aspects of compositionality

Journal of Logic, Language, and Information10: 49–61, 2001.© 2001Kluwer Academic Publishers. Printed in the Netherlands.

49

Aspects of Compositionality

GABRIEL SANDUDepartment of Philosophy, University of Helsinki, P.O. Box 24, 00014 Helsinki, FinlandE-mail: [email protected]

JAAKKO HINTIKKADepartment of Philosophy, Boston University, 745 Commonwealth Ave., Boston, MA 02215, U.S.A.E-mail: [email protected]

(Received 5 October 1998; in final form 21 February 2000)

Abstract. We introduce several senses of the principle of compositionality. We illustrate the dif-ference between them with the help of some recent results obtained by Cameron and Hodges oncompositional semantics for languages of imperfect information.

Key words: Axiom of choice, compositionality, games

1. Compositionality: What Is It?

On the standard account a semantics is defined as compositional if and only if init the meaning of each complex expression is a function of the meanings of itscomponent expressions, in other words, is determined by the meanings of the com-ponent expressions plus the way they are combined into the complex expression.This definition requires some additional explanations.

First, “meaning” is a generic rather than specific term. Instead of this word weshould speak of the different semantic attributes of an expression. But if this isdone, the principle of compositionality becomes ambiguous.

(i) In one sense it can mean that the applicability of a semantic attributeA

to a complex expressionE is completely determined by the applicability of thisparticular attributeA to the component expressions ofE plus the way they arecombined intoA. This sense of compositionality does not necessarily require thatevery subexpression of a meaningful expression be itself meaningful.

(ii) In another sense it can be taken to mean that the applicability ofA toE is determined by the applicability of all and sundry semantic attributes to thecomponent expressions ofE plus the structure ofE in terms of these componentexpressions. For instance, a Tarski-type truth-definition is compositional in thissense but not in the former sense (i). The reason is that Tarski does not specifythe truth-conditions of a sentenceϕ by reference to the truth and falsity of itscomponent expressions, but rather by reference to their satisfaction conditions. Asimilar example is provided by the usual truth-conditional semantics for intensional

50 G. SANDU AND J. HINTIKKA

logics. In this case the truth of a complex sentence is expressed as depending onthe satisfaction of its component expressions in different possible worlds. For yetanother example one may look at dynamic predicate logic (Groenendijk and Stok-hof, 1991) where a formula is interpreted by a set of pairs of assignments, and thenthe truth of a sentence is defined with respect to such a set.

These examples illustrate the fact that the operative sense of compositionalityis (ii). Only in sense (ii) does compositionality go together with the possibility ofthe kind of recursive definitions of semantic attributes that Tarski relied on.

The failure of compositionality in sense (i) creates an example of what hasbeen called anextension problem. It arises when one has an interpretation for thecomplex expressionsE to which the semantic attributeA applies, but not one forthe simpler parts ofE. For instance, in caseA is the truth-predicate, we may havean interpretation for ordinary first-order sentences, but no interpretation for properformulas. Such a partial interpretation is clearly not compositional in sense (i). Butthen the question arises whether it can be extended to a (total) interpretation insense (ii).

(iii) We get a third sense of compositionality from a more constrained versionof (ii). It arises in the context in which one puts some constraints on those othersemantic attributes which determine the applicability of a given attributeA. This isnot, strictly speaking, a new sense of compositionality, but rather a combination ofcompositionality in sense (ii) with additional restrictions on the relevant auxiliarysemantic attributes. The distinctive feature of compositionality here is that the rel-evant meaning entities satisfy some additional requirements which are thought tobe “natural.”

One way to illustrate the interplay between (i), (ii) and (iii) is again via theextensionproblem introduced two paragraphs ago. Suppose we have an interpreta-tion I which computes only the truth (TruthI ) of sentences but which is not definedfor proper subformulas. Such an interpretation is not compositional in sense (i).Suppose we manage to extendI to a compositional interpretation in sense (ii).This means that we are able to find a (total) interpretationI ′ (involving some newsemantic attributes) which applies also to the proper subformulas of the languageand whichagreeswith I on sentences, that is, the interpretation assigned to sen-tences byI ′ is the same as the interpretation assigned to sentences byI . We nowask whether the interpretationI ′ satisfies some further requirements.

