X) c > XThe lower horizontal map is classified by X M K ) > (h ere ex: r0 -> Id i sthe counit of r0 H ). Easy computat ions give

H an d hence ( K A p) = XU / ?. We n ow defin e 0 = : ( ) - * ( ) andcheck, easily, that the couple (0 , D ) with D = y satisfies the following condi-t ions:

(1 ) O H D(2) D < Id < 0(3) D 2 = D, 0 2 = 0(4) 0 ( K K 2) = OK i K 2.A c o up le o f o pe ra t o rs (0 , ) satisfying ( l ) - (4 ) will be called a MAO couple(MAO for "modal adjoint operators").I t is clear th at we can define an o peration 0 5 on predicates of S, just as wedid for D : If AS - + is a predicate of 5, we let A S - ^ > be th e trans-pose of S 0 > t ( y ) > ( ) , where S - ^ ^ ( ) is the transpose of .Remark In the case th at has a left adjoint r0 satisfying F robenius condi-t ion we can simplify th e expression of th e un it s: S - * S as follows:

S - n s _ AS ( H )AS J ^ A S1 x AS > S ( r 0H )rod X S ) >S m .(T Q satisfies Frobenius)7 O ( 1 ) X S - ^ S5 _ _ , s o(l).In other words, s is just the constant m ap. F rom here , we conclude thatA s: AS - > S i r o ( 1 ) is again th e con stant m ap (since preserves expon en tials).


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368 GONZALO E. REYESSimilarly, e 5: AT AS -> AS is just "evaluation at the unit 1 -> r o(l)", since S = 5 o ( 1 ) .

These maps A s and e 5play an important role in [17], where Montaguecalls them "intension" and "extension", respectively.Example 1 (bis)

S = Sets _A+ Sets7 = ,( rwhere / i s a set. In this case ro((-X/)e/) = ll/ e/ ^/ *s ^ e disjoint union of themembers of the family.Fur thermore , the map in the diagram

= 2 j 2 7 = ( )7

is givenby T if K 0 (K) =\\ ifK=0;

i.e., (K) = \\3iel(ieK)\\.The action of 0 5 on predicates may be described as follows: / Ih Os


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R E F E R E N C E AND MODALITY 369Example 3 (bis) - is bounded and locally connected.By a theorem from[1] wecan present 8 as the category of sheaves over a site C of S such that theconstant presheaves are sheaves. In this case, we have the diagram

3uSh (C)rwith AS(C) = S for all C G C, o(F) = l i m ^ F , (F ) = limc


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370 GONZALO E. REYESProposition 2.5.2 Assume that S - S is locally connected and connected.Then the lifting

:( ( ) ) - + ( )

inducesan equivalenceof categoriesbetween ( ( )) and 6.Proof: Simple computations using Proposition 2.5.l(b).

In the presence of Assumption 2, connectivity of leads to further condi-tions on the maps

s !i ; ( ) .7

In fact, y = l s = = l s .Finally, we shall look again at the example of Section 2.1 to see what thiscondition on connectivity means in these particular cases:Example 1

< _ 2 _ = Sets _A> Sets7 = 8 .* rIn this case 7ro(l) = o((l);G / ) = / e/ 1 = / and the connectivity means that/ = 1. Therefore, connectivity brings about a collapse of the "possible worlds"semantics to just one world. Correspondingly, 0 = D = Id.Example 2

S = Sets !A ; Setsp* = 8.* rIn this case ro(l) = set of connected components of the graph P. Therefore,7 o(l) = 1 iff given U,VG there is a finite chain of elements of IP:

Therefore, it is possible to impose a condition of connectivity on "possible sit-ua t ions" semantics without collapsing modal operators to the identity.Example 3 This example is delicate and I shall deal with it elsewhere.

In Section 3 we shall see what further logical rules connectivity imposes on0 and D.

2.6 Sets with coincidence relation and kinds Members of a kind such asD O G , PERSON , etc., may come in and go out of existence, contrary to mem-bers of the set of natural numbers, say. On the other hand, kinds such asP R O P O S I T I O N or PREDICATE OF PERSON have members that may hap-pen to coincide at a given situation, although they are not identical.I t may well happen, for instance, that singing and working may coincide ata situation in which precisely those who are singing are those who are working.Nevertheless, these predicates are not identical.


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R E F E R E N C E AND MODALITY 371This notion of coincidence is essential to understand problems of opacity

which are ubiquitous in natural languages. Keenan and Faltz [8] have given nu-merous examples of lack of substitutivity of "equals for equals" which do notresult from epistemic contexts of the kind "believes th at", "wonders whether",etc . For instance, in the situation envisaged above, those who are singing withFred need not coincide with those who are working with Fred. Of course mo-dal operators create contexts for which we cannot substitute "equals for equals"without altering t ru th values.I t is not cogent to say that we are dealing with identical members of a kindwhich have different properties! Nevertheless, it is quite cogent to consider mem-bers which happen to coincide but are different (and hence they may have dif-ferent properties).

