Contexts in Dynamic Predicate Logic

Journal of Logic, Language, and Information7: 21–52, 1998. 21c 1998Kluwer Academic Publishers. Printed in the Netherlands.

Contexts in Dynamic Predicate Logic

ALBERT VISSERDepartment of Philosophy, Utrecht University, Heidelberglaan 8, 3584 CS Utrecht, The NetherlandsE-mail: [email protected]

(Received in final form 19 February 1997)

Abstract. In this paper we introduce a notion of context for Groenendijk & Stokhof’s DynamicPredicate LogicDPL. We use these contexts to give a characterization of the relations on assignmentsthat can be generated by composition from tests and random resettings in the case that we areworking over an infinite domain. These relations are precisely the ones expressible inDPL if weallow ourselves arbitrary tests as a starting point. We discuss some possible extensions ofDPL andthe way these extensions interact with our notion of context.

Key words: Context, dynamics, Predicate Logic, relation, resetting, register, variable, expressability,definability

Contexts manifest themselves even when notexplicitly definitionally introduced.

1. Introduction

1.1. METAMATHEMATICS OF DYNAMIC PREDICATE LOGIC

Dynamic Predicate Logic (DPL) was invented by Groenendijk and Stokhof (1991),see also our Section 2) as a specification language (or better: as a module fora specification language) of meanings for fragments of natural language. Mostof the research concerningDPL has gone into integrating it with versions ofMontague Grammar (see Muskens, 1991) and into integrating it with Veltman’sUpdate Semantics (see Veltman, 1996; Groenendijk et al. 1996).

DPL is a theory of testing and resetting of variables/registers. These are fun-damental operations in computer science. Thus, apart from its use in LogicalSemantics,DPL is a simple theory of these basic operations.?

DPL is a natural variant of Predicate Logic. It mainly differs in the treatmentof the scope of the existential quantifier. Certain basic truths about variables inPredicate Logic, however, fail inDPL (see Section 2, see also Hollenberg andVermeulen, 1994) for similar observations onDPLE). The study ofDPL and

? DPL is just one theory in a family of alternatives to Predicate Logic. In other alternatives“resetting a register” is replaced by other related actions, like “create a new register”(see Appendix Bfor references).

22 A. VISSER

its kin makes the dependence of these truths on the specific choice of scopingmechanisms in standard Predicate Logic visible.

In the light of the varied interest ofDPL, it seems a good idea to make a closerstudy of its metamathematical properties. We focus on the closely related questions:

� Which relations between assignments are expressible inDPL (in a sense wewill specify later)?

� How doesDPL treat its variables?

To throw some light on both questions a good notion of context inDPL is indis-pensable. In the next subsection we present our notion of context and discuss itsconnections and differences with notions of context in the literature.

1.2. CONTEXTS

Contexts appear in the literature in bewildering variety. We briefly discuss threemeanings ofcontext, viz.

1. contexts as syntactical objects with “holes,”2. contexts as assignments/environments/anchors,3. contexts as (structured) declarations of variables.

The last notion is the one that we are concerned with in this paper.A common occurrence ofcontextis as a “hole” in a syntactical object. This

notion is important in term-rewriting and proof theory. See Troelstra and van Dalen(1988a: 60), Barendregt (1984: 29, 375), Klop (1992: 12) for standard references.The contexts studied in this paper havenothingto do with thisnotion of context.

The second notion of context closer – but not equal – to the notion we areafter is a context as an environment or assignment. This usage is well established inLinguistics. E.g., in the Heim tradition one speaks about “context change potential”for the potential to change assignments. Usingcontextfor assignment is consonantto the way David Kaplan and the Situation Semanticists employ the word. Here acontext is the speaker situation: this situation provides values forI, here, now,: : :in the same way as an assignment provides values for the variables. For the use, inComputer Science, ofenvironmentfor assignment, see, e.g., Mitchell (1991: 383).In Situation Semantics sometimesanchoris used for assignment/environment.

The notion of context that will emerge in the present paper is best viewedas in harmony with a tradition in Proof Theory, Type Theory, Category Theoryand Computer Science in which a context isa declaration of variables. Somereferences for this usage are: Troelstra and van Dalen (1988b: 577); Jacobs (1991:29); Nederpelt (1994: 103, 104, 233, 253), etc. A context in this sense has alsobeen calleda basis. See, e.g., Barendregt (1991: 351) or Barendregt (1992: 149).In this last paper (Barendregt, 1992) the wordcontextis used for alinearly ordered

CONTEXTS IN DYNAMIC PREDICATE LOGIC 23

set of declarations as opposed to such a set simpliciter: see p. 199. Contexts arecalledreferent systems, in Vermeulen (1995). Contexts in our present, third, senseare clearly connected with contexts in the second sense: contexts in the third senseprovide the domains for contexts in the second sense.

We give a brief discussion to introduce some of the ingredients of contexts inthe third sense, which from now on we simply callcontexts. Variables are usuallydeclared as objects of a certaintype. Since, however, we are thinking aboutFirstOrderPredicate Logic here, we may ignore the types for the purposes of this paper.As we will see, what is left is still not completely trivial, especially not in a dynamicsetting.

The simplest form of a context is the context assigned to a formula� of PredicateLogic, viz. the set of its free variables. In the “indexed” treatment of the semanticsof Predicate Logic (see Jacobs, 1991) this context also occurs at the semanticallevel: the meaning of� in a given modelM is the pairhX;F i, whereX is FV(�),the set of free variables occurring in�, andF is a set of assignments fromX to thedomain ofM. Note that on the indexed perspective an assignment can be viewedas a pairhX; fi, whereX is the domain off . Thus, assignments/environments areobjects “in context.”

To develop more dynamical kinds of context, we consider two example sen-tences. Consider the sentenceshe is a mananda man comes in. In the PredicateLogical specification language ofDPL and Kamp’s Discourse Representation The-ory, DRT, we will render these sentences, e.g., as

MAN(x) and 9x:MAN(x):COMES-IN(x):

In the style of the “indexed” semantics we would give the first sentence as contextfxg and the second one;. Viewed dynamically we can interpret the first contextas embodying the information that a value forx is asked from the incomingassignment. This is analogous to the way in which the anaphorhe “asks for” avalue from (the interpretation of) the previous text, and, also, to the way in whichthe present king of Francesignals that a king of France is presupposed. Thus,an “indexed” style context signals the presence of a kind of presupposition. Thesecond context,;, signals that no previous value for any variable is demanded.

However, variables occur also in other ways in our formulas.9x newly intro-duces anx. Consider the Discourse Representation Structures orDRSs associatedto our sentences. We can represent these as:

h;; fMAN (x)gi and hfxg; fMAN (x);COMES-IN(x)gi:

It seems reasonable to consider the first components ofDRSs – these are the topboxes of the box representation – also as contexts. These first components signalwhich variables arenewly introducedor declared. Reflecting on the “indexed”contexts and theDRT-contexts, it is a small step to form combined contextshX;Y i,whereX andY are finite sets of variables. HereX will tell us which variables are

24 A. VISSER

imported from outside andY will inform us which variables are newly introduced.For example,

FARMER(x):9y:OWNS(x; y):DONKEY(y)

will have contexthfxg; fygi. Following up this idea it is not too difficult to developversions ofDRT that are compatible withDPL, i.e., versions ofDRT in whichresetting actions are associated toDRSs in a compositional way.? See, for example,(Visser, 1995) for an implementation of this idea. TheDPL-contexts of this paperare much like such pairshX;Y i. However, they have a third component: the futuredirected mirror image of the presupposition set. That such a mirror image shouldexist follows from the time symmetry of relation composition, the fundamentaloperation ofDPL. So we get contexts of the formhX;Y;Zi, whereX is the set offree or presupposed or input-constrained variables, whereY is the set of variablesthat may be reset and whereZ is the set of output-constrained variables.

To define contexts as objects of a certain sort is a rather empty exercise if we donot specify certain operations on contexts. The most important operation is withoutdoubt themergeor simple addition of contexts. We represent the merge by “�.” Forexample, the indexed contexts for Predicate Logic were just sets of free variables.The corresponding merge is set theoretical union, consonant with the familiar factthatFV(�^ ) = FV(�)\ FV( ). What would be the merge on pairshX;Y i asdiscussed above??? Here more than one choice is possible. One such choice is:

hX;Y i � hX 0; Y

0i = hX [ (X 0 n Y ); Y [ (Y 0 nX)i:

What does this choice mean? To gain better understanding, we must digress for amoment on the subject of variable clashes.

In dynamic semantics for Predicate Logic, we have to agree upon conventionsto handle clashes of variables. For example, what happens whenx is alreadyintroduced and assigned a value and we try to introduce it or a new value for it again?A wide range of solutions is available. We may decide that the new introductionwill bounce. Nothing will happen andx will keep its old value. This is the strategyof classicalDRT (see also Visser, 1995). We may also opt for the contrary strategyof throwing away the old value and making the new value the actual one. Thisis the strategy of classicalDPL. A third possibility is to stack the new value ontop of the old one (see Vermeulen, 1993; Hollenberg and Vermeulen, 1994; Visserand Vermeulen, 1996). A fourth possibility is to distinguish between the variable-qua-identifier, or variable-as-syntactic-object, and the variable-as-underlying-file.If we make this, philosophically plausible, distinction, we can view the secondintroduction as arenaming: the original file named “x” loses its name; a newlyintroduced file is baptized “x” (see Vermeulen, 1995; Groenendijk et al., 1996;Visser and Vermeulen, 1996).? OrdinaryDRT with sets as contexts does not have this property.?? To be precise, we assume at this point, locally, that the intersection of the first and the second

component is empty.