One feature which is common to (i), (ii) and (iii) is semantic context independ-ence. For if compositionality in sense (ii) holds, then the semantic attributes ofE cannot depend on its context. Conversely, if it makes sense to speak of thecomponents ofE and of the structure they form inE, there is little else to determinethe semantic attributes of the components plus the structure.

The distinction between the different senses of compositionality is, in our opin-ion, very well illustrated by the recent results of Hodges (1997a, 1997b, 1998,2001), and Cameron and Hodges (forthcoming), which give compositionality res-ults for languages of imperfect information (IF-languages) introduced by Hintikka

ASPECTS OF COMPOSITIONALITY 51

and Sandu (1989, 1997). We shall interpret Hodges (1997a, 1997b) as showingthat IF-languages have a compositional semantics in sense (ii), Hodges (2001) assolving an extension problem forIF-languages, and Cameron and Hodges (forth-coming) as proving the impossibility of a compositional semantics in sense (iii).We devote the rest of the paper to expounding these results and making someadditional comments pertaining, in particular, to the role of the Axiom of Choice(AC).

2. Independence-Friendly (IF) Languages

One of the things one may want to represent in a logic are the different patterns ofdependence and independence of different variables. Now how are dependenciesand independencies between different variables represented in first-order logic?The obvious answer is: by means of the corresponding relations of dependenceand independence between quantifiers. Indeed, the functional dependence ofy onx to which it bears the relationR(x, y) is expressed by

(∀x)(∃y)(R(x, y) ∧ (∀z)(R(x, z)→ z = y)). (1)

In fact, the great improvement of modern logic as compared with Aristoteliansyllogistics lies precisely in the use of dependent quantifiers. Without them, wecannot express any fundamental relationships.

But the step from syllogistic logic to received (Frege–Russell) first-order logicdid not reach far enough. It enabled us to express certain patterns of dependenceand independence between quantifiers andergovariables, but not all such patterns.

The best known example of a pattern of quantifiers not expressible in receivedfirst-order logic is theHenkinquantifier (Henkin, 1961). It expresses the fact thatthere is a functional dependence ofy on x and a functional dependence ofu on z.Henkin rendered this pattern of dependencies in the obvious branching notation( ∀x ∃y

∀z ∃u). (2)

Henkin interpreted sentences of the form( ∀x ∃y∀z ∃u

)R(x, y, z, u) in a modelM

by a semantic game of imperfect information played byAbelard(∀) andEloise(∃):Abelardchoosesa andc from the universe|M| of M, andEloisechoosesb anddsuch that when she choseb she did not know the choicec of Abelard, and whenshe chosed she did not knowAbelard’s choicea. Player∃ wins iff RM(a, b, c, d)and she has a winning strategy in the game iff there are functionsf andg such thatRM(a, f (a), c, g(c)) for any choicesa, c of Abelard.

Henkin quantifiers are expressed inIF-languages by having an explicit notationfor independent quantifiers. We shall use(∃y/∀x) to express the semantic inde-pendence of the quantifier(∃y) of the quantifier(∀x), within whose syntactical


scope it occurs. Thus the branching prefix (2) will be expressed as

(∀x)(∀z)(∃y/∀z)(∃u/∀x) (3)

or even as

(∀x)(∀z)(∃y/z)(∃u/x) (4)

and the semantic interpretation ofIF-sentences will be in terms of semantic gamesof imperfect information as above.

