Compositionality

1. Clarifications

The principle of compositionality concerns complex expressions. Theseexpressions have a structure and constituents. The principle relatesthe meaning of the complex expression to its structure and themeanings of its constituents. Let’s take the following as acommon reference point.^[1]

(C)

The meaning of a complex expression is determined by its structureand the meanings of its constituents.

The compositionality principle is normally taken to quantify over allexpressions of some languageL.

(C′)

For every complex expressione inL, the meaningofe inL is determined by the structure ofe inL and the meanings of the constituents ofe inL.

The principle can be expanded to apply to all expressions in someclass of languages such as the natural languages for human beings.

These glosses of the principle of compositionality say that themeaning of a complex expression isdetermined by itsstructure and themeanings of itsconstituents. There are different interpretations of theprinciple that depend on different interpretations of the notions of(i) “structure” and “constituency”, (ii)“meaning”, and (iii) “determination”. Thissection examines how the interpretation of the principle ofcompositionality depends on clarifications of these notions.

1.1 Syntax

This section addresses different interpretations of the principle ofcompositionality arising from syntax.

1.1.1 Local and Global Compositionality

A syntax for a language specifies a set of expressions and rules forderiving complex expressions from simpler expressions. Theseexpressions are the immediate constituents of the complex expression.Here is a toy example. Suppose English contains expressions of thecategory noun such as ‘book’ and ‘thieves’,and expressions of the category (intransitive) verb such as‘steal’. One syntactic rule might specify thatconcatenating two nouns such as ‘book’ and‘thieves’ produces a noun phrase ‘bookthieves’. Another syntactic rule might specify thatconcatenating a noun or noun phrase ‘book thieves’ and averb ‘steal’ produces a sentence such as (1).

(1) Book thieves steal.

The expressions ‘book thieves’ and ‘steal’ areimmediate constituents of (1) because there is a rule thatapplies to these expressions to derive (1). The immediate constituentsof any proper constituent of an expression are non-immediateconstituents. Thus, the expressions ‘book’ and‘thieves’ are non-immediate constituents of (1) in virtueof being immediate constituents of one of its constituents.

Each complex expression is generated by some syntactic rule. How doesone identify the syntactic rule that generates the expression? Theevidence for a syntactic theory depends on whether the language isnatural or artificial.

The syntax of anartificial language (such as the languagesof propositional and first-order logic or computer programminglanguages) can be settled by checking the appropriate stipulations.Often, the syntax is designed to compositionally align with thesemantics.

The syntax of anatural language requires empiricalinvestigation. Constituency tests are one source of evidence. Thesetests purport to identify the constituents of a sentence in terms of arange of related grammatical sentences. One constituency test istopicalization. Syntactic constituents of the right sort can normallybe made the topic of the sentence by preposing them. For example, in‘the boy believed the book fell’, the string ‘thebook fell’ can be made the topic. This is clear from the factthat ‘the book fell, the boy believed’ is an acceptablestring of English. However, non-constituents can usually not be madethe topic. For example, the string ‘the book fell’ occursin the longer string ‘the boy who put the note in the bookfell’. However, the string ‘the book fell, the boy who putthe note in’ is not an acceptable string of English. Thisprovides some evidence that ‘the book fell’ is not aconstituent of the latter string. Other coordination tests investigatewhether a putative constituent can be coordinated with another stringusing ‘and’ or ‘or’ or can be replaced by aproform. Again, passing a constituency test isprima facieevidence that the string is a constituent of the larger sentence.Failing a constituency test in the right circumstances can beprima facie evidence that a string is not a constituent ofthe larger sentence. The range and significance of constituency testswill depend on one’s background syntactic theory.^[2]

The constituents and structure of an expression follow from itssyntactic derivation. Different versions of the principle ofcompositionality appeal to different features of these derivations.Local compositionality appeals to the fact that an expressionis theimmediate output of some syntactic rule to itsimmediate constituents. The sentence ‘book thieves steal’is the immediate output of a rule combining ‘book thieves’with ‘steal’. Local compositionality says that the meaningof an expression is determined by the local rule that generates it andthe meanings of its immediate constituents.

(C\(_{\textit{local}}\)) For every complex expressione inL, the meaning ofe inL is determined bythe immediate structure ofe inL and the meaningsof the immediate constituents ofe inL.

According to (C\(_{\textit{local}}\)), two expressions of a languagethat have different meanings must either derive from applyingdifferent local syntactic rules or must have corresponding immediateconstituents with different meanings.