Strategies introduced to avoid clashes of variables are reflected by the definitionsof contexts and their merge. Consider the merge that we introduced for pairs of setsof variables. Ifx is both inX and inY 0, thenx will be in X [ (X 0 n Y ), but not inY [ (Y 0nX). The reason is that our example corresponds to a “bouncing” strategy:if x is already present, any attempt to newly introduce it will fail. Another example:the contexts, calledreferent systems, studied in Vermeulen (1995) and Groenendijket al. (1996) contain both files and file names. Thus, the fundamental distinctionis already present at the contextual level. In Visser and Vermeulen (1996) it isattempted to handle the variable clashes fully at the level of the contexts.

As this point, sufficiently many ideas have been touched upon to sketch ageneral picture of the role that in my view contexts have to play in dynamics. Thispicture is more extensively described in Visser and Vermeulen (1996). A contextis essentially a set of files or discourse referents. These files are best thought of asfeatureless objects. Also part of the contexts is machinery that “programmes” whathappens to files when two contexts are merged: whether various files are fused,whether a given file is aborted, whether a file is renamed, whether a file is relocatedin a structure, etc. So the context of a formula� will contain the files used in� plusthe information about how the files are “handled” by�: are theyfree, fresh: : : ?

In dynamic semantics contexts will be an important part of the larger picture.They may be enriched with contents that could be as varied things as sets of assig-ments, pictures or relational databases. The contexts determine what happens to thecontents in merging full meaning objects. Thus, contexts are “demi-mondain:” theycan be directly assigned to syntactical objects like terms and formulas, but are alsonaturally paired with the semantical objects, like assignments, sets of assignments,resetting relations, update functions. All these semantical objects are objects “incontext.”

Updating does not just result in growth of contentual information, but alsoin extension of context. In this sense, information growth is many dimensional.It is important to realize what this means for improved versions ofDPL. Theassignments that are modified will be not assignments on a rigid set of variablesbut on a changing set of files. These files will be enriched with certain informationto regulate their interactions in merging. Such an “assignment in context” can beviewed as a special case of an information state in update semantics, which is a setof assignments or a set of assignment-world pairs in context.?

We mention two successes of the use of contexts in dynamics. The first successis the clarification of the relationship betweenDPL-style semantics andDRT-style semantics. See, e.g., Vermeulen (1995) and Visser (1995). The second is thesolution of the problem to integrateDPL and Update Semantics, including thesemantics formaybe, by Groenendijk et al. (1996).

? DRT-meanings can be viewed as information states in context. Remember thatDRT-meaningscan be interpretations of formulas with free variables. Thus their contexts will witness a presupposi-tional structure.DRT-meanings are partial states or states under presuppositions. (See Visser, 1994,for a first attempt to come to grips with the idea of a partial state.)

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Contexts in this paper will be partly atypical. In various versions ofDPL andDRT as introduced in Vermeulen (1993, 1995), Visser (1995), Visser and Ver-meulen (1996), Groenendijk et al. (1996), contexts make their appearance in thevery definition of the semantics: they are part of the design of the semantics.Moreover, they admit a realistic interpretation: they could model real mechanismsof handling files. In this paper, however, we study classicalDPL as introducedin (Groenendijk and Stokhof, 1991). No contexts occur in the definition of clas-sical DPL. Thus, in this paper, contexts can only appear asobjets trouves: theyemerge as a useful device to characterize expressibility inDPL. Contexts resultfrom “abstracting away” or “erasing” certain properties of the predicate logical lan-guage (both vocabulary and structure), thus yielding an underspecified language.Underspecification simply means here that the denotations or meanings of thecontexts are properties of relations, rather than relations.

A second feature of contexts will be lacking in this paper: assignments willhave no associated context. ClassicalDPL is a theory oftotal assignments. Thus,the only possible context these assignments could have is the fixed set of allvariables. Explicitly carrying this context along is pointless. Thus, our contextswill be exclusively associated to the resetting relations.

The present paper illustrates that contexts are useful, not just for their philo-sophical plausibility, but also because they are technically fruitful.

1.3. THE CORNERSTONEARGUMENT

Consider the sentencePx:9x:Qx. The leftmost occurrence ofx is in some sense“used” in a different way from the rightmost occurrence. This fact gives rise to theproblem how these two occurrences have to be treated in the semantics.

It is often argued that one of the most salient and exciting aspects of theorieslike DPL andDRT, viz. the treatment of the problem of multiple uses of variables,is irrelevant for linguistics. We give the argument (for the specific case ofDPL) asit occurs in an anonymous reviewers comments.

Consider a typical example of multiple use, say,Px:9x:Qx. Such formulasare never needed when one translates natural language sentences intoDPLformulas. When translating an indefinite noun phrase, one can always use afresh variable, so that in any subformula of the translation, no variable occursboth free and bound. Thus, instead of a formula like the above, one could usePx:9y:Qy with no loss. The existence of a formula like the first one is anartifact of the formalization, and the study of such formulas does not revealanything about natural language.

I call this the cornerstone argument, since it denies a place to what I consider oneof the central issues in the development of dynamic versions of Predicate Logic. Ihave two objections to the argument, a minor and a major one. Before giving theseobjections, let me point out that not all sources of interest of theories likeDPLderive from the study of Language. The treatment of variables is, e.g., an important


subject in Computer Science. The “overloading” of variables in large programmesis often nearly unavoidable. Moreover, the precise treatment of variables is almostdefinitionally of logical interest.

We turn to my minor objection. Our objector is certainly right that “one canalways use a fresh variable, so that in any subformula of the translation, no variableoccurs both free and bound.” But this is only obvious for cases where the variable-qua-syntactic-object is really extrinsic to natural language. I am thinking here ofcases where natural language clearly employs different mechanisms for referentlinking than the linking by label ofDPL. Natural language simply does not employvariables-qua-syntactic-objects to realize the linking of the referents associated toanaphors. However, in Visser and Vermeulen (1996) the idea is explored to treatroles/casus/items-of-feature-information as dynamic variables. Here re-using thesame “variable,” likesubject is quite plausible. Now, there must be somethingin the semantics to stop the unification of all subjects: : : . Our counterargumentleaves open the possibility that the cornerstone argument convincingly shows thatthe study of multiple uses of variables is unilluminating for the study ofanaphors.

My main objection is that the cornerstone argument presumes that the only wayin which theories likeDPL could reveal anything about natural language is by beinga specification language whose formulas are assigned to natural language texts bya certain translation procedure. I submit that having such a specification languageand such a translation procedure (for a reasonable fragment) is not something wemust rest content with. The cornerstone argument itself indicates that the variablesof the specification language (in the case of the treatment of anaphors) are extrinsicto the natural language. The variables are added in the translation procedure bythe translator. This means that the translation procedure could very well producethe correct meaning objects, but can never be considered as a model of the processof understanding, since the resolution of anaphors is clearly partially based oncontentual considerations. The ordinary regulative principle of compositionality asemployed in the Montague tradition already shows that not just the end result oftranslation is important, but also the way the result is obtained. The programme ofDynamics, as I see it, adds a further constraint. Dynamics is directed at providingmodels of interpretation processes. Thus, we want our semantical interpretationto reflect the interpretation process studied. The translation procedure relied uponin the cornerstone argument does not provide any insight in the dynamics ofinterpretation.

So what is the role of theories likeDPL in the study of natural language? Thealternative way, of viewing the relationship of dynamical systems and natural lan-guage that I would like to push, is this. I want to view these systems asanaloguesofnatural language. The systems (languages with their given semantics) are analogousto natural language in certain respects. E.g., they allow long distance linking ofvariables/anaphors. We study these systems as fruitflies to receive hints about whata semantics for (a reasonable fragment of) natural language should look like.DPLand natural language have long distance linking, but the detailed mechanisms seem

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to be quite different. The one,DPL, has linking by label and the other: : : . Stillthe careful study of the linking machinery ofDPL could help us understand otherlinking strategies. Analogy is helpful not only by providing points of agreement,but also by giving us a good understanding of points where the analogues differ.

Let us look at the matter from a different perspective. What is the business ofSemantics? Well, it is to associate objects from a semantical algebra to a syntacticalalgebra. The semantical objects and the syntactical objects differ in some salientrespects. This is what gives the exercise a point. The semantical objects are bothricher and poorer than the syntactical ones. E.g., aDRS lacks the linear organisationof the linguistic string. E.g., a set of possible worlds contains all kinds of stuff notpresent in a string of letters. Now it seems to me that we should be able to understandthe semantical algebra as a thing on its own. Part of the essence of the semanticalobjects is revealed by the natural interactions between these objects, in other words,what makes the algebra into an algebra. The cornerstone argument asks us to leavea substantial part of these interactions unspecified or arbitrarily specified, sincethey are supposed not to matter.

Let me end with a question: how can one expect to understand natural language,other than as one way of doing thingsamong others?