The idea of independence can be extended to cover also connectives. But herewe face some problems. Consider a sentence like

(ϕ1 (∧/∨) ϕ2) ∨ P(a), (5)

where conjunction is independent of disjunction. Hodges (1997a) pointed out thatthis sort of independence does not make sense in the game-theoretical setting. Forafter player∃ has made the first move choosing one of the disjuncts, player∀mustknow whether it is his turn to move or not. If it is not his turn to move, then he mustbe able to infer that∃ has chosen the right-hand disjunct, which being an atomicformula, does not call for a move from any of the players. If again, it is his turnto move, then he must know what move it is that he is supposed to make. And ifthe move he is supposed to make is to choose one of the conjunctsϕ1 or ϕ2, thenhe must be able to infer that∃ has chosen the left-hand disjunct earlier on. In otherwords, the fact of∀′s knowing what moves he is supposed to make at a particularstep implies that he knows what stage the game has reached, from which he mustbe able to infer all the choices∃ has made earlier on. But then the effect of theconjunction being independent of disjunction is cancelled! Notice that this problemdoes not arise with quantifiers. So one could have independent connectives, butthen these connectives should be construed as restricted quantifiers.

We can illustrate what we have in mind with the following example. In theHenkin prefix (2) we replace the two existential quantifiers∃x and ∃u by thedisjunctions (restricted or Boolean quantifiers)

∨i∈{1,2}and

∨j∈{1,2}, respectively.

Then the game-theoretical interpretation of the sentence( ∀x ∨i∈{1,2}

∀z ∨j∈{1,2}

)Rij (x, z) (6)

is completely identical to the one presented above for the Henkin quantifier, exceptthat Eloisechooses an indexi ∈ {1,2} and an indexj ∈ {1,2}. The strategy ofEloisewill now consist of two functionsf, g from the universe of the modelM tothe set{1,2}. This strategy is a winning one iff(M, f, g) |= Rf(x)g(z)(x, z) for anyx andz picked up byAbelard.

The idea of branching connectives goes back to Blass and Gurevich (1986);some of their basic model-theoretic properties have been investigated in Sandu andVäänänen (1992).


An alternative way to express branching connectives is by using slashed quanti-fiers. For instance, (6) will be rendered as an informational independent variant ofa so-called Vaught sentence:

∀x∀z ∨i∈{1,2}

/z

( ∨j∈{1,2}

/x

Rij (x, z). (7)

Before addressing the question of compositionality forIF-languages, let usbriefly review the game-theoretical interpretation of ordinary first-order languages.

3. Games of Perfect Information

Let ϕ(x0, . . . , xn−1) be a first-order formula. We recall the definition of a se-mantical gameG(M,ϕ, a) played with the formulaϕ in the modelM with respectto the sequencea = (a0, . . . , an−1) assigned to the free variablesx0, . . . , xn−1,respectively. The game is played by two playersAbelard(∀), andEloise(∃) and isdefined in the following way:

− ϕ is an atomic formula: ifM |= ϕ[a0, . . . , an−1], then∃ wins right away;otherwise∀ wins.

− ϕ is (ϕ1 ∨ ϕ2): player ∃ choosesi ∈ {1,2}, and the game goes on as inG(M,ϕi, a).

− ϕ is (ϕ1∧ϕ2): identical with the previous case, except that player∀makes thechoice.

− ϕ is ¬ψ : the same as the gameG(M,ψ, a), except that the players∀ and∃are transposed.

− ϕ is ∃xψ(x): player∃ chooses an elementa of the universe ofM. The gamesgoes on as inG(M,ϕ, aa)

− ϕ is ∀xψ(x): identical to the previous case, except that player∀ makes thechoice.

We notice that each formulaϕ(x) and modelM determine a familyG(M,ϕ)of games. We get a gameG(M,ϕ, a) in this family by specifying a sequencea tobe the interpretation of the free variablesx. (Actually the sequencea will containnot only elements from the universe of the model, but also indices from a fixedset; and correspondingly, the formulas of the language will have subindexes, asindicated in the rules for disjunction and conjunction. We will ignore, however,these complications here.) Obviously, whenϕ is a sentence, the familyG(M,ϕ)reduces to the single gameG(M,ϕ, 〈〉), where〈〉 is the empty sequence.