In this section, we shall provide means to represent these facts in our theory,by defining, for any topos , and any complete Heyting algebra / / in , thecat-egory of sets with coincidence relation ( (/ / )).T he objects of (/ / ) are couples ( E , E ) where E : E x E -> / / satisfies

(i) E(e,e') = E(e'9e)(ii) E(e$e') E(e',e")* E(e9e").O n the other hand, a morphism ( E , E ) - A (F, F) is a map E - A F o f which satisfies

E(e9e')


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372 GONZALO E. REYESFinally, the operators may be described as follows: if (E, E) - A (F, F) E

/ (G9 G)\ / (P9 p)\ /P\ /G\ , . | - [ . where | - , , j .\ ( E , E)/ \ (F , F)/ W WTo describe p we need some preliminaries. Recall that in th e category ofsets, P = beFHaGf- b(x) = )}and the map P^Fis just the (restriction) of the first projection. We now define

M H defined by o(y,y') =V f{x)=y,Ax')=y'&E(x>x')-


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R E F E R E N C E AND MODALITY 373Images, however, are not stable under pull- backs, as the following exampleshows in Sets ( o), where o = {0,^,1} with obvious operations.Let E = (0,1,2,3), F = {0, 1,2},^ = (0,1) and let F = , x = , where is the coincidence relation whose value is always equal to T E , E(0,2) = E(l,2) = E(l,3) = E(0,3) = \ and T elsewhere and (0, l) = ( l,0) = \ an dT elsewhere. Then the diagram

( E , E ) - - > (F, F) I !(X,&) ^ (X, x)

is a pull-back, where (0) = 0, (3) = 2, a (I) = (2) = 1, w(0) = 0, u(l) = 2,v(0) = 0, v(l) = 3. Clearly a is an image, but Id is not! Similarly, (H) hassuprema of subobjects which are, in general, not stable.From ourdefinition of morphisms in 8( / / ) we obtain the forgetful functor/:8( /J) -> 8 defined by

/ ( E , E ) = E and / ( / ) = / .Proposition 2.6.2 Theforgetful functor U is locally cartesian closedand hasboth a left and a right adjoint: L- \U \R.Proof: The first statement is obvious from the description of the operations ofth e category. As for th e second,

I ( E ) = ( E ) , (E) = ( E ) . andL ( / ) = * ( / ) = / ,where x(e,e') = 1E H, o(e,e') = 0 E H.

A particular case which is of great importance for us is H = E 8.In this case S( ) can be described as the category PER (8) of partial equiv-alence relations, i.e., an object of PER (S) is a couple (E,R) where i ? ^ E x Eis a symmetric and transitive relation. A morphism/ : (E,R) -> (F, S) of PER(8) is a m a p / : E - >F o f 8 such that (e, e') E Z? - (f(e),f(e')) E S. In fact, thisfollows immediately from the bijection

ExE-^QR^E X E

given by definition of .

3 The language of many- sorted modal theory and its interpretations Inour topos theoretic semantics wehave variable sets (arbitrary objects of 8) andconstant sets (objects of 8 of the form AS, with S E S) . It is natural to intro-duce types and sorts to be interpreted as variable and constant sets, respectively.However, to simplify our exposition and bring forth what is new in our ap-proach, we shall concentrateon sorts only. Furthermore the theory of typeshasalready been dealt with at great length by several authors.

We define sorts and terms by recursion:


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374 GONZALO E. REYESSorts

(a) Basic sorts are sorts: passenger, person, boy, bachelor, reading, river,etc .(b) 1, PROP are sorts(c ) If X9 Yare sorts, so are X x Yand Yx(d) Nothing else is a sort.

Terms In what follows, wewrite "t: X" for "t is a term of sort X".(a) Basic constant terms c G C on ^ are terms of sort X: * G Con! John G

C o n p e r s o n ; run G ConpROP person I meet G C o n p R o p p e r s o n x p e r s o n > etc.(b) If x G Var , then x is a term of sort X, where Var^ is an infinite set ofvariables, for each sort X.

(c) If t:Xands:Y9 then : * x Y.(d) If x G Varx and t: F, then \xt: Yx.(e) If t:Yx and s:X, then t(s):Y.(f) T,_L :PROP.(g) If t,s:X, then/ = s and t Xs: PROP(h) If 0>, ^ :PROP , then * : PROP, where * G { ,V,-> }(i) If :PROP and x G Var , then lx ,Vx : PROP(j) If :PROP , then U , : PROP(k) Nothing else is a term.A formula is a term of sort PROP. Weuse the following abbreviations:I f t:X, w e l e t E ( t ) = t X tIf is a formula, we let - = - > .We shall assume that we have defined the usual notions like "substitution of

a variable by a term", "free variable of a term or a formula", "a term is free fora variable in a term or formula", etc.