1.4. CONTENTS OF THEPAPER

Section 2 is a straightforward introduction to Dynamic Predicate Logic. It containsall the technicalmaterial the reader needs to know. For a discussion of the appli-cations to discourse phenomena, however, the reader should consult Groenendijkand Stokhof (1991). Section 3 presents our theory of contexts inDPL. We studycontexts as mathematical objects in their own right and establish their connectionsto language and semantics. Some materials concerning the information ordering oncontexts are placed in an appendix. In Section 4, we treat the Switching Property.This property is characteristic for theDPL-expressible relations. Section 5 containsthe main result of the paper: a relation over an infinite domain isDPL-expressibleiff it “has” a context and satisfies the Switching Property. In the next section, wetouch on the subject of extending theDPL-language with new operations such asconjunction and disjunction. We consider the question whether such extensionssupport a good theory of contexts. We will produce two extensions that arecom-plete for all relations that have a context. In other words: all such relations areexpressible in those extensions. In Appendix B we have a brief look at the idea ofmaking the contextpart of theDPL-semantics.

2. What Is Dynamic Predicate Logic?

We provide the basic definitions ofDPL. Nothing in this section pretends to beoriginal. We start by introducing some basic relational notions.


DEFINITION 2.1. LetX be any non-empty set.Rel(X) is the set of binary rela-tions onX, i.e.,Rel(X) := }(X �X). LetR;S 2 Rel(X). We define:

1. The compositionR � S of R andS is defined by:x(R � S)y :, 9z xRzSy.Note that composition is in the order of application.

2. The dynamic implication(R! S) betweenR andS is defined by:

x(R! S)y :, x = y and8z(xRz ) 9u zSu):

Our use of! here overloads the symbol, since we also use it for implicationin the object language. We write:(R) for (R! ;)

3. idX is the identity relation.R is aconditionor testif R � idX .4. ConsiderY � X. We definediag(Y ) := fhy; yi 2 X �X j y 2 Y g.5. dom(R) := fx 2 X j 9y xRyg andcod(R) := fy 2 X j 9x xRyg.

The notion of dynamic implication was first introduced by Kamp in his pioneeringpaper (1981). Note that:::(R) = (idX ! R) = dom(R). The relations in therange ofdiag are precisely the conditions. Writing(Y ! Z) := (XnY )[Z, wehave:

� diag(X) = idX� diag(;) = ;

� diag(Y \Z) = diag(Y ) � diag(Z) = diag(Y )\ diag(Z)� diag(Y [Z) = diag(Y )[ diag(Z)� diag(Y ! Z) = diag(Y )! diag(Z).

Thus,diag embeds the structureh}X;X; ;; \ ; \ ; [ ;!i homomorphically intothe structurehRel(X); idX ; ;; �; \ ; [ ;!i. We will sometimes confuse, in therelational context, the setX with the relationdiag(X). We need some furtherrelational notions specifically concerned with relations between assignments.

DEFINITION 2.2. LetD be a non-empty domain and letVar be a set of variables.LetR 2 Rel(DVar), Y � D

Var, f 2 DVar andV � Var. We define:

� fd1;:::;dnv1;:::;vn

is the result of changing the values off on thevi to di.� fIV g :, for all v 2 V f(v) = g(v).� [V ] := IVarnV . We write[v] for: [fvg].� Y is a hV i-set if f 2 Y andfIV g ) g 2 Y . R is a hV i-condition if it is

thediag-image of ahV i-set.Y is finitely restrictedif Y is ahIi-set for somefinite I. A condition is finitely restricted iff it is the image underdiag of afinitely restricted set.

If we want to make the dependence ofI or [:] onVar orD visible, we add them assubscripts.

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We collect some simple facts concerning these notions. We have:

� I; = [Var] = DVar �D

Var, IVar = [;] = idDVar

� IV � IW = IV \W , [V ] � [W ] = [V [W ]� IV \IW = IV [W , [V ]\ [W ] = [V \W ]� IV [IW � IV \W , [V ][ [W ] � [V [W ].

Note that the classical meaning of the existential quantifier as a “cylindrification”can be given as:�x(Y ) := dom([x] � diag(Y )). We turn to the definition ofDPL.

DEFINITION 2.3. We define aDPL-languageL as follows.L is a structurehPred;Ar;Var;Coni. HerePred is a set of predicate symbols;Ar is a functionfrom Pred to the natural numbers (including 0);Var is a possibly empty set ofvariables andCon is a, possibly empty, set of constants. LetRef := Var [Con bethe set ofreferents. We will usev; w; : : : for variables,c; c0; : : : for constants andr; s; : : : for referents. The set ofL-formulas, ForL, is the smallest set such that:

� P (r1; � � � ; rn) 2 ForL, for P 2 Pred, Ar(P ) = n andr1; : : : ; rn 2 Ref

� >,?, r = s, 9v are inForL for r; s 2 Ref andv 2 Var

� If �; 2 ForL, then so are�: and(�! ).

I feel that it is more faithful to the semantics to leave out the brackets in theformation rule for the dotofficially, but nothing important hangs on this choicein this paper. We get an ambiguous syntax, but still unique meanings, since theoperation of composition – the semantic counterpart of “.” – is associative. Analternative notation for9v, is [v :=?] (random reset). We use:(�) and8v(�) asabbreviations of, respectively,(� ! ?) and(9v ! �). If x 2 Var andr 2 Ref

andx andr are distinct, we write[x := r] for: 9x:x = r.

DEFINITION 2.4. A DPL-modelM for a DPL-languageL is a structurehD; Ii,whereD is a non-empty set, thedomainofM; I is a function which assigns to eachpredicate symbolP of PredL anAr(P )-ary relation onD and to each constantcan element ofD. AssM, the set ofassignmentsforM, isDVar. Considerr 2 Ref.We define:

jrjM;f :=

(f(r) if r 2 Var

I(r) if r 2 Con

The interpretation function[:]M : ForL ! Rel(AssM) is given as follows.

� [P (r1; � � � ; rn)]M := diag(ff 2 DVar j hjr1jM;f ; � � � ; jrnjM;f i 2 I(P )g)� [>]M := id

DVar, [?]M := ;

� [r = s]M := diag(ff 2 DVar j jrjM;f = jsjM;fg)


� [9v]M := [v]D� [�: ]M := [�]M � [ ]M� [(�! )]M := ([�]M ! [ ]M).

We write� �M for [�]M = [ ]M. We define validity inDPL by:

� j=M :, 8f; g(f [�]Mg ) 9h g[ ]Mh):

As usual,� j= iff � j=M for all modelsM appropriate for the given language.A binary relationR is definablein aDPL-modelM for a languageL if there is

anL-formula� which definesR, i.e.,R = [�]M.

We will often suppress the subscriptM, when the model is clear from the context.We could extend theDPL-language with function symbols by copying the way thisis done in ordinary Predicate Logic. However, for the kind of result we are aftersuch an extension is immaterial, since the usual trick to eliminate function symbolsworks also inDPL – with a small twist. E.g.,P (f(g(x))) will be translated to:::(9u:G(x; u):9v:F (u; v):P (v)).

We remind the reader of Geach’s Donkey Sentence:If a farmer owns a donkey,he beats it. This sentence can be translated intoDPL in a compositional way as:

(9x:farmer(x):9y:donkey(y):owns(x; y)! beats(x; y)):

One striking feature ofDPL is that it is not “structural:” the values the predicatesymbols may assume are not all the possible meaning objects provided by thesemantics; we only allow tests. A second striking feature is thetime symmetryof resetting and composition, which contrast strongly with ourtime asymmetricintuition about, say, the meaning ofP (x):9x:Q(x). The asymmetry of our intuitionmay be explained by the fact that we tend to think more in terms ofsuccessfulresetting, i.e.,M j= P (x):9x:Q(x), than just in terms of what the resetting relationis.

Ordinary Predicate Logic can be interpreted inDPL as follows. We supposethat the predicate logical language has as connectives and quantifiers:>,?, ^,!,9x. We translate as follows:

� (:)� commutes with atomic formulas and with!� (�^ )� = �

�:

�

� (9x(�))� = ::(9x:��).

We find: [��]M = diag( [[�]]M), where [[:]] is the usual valuation function ofPredicate Logic. Our translation is compositional. It shows that we may considerPredicate Logic as a subsystem ofDPL.

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3. Contexts for Dynamic Predicate Logic

In this section, we study the notion ofcontextand its connections with relationsand language. We placed some materials concerning the information ordering oncontexts in an appendix, since, on the one hand, they are conceptually relevant andhave a clear place in the total picture, but, on the other hand, they have no directbearing upon the main results of the paper.

3.1. INTRODUCTORYREMARKS

To motivate our notion of contexts, we first give an intuitive discussion of sub-stitution and kinds of variable occurrences inDPL. In Predicate Logic variablesmay occur in a formula in two ways: freely and bound. The free variables admit(under certain conditions) substitution. The bound variables may be renamed salvasignificatione (�-conversion). Let us write�tx(�) for: the result of substitutingt forx in �. In Predicate Logic we have, e.g.,

fI(c)x 2 [[�]]M , f 2 [[�cx(�)]]M:

What is the proper analogue of this fact forDPL? To simplify the discussion wewill only treat a special case and refrain from giving official definitions. ConsidertheDPL-formulaP (x):9x:Q(x):9x:R(x). We have:

1. f I(c)x [P (x):9x:Q(x):9x:R(x)]Mg , f [P (c):9x:Q(x):9x:R(x)]Mg

2. f [P (x):9x:Q(x):9x:R(x)]MgI(c)x , f [P (x):9x:Q(x):9x:R(c)]Mg.

Meditation upon (1) and (2) suggests, that, inDPL, we have to distinguish two kindsof substitutionleft substitution andright substitution and corresponding to thesekinds two kinds of “free occurrence:”left free andright free. We also speak ofinputoccurrences andoutputoccurrences. Following temporal intuitions – ignoring theessentially time-symmetric character of resetting and composition – we may alsocall the left free occurrences simplyfree and the right free occurrencesactivelybound. Now consider the following formula, call it�0, in which we have taggedoccurrences ofx with superscript numerals.