We definestrategiesfor the players not with respect to a particular gameG(M,ϕ, a) but with respect to the familyG(M,ϕ). A strategy for a player inthe familyG(M,ϕ) of games is a set of functionsfQ, one for each logical con-stantQ which prompts a move of that player in the game. For instance, ifϕ is


∃xnψ(x0, . . . , xn−1, xn), thenf∃xn is defined on all the sequences of|M| of lengthn with values in|M|. If a sequencea is given, then player∃ follows her strategyfunctionf∃xn in the gameG(M,ϕ, a) if her corresponding move isf∃xn(a); simil-arly for all the other logical constants. In a play ofG(M,ϕ, a) a playerf ollowshis/her strategy if he/she follows the corresponding strategy function at each ofher/his moves.

A strategy for playerP in the familyG(M,ϕ) of games iswinning for P in thegameG(M,ϕ, a) if P wins every play ofG(M,ϕ, a) in which he/she follows it.(From now on we omit reference to the familyG(M,ϕ).)

Now we can take the interpretationϕM of the formulaϕ in the modelM to be aset of sequences of individuals from|M|:

ϕM = {a : player∃ has a winning strategy inG(M,ϕ, a)}. (8)

Whenϕ is a sentence we say thatϕ is true if and only ifϕM = {〈〉}.Is this semantics compositional (in the sense (ii) of the first section)? In other

words, isϕM definable in terms of the meanings of its subformulas? The answer ispositive ifAC holds inM. More exactly, if we assume thatAC holds inM, thenthe following hold too:

(¬ψ)M = |M|n − ψM,

(χ ∧ ψ)M = χM ∩ ψM,

(∃xψ)M = {a : there isa ∈ |M| such thataa ∈ ψM},(∀xψ)M = {a : for all a|M|, aa ∈ ψM}. (9)

AC is needed in the last clause of (9) when showing that

{a : for all a ∈ |M|, aa ∈ ψM} ⊆ (∀xψ)M.Actually, AC is not only a sufficient but also a necessary condition for the

equivalence between (8) and (9), as shown in Krynicki and Mostowski (1995).There is also another way (Hodges, 1997a) to show that the game-theoretical

semantics (8) is compositional. There one proves by induction on the length of aformula ϕ(x) that ϕM = ϕM,T , whereϕM,T is the Tarskian interpretation of theformulaϕ in the modelM defined by

ϕM,T = {a : M |= ϕ[a]}.Since the Tarskian interpretation is compositional (again, in the sense (ii) ofcompositionality), it can be proved that the game-theoretical semantics (8) iscompositional too. The proof also makes use ofAC.

Thus, one way to see the difference between the game-theoretical interpretation(8) and the Tarski type interpretation is that the former’s compositionality, unlikethe latter’s, commits one to stronger set-theoretical principles likeAC.


Another way to see the difference between them was suggested by Hodges(private communication). According to him the difference is that between determ-inistic and non-deterministic strategies. In order to understand Hodges’s proposal(if we got him right), we have to consider a different game-theoretical interpretationthan the one discussed above. (Cf. Kolaitis (1985), in a different setting.)

In this interpretation, the notion of strategy in (8) is replaced by that ofquasistrategy

ϕM = {a : player∃ has a winning quasistrategy inG(M,ϕ, a)}. (10)

A quasistrategy for a player in the family of gamesG(M,ϕ) is defined analogouslyto the notion of a strategy, except that each strategy functionfQ is replaced by astrategyrelation RQ. For instance, ifϕ is ∃xnψ(x0, . . . , xn−1, xn), thenR∃xn is aset ofn-tuples from the universe of the model which satisfies the condition:

For each(a0, . . . , an−1) ∈ |M|n there isa ∈ |M| such that(a0, . . . , an−1, a) ∈ R∃xn .

When a sequencea is given, then player∃ follows her quasistrategy functionR∃xn in the gameG(M,ϕ, a) if she chooses somea such that(a0, . . . , an−1, a) ∈R∃xn. The term “chooses” may be misleading here. The real difference betweenusing a strategy in a game and using a quasistrategy is that in the former case thechoices of the players at each move are completely determined by the earlier courseof the game, while in the latter case this is not the case.