We now interpret this language in a topos - S. The main idea, already dis-cussed, is that sorts are interpreted as couples consisting of a constant set of an d a coincidence relation.

O ur interpretation proceeds in two steps: we first interpret sorts as kinds i.e.,as objects of the category ( ) of the form (S, 5) .Via this interpretation, we shall interpret terms and formulas in the topos in the usual way (cf. Makkai and Reyes [15] and Lambek and Scott [10]).To interpret sorts in ( ), it is enough to interpret basic sorts. In fact, oncethese sorts have been interpreted, weextend the interpretation / to all sorts asfollows:

7(1) = ( l f r )/ ( P R O P ) = ( , )

where ( ) : ( ) x ( ) - ( ) is defined as (< + ).Fur thermore , if X and Yhave been interpreted, thenI{Xx Y) =I(X) x/(y)and

I(YX) = / ( 7) / w ,


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R E F E R E N C E AND MODALITY 375where products and exponentials in the right- hand sides refer to the cartesianclosed structure of 8( ). Furthermore, constant sets of 8 are exponentiable byour assumption on S - S.

We recall that we have a forgetful functor, ( ) - U 8,in terms of which wedefine the interpretation of sorts as follows: if X is a sort,then \ \ X \ \ =\JI(X). Notice that || A'fl is a constant set of the form AA. In thesequel, we shall let t r ( . . . ) be the transpose of ( . . . ) given by the adjunction H .

F or each term / : X and each sequence x = \ \ X \ \ E . If x is any sequence of (distinct) variables of sortsA ,...,Xn, we let ||jf:c|| : \\X\\x . . . x||A || -> 1 - ^ \\X\\ be the com-posite of the unique morphism ||X\ || x . . . x ||A^ || - 1 with || c ||.

b. If Xi E Var , then ||x:x/ || = T ,-, the ith projection.c. If t Xands: Y, then

||J :< 5 > I = < | |x : / | | , | | je : 5 | |> .d . If xG Var^and t:Y, then \\x:\xt\\ - .\\X4x.. .x\\Xn\\ -> \\Y\\m is

defined to be exponential transpose of the map||xx:^||:||^1||x...x|| l|x|| ll- |I (we assume that xx consists of distinct variables).e. If t: F ^ a n d s : ^ , then \\x:t(s)\\ = ev < ||x: t\ \ ,\\x s\ \ >, where ev:\\Y\\^x\\X\\- +\\Y\\.f. ||jc: T || = ||^ 1 | |x . . .x | |A r / l ||^l ^ l - ^

||j c : | | = WX^x... x l ^ l -> 1 - l - ^ .g. ||x: t = s || is the composite

i r 1 |x.. .x|r l l | ) \\x\\ \\x\\ t r ( | l"l |) ) ,where ^i classifies the diagonal || | >+ \ \ X\ \ x \ \ X \ \\\x:t s\\ is the composite

l l ^ l l x . ^ x H ^ I I < ' ' " : / '^"> ) ||A ||x||A || t r ( | l " l | ) > ,where ( | | , m ) = / ( * ) .h. ||x : * \ \ : ||A ||x . . . x||JV || - > is the composite| | X l | | x . . . x i i ^ i < 1* 1 > 1^1 >, x JtElU A where * : x -> is one of the operations V, ,- > on .

i. ||x:3 :^||:||Ar1||x...x||Ar ||- ^ isdefinedtobetr(3 (e o||Jcx:^|))I jc: V II : | i||x . . . x\\Xn\\ - ^ is defined to betr(V7 (e o|j? ::^l)),where T : (A Ix . . . x | | ^ | x (ATI - ||A ix . . . x\\Xn\\ is the canonicalprojection.


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376 GONZALO E. REYESj . \\x: \\: ll^i |x . . .x ^ l - * isthe composite

IJS JX . . .x\\Xn\\ - l ^U - ^ 1 x:0 \ \ :\XX |x. . .x\\Xn\\ -> isthe composite

||;r1||x...x||jr || J I ^ ^ .Remark The reader has certainly noticed that wehave defined a nonstandardinterpretation offormulas asmaps into , rather than . However, given sucha nonstandard interpretations of a formula , \\X ||x . . .x|- Xi,| l * ^ we obtain astandard interpretation of , namely, ||X\ ||x. . .x\Xn || 6Q > simply bycomposing ||x : ^>|| with the counit Q : -> .