A weaker principle,global compositionality, appeals to thefact that the constituents of an expression may themselves be complex.In the example, ‘book thieves’ is a complex constituent of(1). Global compositionality says the meaning of an expression dependsnot only on theimmediate syntactic rule that generates itand on the meanings of itsimmediate constituents, but alsoon the syntactic rules that generate its constituents and on themeanings of the constituents of its constituents, etc. So, the meaningof (1) will depend not merely on the rule that combines a noun phraseand verb phrase and on the meanings of ‘book thieves’ and‘steal’, but also on the rule that generates ‘bookthieves’ and on the meanings of ‘book’ and‘thieves’. Unlike (C\(_{\textit{local}}\)), globalcompositionality allows two expressions to have different meaningseven though they are generated by applying the same rule to immediateconstituent with the same meanings. Unlike local compositionality,global compositionality is compatible with the sentences (1) and‘biblioklepts steal’ having different meanings even if‘book thieves’ and ‘biblioklepts’ have thesame meaning.

1.1.2 Ambiguity, Hidden Structure, and Direct Compositionality

The statement (C) presupposes that the ordinary expression, thesurface form, is the product of the syntactic derivation and is thebearer of meaning. The surface form is construed as a string of words.Thus, the surface form of (1) is the string ‘book thievessteal’. This assumption accords with an approach to semanticsknown asdirect compositionality. Direct compositionalitysays that the surface form is the bearer of meaning. Advocates ofdirect compositionality usually deny that an expression can have emptyor invisible constituents; cf. Jacobson 2002, 2012 and Kracht2007.

Many approaches to semantics reject direct compositionality. On theseviews, the principle of compositionality applies to the derivation orsome underlying “logical” form (or LF) rather than thesurface forms. One reason these approaches depart from directcompositionality arises from ambiguity. As it is glossed in (C),compositionality is a thesis about “the” meaning of anexpression. This assumes that each expression has exactly one meaning.In cases of ambiguity, some surface expression is associated with twomeanings. The ambiguity might be lexical, arising from a single basicconstituent having two meaning (i.e. ‘bank’ can mean afinancial institution or the lining of a river). There are alsostructural ambiguities, arising from one complex expression havingmultiple derivations (i.e. ‘big dogs and cats’ can bederived by combining the adjective ‘big’ with theconjoined noun phrase ‘dogs and cats’ or from conjoiningthe noun phrases ‘big dogs’ and ‘cats’).

Some conclude that the surface forms are not the bearers of meaning.The bearers of meaning are the syntactic derivations themselves orsome disambiguated underlying LFs; cf. Pagin and Westerståhl2010a (252). Advocates of direct compositionality respond by framingtheir semantics in ways that allow for one surface expression to havemultiple meanings; cf. Kracht 2003 (§3.1); 2007; Jacobson 2014;and Pagin and Westerståhl 2010a (footnote 27).

Other departures from direct compositionality posit syntacticconstituents that do not appear in the surface form. Quantifier scopeambiguity is one prominent example. A sentence such as ‘somebodyloves everybody’ is ambiguous between saying that someone is auniversal lover and saying that everyone is loved. One explanation ofthis ambiguity says that the surface sentence corresponds to twounderlying LFs in which the quantifiers ‘somebody’ and‘everybody’ enter the derivation at different points. Thequantifiers bind “traces” that serve as the subject anddirect object of ‘love’. The two readings might berepresented respectively as ‘[Somebody 1][[Everybody 2][1 loves2]]’ and ‘[Everybody 2][[Somebody 1][1 loves 2]]’.May (1977, 1985) offered the classic syntactic arguments in favor ofthis type of approach, with some arguments figuring in earlier sourcesincluding Lackoff 1970; cf. Chierchia and McConnell-Ginet 1990 (Ch.3); Larson and Segal, 1995 (Ch. 7); Heim and Kratzer 1998 (Ch. 5).

Advocates of direct compositionality argue that quantifier scopeambiguities can be handled at the surface level, without appealing tohidden constituents. Instead, they appeal to complexity in thesyntactic categories and semantic types. Using this additionalcomplexity, direct compositionality can avoid the complexity involvedin positing unvoiced constituents or in making use of more complicatedsyntactic derivation rules (Jacobson 2014; Heim and Kratzer 1998, Ch.7). It is unlikely that any one data point will decide between the twoapproaches, and the choice will have to be made by high-leveltheoretical considerations once the theories are fully developed.

1.1.3 Syntax and Non-Linguistic Representations

Compositionality primarily concerns language. However, it may beextended to other kinds of representation provided that the syntacticnotions of constituency and structure can be extended to theserepresentations. What would such an extended sense be? Consider roadsigns such as the No-Left-Turn sign:

This could be viewed as a complex sign decomposable into meaningfulfeatures—the shape, the color pattern, the arrow, etc. Thesefeatures are the analogues of simple expressions: they appear in manyother complex signs, and they appear to contribute more or lessuniformly to their meanings.

Once the relevant notion of constituency has been fixed, it ispossible to ask the question of whether this system of representationsis compositional.^[3]

There have been many attempts to extend formal semantics to othersorts of representation, including maps (Casati and Varzi 1999, Camp2007, Rescola 2009, Blumson 2012), images (Kulvicki, 2006, Greenberg2021), and Venn and Euler diagrams (Shin 1994, Greenberg 2024). Ineach case, the issue of compositionality can be raised.