Now it is obvious that if player∃ has a strategy inG(M,ϕ) that is winning inthe gameG(M,ϕ, a), then she also has a quasistrategy inG(M,ϕ) that is winningin the gameG(M,ϕ, a), but the converse may fail. If, in addition,AC holds inM, then the converse can be shown to hold too, and (8) and (10) thus becomeequivalent.

Unlike the strategy interpretation (8), the quasistrategy interpretation (10) canbe shown to be compositional even withoutAC, that is, the quasistrategy versionof (9) can be shown to hold without any additional set-theoretical assumptions.And as with the strategy interpretation (8), a proof by induction shows that thequasistrategy interpretation (10) is equivalent to the Tarski type interpretation, butin the proof one does not any longer needAC. So one can say that the interpretation(10) is nothing more than the Tarski interpretation in the game-theoretical jargon.

However, if we want the interpretation (10) to satisfy certain further constraints,like for instance the quasistrategies of the players being deterministic, then this stepcommits one toAC. We shall return to this matter at the end of the paper, but beforemoving to the next section one clarification is still in order.

The way strategies of the players have been defined in this section is not actuallythe way they are defined in Hintikka’s work (cf. Hintikka and Sandu, 1997, for anoverview). There the strategies of the players in a game are not defined on the setof all possible sequences chosen earlier in the game, but only on those sequences


which represent possible choices of theopponent. For this reason, the strategyfunction f∃xn corresponding to∃xnψ(x0, . . . , xn−1, xn) is not defined on all thesequencesa of |M| of lengthn with values in|M|, but only on those subsequencesof a whose elements have been chosen byAbelard. But in order to specify thosesubsequences, one needs to know the larger formula into which the subformula∃xnψ(x0, . . . , xn−1, xn) is embedded, so that one can keep track of the universalquantifiers in whose scopes∃xn occurs. This is one of the reasons why Hintikka’ssemantic games are played only withsentencesand also one of the reasons whystrategy functions are not defined from the inside out, as above, but on the possiblesequences chosen by the opponentearlier in the game. Then a sentenceϕ is definedas true inM if and only if Eloisehas a winning strategy in the game associated withϕ andM.

Clearly such an interpretation, that we shall call aHintikka interpretation, is notcompositional in sense (i). So an interesting question that was mentioned in the firstsection is whether this partial interpretation can be extended to a compositional onein sense (ii). We shall discuss its answer in Section 5.

4. Games of Imperfect Information (Hodges’s Results)

In semantic games of imperfect information the players might not alwaysknow what individuals from the model are assigned to the free variables ofthe relevant formula. For suppose we play the gameG(M,ϕ, 〈〉) where ϕ is∀x0∀x1(∃x2/x0)S(x0, x1, x2). After Abelard’s choosing elementsa0, a1 from theuniverse ofM (the rules for the moves of the players in a game are identical withthe earlier ones), the game continues asG(M, (∃x2/x0)S(x0, x1, x2), 〈a0, a1〉) withEloisenot “knowing” which a0 has been chosen. So one central task is to makeprecise this lack of knowledge. By analogy with the game of perfect informationcase, one possibility would be to take the function strategyf(∃x2/x0) which construesthe appropriate choice ofEloiseto be a function of only one argument instead oftwo. In other words, in a play〈a0, a1, a2〉 of the gameEloise follows a strategyfunction f(∃x2/x0) if a2 = f(∃x2/x0)(a1). Then we could proceed as in the previoussection and define:

ϕM = {a : player∃ has a winning strategy inG(M,ϕ, a)}. (11)

Whenϕ is a sentence, we say thatϕ is true inM iff the right-hand side of theabove definition contains the empty sequence.