To finish this section, we shall describe aformal system MAO (for"modaladjoint operators") based onGentzen's sequents. These sequents, followingBoileau and Joyal [2] will be expressions ofthe form Vx , where is a finiteset offormulas (ofthe language already described), asingle formula andXa finite setofvariables containing all the free variables of and . We shallas-sume that sequents satisfy 1-8below. This system partly follows [10], p. 134:1. Structural rules:1.1 pVxp1 2 ThxpTUlp] Yxg

T\-Xg13 h * *TUlp}hxq1.4 lV*

\J{y\ Q5 \ - \ j\y) ' T[t/y]\- x [t/y]9

where tisfree foryin and .2. Logical rules:2.1 pVxT and_L \- xp.2.2 rhxp qiffr Vxpand rYxqpv q Vxriffp \ - xrand qVxr2.3 p\- xq- *r iffp / \q Vx r2.4 p VxVX iffpYX\ j x] 3X \ - pif hX\ j[X} Aprovided that xX.3. Identity rules:3.1 \-xt = t3.2 t = S9 [t/x] \~x [s/x],provided that tand sare free forxin .


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REFERENCE AND MODALITY 3774. Rules on special symbols:4.1 hMx=* (xeVari)4.2 (a,b) = (c,d) V

xa= c

(a,b) = (c,d)\- xb = d4 3 ,z= \- u[x,y,z\

provided that xand yare not free in or .5. Coincidence rules:5.1 h *X*5.2 (a,b)X


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378 G O N Z A L O E . R E Y E St h a t a rule of inference is sound u n d e r ||. || if the conc lus ion is valid whenevert h e premises are.Theorem 3.1 (S o u ndness of MAO) Assume that the geometric morphism - ^ S / locally connected. T hen allsequents of M AO arevalid and allrules ofinference of M AO aresound, under any interpretation.Proof: To simplify ou r c o m p u t a t i o n s , we first define a forc ing re la t ion as fol-lows: C h [au. . . ,a n] iff C G t r ( e ||x: \ \ )(a u. .. 9an) whe r e t r ( . . . ) is thet r a n s p o s e of ( . . . ) for the ad j u nc t io n H .Lemma 3.2 Ih satisfies the usual clauses for theforcing of Beth- Kripke-JoyaLProof: We shal l d o o n l y th e clause V, th eo t h e r s being s imilar . Let A xAB l l (x> y ) : s p | l > . Wehave th e following equivalences:

C\hvy [a] (defini t ion of Ih)Cetr(eQo\\x:vy \\)(a) (definition of II x: vy II)C e t r ( V ( c 0 o |(x , jQ:g|) ) ( ) . .(Proposition A.4, Appendix)(v en\$x,y): \$)c(a) = T c

By definition of v , the last line is equivalent to VC - > C G C vJ G 5 ( e f l "Using Proposition A.4 of the Appendix once again, this is equivalent to

v C ' ^ C G C v i e ^ C tr(e || (x, y): \\) (a, b)which, in turn, is equivalent to

V C ^ C G C v E ^ C h [a9b]by definition of Ih.

By the argument used in Proposition 4.1 (4), one can easily conclude that 0 | 2 |


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R E F E R E N C E AND MODALITY 379Connectivity axiom:

{ v )9O 9O \- xO( ).Indeed, we have the following result whose proof will appear elsewhere:Theorem3.3 (Soundness of MAO with Connectivity axiom) Assume thatthegeometric morphism -> S is locally connected and connected. Then all se-quences of MAO aswellas theConnectivityAxiom are validand all rulesof in-ference are sound, under any interpretation.

4 Kinds andvariable sets with coincidence relation The main result of thissection concerns the relation between ( ( )) and S( ). We first "lift" thefunctor to a functor

:( ( ) ) - > ( )which sends (X, x) into (AX, AX), where AX is the transpose of x via theadjunction H , i.e ., AX = A x and sends (X, x) - ^ (Y, y) into / .Proposition4.1(a) bx is a coincidencerelation iff AX is a coincidencerelation.(b) f is a morphismo/ ( ( )) iff Afis a morphism o/ 8( ).Proof: Easy exercise in transposition. As an example, the assertion that ;r(i>2) AX( 2,3) ^ AX( u 3) is equivalent to the existence of thedotted arrow in the commutative diagram:

AXxAXx AX < ( A (7ri> 2)> ô( 7 2 > 7 3)> ô( 7ri>7 3)>) 0 0

< > > x ,which in turn is equivalent (bytaking transposes) to the existence of the dottedarrow of the commutative diagram

X x X x X < ^(^'7r2), ^(7r2, 7r3)> ô( 7 l, 3)>) ( f i ) ( ) x ( )! I ( )X ( Q)i I

( ^ x .But this last statement is equivalent to x(x,x2) x(Xi,Xi) ^ ^(xi, r3).Similarly, the assertion that / is a morphism is equivalent to the existenceof the dotted arrow in the commutative diagram:

AX X AX < > r/ x/ > )


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380 GONZALO E. REYESBy taking transposes, this is equivalent to the distance of the dotted arrowin

XxX , ( ) x ( )

^ ( < ) .But this last condition says t h a t / is a morphism.Proposition 4.2 Assume that 8 -> is ageometricmorphism ( H ) . Then:S( ( ) - ( ) preservesfinite limitsand has a right adjoint such thatthefollowing diagram is commutative:

S( ( )) t= =t8(0)

s ^ Proof: The fact that preserves finite limits is easy an d left to the reader.