P (x1):9x2:Q(x3):9x4

:::(9x5:R(x6)):S(x7):

We see thatx1 is a (left) free or input occurrence. Left substitution forx will causeit to be replaced. If we formT (x0):�0, in the semantics the values assigned tox

0

andx1 will be unified. If we form9x:�0, x1 will be “bound” or “initialized” bythe new “quantifier.” Symmetrically,x7 is right free or actively bound. It will bein the scope of right substitution. If we form�0:T (x

0), the values ofx7 andx0

will be unified, If we form�0:9x, x7 will be “aborted.” Neitherx1 norx7 are opento �-conversionsalva significatione. x3 is not accessible for substitution, nor isit �-convertible: replacingx2, x3 andx4 by, say,y, will result in a formula that


resetsy, which�0 does not do. We callx3 a garbageoccurrence: it is somethingthat “exists,” but is no longer “used.”? x6 is also inaccessible for substitution, butin addition it can be�-converted: replacingx5 andx6 by y does not change themeaning of�0. We say thatx6 is classicallybound. Finally, we considerx2, x4 andx

5. These are “occurrences” in a purely syntactical sense only: they do not represent“files” carrying information, but just signal that incoming files labeledx shouldnot be “unified” with outgoing files labeledx. We say that these “occurrences” areblockers. x5 is not a blocker in�0 as a whole.

Contexts, in our present set-up, signal the presence of input occurrences, ofblockers and of output occurrences.?? They are abstract – in comparison withformulas – in the sense that they contain no information about the number orthe place of these occurrences. Contexts can be studied independently from theirconnection with the logical language.

Contexts are familiar from Predicate Logic. There the context associated with aformula� is simply the setX of free variables of�.z A salient property of contextsin Predicate Logic is as follows. SupposeX is a context for�, then [[�]]M isX-restricted, i.e.,f 2 [[�]]M andfIXg, impliesg 2 [[�]]M. Most of the workin this section will be devoted to proving the appropriateDPL-analogue of thisproperty of Predicate Logic.

3.2. CONTEXTS, CONSIDERED BYTHEMSELVES

In this subsection, we treat contexts as mathematical objects in their own right. Thenatural connection withDPL will surface in the subsequent subsections.

DEFINITION 3.1. ADPL-context is a triplehI;B;Oi, whereI,B andO are finitesets of variables and whereInB = OnB, or, equivalently,I [B = O [B. The setI is theinput set, i.e., the set on which the incoming assignments are constrained.The setO is theoutput set, i.e., the set on which the outgoing assignments areconstrained. Finally, the setB is the set of blocks. This is the set of variables forwhich the identity between input and output value is cut through. The “block” is abarrier between past and future, breaking the link between input- and output-value.We write# for is definedor converges, and" for is undefinedor diverges. Define:

� id := h;; ;; ;i

� hI;B;Oi � hI 0; B0; O

0i := hI [ (I 0nB); B [B0; (OnB0)[O0i

� hI;B;Oi ! hI 0; B0; O

0i := hI [ (I 0nB); ;; I [ (I 0nB)i

? The notion ofgarbageis studied in Vermeulen (1995) and Visser and Vermeulen (1996).?? In fact, there are good reasons also to put witnesses ofgarbageinto the contexts. We will not

do this in the present paper, since it is not necessary for our results here. Moreover, adding garbageleads to considerable complication of the framework and it necessitates bringing in Category Theory.We refer the reader further to Visser and Vermeulen (1996).

z The reader is referred to Jacobs (1991) for a category-theoretical framework appropriate for thestudy of contexts in Predicate Logic.

34 A. VISSER

� hI;B;Oi � hI 0; B0; O

0i , I � I0; O � O

0; B � B

0 � B [ (I 0 \O0)� \ is a partial operation on contexts, defined by:

hI;B;Oi\ hI 0; B0; O

0i :=

8>><>>:hI \ I 0; B \B

0; O \O

0i ifB � B

0[ (I \O);

B0 � B [ (I 0 \O0)

" otherwise

We will usec,d, : : : as variables over contexts. We writeIc for the first componentof c, etc.

The meanings of these objects, relations and operations will become apparent inSection 3.3.

LEMMA 3.2. The operations�,! and \ are well defined.Proof. To see that� is well defined, note that:

(I [ (I 0nB))[ (B [B0) = (I [B)[ (I 0 [B0)

= (O [B)[ (O0[B

0)

= (OnB0)[O0[ (B [B

0):

It is trivial that! is well defined. For the proof that\ is well defined we refer thereader to the Appendix. 2

THEOREM 3.3.The contexts withid and � form a monoid. Moreover,� is apartial ordering.

The proof of the theorem is easy. In the appendix we will show that\ , if defined,is the infimum, w.r.t.�.

Consider the monoid of contexts. It can be represented in an alternative way,as follows. The monoidA is the monoid on two generatorsa andb, given by theequations:a�a = a, b�b = b andb�a�b = b. The table of the monoidal operation� is as follows.

� e a b ab ba aba

e e a b ab ba aba

a a a ab ab aba aba

b b ba b b ba ba

ab ab aba ab ab aba aba

ba ba ba b b ba ba

aba aba aba ab ab aba aba


The monoid of contexts is now given as the set of functions fromVar toA that areon all, but finitely many arguments equal toe. We put:(f � g)(v) := f(v) � g(v).A triple hI;B;Oi “translates” to a functionf with, e.g.,f(x) = ab iff x 2 I,x 2 B andx 62 O, etc. A functionf translates to a triplehI;B;Oi with, e.g.,x 2 I iff f(x) 2 fa; ab; abag, etc. It is easily seen that these “translations”give us an isomorphism of monoids between the representations.A is in factisomorphic to the monoid of contexts in the case thatVar = fxg. The alternativerepresentation is possible by the fact that in our monoidal operation treats allvariables “independently.” It is not difficult to extend the structure ona; b; : : :, toget a function representation also for!,� and\ .

In this paper we will stick to the set representation, since this representationis closest to the relational notions we will need to formulate our theorem oncontexts and relations – the theorem that tells us what the contextsdo. The functionrepresentation, however, has two advantages. First, it is easier to use for doingcomputations “in the head.” Secondly, its connection with the framework developedin (Visser and Vermeulen, 1996) to study contexts, is more perspicuous.

3.3. CONTEXTS AND RELATIONS

We turn to the connection between contexts and relations. We show that thisconnection “commutes” w.r.t.�=�,! =! and�=�. We fix a non-empty domainD.

DEFINITION 3.4. Consider a relationR onDVar. R is anhI;B;Oi-relation ifR = (II � R � IO)\ [B]. We say thatc is a contextfor R, if R is a c-relation.Ris an IBO-relation if R is anhI;B;Oi-relation for somehI;B;Oi. We assign toeach contextc the property or “meaning”fcgD, the set of allc-relations onDVar.(We will often suppress the subscriptD.)

The heuristic for this property is as follows.R is hI;B;Oi means thatR is onlyconcerned with the values of the incoming assignment onI;R only cares about thevalues of the outgoing assingment onO; all this under the constraint that in goingfrom input to output only values of variables inB are changed. Before provingsome facts about the notion introduced above, we sample some immediate insights.

� f;; [Bc]g � fcg

� fh;; B; ;ig = f;; [B]g� R is ahI; ;; Ii-relation precisely ifR is anhIi-condition.

We show that� on contexts describes the “information ordering” on contexts. Theidea is simply thatc is moreinformative thand if fcg � fdg .

THEOREM 3.5.1. c � d) fcg � fdg .

36 A. VISSER

2. SupposejDj � 2. Then:fcg � fdg ) c � d.

Proof. Let c = hI;B;Oi andd = hI 0; B0; O

0i. We prove (1). Letc � d andR 2 fcg . We want to show:R = (II0 �R �IO0)\ [B0]. Trivially, R is contained in(II0 �R�IO0)\ [B0]. We show(II0 �R�IO0)\ [B0] � R. Supposef 0II0fRgIO0g

0

andf 0[B0]g0. It is immediate, thatf 0IIfRgIOg0. We show thatf 0[B]g0. Supposev 2 B

0nB. Then,v 2 I0\O

0. We find:f 0(v) = f(v), sincev 2 I0. Moreover,

f(v) = g(v), sincefRg,R � [B] andv 62 B. Finally,g(v) = g0(v), sincev 2 O0.

SinceR is ac-relation, we may conclude thatf 0Rg0.We prove (2). We writeXc for VarnX. Let jDj � 2 andfcg � fdg . Since,

[B] 2 fcg , we have[B] 2 fdg . Hence, using the facts of Section 2, we find:

[B] = (II0 � [B] � IO0)\ [B0]

= [(I 0c[B [O

0c)\B0]:

SincejDj � 2, it follows that:B = (I 0c [B [O0c)\B0. Now it is immediate that:

B � B0 � B [ (I 0 \O0).