We now ask the same question as before: is this semantics compositional (insense (ii))? The answer, due to Cameron and Hodges (forthcoming) is: No! Unlikein the earlier case, compositionality cannot be rescued with the help of the Axiomof Choice. Actually Cameron and Hodges’s results are more general, showing thatthere is no compositional interpretation which agrees with the interpretation (11)on sentences, and in addition is such that every formula is interpreted by a set ofsequences.


Hodges’s strategy for giving a compositional interpretation forIF-languageswhich agrees with (11) on sentences is to interpret the lack of information of theplayers as requirements ofuniformityon their strategies.

EachIF formula ϕ introduces an equivalence relationE(ϕ,M) on the modelM. For instance,E(∃x7/x0, x2)R(x0, . . . , x7),M) is

E((a0, . . . , a7), (b0, . . . , b7)) ⇔ ai = bi wheneveri 6= 0,2.

With this equivalence relation fixed, the corresponding strategy functionf(∃x7/x0,x2) is said to beE-uniform, if

E(a, b) ⇒ f(∃x7/x0,x2)(a) = f(∃x7/x0,x2)(b),

for all a andb.Consider the family of gamesG(M,ϕ). A trumpof ϕ is defined as a nonempty

setT of sequences such that some uniform strategy for player∃ is winning in everygameG(M,ϕ, a), with a ∈ T . The interpretationϕM is taken to be the set of alltrumps. (We assume that the negation sign occurs only in front of atomic formulas).A sentence is true if and only if it has as its only trump{〈〉}.

Thus in comparison with the games of perfect information, the interpretationof a formula of the givenIF-language is no longer a set of sequences, but a set ofsets of sequences satisfying certain conditions (i.e., a set of trumps). Then Hodges(1997a) proves that the trump semantics is compositional and agrees with theinterpretation (11) on sentences.

The Hintikka interpretation forIF-languages is completely analogous to that forordinary first-order languages: semantic games are played only withIF-sentences,and strategies are defined on the possible sequences of choices of the opponentearlier in the game. For the same reasons as before, this interpretation is not com-positional in sense (i), and thus we raise the same question again: can it be extendedto a compositional one?

5. Compositionality and Extensions

We return in this section to the two questions raised earlier: can the Hintikka inter-pretation for first-order sentences and the Hintikka interpretation forIF-sentencesbeextendedto a compositional interpretation in sense (ii)? As we pointed out at theend of Sections 3 and 4, in Hintikka’s game-theoretical semantics onlysentencesof a language receive an interpretation and for this reason this semantics is notcompositional.

The answer to the first question is straightforward. The interpretation (8) inSection 3 agrees with the Hintikka interpretation on sentences, because it does notactually matter how the “skolemization” is done, that is, whether the argumentsof the Skolem functions (which are the strategies of the players in the game) aredefined as in Hintikka’s interpretation or as in the interpretation described in Sec-tion 3 (which is, by the way, the original interpretation of Skolem, a point we owe


to Hodges), where the Skolem functions are defined over all the possible sequenceschosen earlier. To take an example, the two ways of defining strategies in the gameassociated with the sentence∀x∃y∃zS(x, y, z) yield respectively the second-ordersentences

∃f ∃g∀xS(x, f (x), g(x)) and ∃f ∃g∀xS(x, f (x), g(x, f (x)))which are provably equivalent.

The answer to the second question is also positive, but proved to be more diffi-cult. As pointed out in Hodges (1997a), unlike in the games of perfect informationcase, the two ways of skolemizing are no longer equivalent. This can be seen froma simple example like∀x(∃y/x)(x 6= y), where both ways of skolemizing yieldthe logically false sentence∃y∀x(x 6= y). However, if we add a dummy variableand obtain∀x∃z(∃y/x)(x 6= y), then the skolemization as done in the Hintikkainterpretation yields∃y∀x(x 6= y), while the other way of skolemizing yields thesentence∃f ∃g∀x(x 6= f (g(x))), which is true in every model with at least twoelements.

Hodges (1997b) produces a trump semantics forIF-languages which is com-positional and agrees with the Hintikka interpretation on sentences.

The answers to both questions may be also derived as corollaries to a moregeneral Extension Theorem contained in Hodges (2001).