We define as t he functor which sends (E, E) into ( ( E ) , ( E ) ) , where r(E) = ( E ) , an d which sends a morphism (E, E) - U (F9 F) into ( ) : ( E ) - ^ ( F ) .Proposition4.3(a) ( E ) is a coincidence relation(b) (o ) is a morphism o/ ( ( )).Proof: Let us prove (b) only, since (a) is quite similar.T he assertion that a is a morphism is equivalent to the factorization of thehorizontal map as indicated:

E x E ( g E ' f X ) > x 12

\ i^

X of t h e m a p < ( E ) , ( F ) o (ce) X ( )> , i.e., we obtain that ( ) isa morphism.We now check that H . In other words, we have to check that wehave ana tu ra l bijection:

(AX,AX

)^^ ( E , E)E ( )

( ^ ^) ^ ( (E ), ( E )) G S( ( ))where tr( ) is the transpose of a via H . Since this bijection exists at thelevel of maps of 8 and maps of S, it is enough to show that a is a morphism ifft r ( ) is a morphism. Once again we have the following equivalences:


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REFERENCE AND MODALITY 381a is a morphism

AX X AX < * * * > * * ' * * * > , X

XxX S satisfies the Frobenius's condition, then so does7fo: ( )- ^S ( ( )).

Proo/ : Wedefine o: ( )^S ( ( ))asthemapwhichsends (E, E)into(7r0(E), TroE)) where 0(E) will be defined presently, and a morphism/ : (E, E) ->(F, F) into ro( / ): ro(E)- 7 o(F). Later on, we shall check that r0 is a functor.We first define the transpose of T0(E), namely 7roE: ( r0E)2 - , as the fi-nal structure on r0E for which E: E 0E is a morphism of ( ). In otherwords, o E is the smallest coincidence relation on 7r0E such that E < ;E.To make this definition precise, let D = { G ( o E)21 is a coincidence rela-tion and E -> 2 x( r0E)2 -2U be the restriction of the evaluation to functions in D. If e:( T O E )

2->

Dis its exponential transpose, we define 7roE to be the composite

o E : ( * o E ) 2 - ^ Q I ) ^ .Using set- theoretical notation

*oE= n{ e Q ( o E ) 2 | e D } .


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382 GONZALO E. REYESClaim(a) dE* AvoEo l(b) If d and df are coincidencerelationson X, then d < d' iff t (d) < tr(tf')(c) Ifd is a coincidencerelation on ro(E), then oE < d iff E < tr(rf) o \.Proof:

Ad(a): E < \forall G D (by the very definition of D ). Therefore, E < { o r 7 2 E | G D j= 7 r o E o r 7 2 E .Ad(b): simple exercise in transposition as in Proposition 4.1.Ad(c): =>: from (a),

E tr E)) i|i tr(cf) i|l ro(F) is amorphism in S( ( )), provided t h a t / is amorphism in8 ( 0 ) .In fact, consider the coincidence relation d = o(F) ( r 0 / ) 2 . Since tr(rf) ill = tr( o(F) o ^ o / 2 , we have that

E < Fof2 < tr( o(F))o 2F o/ 2 = tr(rf) or;2.By applying (c), we conclude that 7 o (E ) < d. We now check that 7r0 H , i.e.,tha t we have a natural bijection

r o ( E , E ) - ^ ( ^> J f)S( ( Q)(E ) (%G( ).Since tr( ) gives such a bijection for maps, we need only show th a t / is a mor-phism iff t r ( / ) is a morphism. But we have the following equivalences:

r o ( / ) is a morphismfi*o(E)SfiW2 E < t r ( o / 2 ) o r y 2 E E < o ( / ) 2 o r ? 2 E E < ^ o t r ( / ) 2

Assume now that r0: -* S satisfies the Frobenius condition. We shall provetha t its lifting satisfies the same condition, i.e., we shall prove that if( F , F ) =( E , E ) X (y, y) , then ( o ( F ) , o ( F ) ) = ( o(E)AO ( E > ) X(^W-By assumption, o(F) = ro(E) x yand we need only prove that

Ko(F)((a9y) a',y'))=Ko( Aa>


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R E F E R E N C E AND MODALITY 383In fact, this follows from the last claim (b) and the fact that the transposeof

^r f')] asfollows from the equivalence o ( E ) 2 2 ) ( ) ( Q ) _ r ( ^ ( )

(7 0E)2 X AY2 ^ ^ eXe e


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384 G O N Z A L O E. REYESSince 7 o E isthe smallest coincidence relation satisfying the second condition, = ^AT OE-