The arguments thatI � I0 and thatO � O

0 are analogous to one another. Wegive the argument for theI-case. Suppose, for a reductio, thatv 2 InI 0. Letd ande be distinct elements ofD. We write[v = d] for the test:is f(v)=d?. Consider therelationR := [v = d] � [B]. Clearly,R 2 fcg and, so,R 2 fdg . Considerf withf(v) = d. In casev 2 B

0, we have:f evII0fRgIO0g. Since, as we have alreadyshown,B � B

0, we find:f ev [B0]g. Hence, sinceR 2 fdg , we havefdvRg. Quod

non. We turn to the case thatv 62 B0. SinceI 0nB0 = O0nB0, we find:v 62 O

0. So:fevII0fRgIO0g

ev. SinceB � B

0, we findf ev [B0]gev . Thus, sinceR 2 fdg , we have

fdvRg

dv . Quod non. 2

Theorem 3.5 shows clearly that contexts stand in a many-many relation to relations.Thus, the questionWhat is the context ofR?has no definite answer. In the appendixwe show that, but for one notable exception, every relation has a most informativecontext. The next lemma may be used in some cases to simplify the verificationthat a relation ishI;B;Oi.

LEMMA 3.6. R is a hI;B;Oi-relation iffR = (II �R � IO\B)\ [B].?

Proof. It is clearly sufficient to show that foranyR � [B]:

(II � R � IO)\ [B] = (II �R � IO\B)\ [B]:

From left to right is immediate, sinceIO � IO\B. For the converse, supposef0IIfRgIO\Bg

0 andf 0[B]g0. We have to show:gIOg0. Considerv 2 O. In casev 2 B, we haveg(v) = g

0(v). Supposev 2 OnB. Then, alsov 2 InB. Wehave:g0(v) = f

0(v), sincev 62 B. Moreover,f 0(v) = f(v), sincev 2 I. Also

? Note that, due to our design choice that in contextsI n B = O nB, the triplehI;B;O\Bi isnot generally a context.


f(v) = g(v), sincefRg, R � [B] andv 62 B. Putting the identities together, wefind g0(v) = g(v). 2

The following lemma is quite useful in applications (see Example 3.14). We writeA := I [B.

LEMMA 3.7. SupposeR is an hI;B;Oi-relation andf 0IIfRg. Then there is auniqueg0, such thatf 0Rg0IBg. Thisg0 has the following property: for any set ofvariablesJ , if f 0IJf , theng0IJg. As a consequence, we find thatg

0IIg and, hence,g0IAg. So, also,g0IOg.

Proof. LetR be anhI;B;Oi-relation withf 0IIfRg. Any g0 with f 0Rg0IBg,must satisfy:f 0[B]g0IBg. So the only possible choice of such ag0 is: f 0 �

(VarnB)[ g � B. We verify thatg0, thus defined, satisfiesf 0Rg0. It is sufficient toshow thatgIOg0. This, in its turn, follows immediately from the property in thelast part of this lemma.

Consider any setJ such thatf 0IJf . Supposev 2 J . In casev 2 B, we haveg(v) = g

0(v). In casev 62 B, we have,g(v) = f(v), sincev 62 B, Moreover,f(v) = f

0(v), sincev 2 J . Finally f 0(v) = g0(v), sincev 62 B. Putting things

together, we findg(v) = g(v0), as desired. 2

Note that in the lemmaO plays no significant role. Due to the “forward looking”and time asymmetric nature, however, of the definitions of implication and validityin DPL, it is sufficient for most applications. An immediate consequence of thelemma is that ifR is anhI;B;Oi-relation, thendom(R) is anhIi-condition and(by symmetry)cod(R) is anhOi-condition.

THEOREM 3.8.SupposeR is a c-relation andS is a d-relation. ThenR � S is ac � d-relation.

Proof. Let c = hI;B;Oi, d = hI 0; B0; O

0i andc�d = hI;00B;00O00i. It is easy tosee that:R�S � (II00�(R�S)�IO00 )\ [B00]For the converse, suppose thatf

0[B00]g0,f0II00f , f(R�S)g andgIO00g

0. We have to show:f 0(R�S)g0. For someh, we havefRhSg. We partitionVar into three setsX1 := O [ I

0, X2 := Bn(O [ I0[B

0)andX3 := Varn((BnB0)[O [ I

0). Define:h0 := h�X1[ g0�X2[ f

0�X3. We showthatf 0Rh0Sg0. We first prove thatf 0Rh0. We check the conditions for applying thefact thatR is anhI;B;Oi-relation.

1. f 0II00f and, hence, sinceI � I00, f 0IIf .

2. fRh.3. hIO[ I0h0 and, hence,hIOh0.4. We show thatf 0[B]h0. Consider a variablev not in B. We have to showf0(v) = h

0(v). We can only run into trouble in casev is not inX3, i.e., if vis in BnB0 or in O [ I

0. The first possibility is excluded, by the fact thatvis not inB. Supposev is in O [ I

0. Then:h0(v) = h(v), by definition.R is

38 A. VISSER

anhI;B;Oi-relation, soR � [B]. We may conclude thath(v) = f(v), sincev 62 B. In casev 2 O, we find:v 2 OnB = InB � I � I

00. In casev 2 I0,

we find:v 2 I0nB � I

00. So in both cases:v 2 I00. Sincef 0II00f , it follows

thatf(v) = f0(v). Composing the identities, we findh0(v) = f

0(v), as desired.

By (1–4), we may conclude thatf 0Rh0. Next, we check the conditions for applyingthe fact thatS is anhI 0; B0

; O0i-relation.

1. We havehIO[ I0h0 and, hence,h0II0h.2. hSg.3. We havegIO00g

0 and, hence, sinceO0 � O00, gIO0g

0.4. We show thath0[B0]g0. Consider a variablev not in B0. We have to showh0(v) = g

0(v). Inspecting the definition ofh0, we see that our desired iden-tity can only fail if eitherv 62 B or v 2 B andv 2 O [ I

0[B

0. We con-sider the case thatv 62 B. We already showed thatf 0[B]h0. We assumedf0[B00]g0. Hence:h0[B]f 0[B [B

0]g0. So,h0[B [B0]g0. Sincev 62 B [B

0, wefind:h0(v) = g

0(v). Next, we consider the case thatv 2 B andv 2 O [ I0[B

0.Since, we choosev outside ofB0, we need only consider the possibility thatv 2 O [ I

0. We have, by definition,h0(v) = h(v). Sincev 62 B0, we find

h(v) = g(v). By our early assumption,gIO00g0, whereO00 = (OnB0)[O0. If

v 2 O, then clearlyv 2 OnB0, and we findg(v) = g0(v). If v 2 I

0, we havev 2 I

0nB0 = O0nB0 � O

0, hence, again,g(v) = g0(v). Putting the identities

together, we findh0(v) = g0(v), as desired.

By (1–4), we have:h0Sg0. 2

THEOREM 3.9.SupposeR is a c-relation andS is ad-relation. ThenR! S is ac! d-relation.

Proof. SupposeR is a c-relation andS is a d-relation. Let c = hI;B;Oi,d = hI 0; B0

; O0i and(c ! d) = hI;00 ;; I 00i. Trivially, (R ! S) � [;]. Moreover,

sinceid � II00 , (R! S) � (II00 � (R! S) � II00)To prove the converse, supposef 0II00f andf(R! S)f . We have to show that

f0(R! S)f 0. Supposef 0Rg0. By Lemma 3.7, there is ag such thatfRgII00[Bg0.

It follows thatg0II0g. Sincef(R ! S)f , we can find anh, such thatgSh. Againapplying Lemma 3.7, we find anh0 with g0Sh0. 2

We close this subsection with a language-free soundness result.

DEFINITION 3.10. A relation onDVar is DPL-expressible overD iff it can begenerated by composition from resettings[v] and finitely restricted conditions overD.

THEOREM 3.11.EveryDPL-expressible relation overD is an IBO-relation.


The proof is an obvious induction on the way the relation is generated.

3.4. CONTEXTS AND LANGUAGE

We turn to our discussion of how contexts are connected to formulas.

DEFINITION 3.12. We assign to everyDPL-formula� a contextc�. Define:

� cP (r1;��;rn) = hV; ;; V i, whereV = fr1; � � � ; rng\Var

� c> = c? = h;; ;; ;i, cr=s = hfr; sg\Var; ;; fr; sg\Vari, c9v = h;; fvg; ;i

� c�: = c� � c andc(�! ) = (c� ! c ).

We writeI� for Ic� , etc.

Note that the definition correctly defines a function, by the associativity of�. Wenow prove the main theorem of this section.

THEOREM 3.13.For every formula�, [�] is a c�-relation onDVar.Proof. The proof is by induction on� using Theorems 3.8 and 3.9. The atomic

cases are easy. 2

We obtain the following picture of the way contexts work:� is mapped toc� byabstracting both from part of the vocabulary and part of the structure.c� is mappedto fc�gD, a property of relations. Via a different route� is mapped to the relation[�]M. The two routes are connected by the theorem that[�]M 2 fc�gD.

Note thatc� is not always the�-minimal context of[�]M, as is witnessed bythe fact thatcx=x = hfxg; ;; fxgi and that[x = x]M is the identity onDVar, thusadmitting the contexth;; ;; ;i.

Example 3.14. We provide two examples of how Theorem 3.13 in combinationwith Lemma 3.7 can be used to verify a valid principle forDPL. We first prove:

�:� j= �, if B� \ I� = ;:

SupposeB� \ I� = ; andf [�:�]g. We have to produce anh with g[�]h. Sincef [�:�]g, we can find aj with j[�]g. By Theorem 3.13:j[B�]g. SinceB� \ I� = ;,we find:gII�j. So,gII�j[�]g. By Theorem 3.13 and Lemma 3.7 we can find anh

with g[�]h.As a second example we prove:

� j= �: ) � j= , if B� \ I = ;:

SupposeB� \ I = ;, � j= �: andf [�]g. We have to produce anh with f [ ]h.By our assumptions, there arei andj such thatf [�]i[ ]j. Hence,f [B�]i and sofII i. Thus,fII i[ ]j. We may conclude that there is anh with f [ ]h.