Hodges defines a grammar to be a triple(E,A,6), whereE is a set of expres-sions,A is a set of atomic expressions (a subset ofE) and6 is a set of syntacticrules. An interpretation for a grammar is a function which need not be defined onall the expressions ofE. In other words, the interpretation may be partially defined.Two expressions aresynonymous(with respect to an interpretation) if they receivethe same semantic value with respect to that interpretation. Ifu and v are twointerpretations, thenv is said toextend u, if v(p) equalsu(p) wheneveru(p) isdefined.

Hodges’sExtension Theoremshows that, when certain conditions obtain, anypartial interpretation for a grammar can be extended to a total compositionalinterpretation. The conditions in question are the following:

(a) 1-compositionality: if a subexpression of a meaningful complex expression eis replaced by a synonymous expression, and the result of that replacement,say e′, is again meaningful (i.e., it is defined), then e and e′ are synonymous.

(b) The condition of beingHusserlianrequires, roughly, that whenever two expres-sionsp, q are synonymous, they can be substituted in any complex expressions(ξ), so thats(p/ξ) is meaningful (according to the interpretation in question)if and only if s(q/ξ) is meaningful. (Hereξ is the free variable of the terms.)

Now Hodges’s result also yields an answer to the extension problem for first-order andIF-languages. For instance, it is obvious that anIF-language has agrammar: we take the expressions of the grammar to beIF-formulas, and its atomic


expressions to be theIF-atomic formulas. The syntactical operations of the gram-mar are just the connectives and quantifiers (negation signs occur only in frontof atomic formulas). Since proper formulas do not receive an interpretation, thegrammar of this language is only partially interpreted. Conditions (a) and (b) canalso easily be shown to hold. Then the Extension Theorem shows that there is atotally defined compositional Husserlian interpretation for any givenIF-languagewhich extends the Hintikka interpretation. (Here the term “compositional” standsfor the requirement corresponding to (a) when substitution is allowed to concernseveral subexpressions at once.)

6. Is the Principle of Compositionality Methodologically Empty?

Some of Hodges’s results (1998), together with results in the same spirit by Janssen(1997) and Zadrozny (1994), which show that a formal language which is recurs-ively built up always admits of a compositional interpretation, have been taken toimply that the principle of compositionality is methodologically empty: as a factof logic, it can always be implemented. This is true, but only if compositionalityis taken in sense (ii). In other words, given an interpretation, one can always findanother one which is compositional in sense (ii) and which is related somehow tothe original one. However, compositionality in sense (ii) is not of much help tolinguists or philosophers of language, for an obvious reason. Most typically oneis not interested inany compositional interpretation, but in aparticular one, thatis, in an interpretation where the meaning entities have some additional properties.In other words, one is interested not so much in compositionality in sense (ii) butrather in compositionality in sense (iii). And as we have seen, there is no guaranteethat when extra requirements are put on the semantic attributes needed to obtaincompositionality in sense (ii), these requirements can be consistently implemented.We take Westerståhl (1998) to make very much the same point with respect to someof Janssen’s and Zadrozny’s results.

An example which illuminates the different senses of compositionality was con-sidered in Section 3. There one possible interpretation for an ordinary first-orderformulaϕ was

ϕM = {a : player∃ has a winning quasistrategy inG(M,ϕ, a)}.This interpretation is compositional in sense (ii). We also asked whether it is com-positional in sense (iii), by imposing some extra requirements on quasistrategies,i.e., that they are deterministic. It was pointed out in Section 3 that the answer ispositive, but only in the presence ofAC. This is an example of how one can movefrom compositionality in sense (ii) to compositionality in sense (iii) by assumingsome strong set-theoretical principles, likeAC.