Recalling from the remark after Proposition 2.6.2 (2.6) tha t 8( ) can bealternatively described as the category of partial equivalence relations on objectsof , we may formulate the previous lemma as follows:Lemma 4.6 Let( E , E ) G 8( ). Then loE(T ) isthe smallest partialequivalence relation on r0E containing S >> ( r 0E ) 2, whereSis given as theimage factorization

E 2 ^ - > ( r 0 E ) 2

ofuiLemma 4.7 Assume that 0: -* S satisfies the generalized Frobenius con-dition.If

(F, F) ( E , E )1 l(AY, AY)^(AX, AX)

is apull- back, then forall (a, f), ( ', f') e7 o ( , ') y(f, f') = ((o, f) , (o ', D ) . w / o ( , ' ) r (f, r) = V,E(e)= ., 6(e')= 'E(


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R E F E R E N C E AND MODALITY 385th e lifting of r0 satisfies the generalized Frobenius condition, since ^xi^i(Xi)i - (II/-Y,-)/ is monic.O n the other hand, the following example (particular case of Example 2(2.2)) shows that the lifting of 0 does not always satisfy this generalizedFrobenius condition (even when r0does). Consider 2 = {0 - 1} and

Set ! Set**.( rIn this case, AS = (S -**> S), 7ro(Eo - A E ) = E l9 and (E0 - A EO = E o,with obvious actions on morphisms. The object of truth- values is = ( o - ^ O, where o = {0, ,1}, j = (0,1), e(0) = 0, e ( |) = e(l) = 1. Using our iden-tification 8( ) = PER (S) , define (E,R) as follows:

E = ( E 0 = {0, 1, 2, 3}^, {0,1,2} = E 0with (0) = 0, ( l) = (2) = 1, (3) = 2;

i? = (R0M R {) wherei?o = {(0,0),(1,1),(2,2),(3,3),(0,1),(1,0),(2,3),(3,2)}

an d is the restriction of a2.Define (F, S) by means of the pull- back diagram

(F,S) > (E, )i i({0},({0})2) ^ ^ - > ( {0, l), ({0, l}) 2)

where / = {0} {0,1} is the inclusion, uo(O) = uo(3) = 0, uo(\) = wo(2) = 1,u (0) = U\ (2) = 0 and ux (1) = 1. Obviously, wis a morphism. F rom the defi-nition of ( F , S ) ,F = ({0,3} ^+ {0, 2})S=(S 0J^S 1)9

where 5 0 = {(0,0), (3,3)}, S = {0,2}2, a' is the restriction of a and /?' is the re-striction of 2.T he map E - -> r0E is given by the diagram

0 * ti | I M E ,

We compute the image of i? under iy| to obtain R' = (i? - ^ i? ) where/?) = {(0,0),(1,1),(2,2),(0,1),(1,0),(1,2),(2,1)}, R\ = E? and 7 is the restrictionof the identity.In other words, the smallest partial equivalence relation containingR'9i i o E ( ) = E xE . Thus, o E = .O n the other hand, F - ^ 7 OF is given by


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386 G O N Z A L O E . REYES

The image of S under jris S' = {S'o - * - > S [) 9 where So = {(0,0),(2,2)}, Si ={0,2}2, and ' is the restriction of the identity. But then S' is already th e partialequivalence relation loF( T ) .Since in this case ^ j?) = ( A o F) o, we obtain tha t o ( F )(0,2) T andhence tha t the diagram

( o( F ) , o ( / 7 ) ) > ( i r o ( E ) , o T o ( E ))I tr()0}, )> j > ({0,1}, )

is not a pull- back.We finish this section with a reformulation of Frobenius conditions for r0in terms of properties of .

Proposition 4 .9 L et - be a geometric morphism H . Assume thatA: S( ( )) -> ( ) has a left adjoint 0 H . Then preserves exponentials{respectively / operations) iff 0:8( ) - >( ( )) satisfiesth e Frobenius con-dition (respectively th e generalized Frobenius condition).Proof: This is just a formal computa t ion (cf. Theorem 5 of Barr- Pare [1]): forthe case of exponentials, e.g.,

[ ro(E ) X X, S ( 0 ) - [ ro(E), Yxh( V) - [ E , ( 7 * ) ] ( )[ fo(E X AX), y ] S ( 0 ) - [ E x AX,AYhm - [E,AY* xhm