A related issue is whether thought is compositional. If thought is akind of language, it is intelligible to ask whether it iscompositional. Thought would not have to bemuch likeEnglish, Korean, or Swahili for the question of compositionality tomake sense, but thoughts must have meanings and have meaningfulconstituents. These assumptions follow fromthe language of thought hypothesis (see entry).

Even if thought is not a language, it might be compositional in anextended sense. Compositionality demands that some complexrepresentations be derivable from other representations. However, thisderivability does not immediately entail that thought islanguage-like; cf. van Gelder 1990. If the representational properties(or meaning) of the complex representation are determined by therepresentational properties of the simpler representations and theirstructures, then the complex representation is compositional in theextended sense.

These issues arise in the major debate within the philosophy of mindbetween proponents of classical cognitive architecture and proponentsofconnectionism (see entry). The debate is sometimes presented as a debate aboutcompositionality. However, the issue tends to be whether there aresuch things as meaningful constituents of thought, and if there are,whether these contribute the same thing to all thoughts in which theyoccur. If the answer to the first question is negative, the questionof compositionality does not arise. If the answer to the firstquestion is positive, the second is independent of compositionality.The debate about connectionism also raises the question about whethersemantic composition is reversible. This notion ofreversecompositionality is discussed below in section 1.4.2.

1.2 Meaning

Compositionality says that the “meaning” of an expressionis determined by the “meanings” of its constituents and bytheir arrangement in the sentence. What notion of meaning is governedby the principle? Certainly, some notions of meaning such asconversational implicatures are not determined compositionally.However, logicians, philosophers, and linguists have proposed thatvarious notions of meaning can be compositionally determined in anatural or artificial language. This section will briefly review someof these notions. In each case, there have been challenges to theirclaims of compositionality.

1.2.1 The Compositionality of Reference

Frege constructed an artificial language, theBegriffsschrift, to model reasoning in logic and arithmetic.Frege’s explanation of theBegriffsschrift is amilestone of compositional semantics. InThe Basic Laws ofArithmetic, Frege most clearly laid out his syntax and semantics.Frege enumerated basic expressions ofBegriffsschrift. Thelanguage includes proper names such as ‘Caesar’,‘Cicero’, and ‘Gaul’. There are alson-ary predicates such as the binary predicate‘conquered’. Finally, Frege specified rules for derivingcomplex expressions out of simpler expressions. A sentence is formedby combining ann-ary predicate andn names. So,sentence (2) can be formed by combining the binary predicate‘conquered’ with two names ‘Caesar’ and‘Gaul’.

(2) Caesar conquered Gaul.

Frege also proposed meanings for the basic expressions and rulesspecifying the meaning of a complex expressions in terms of itsstructure and the meanings of their constituents. The relevant notionof meaning is determined by what is important for logic. Frege callsthis notion of meaningBedeutung, orreference.

For Frege, a proper name refers to an object: ‘Caesar’refers to Caesar; ‘Cicero’, to Cicero; and‘Gaul’, to Gaul. Ann-ary predicate refers to afunction fromn objects to truth-values. Since‘conquered’ is a binary predicate, it refers to a functionfrom two objects into truth-values.

The sentence that results from combining ann-ary predicatewithn proper names refers to the value of the functionreferred to by the predicate for the arguments referred to by thenames. For example, (2) refers to the value of the function referredto by ‘conquered’ for the arguments referred to by‘Caesar’ and ‘Gaul’. So, a sentence such as(2) refers to its truth-value, the true, and a sentence such as (3)refers to its truth-value, the false.

(3) Cicero conquered Gaul.

For Frege, the truth-value is the referent of a sentence because it issentence’s preeminent logical property.

This fragment ofBegriffsschrift is compositional. Indeed, itis widely held that Frege wanted reference to be compositional for thewhole language ofBegriffsschrift (including logicalconnectives and the quantifiers). The evidence for this is that Fregeclearly holds that substituting expressions with the same referencepreserves reference; cf. (Pelletier 2001, 101-2). However, even ifFrege aimed for a compositional theory of reference, there is debateabout whether he achieved it, for reasons having to do with quantification.^[4]

1.2.2 The Compositionality of Propositional Content

Frege observes that two sentences with the same truth-value can havedifferentcognitive values. For example, the sentences (2)and (4) are both true.

(4) Cicero is identical to Cicero.

However, (2) is significant while (4) is trivial. Frege posits anadditional level of meaning, sense, to explain this difference. Thesense of a sentence is thethought that the sentenceexpresses. So, (2) has as its sense the thought that Caesar conqueredGaul, and (4) has as its sense the thought that Cicero is identical toCicero. A Fregean thought is the object of thinking, belief, and otherattitudes. Today, Fregean thoughts are commonly known aspropositions (see entry).