This move does not always succeed, however. The extension problem forIF-languages shows that one cannot obtain a compositional semantics in sense (iii),even assuming stronger set-theoretical principles. In reviewing Hodges’s results,


we pointed out that the trump interpretation forIF-formulas is compositional (insense (ii)) and agrees with the Hintikka interpretation on sentences. However,one may want a compositional interpretation which does not only agree with theHintikka interpretation on sentences, but in addition assigns to every formula not aset of sets of sequences, but a set of sequences. This is in our opinion a naturalrequirement, sinceIF-languages are a natural extension of ordinary first-orderlanguages, and the latter are interpreted compositionally by sets of sequences.Cameron and Hodges (forthcoming) show that this extra requirement cannot besatisfied, i.e., there is no interpretation in whichevery formula of a givenIF-language is interpreted by a set of assignments, which is also compositional andwhich agrees with the Hintikka interpretation on sentences. This takes us to a claimmade by Hintikka:

. . . there is no realistic hope of formulating compositional truth-conditionsfor [sentences ofIF], even though I have not given a strict impossibility proofto that effect (Hintikka, 1996: 110ff).

If we interpret compositionality in this passage in sense (iii), then Cameron andHodges’s results show that, indeed, the claim is true.

Acknowledgement

We are grateful to Dag Westerståhl for very useful comments and corrections toearlier drafts of this paper.

References

Blass, A. and Gurevich, Y., 1986, “Henkin quantifiers and complete problems,”Annals of Pure andApplied Logic32, 1–16.

Cameron, P. and Hodges, W., “Some combinatorics of imperfect information,” forthcoming.Groenendijk, J. and Stokhof, M., 1991, “Dynamic predicate logic,”Linguistics and Philosophy14,

39–100.Henkin, L., 1961, “Some remarks on infinitary long formulas,” pp. 167–183 inInfinitistic Methods,

Oxford: Pergamon Press.Hintikka, J., 1996,The Principles of Mathematics Revisited, Cambridge: Cambridge University

Press.Hintikka, J. and Sandu, G., 1989, “Informational independence as a semantical phenomenon,”

pp. 571–589 inLogic, Methodology and Philosophy of Science VIII, J.E. Fenstad, I.T. Frolov,and R. Hilpinen, eds., Amsterdam: Elsevier.

Hintikka, J. and Sandu, G., 1997, “Game-theoretical semantics,” pp. 361–410 inHandbook of Logicand Language, A. ter Meulen and J. van Benthem, eds., Amsterdam: Elsevier.

Hodges, W., 1997a, “Compositional semantics for a language of imperfect information,”Journal ofthe IPGL5, 539–563.

Hodges, W., 1997b, “Some strange quantifiers,” pp. 51–65 inStructures in Logic and ComputerScience, J. Mycielski, G. Rozenberg, and A. Salomaa, eds., Lecture Notes in Computer Science,Vol. 1261, Berlin: Springer-Verlag.

Hodges, W., 1998, “Compositionality is not the problem,”Logic and Logical Philosophy6, 7–33.


Hodges, W., 2001, “Formal features of compositionality,”Journal of Logic, Language, and Informa-tion 10, 7–28.

Janssen, Th., 1997, “Compositionality,” pp. 417–473 inHandbook of Logic and Language, A. terMeulen and J. van Benthem, eds., Amsterdam: Elsevier.

Kolaitis, Ph., 1985, “Game quantification,” pp. 365–421 inModel-Theoretic Logics, J. Barwise andS. Feferman, eds., New York: Springer-Verlag.

Krynicki, M. and Mostowski, M., 1995, “Henkin quantifiers,” pp. 193–262 inQuantifiers: Logics,Models, and Computation, M. Krynicki, M. Mostowski, and L.W. Szczerba, eds., Dordrecht:Kluwer Academic Publishers.

Sandu, G. and Väänänen, J., 1992, “Partially ordered connectives,”Zeitschrift für MathematischeLogik und Grundlagen der Mathematik38, 361–372.

Westerståhl, D., 1998, “On mathematical proofs of the vacuity of compositionality,”Linguistics andPhilosophy21, 635–643.

Zadrozny, W., 1994, “From compositional to systematic semantics,”Linguistics and Philosophy17,329–342.

aspects of compositionality

Documents