Therefore, (by Yoneda) o(E x AX) - o(E ) x XifA(Yx) ^ AYKx. The caseof I lf operations is similar.Corollary 4.10 Let & - S be a geometricmorphism A H . Assume that Ahas a left adjoint r0 satisfying th e Frobenius condition. Then th e lifting : ( ( ) - + ( ) preservesexponentials.Remark The cou n t e rex am ple Se t 2 o / ? -> Set together wi th P rop os i t ion 4 .9sh ows th a t : S e t ( ( ) ) - * S e t xP ( ) does no t p re se rve t he Uf opera tors a l -though 7r 0: St2P -> Set sat isfies th e generalized F robe n ius con dit io n .Acknowledgm entsAmon g the man y people who have helped me in on e way or anoth erwith the research repo rted in this paper, I owe a special debt of gratitude to Joh n M ac-namara and M arie Reyes. They first directed my attention to the G each- Bressan- G upta-Macnamara approach to intensional logic with great patience and considerableenthusiasm, gave m e advice and critized earlier drafts o f this pape r. As m entioned in theintroduction, the first part of this paper is taken, alm ost unch anged, from unpu blishedwork d on e in collaboration with M arie Reyes. I am grateful to her for letting me use ourresults an d for num erous discussions an d/ or suggestions on the subject of this paper.I would also like to m en tion fruitful discussions with M . H allett, F. W. Lawvere, R.Lavendhom m e, Th. Lucas, M . M akkai, R. P are, R. Wood, and M . Zawadowski. The


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R E F E R E N C E AN D M OD ALITY 387whole paper has been reorganized following useful suggestions of a referee unknown tom e.

Earlier versions of this paper were presented at several places: Montreal, Pittsburgh,Louvain- la- Neuve, and Halifax and I am grateful to the people who attended these talksfor their comments.

Finally, I would like to thank the Natural Sciences and Engineering Research Coun-cil of Canada and the Ministere de Education du Gouvernement du Quebec for thefinancial support that made this research possible.

R E F E R E N C E S[1] Barr, M . and R. Pare, "Molecular toposes," Journalof Pure and Applied Algebra,

vol. 17 (1980), pp. 127- 152.[2] Boileau, A. and A. Joyal, "La logique des topos, " TheJournalof Symbolic Logic,

vol. 46 (1980), pp. 6- 16.[3] Bressan, A., A GeneralInterpreted Modal Calculus. Yale University P ress, New

Haven and London, 1972.[4] Fourm an, M. P. and D . S. Scott, "Sheaves and logic," in Applications of Sheaves

(Proceedings, Durham 1977), edited by M. P. Fourman, C. J. Mulvey, and D. S.Scott, Lecture Notes in Mathematics 753, Springer- Verlag, 1979.

[5] Geach, P., Reference and Generality (Third edition), Cornell University Press,I thaca and London, 1980.[6] G upta, A. K., TheLogic of Common Nouns, Yale L aiversity Press, New Haven,

1980.[7] Higgs, D., "A category approach to Boolean- valued set theory," unpublished

manuscript, Waterloo, 1973. (See also [4].)[8] Keenan, E. L. and L. M . Faltz, Boolean Semantics for Natural Language, Synthe-

sis Language Library, D. Reidel Publishing Company, Dordrecht, Boston, Lan-caster, 1985.

[9] Kripke, S., Naming and Necessity (Revised and enlarged edition), Basil Blackwell,Oxford, 1980.

[10] Lambek, J . and P. J. Scott, Introduction to higher- order categorical logic, Cam-bridge Studies in Advanced Mathematics, Cambridge University Press, London,New York, New Rochelle, Melbourne, Sydney, 1986.

[11] Lavendhomme, R., Th . Lucas, and G. E. Reyes, "Formal systems for topos-theoretic modalities," Actes du colloqueen honneurdu soixantieme anniversairede Rene Lavendhomme (F. Borceux, editeur), Bulletin Soc. Math, de Belgique(Serie A), XLI (2), (1989), pp. 333- 372.

[12] Lawvere, F. W., "Equality of hyperdoctrines and the comprehension schema as anadjoint functor," in ProceedingsNew York Symposium on Applications of Cate-gorical Algebra, AMS, Providence, RI, 1970.

[13] Macnamara, J., A BorderDispute: ThePlaceof Logic in Psychology,A BradfordBook, The MIT Press, Cambridge, Massachusetts and London, England, 1986.


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388 GONZALO E. REYES[14] Macnamara, J. and G. E. Reyes, "The learning of proper names and count nouns:F o u nd a t i o na l and empirical issues," to appear.[15] Makkai, M. and G. E. Reyes, First- order categorical logic, SpringerLecture Notes

in Mathematics 611, Springer- Verlag, Berlin, Heidelberg, New York, 1977.[16] Moerdijk, I., "Lectures on topos theory at Sydney," unpublished, 1986.[17] Montague, R., "The proper treatment of quantification in ordinary English," For-

mal Philosophy:Selected Papersof RichardMontague (Ed. Richmond H. Thoma-son) , Yale University Press, New Haven and London, 1974.[18] Quine, W. V., "On what there is," in From a Logical Point of View, Harvard Uni-

versity Press, Cambridge, Massachusetts, 1953.[19] Reyes, G . and M. Zawadowski, "Formal systems for modal operators on locales,"

unpublished manuscript.[20] Reyes, M., A Semantics for L iterary Texts, Ph.D . thesis, Concordia University,Montrea l , 1988.[21] Reyes, G ., "Non- standard truth values and modalities," Proceedingsof the 1989