Frege also posits senses for subsentential expressions such as namesand predicates. One reason is that substituting co-referential namesin a sentence does not always preserve its sense. The names‘Cicero’ and ‘Tully’ both refer to the Romanorator Cicero, but sentences (4) and (5) differ in cognitivesignificance.

(5) Cicero is identical to Tully.

Frege concludes that names like ‘Cicero’ and‘Tully’ have distinct senses even though they share areferent.

Frege seems to be arguing that because (4) and (5) express distinctthoughts but differ only in the substitution of ‘Cicero’for ‘Tully’, these subsentential expressions must havedifferent cognitive values. Although there are disagreements on thispoint, arguments of this sort suggest that Frege holds that the senseof a complex expression is determined by the senses of itsconstituents; cf. Dummett 1973, Heck and May 2011, and Pickel 2021.For dissent, see Janssen 1997 and Pelletier 2001.

1.2.3 The Compositionality of Intension

Another milestone in the discussion of compositionality isCarnap’sMeaning and Necessity. Carnap develops anartificial language to model reasoning about possibility andnecessity. In this language, substituting one sentence of another withthe same truth-value does not always preserve truth-value. Thesentences (2) and (4) have the same truth-value, but the sentences‘Necessarily: Caesar conquered Gaul’ and‘Necessarily: Cicero is identical to Cicero’ havedifferent truth-values. In Carnap’s language, if a constituentin a complex expression is substituted for another constituent withthe same intension, then the new complex expression will have the sameintension as the original. Thus, there is a clear sense in which theintension of a expression is determined by its structure and by theintensions of the expression’s constituents. (See Carnap 1947,§§11-12.)

But what are intensions? The intension of an expression determineswhat Carnap calls anextension for each possibility. Anextension is very close to what Frege takes to be the referent of anexpression. The extension of a sentence is a truth-value; theextension of a name is its referent; and the extension of a predicateis a set of individuals (or its characteristic function, as in Fregeanreference). So, the intension of an expression can be modeled as afunction that takes a possibility as argument and returns itsextension (or Fregean referent) as value. In Carnap’s own work,possibilities are given by state descriptions, which are sets ofsentences defining a possible situation. Today, these possibilitiesare taken as possible worlds.

The appeal of intensional semantics is not limited to artificiallanguages like Carnap’s. Compositional intensional semantics hasbecome a dominant approach in contemporary formal semantics fornatural language. See Montague 1970; Lewis 1970; Stanley2009.

1.2.4 Meaning and Context

In contrast to artificial languages, natural languages containindexicals, which require the distinction between two notions ofmeaning. On the one hand, expressions have astanding meaningfixed by convention and known to those who are linguisticallycompetent. A sentence such as ‘I conquered Gaul’ is anunambiguous sentence of English. On the other hand, expressions in useare associated withoccasion meanings which is discerned byinterpreters in part on the basis of contextual information. Theutterances of ‘I conquered Gaul’ by Caesar and by Cicerodiffer in all of the respects covered thus far: truth-value,propositional content, and intension. The terminology ofstandingmeaning andoccasion meaning comes from Quine (1960).Kaplan (1977) uses the termscharacter andcontent.

Corresponding to these two notions of meaning, there are two versionsof the principle of compositionality. Since occasion-meaning isdetermined, in part, by context (C\(_{\textit{occ}}\)) must berelativized to context:^[5]

(C\(_{\textit{stand}}\))

For every complex expressione inL, thestanding meaning ofe inL is determined by thestructure ofe inL and the standing meanings of theconstituents ofe inL.

(C\(_{\textit{occ}}\))

For every complex expressione inL and everycontextc, the occasion meaning ofe inLatc is determined by the structure ofe inL and the occasion meanings of the constituents ofeinL atc.

Recently, a number of arguments have purported to show that (C) isfalse. These arguments pertain to the features that are often modeledas variables: bound pronouns, tenses, and modalities. Schlenker (2002,64) offers a natural language example. A sentence such as (6) has asits occasion meaning that the referent of ‘he’ in contextleft two days prior to the context of utterance. So, an utterance of(6) on 4 January 2000 demonstrating John has the same occasion meaningas (7).

(6) He left two days ago.

(7) He left on 2 January 2000.

However, (6) and (7) seem to have different occasion meanings whenembedded (even when they are both uttered on 4 January 2000). Contrastthe following: ‘Over the years, John told me that he left twodays ago’ and ‘Over the years, John told me that he lefton 2 January 2000’. Schlenker judges that these two sentenceshave different occasion meanings. The former sentence says that Johnrepeatedly told the speaker that John left two days beforeJohn’s speaking. This suggests that the expression ‘twodays ago’ contributes something other than its occasionmeaning. One standard response is that an expression such as‘two days ago’ does not contribute its occasion meaning,but only its standing meaning to determining the occasion meaning ofthe longer sentence. A rival response appeals to internal dynamics ina sentence to find an appropriate occasion meaning. On these views,the context relevant to determining the occasion meaning of ‘twodays ago’ depends on its position in the sentence. Fordiscussion of related cases, see: Rabern 2012, Yli-Vakkuri 2013, Kingand Glanzberg 2020, and Stonjić 2021.^[6]

1.3 Determination

This section will discuss interpretations of compositionality thatdiffer in their understanding ofdetermination. In what senseis the meaning of a complex expression determined by its structure andthe meanings of its parts?