40 A. VISSER

The examples demonstrate the role contexts must play in the formulation ofschematic principles forDPL.

4. The Switching Property

In Section 3 we introduced contexts orIBOs as properties of relations and showedthat every[�] is an IBO-relation. A first conjecture for characterizing theDPL-expressible relations would be that these are precisely theIBO-relations. We willsee, however, that this conjecture is false. To characterize theDPL-expressiblerelations we need one extra property: the Switching Property. In the present sectionwe will prove that theDPL-expressible relationsdo have the Switching Property(soundness). In Section 5 we will show that everyIBO-relation on an infinitedomain that has the Switching Property isDPL-expressible (completeness).

DEFINITION 4.1. A relationR onDVar has theswitching propertyif it is eithera condition or there are variablesx and y (not necessarily distinct), such thatR = dom(R) � [x] � R � [y] � cod(R). If the second case obtains, we call thevariablesx andy involved a pair ofswitching variables. There might be more thanone pair of switching variables.

There are various other ways to define the Switching Property, but, I submit, theone presented here is the most natural one. In the lemma below, we collect somehelpful insights.

LEMMA 4.2. SupposeR is a relation onDVar

1. R = dom(R) � R � cod(R).2. SupposeR is C � T � C 0, whereC andC 0 are conditions and whereT is a

relation. Thendom(R) � C = dom(R) andC 0 � cod(R) = cod(R).3. SupposeC is a condition. Then,C � [x]�C � [x] = C � [x] and[x]�C � [x]�C =

[x] � C.

The easy proof is left to the reader.

THEOREM 4.3.EveryDPL-expressible relation overD has the switching prop-erty.

Proof. In caseR is a condition, we are done. SupposeR is not a condition. Asis easily seen,R must be of the formC � [x] � S � [y] �C 0, for some variablesx, y,some conditionsC, C 0, and some relationS. (In a formula� definingR, x wouldcorrespond to the first existential quantifier occurring in�, y to the last. Note thatwe allow x andy to be the same variable and even the first and last existentialquantifier occurrence to be the same occurrence.) We have, using Lemma 4.2:


dom(R) � [x] � C � [x] � S � [y] � C 0 � [y] � cod(R) =

dom(R) � C � [x] � C � [x] � S � [y] � C 0 � [y] � C 0 � cod(R) =

dom(R) � C � [x] � S � [y] � C 0 � cod(R) =

dom(R) � R � cod(R) = R:

2

The results of Section 3 and of this section combine to the obvious “soundness”-result:

THEOREM 4.4.EveryDPL-expressible relation overD is an IBO-relation withthe Switching Property.

Example 4.5. We show how to use the Switching Property to prove that certainrelations are notDPL-expressible. SupposejDj � 2.

� LetR := [x := y; y := x], wheref [x := y; y := x]g :, g = f

f(y);f(x)x;y :

R is anIBO-relation, with contexthfx; yg; fx; yg; fx; ygi. SupposeR has theSwitching Property.R is evidently not a condition. Letv; w be a pair of switch-ing variables. LetfRg, f(v) = d, andd 6= e. Using the fact that the domain ofR is the set of all assignments, we find:f

ev (dom(R))f ev [v]fRg[w]g(cod(R))g.

Hence, by the switching property:f evRg. ButR is obviously injective. So wehave a contradiction.

� Suppose our modelM is the usual structure of the natural numbers. LetS := [x := x + 1], wheref [x := x + 1]g :, g = f

f(x)+1x . S is an IBO-

relation with contexthfxg; fxg; fxgi. S does not have the Switching Propertysince:S is not a condition;S has as domain the set of all assignments;S isinjective.

� Let T := [(9x_9y)], where[(9x_9y)] := [9x][ [9y]. T is an IBO-relation.The best possible context for it ishfx; yg; fx; yg; fx; ygi.? Suppose thatThas the Switching Property.T is not a condition, so we can find switchingvariablesv andw. By symmetry we may assume thatv 6= x. We can findfandg with fTg andf(x) 6= g(x). Choosed with d 6= g(v). Using the factthat the domain ofT is the set of all assignments, we find:

fdv (dom(T ))fdv [v]fTg[w]g(cod(T ))g:

By the Switching Property:fdvTg. But we have two distinct variablesx andv such thatfdv (x) = f(x) 6= g(x) andfdv (v) = d 6= g(v). This is clearlyimpossible.

? We will discuss this phenomenon in more detail in Section 6.3.

42 A. VISSER

5. The DPL-Expressible Relations on an Infinite Domain

In Sections 3 and 4 we have seen that theDPL-expressible relations areIBO-relations with the Switching Property. Here we show the converse – for the casethat the domain,D, is infinite.

THEOREM 5.1.LetD be aninfinite set. Then theDPL-expressible relations overD are precisely the IBO-relations with the switching property onD

Var.Proof. One direction is immediate by our previous results. LetR be an

hI;B;Oi-relation with the switching property. In caseR is a condition we aredone. SupposeR is not a condition and letx, y be a pair of switching variables. Bythe switching property

R = dom(R) � [x] � R � [y] � cod(R):

Note thatdom(R) is anhIi-condition and thatcod(R) is anhOi-condition. Thus,it is sufficient to show that[x] � R � [y] is DPL-expressible.[x] � R � [y] is anhInfxg; B;Onfygi-relation, wherex; y 2 B. After renaming, we see that it issufficient to prove that anyhI;B;Oi-relationR, with x; y 2 B andx 62 I andy 62 O is DPL-expressible.

We will assumex 6= y. In casex = y, the proof is simpler. To increasereadability, we will specifyR in aDPL-language that we introduce along the way.SupposeI = fi1; � � � ; img,BnO = fb1; � � � ; bng andO \B = fo1; � � � ; opg. Herethe ik are supposed to be mutually distinct and similarly for the other sets. SinceD is infinite there is a coding of finite sequences of elements ofD in D. Par abusde langage, we will confuse this coding with our ordinary sequences of elementsof D. Our language has an(m+ 1)-ary predicate symbolP , where:

hd1; � � � ; dm; ei 2 I(P ), 9f; g fRg and f(i1) = d1; : : : ; f(im) = dm

and e = hg(o1); � � � ; g(op)i:

Remember that[y := x] is short for9y:y = x. The formula� is given by:

9x:P (i1; � � � ; im; x):[y := x]:9o1: � � � :9op:(y = ho1; � � � ; opi):9b1: � � � :9bn:

Here “y = ho1; � � � ; opi” stands for the obvious condition. Note that[�] is anhI;B;Oi-relation. We claim thatR = [�]. Suppose first thatfRg. Take:

� h1 := fhg(o1);��;g(op)ix .

Remember thatx 62 fi1; � � � ; img. We have:

h1[P (i1; � � � ; im; x)]h1 , hh1(i1); � � � ; h1(im); h1(x)i 2 I(P )

, hf(i1); � � � ; f(im); hg(o1); � � � ; g(op)ii 2 I(P ):

Clearly, f and g witness thathf(i1); � � � ; f(im); hg(o1); � � � ; g(op)ii is in I(P ).Next we set:


� h2 := (h1)hg(o1);��;g(op)iy

� h3 := (h2)g(o1);:::;g(on)o1;:::;on

� h4 := (h3)g(b1);:::;g(bn)b1;:::;bn

.

We find (usingy 62 fo1; � � � ; opg):

f [9x]h1 h1[P (i1; � � � ; im; x)]h1 h1[[y := x]]h2

h2[9o1: � � � :9op]h3 h3[y = ho1; � � � ; opi]h3 h3[9b1: � � � :9bn]h4:

Sincef [B]g, it is easy to see thatg = h4.For the converse, supposef 0[�]g0. Leth1, h2, h3, be such that:

f0[9x]h1 h1[P (i1; � � � ; im; x)]h1 h1[[y := x]]h2

h2[9o1: � � � :9op]h3 h3[y = ho1; � � � ; opi]h3 h3[9b1: � � � :9bn]g0:

We are going to apply the fact thatR is an hI;B;Oi-relation. By the fact thath1[P (i1; � � � ; im; x)]h1 and by the definition ofP , we can findf andg such thatfRg, f(i1) = h1(i1),: : : f(im) = h1(im), andhg(o1); � � � ; g(op)i = h1(x). Sincex 62 fi1; � � � ; img andf 0[x]h1, it follows thatf 0IIf . Collecting what we have, wesee:

� fRg.� f

0[B]g0

� f0IIf .

By Lemma 3.6 we need to check:

� gIO\Bg0.

Considerv 2 O \B. We have:h1(x) = hg(o1); � � � g(op)i and, hence,h2(y) =hg(o1); � � � g(op)i. We havey 62 O, and, thus, we get:h3(y) = hg(o1); � � � g(op)i.Sincev 2 fo1; � � � ; opg andh3[y = ho1; � � � ; opi]h3, we find:h3(v) = g(v). Finally,v is not among theb1; � � � ; bn, and thus:g0(v) = g(v).