Siena Meeting on Modal and Tense Logics, unpublished.[22] Reyes, G . and H . Zolfaghari, "Topos- theoretic approaches to modality," Proceed-ings of the 1990 Como Meeting on Category Theory, forthcoming inSpringer-VerlagLecture Notes in Mathematics.[23] Reyes, G . and H . Zolfaghari, "Bi- Heyting algebras, toposes and modalities," un-

published.Department of MathematicsUniversityof MontrealCP 6128 Succ "A"Montreal QC CanadaH3C 3J7

Appendix In this Appendix, we derive the main result of Example 3 (2.2),namely the relation between forcing relation for in terms of forcing relationfor .Recall that we had a topos -> bounded. This means that 8 may be pre-sented as the category of sheaves over a site C in S. We have the diagram

S h ( C ) = >//r . | | ,

^ c - o

where a is the "associated sheaf functor", o is the constant presheaf 0(S ) (C) = S VC E C, 0(F) = lim c


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R E F E R E N C E AN D M O D A LI T Y 389LemmaA.I ( ) = setofclosed sieves (p)=cl{C Uo( )Fo(A) o ( ;ok

and counit given bye F^oUoU ldc

C ~ (to)c= (e coFiUoUiio)-Proof: Simple com putat ion .Corollary A.3 T he following diagram is commutative


32/33

390 GONZALO E. REYES

FoUoUAC) ( 7 l W ' ( C ) > UiFiFoUoUUC)e o c / ( C ) \ y/ U ec)

U (C)Proof: Look atitstranspose and useLemma A.2.

We now apply these resultstos c Shs(C). o i

We letI d s - ^ 0>, I d s c - -2J-> tobet h e units an d 0 0 - ^ I d s c - , ai - ^Idsh s(o be the counits ofthe corresponding adjunctions. F u r t h e r m o r e , we de-fine = tf0, = o/ an d let ,ebe th e unit an d counit, respectively.Let AS- > G begiven and consider the diagram (*)

Aovs 0 S o I V > 0 (*) 0 5 < ^ (eo)/s| I (eo)/

( I ) A 0 N / S : i QiThe square comm utes bynaturality of e0and the triangle also commutes:

A0S ^ ^ > 0 5( ? i) os ( 7 I ) O S

^~~* /SI n fact, S o (TJI)O S = (ei)i S bythe corollary and UAS/ s = / (e SA s) =/ ( Id ) = Id.

We compute (*)atCe C toobtain (in S) TAS -^ ( O)= o( / Q)

S \ I If ), )CViKs)cX i iAS(C) (i )c > (, )(C)

The m ap (ei), isjust what wecalled ( (K)C inLemma A . I .Let / s= (/I) OS tosimplify the notation.PropositionA4 Ct [(ls)c(s)] iffCeT( )( s(s)).Proof: Wehave the following equivalences:

CU- [Us)(s)] (def of Ih)(i )c(ds)c(s) = Tc((e) iQ)cT( )(ys(s)) = TC (def of(ei),a).CeT( )( S(s))


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R E FE R E N C E AN D M O D AL IT Y 391The fo llowing answers the questio n about Ih for D :Corollary A.5

(1) c t [(s)

c(s)] /# 3{c,-*C},

e/eCov(C) v/ e/ vcecc'h

lUs) ( s) ](2) Clh [ ] iff 3 { C, - + C} ieI E Cov(C) Vi /3J , 5{I C,=(4)c,(*,) 3{Cy,- * Ci}j eCov(C,)VyVC e C C'\ \ - [ ( l s)c' (Si)] Proof:(1) FromProposition A.4 and the definition ofD, we obtain

CK - [ ( l s) c(s )] iffC T( ) ( s( S) ) = y( T( ) ( s( s ) ) )if c e ci{c ec\\\v e c(C er( ) ( s( s ) ) \ \ }iff a{c,,-+c }, e / eCov(C) v/ 6 / vc ' e ciff 3 {C,-,-* C ) / e / E Cov(C) V 6 / VC 'e CC \ - [ Us) c( s) ]

(2) C Ih D ^[ ] iff3{C, - C}, e/ Cov(C) V E / 3, E 5 IC, = Us)c,( s) Q Ih D^[ ( / s)C / (s , ) ]iff 3 {C, ^C} ieI E Cov(C) V E / 3s ; E S

E Cov(C,) VyVC ECC'\ \ - [ Us) c - ( S i ) ] .

a topos-theoretic approach to reference and modality

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