1.3.1 Substitutional Compositionality

If the meaning of a complex expression isdetermined by itsstructure and the meanings of its constituents, then any expressionwith the same structure and the same meanings assigned to itscorresponding constituents must have the same meaning. One way ofimplementing this idea is to hold that if two complex expressionsdiffer only by the substitution of synonyms, then they are synonymous.The antecedent of this principle is meant to hold fixed the structureof the complex expression and the meanings of its pairwiseconstituents. The only thing that differs between the complexexpressions are the words used, not their meanings. Thus, the complexexpressions must have the same meaning.

This interpretation assumes that compositionality islocal.If the expressions ‘book thieves’ and‘biblioklepts’ are synonyms, then local compositionalitydemands that expressions that differ by substituting ‘bookthieves’ for ‘biblioklepts’ have the same meaning.Global compositionality says that the meaning of a complex expressionis determined not merely by its immediate structure and the meaningsof its immediate constituents, but by its total structure and themeanings of all of its constituents (including both immediate andnon-immediate constituents). If global compositionality holds, thensubstituting basic constituents with the same meaning will alwayspreserve meaning. Substituting non-basic constituents can changemeaning. Because local compositionality is more relevant forlinguistics, the following discussion will be centered around thisprinciple.

The interpretation also assumes that constituents (rather than meresub-strings) are substituted. Otherwise, as Geach pointed out, thesynonymy of ‘Plato was bald’ with ‘Baldness was anattribute of Plato’ would guarantee the synonymy of ‘Thephilosopher whose most eminent pupil was Plato was bald’ and‘The philosopher whose most eminent pupil was baldness was anattribute of Plato’ (Geach 1965, 110).

The principle that synonyms are substitutable comes in twoversions:

(S\(_{\textit{singular}}\))

If two meaningful expressions differ only in that one is theresult of substituting a synonym for a constituent within the other,then the two expressions are synonyms.

(S\(_{\textit{plural}}\))

If two meaningful expressions differ only in that one is theresult of substituting some synonyms for some constituents within theother, then the two expressions are synonyms.

The principle (S\(_{\textit{singular}}\)) says that substituting onesynonym at a time preserves meaning. The principle(S\(_{\textit{plural}}\)) says that substituting several synonyms at atime preserves meaning. The two principles are equivalent given abackground assumptions about substitution.

The background assumption concerns whether substitution of synonymscan turn a meaningful expression into a meaningless one, and whetherit can turn a meaningful expression into an expression with adifferent meaning. The principle that rules out the former possibilitywas first proposed by Husserl (1913, 318), and it is usually stated interms of the notion of asemantic category. Two expressionsbelong to the same semantic category just in case they areintersubstitutable within any meaningful expressionsalvasignificatione (without loss of meaningfulness). According toHusserl’s principle:

(H)

Synonyms belong in the same semantic category.

Assuming in addition that the language satisfies(H), (S\(_{\textit{singular}}\)) is equivalent to(S\(_{\textit{plural}}\)); cf. Hodges (2001: Theorem 4). The reasonthat the two principles are not equivalent without (H) is that it ispossible that a double substitution of expressions yields a meaningfulexpression while individual substitutions do not. Hodges (2001, 13)offers the example that starting with ‘fast asleep’ andjointly substituting ‘fast’ for ‘deeply’ and‘sleeping’ for ‘asleep’ yields a meaningfulexpression ‘fast asleep’. On the other hand, theindividual substitutions yields expressions that may not bemeaningful: ‘fast sleeping’ and ‘deeplyasleep’.

(H) is a controversial for natural language—intuitively, thereare many synonyms that are not everywhere intersubstitutable. Forexample, ‘likely’ and ‘probable’ mean prettymuch the same even though ‘Jacques is likely to leave’ ismeaningful while ‘Jacques is probable to leave’ isarguably not; cf. Gazdar (1985, 32).^[7] And—more controversially—there might be synonyms that arealmost nowhere intersubstitutable: ‘quick’ and‘quickly’ are good candidates.

1.3.2 Rule-to-Rule Compositionality

Sometimes the claim thatL is compositional is presenteddirectly as a claim about the relationship between its syntax andsemantics. The following thesis is often calledrule-to-rulecompositionality:

(RR)

To every syntactic rule corresponds a semantic rule that assignsmeanings to the output of the syntactic rule on the basis of themeanings of its inputs.