Putting the itemized insights together, we may conclude:f0Rg

0. 2

6. Extensions of the DPL-Language

We consider three extensions of theDPL-language. One with conjunction interpret-ed as intersection of relations, one with a new quantifier99 and one with disjunctioninterpreted as union of relations. We will show that our contexts work for each ofthese extensions. The contexts provided for disjunction are not optimally informa-tive and intuitively queer, however. We will give some hints on how we think this

44 A. VISSER

apparent defect should be repaired. We show that99 is definable using and that inthe system with99 all IBO-relations are expressible.

6.1. CONJUNCTION

We study the effect of adding intersection of relations to theDPL-repertoire. Oneway of of thinking aboutR\S is as:reset simultaneously via R and via S, andcompare the results. If they are equal, make the output of our new relation theshared output, otherwise abort. At the syntactical level, we reflect the new opera-tion by extending the language ofDPL by adding the clause:

� If �; 2 L, then(�^ ) 2 L.

We will call the new language:L(^). The semantic clause is:[�^ ] := [�]\ [ ].We define intersection of contexts as follows.

� hI;B;Oi^hI 0; B0; O

0i :=hI [ I 0 [ (O [O

0)n(B \B0); B \B

0; O [O

0[ (I [ I 0)n(B \B

0)i:

Note that:

I [ I0[ (O [O

0)n(B \B0)[ (B \B

0) = I [ I0[O [O

0[ (B \B

0)

= O [O0[

(I [ I 0)n(B \B0)[ (B \B

0):

So^ is a well-defined operation on contexts. We define:

� hI;B;Oi � hI 0; B0; O

0i :, I0 � I andB � B

0 andO0 � O.

Note the difference between� and�. It is easy to see thatis precisely theinfimumwith respect to�.

THEOREM 6.1.LetR be ac-relation and letS be ad-relation. Then,R\S is a(c^d)-relation.

Proof. Suppose that the conditions of the theorem are fulfilled. Letc =hI;B;Oi, d = hI 0; B0

; O0i, (c^d) = hI;00B;00O00i. Supposef(R\S)g. Clear-

ly, fII00f(R\S)gIO00g. We may conclude thatf([B]\ [B0])g, i.e., f [B \B0]g.

So:(R \S) � (II00 � (R\S) � IO00)\ [B00].For the converse, supposef 0II00f , f(R\S)g, gIO00g

0, andf 0[B00]g0. Evidently,f0IIf , fRg, gIOg0, andf 0[B]g0. Ergof 0Rg0. Similarly f 0Sg0. Hencef 0(R\S)g0.

2

We extend the definition ofc� to the new language by adding the clausec(�^ ) :=c�^c . Theorem 6.1 immediately yields the next theorem.


THEOREM 6.2.[�] is a c�-relation for every� 2 L(^).

We consider an example. Letc� = hI;B;Oi. SupposeB = fb1; � � � ; bng. Let := (�:9b1: � � � :9bn^>). We have:[ ] = [::(�)] = [(> ! �)]. We computec .

c = (hI;B;Oi � h;; B; ;i)\ h;; ;; ;i

= hI;B;OnBi\ h;; ;; ;i

= hI [ ;[ (OnBn;); ;; OnB [ (In;)i

= hI; ;; Ii:

Thus, in this example our conjunction on contexts gives us the “intuitive result,”i.e., [ ] is anI-condition.

Let us say that theDPL(^)-relations over a given domainD are the relationson this domain generated by finitely restricted conditions and resettings usingcomposition and intersection. The results of the present section show that theDPL(^)-relations are allIBO-relations. The results of the next section, will implythat, conversely, everyIBO-relation isDPL(^).

6.2. A NEW EXISTENTIAL QUANTIFIER

We define:99x(R) := ([x] �R � [x])\ Ifxg. In caseR is anhI;B;Oi-relation, wesee that99x(R) := ([x]�R� [x])\ [Bnfxg]. It follows – by the result of Section 6.1– that99x(R) is anhInfxg; Bnfxg; Onfxgi-relation.

We extend the language ofDPL by adding the clause:

� If � 2 L andx 2 Var then99x(�) 2 L.

Note the overloading of notations. The new language will beL(99). The newsemantical clause is the obvious:[99x�]M = 99x[�]M. 99x is definable inL(^) asfollows. Supposec� = hI;B;Oi. Let Bnfxg = fb1; � � � ; bng, then we can put(9x:�:9x ^ 9b1: � � � :9bn) for 99x(�).

THEOREM 6.3.For anynon-empty domainD, theDPL(99)-expressible relationsare precisely the IBO-relations.

Proof (Sketch). We have already seen that everyDPL(99)-expressible relationis IBO.

For the converse, suppose thatR is anhI;B;Oi-relation. LetI = fi1; � � � ; img,BnO = fb1; � � � ; bng andO \B = fo1; � � � ; opg. Here theik are supposed to bemutually distinct and similarly for the other sets. Take aDPL(99)-language with an(m+ p)-ary predicate symbolP , where:

hd1; � � � ; dm; e1; � � � ; epi 2 I(P ) , 9f; g fRg and

f(i1) = d1; : : : ; f(im) = dm and

f(o1) = e1; : : : ; f(op) = ep:

46 A. VISSER

Let u1 : : : up be variables disjoint fromI [B [O. Let� be given by:

99u1(� � � (99up(P (i1; � � � ; im; u1; � � � ; up):9o1: � � � :9op:

o1 = u1: � � � :op = up:9b1: � � � :9bn) � � �):

Clearly, that[�] is anhI;B;Oi-relation. The verification thatR = [�] is along thelines of the proof of Theorem 5.1. 2

Example 6.4.We show how to define the three relations of Example 4.5. We doa bit more than the theorem promises, because we give explicit descriptions of thepredicate “P ” employed in the proof of Theorem 6.3.

� [x := y; y := x] can be defined by:

99u(x = u:9x:x = y:9y:y = u):

Note that this gives us the expected context:hfx; yg; fx; yg; fx; ygi.� [x := x+ 1] can be defined by:99v(x = v:9x:x = v + 1). (Strictly speaking

we are working in a relational language, sox = v+1 is a suggestive notationfor, say,S(v; x).) The context produced by the formula is as expected.

� We can define[(9x_9y)] by:

99u(99v(:(:(x = u)::(y = v)):9x:9y:x = u:y = v)):

This gives us the contexthfx; yg; fx; yg; fx; ygi. We will discuss this contextin the next subsection.

Since99 is definable using , the “expressive completeness” of99 w.r.t. theIBO-relations implies the “expressive completeness” of^. Finally, we can translatePredicate Logic intoDPL(^), by changing the9-clause of our earlier translationto: (9x(�))� := 99x(��). Remarkably, the old and the new translation produceprecisely the same relations at the semantical level.

Operations like!, ^ and99 are not themselvesactions in the sense of oursemantics. They are transformers of actions. Yet there is a tendency to understand99x(�) dynamically as a sequence of actions:reset x; execute�; set x back to itsoriginal value. The problem with this way of viewing things is summarized withthe question: where do we store the original value ofx, so that we can restore itat the end?DPL-semantics does not supply the right kind of “memory” to realize99x(�) as a sequence of actions. We can do that (or, rather, something very muchlike it) in the richer semantics of Vermeulen’sDPLE (see Vermeulen, 1993), whereunder avariable namewe do not store just one value, but a stack of values. Herethe original value ofx is simply stored “under” the new one.


6.3. DISJUNCTION

In this section we have a brief look at the problem of adding disjunction/unionto DPL. Adding disjunction/union evokes problems that are definitely beyond thescope of the present paper. So we can only offer some tantalizing remarks.

One way of of thinking aboutR[S is as:Choose betweenR andS, and resetvia the relation chosen. At the syntactical level, we reflect the new operation byextending the language ofDPL by adding the clause:

� If �; 2 L, then(�_ ) 2 L.

We will call the new language:L(_). The semantic clause is:[�_ ] := [�][ [ ].What could be a context for[x][ [y]? Some experimentation shows that the

best we can do is:hfx; yg; fx; yg; fx; ygi. This seems a wasteful way to representthe variable handling of this relation. Our intuition tells us that[x][ [y] is a pureresetter and not something that “constrains” w.r.t.x andy. The resetting part of ourcontexts is somehow too crude to represent “choice” well. The example does nottell us that in any strict sense our present framework is wrong. It just suggests that,possibly, we could do better. We might try out richer notions of context. The mostobvious proposal is to take as a context in the new sense a set of contexts in the oldsense, where the set is given “disjunctive reading.” So, e.g., we would have:

c(9x_9y):P (x;y) = fhfyg; fxg; fx; ygi; hfxg; fyg; fx; ygig:

Note that, e.g., the second occurrence ofx in (9x_9y):P (x; y) seems to be ambigu-ous between free and actively bound. So what is an ambiguous occurrence and howdo we handle it theoretically? We propose to addres this question elsewhere.

7. Concluding Remarks

In this paper we introduced a notion of context and specified its connections withrelational semantics and language. We used these contexts to prove a characteri-zation of theDPL-expressible relations. Moreover, we illustrated the usefulness ofcontexts both in formulating and in verifying valid sequents ofDPL. We illustratedthe fact that “understanding of what is going on” is not automatically preserved ifwe extend theDPL-language. E.g., adding disjunction leads to ambiguous occur-rences of variables. This observation tells us that the study of extensions willprovide us clues regarding the question:what is it to be a variable occurrence of acertain kind?