Consider a rule of a toy syntax that allows one to combine a nounphrase (such as ‘Caesar’, ‘Cicero’,‘Tully’) with a verb phrase (‘conquered Gaul’)to form sentences (‘Caesar conquered Gaul’, ‘Ciceroconquered Gaul’, ‘Tully conquered Gaul’, and so on).Rule-to-rule compositionality says that there is a corresponding ruleG that specifies the meanings of the sentence in terms of themeanings of its constituents. The meaning of ‘Caesar conqueredGaul’ would be the result of applying the operation G to themeanings of ‘Caesar’ and ‘conquered Gaul’. Themeaning of ‘Cicero conquered Gaul’ would similarly be theresult of applying G to the meanings of ‘Cicero’ and‘conquered Gaul’.

Montague (1970) suggested a perspicuous way to formally capturerule-to-rule compositionality: compositionality requires the existenceof ahomomorphism between the expressions of a language andthe meanings of those expressions. Consider the expressions of alanguage (including both simple and complex expressions) as a setE upon which a number of operations (syntactic rules) aredefined. There is a set \(\Gamma \) of syntactic rules for thelanguage. Each of the rules \(F_{\gamma}\) is characterized by a(partial) function that takes a fixed number of expressions fromE and returns another expression ofE. For example,one rule might combine a noun phrase (‘Caesar’) and a verbphrase (‘conquered Gaul’) to form a sentence(‘Caesar conquered Gaul’). The expressionsE andrules in \(\Gamma \) define a partial algebra \(\mathbf{E} = \langleE, (F_{\gamma})_{\gamma \in \Gamma}\rangle\). Call this thesyntactic algebra. This is just the set of expressions in thelanguage and the set of formation rules relating them.

Compositionality concerns the relationship between the language (thesyntactic algebra) and the set of meanings. A meaning-assignmentm is a function that maps expressions inE toM, the set of available meanings for the expressions ofE. On Montague’s implementation, compositionalitydemands that the meaning assignmentm is a homomorphism.

Consider a specific syntactic operationF which takeskexpressions, \(e_1 ,\ldots ,e_k\), as input and specifies acomplex expression, \(F(e_1 ,\ldots ,e_k)\).m isahomomorphism for operation F if and only if there is ak-ary partial functionG onM such thatwhenever \(F(e_1 ,\ldots ,e_k)\) is defined,

\[m(F(e_1 ,\ldots ,e_k)) = G(m(e_1),\ldots ,m(e_k)).\]

In English: this means that there is a partial functionGfrom the meanings of \(e_1 ,\ldots ,e_k\) to the meaning of theexpression built from \(e_1 ,\ldots ,e_k\) through an application ofthe syntactic ruleF. To return to the example, for thesyntactic rule that forms a sentence by concatenating a noun phrase(‘Caesar’) with a verb phrase (‘conqueredGaul’), there is a semantic ruleG such that themeaning of the sentence (such as its truth-value or intension) isdetermined by applyingG to the meaning of the noun phrase(say, Caesar) and the meaning of the verb phrase (such as the set ofpeople who conquered Gaul).

Finally, the meaning assignmentm is compositionalsimpliciter just in casem is a homomorphism foreach syntactic operation inE. Wheneverm iscompositional, it induces the semantic algebra \(\mathbf{M} = \langleM, (G_{\gamma})_{\gamma \in \Gamma}\rangle\) onM, and it isa homomorphism betweenE andM; cf.Westerståhl (1998). (For details, variants, and formal results,see Janssen 1983, 1997, Hodges 2001, and Pagin & Westerståhl2010a. For generalizations that cover languages with various sorts ofcontext-dependence, see Pagin 2005, Pagin & Pelletier 2007, andWesterståhl 2012.)^[8]

1.3.3 Compositionality Across Languages

The substitutivity of synonyms and rule-to-rule compositionality eachseem to be entailments of the principle of compositionality. However,they are both language-bound in ways that the compositionalityprinciple itself is not. Each principle rules out the existence of apair of non-synonymous complex expressions with identical structureand pairwise synonymous constituents within asinglelanguage. However, the principles say nothing about how to comparemeanings across languages. There are two issues with this.

One issue is that both implementations are compatible with theexistence of non-synonymous complex expressions in different languageswith the same structure and whose constituents have the same meaning.This is a violation of the idea that the meaning of a complexexpression is determined by its structure and by the meanings of itsconstituents. Here is an illustration from Szabó (2000b).Suppose English is compositional. We define a new languageCrypto-English that is just like English one small difference. So,Crypto-English has the same expressions and the same syntax asEnglish. Semantically, all expressions of Crypto-English have the samemeanings with one important exception. Take two of its non-synonymoussentences, say, ‘Elephants are grey’ and ‘JuliusCaesar conquered Gaul’. Theonly difference between thetwo languages is that if a sentence is synonymous in English with oneof the two designated sentences, then it is synonymous with the otherin Crypto-English. Given the assumption that English is compositional,there is no pair of non-synonymous complex expressions in English withidentical structure and pairwise synonymous constituents. Trivially,the same must hold for Crypto-English as well. But intuitively,Crypto-English isnot compositional. The structure and themeanings of constituents of the Crypto-English sentence‘Elephants are grey’ cannot determine what this sentencemeans in Crypto-English—if they did then the structure and themeanings of constituents of the English sentence ‘Elephants aregrey’ would have to determine what ‘Julius Caesarconquered Gaul’ means in English. If there is a language such asCrypto-English in addition to English, it would follow that themeaning of a complex expression is not determined only by itsstructure and by the meanings of its constituents, but also by thelanguage in which it occurs.

We have understood compositionality as saying that the meaning of acomplex expression is determined by its structure and the meanings ofits constituents. The problem is that this formulation is compatiblewith different languages having different functions from structuresand the meanings of the parts to the meaning of the whole. This can beavoided by requiring that in every possible natural human language,the meaning of a complex expression is determined by the same functionfrom structures and the meanings of the constituents.

(C\(_{\textit{cross}}\))

For every complex expressione inL, the meaningofe inL is functionally determined through asingle function for all possible human languages by the structure ofe inL and the meanings of the constituents ofe inL.

It was assumed that English and Crypto-English have the same syntacticrules (and thus the same structures) and that the meanings of thebasic expressions are the same. If the function that determines themeanings of the complex expressions must be the same in all languages,then the possibility of a language such as Crypto-English, whichdiffers from English only by switching the denotations of certaincomplex expressions, is ruled out.

The other issue arises from the fact that the principle ofcompositionality is meant to abstract away from the particularfeatures of the words (e.g. spelling, pronunciation) in determiningthe meanings of complex expressions. For example, if‘biblioklept’ and ‘book thief’ have the samemeaning, then complex expressions that differ only by the choice ofone expression for the other have the same meaning. The identities ofthe words don’t matter to determining the meaning of the complexexpression. Suppose that there is a natural language that contains aword meaning the same as ‘Caesar’ and a phrase meaning thesame as ‘conquered Gaul’. Suppose that there is asyntactic rule that combines these expressions into a sentence. Doesthe principle of compositionality (understood as either thesubstitutivity of synonyms or as rule-to-rule compositionality) entailthat this sentence has the same meaning as the English sentence (2)‘Caesar conquered Gaul’? It does not. The principle ofcompositionality says that if two sentences havethe samestructure and their corresponding constituents have the samemeanings, then the two sentences have the same meaning. In order toinfer that the meaning of the sentence of the new language is the sameas the meaning of (2), one needs the additional premise that the twosentences have the same structure. In the context assumed so far, thiswould mean that the syntactic rule from which the new sentence isderived is the same as the syntactic rule from which the Englishsentence (2) was derived. This (toy) English rule said that a sentencecan be derived by combining a noun phrase and a verb phrase. However,the new language might not even have the same syntactic categories asEnglish.

One response strengthens rule-to-rule compositionality. Rule-to-rulecompositionality says that for every rule generating a complexexpression out of some constituent expressions, there is a functionfrom the meanings of the constituent expressions to the meaning of thewhole. In other words, for every syntactic composition rule, there isa corresponding semantic composition rule. The new response restrictsthe range of admissible semantic composition rules. There are only afew functions from the meanings of the constituents to the meanings ofthe complex.

Some have held that a single semantic evaluation rule specifies themeaning of a complex in terms of its structure and the meanings of itsconstituents and that this rule applies in every natural language.This type of view might be seen in Lewis’s (1970, 29)description of the appeal of a “uniformfunction-and-argument” rule. On this view, every complexexpressionE in any natural language has a constituent whosemeaning is a functionf, and the meaning ofE is thevalue off when it takes the remaining constituents asarguments. So, if (2) is composed of ‘Caesar’ and‘conquered Gaul’, then the meaning of ‘conqueredGaul’ is a function and the meaning of ‘Caesar’ isan argument for the function. The meaning of (2) would be the resultof applying this function to the meaning of ‘Caesar’. Inany language, if there is a sentence with one component meaning thesame as ‘Caesar’ and the other component meaning‘conquered Gaul’, the meaning of this sentence will bedetermined in exactly the same way. Other approaches agree that thereis a single uniform semantic evaluation rule for all natural languagesbut deny that it is functional application. For instance, Pietroski(2005, 2012, 2018) advocates a relatively uniform rule of conjunction.The meaning of any complex expression is the result of conjoining themeanings of its constituents.

Many standard works opt for a minimum set of rules rather than asingle semantic rule. On these views, it is not the case that everysyntactic formation rule corresponds to the same function in everynatural language. Even within a language, different syntacticformation rules correspond to different function. However, each newsemantic rule is considered a cost to the theory; cf. Heim and Kratzer1998 (65-67).