So – apart from the concrete results – what general conclusions may we drawfrom the paper? A first one is, surely, that a study of the elementaria ofDPL isboth necessary and rewarding. Questions on the nature of variable occurrences,the proper notion of syntactic substitution, etcetera, appear in a new light. Thefruitfulness of the study ofDPL is independent of the question whetherDPL is

48 A. VISSER

really the best choice as a medium for representing dynamic phenomena. Onereason is that, in a sense, the relational semantics ofDPL is very simple and that itis, therefore, easier to make progress. A second conclusion is that it is rewardingto engage in a study that stresses thedifferencesbetweenDPL and PredicateLogic. Much effort has gone into integratingDPL into the classical Montagueframework. This project has unavoidably a conservative flavour. The result has beenthat the unfamiliarity, the strangeness ofDPL has been underadvertised. Preciselymastering the strangeness provides us with better insight into the formerly familiarnotions. My third conclusion is simply: contexts are essential in the study ofDPLand its kin. We may want to vary the contexts, e.g., we may want to add “garbageelements” or to ignore theO-component, but contexts per se are there to stay. Ourthird conclusion points to a larger programmatic point: the study of contexts andthe way they are contexts of their contents should be one of the central endeavorsof the study of Information Processing and Dynamics.

Acknowledgements

I thank Kees Vermeulen for many enlightening conversations and for his carefulreading of two versions of this paper. I am grateful to the students of various classesin Dynamic Logic both in Amsterdam and in Utrecht for providing a critical forumfor some of my ideas onDPL and Dynamics. I thank the two referees for theiracute observations.

Appendix

A. Intersection of Contexts

In this appendix we treat the properties of intersection of contexts.

LEMMA A.1. \ is well defined.Proof. We have to prove:(I \ I 0)n(B \B

0) = (O \O0)n(B \B

0) on theassumption thatB � B

0[ (I \O) andB0 � B [ (I 0 \O0). We prove

(I \ I 0)n(B \B0) � (O \O

0)n(B \B0):

The converse direction is dual. Supposev 2 I \ I0 andv 62 B \B

0. We want toprove:v 2 O \O

0. We have:v 62 B or v 62 B0. Supposev 62 B. It follows that

v 2 InB and, hence,v 2 OnB, sov 2 O. If v 62 B0, we findv 2 O0, and we are

done. Supposev 2 B0. We assumed thatB0 � B [ (I 0 \O0). Sincev 62 B, we get:v 2 I 0 \O0, and, hence,v 2 O0. The case thatv 62 B0 is similar. 2

Next we prove that\ produces the� minimum, whenever there is one.

THEOREM A.2.1. z � c andz � d , c\ d #; z � c\ d.


2. c\ d, whenever it exists, is the infinimum ofc andd.

Proof. Let c = hI;B;Oi, d = hI 0; B0O0i andz = hJ;C; P i. Supposez � c and

z � d. We have:J � I \ I0 andP � O \O

0 andC � B \B0 andB � C [ (I \O)

andB0 � C [ (I 0 \O0). It follows thatB � (B \B0)[ (I \O), and, thus,B �

B0[ (I \O). Moreover,B0 � (B \B

0)[ (I 0 \O0), and soB0 � B [ (I 0 \O0).We may conclude thatc\ d #, c\ d � c andc\ d � d.

For the converse, it is sufficient to prove that ifc\ d #, then c\ d � c andc\ d � d. So, supposec\ d #. We verify c\ d � c, the case ofd being similar. Theonly problematic case to check is:B � (B \B

0)[ (I \O). But this is immediatefrom:B � B

0[ (I \O). 2

In the following theorem we show that one can always find a “best” context for agiven non-emptyIBO-relation.

THEOREM A.3.SupposejDj � 2. We have:

1. Suppose; 6= R 2 fcg \ fdg . Thenc\ d # andR 2 fc\ dg.2. c\ d #) fc\ dg = fcg \ fdg.3. Suppose; 6= R. Let X := fx jR 2 fxgg. SupposeX 6= ;. ThenX has a

minimum.

Proof. Let c = hI;B;Oi and d = hI 0; B0; O

0i. We prove (1). Suppose; 6=R 2 fcg \ fdg . SupposefRg. We first prove thatc\ d #, i.e.,B � B

0[ (I \O)

andB0 � B [ (I 0 \O0). By symmetry, we only need to treat the first desideratum.Suppose, to obtain a contradiction, that for somev: v 2 B, v 62 I \O andv 62 B0.By the duality betweenI andO, we can restrict ourselves to the case thatv 62 I.Pick d with d 6= g(v). We have:fdv IIfRgIOg and, sinceR � [B] andv 2 B,fdv [B]g. HencefdvRg. It follows thatR 6� [B0], a contradiction.

We show thatR 2 fc\ dg . Suppose that

f0II\ I0fRgIO\O0g

0 and f0[B \B

0]g0:

We have to show:f 0Rg0. Define:

� f� := f � I [ f

0 � (VarnI).� g

� := g � O [ g0 � (VarnO).

Clearly,fIIf�, and, sincef 0II\ I0f , f�II0f 0. Similarly we find thatgIOg� andg�IO0g

0.We showf�[B]g�. Considerv 62 B. In casev 2 I, we havev 2 InB, and, hence,

v 2 OnB and, thus,v 2 O. We find:f�(v) = f(v), sincev 2 I. f(v) = g(v),sincefRg andR � [B]. g(v) = g

�(v), sincev 2 O. Sof�(v) = g�(v), as desired.

In casev 62 I, we also havev 62 O, since, otherwisev 2 OnB = InB. By the

50 A. VISSER

definitions off� andg� we find:f�(v) = f0(v) andg0(v) = g

�(v). Moreover, bythe fact thatf 0[B \B

0]g0, we getf 0(v) = g0(v). Hence,f�(v) = g

�(v).Sincef 0[B \B

0]g0, we have,a fortiori, f 0B0g0. Collecting all previous insights,

we may conclude:f�IIfRgIOg� and f�[B]g�. Hencef�Rg�. It follows thatf0II0f

�Rg

�IO0g0 andf 0[B0]g0. Hencef 0Rg0.

We turn to (2). Supposec\ d #. By Theorem 3.5 and the fact thatc\ d � c andc\ d � d, we have:fc\ dg � fcg \ fdg . For the converse, apply (1). Finally toprove (3), note that, since contexts are finite,� is well founded. Hence,X has aminimal element. Moreover, (1) implies thatX is closed under\ . So the minimalelement must be the minimum. 2

Our last theorem corresponds to the familiar fact of ordinary Predicate Logic thatif a set of assignmentsF is finitely restricted, then one can find a�-minimal I,such thatF is hIi-restricted. Note that; in the DPL case corresponds to many�-incomparable contexts. Thus, it is hopelessly ambiguous, in contrast to thepredicate logical case.

B. Relations in Context

In DPL meanings are relations. The contexts we studied appear as properties ofthese relations. We could give an alternative semantics forDPL by building thecontext into the meaning. Thus we take as meanings pairshc; Ri, whereR 2 fcg .Let us call such a pair ac-relation. We define: [[�]]M := hc�; [�]Mi. The newdomain of meanings is, on the one hand, essentially richer than the old one, sincethe same relation falls under several contexts. On the other hand, we threw all non-IBO-relations away. We can “lift” the notions intoduced in this paper toc-relations:

� hc; Ri � hd; Si = hc � d; R � Si.� A c-condition is ac-relations of the formhhI; ;; Ii; Ri.� If R = hhI;B;Oi; Ri, thendom(R) := hhI; ;; Ii; diag(dom(R))i. Similarly

for cod.� [[B]] := hh;; B; ;i; [B]i. We write [[v]] for [[fvg]] .� A c-relationR has the Switching Property if it is either ac-condition or there

are variablesv andw such that:

R = dom(R) � [[v]] � R � [[w]] � cod(R):

� A c-relation isDPL-expressible (over a given domainD) if it can be generatedusing� from c-conditions and resettings[[v]] .

In a similar way we can upgradeand99. Inspection of the proofs in this papershows that, in caseD is infinite, theDPL-expressiblec-relations are precisely theones with the Switching Property. Moreover, allc-relations over the given domain


– infinite or not – areDPL(99)-expressible. We consider an example. Rememberthat:

h;; ;; ;i � hfxg; ;; fxgi � hfxg; fxg; fxgi:

Let a model with domainD be given. We assume thatD has at least two elements.Let id := id

DVar. We consider threec-relations with associate relationid.

1. [[>]] = hh;; ;; ;i; idi,2. [[x = x]] = hhfxg; ;; fxgi; idi,3. [[99u(u = x:9x:u = x)]] = hhfxg; fxg; fxgi; idi.

(1) and (2) arec-conditions and, henceDPL-expressible. In fact, they can be definedin the language by>, respectivelyx = x. In contrast, (3) is not a condition. It iseasy to see that (3) does not have the Switching Property, since the domain of itsinternal relationid consists of all assignments andid is injective. Hence (3) is notDPL-expressible. Note thathh;; fxg; ;i; idi is not ac-relation at all.

A further step in modifying our semantics is to make the assignments “local.”The idea is that the context “provides” the files/discourse referents/variables onwhich the variables are defined. Thus, our meaning objects would be of the formhhI;B;Oi; Ri, whereR would be a relation taking input assignments defined onI

and yielding output assignments defined onO. This approach leads to a semanticsvery much like Vermeulen’s Referent Systems (see Vermeulen, 1995). One effectof this further modification is that it leads to a somewhat different view of contexts.In the local approach, contexts are the central “engines” that manage the flow ofthe files in the interactions of meanings. This more dramatic view of contexts iselaborated in Visser and Vermeulen (1996).

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Contexts in Dynamic Predicate Logic

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