These approaches can be freed from dependence on features of aparticular language by embracingtype-drivencompositionality, which is another strengthening of rule-to-rulecompositionality. Type-driven compositionality assumes that meaningsdivide into a variety of semantic types just as expressions divideinto syntactic categories. There are certain admissible rules forputting meanings of some given types together in order to determineanother meaning. Type-driven compositionality says that the meaning ofa complex expression is determined by the meanings of its immediateconstituents alone. Type-driven compositionality is meant to explainwhy speakers do not need to learn the particular identity of thesyntactic rules of a language in order to derive the meaning of acomplex expression from their knowledge of the meaning of itsconstituents. Type-driven compositionality is normally combined withthe thesis that syntactic branching is binary so that every complexexpression has exactly two immediate constituents. It would followthat the meaning of any expression is wholly determined by themeanings of its two immediate constituents. The original sources ofthis idea are Klein and Sag 1982 and Jacobson 1982. Some moderndiscussion includes (Heim and Kratzer 1998, §3.1) and (Jacobson2014, chapter 6).

2 Related Principles

This section compares(C) to some principles often discussed alongside compositionality.

2.1 Collective Compositionality

Compositionality says that the meaning of a complex expression isdetermined by its structure and the meanings of its constituents. Oneplausible consequence was the substitutivity of synonyms: that if twocomplex expressions differ by the substitution of one term for asynonym, then the complex expressions have the same meaning sentence.However, to derive this consequence, compositionality must beconstrued asdistributive. Each occurrence of an expressioncontributes its meaning individually to expressions that containit.

Distributive compositionality has been challenged, however. Putnam(1954), Kaplan (1990, 95, footnote 6), and Fine (2007) have held that‘Cicero’ and ‘Tully’ are synonyms, but‘Cicero is Cicero’ and ‘Cicero is Tully’ arenot. They argue that the difference in meaning arises from the factthat the former sentence encodes semantic co-reference between‘Cicero’ and ‘Cicero’, but the latter does notencode coreference between ‘Cicero’ and‘Tully’. They thereby propose principles weaker thancompositionality such as collective compositionality(C\(_{\textit{coll}}\)):

(C\(_{\textit{coll}}\)) For every complex expressione inL, the meaning ofe inL is determined bythe structure ofe inL and the meanings of theconstituents ofe inL collectively.

On Fine’s view, what the constituents of ‘Cicero isCicero’collectively mean goes beyond what theconstituents of ‘Cicero is Tully’ do. The collectivemeaning comprises the individual meanings plus certainmeaning-relations that hold among them. Call the weak principle(C\(_{\textit{coll}}\))collective compositionality;following usual practice(C) will be understood asdistributive compositionality.

Fine thinks the meaning of ‘Cicero is Cicero’ depends onits structure (this is why it is not synonymous with ‘Is CiceroCicero?’), on the individual meanings of its constituents (thisis why it is not synonymous with ‘Cicero is Caesar’), andin addition on the intended co-reference relation between thesubject and the object, which is an aspect of the collective meaningof its constituents (this is why it is not synonymous with‘Cicero is Tully’).

Fine (2007, 38) is explicit that this his approach is meant to be adeparture from standard conceptions of compositionality such as localand global compositionality glossed above. This is certainly true ofthe views of Putnam and Kaplan. It is debatable whether Fine’sown approach abandons many standard forms of compositionality. Bymaking the co-reference relation a separate input to the semantics,Fine’s view may preserve global compositionality; cf. Pickel andRabern 2017.^[9]

2.2 Reverse Compositionality and the Building Principle

A commitment of compositionality as understood by(C) is that if two expressions havedifferent meanings, thenthey must either havedifferent structures or somecorresponding pair of constituents must havedifferentmeanings.Reverse compositionality say that this claim isreversible: if two expressions have thesame meaning, thenthey must have thesame structure, and all of theircorresponding constituents must have thesame meaning. Forexample, ‘book thief’ and ‘biblioklept’ havedifferent structures. So, by reverse compositionality, they must havedifferent meanings. Reverse compositionality neither entails nor isentailed by compositionality understood as(C). While(C) is a widely assumed principle in natural language semantics, reversecompositionality is not. However, many debates in philosophy of mind(especially those pertaining to connectionism) concern reversecompositionality. (Fodor (1998b), Fodor & Lepore (2001), and Pagin(2003) advocate reverse compositionality; Patterson (2005), Robbins(2005), and Johnson (2006) are among its opponents. The debate iscomplex, in part because at least some proponents of reversecompositionality advocate it only for the language of thought; cf.Fodor 2001.) The issue is also discussed in connection with therepresentational properties of images; see Isaac 2017.

The claim thatL is compositional is sometimes taken to meanthat the meaning of an arbitrary complex expression inL isbuilt up from the meanings of its constituents inL—call this thebuilding principle forL. This is a fairly strong claim, at least if the buildingmetaphor is taken seriously. For then the meanings of complexexpressions must themselves be complex entities whose structuremirrors that of the sentence; cf. Frege 1892, 1919. This presumablyentails but is not entailed by compositionality as understood by(C); cf. Keller and Keller 2013 and Keller 2022. It also seems to entailreverse compositionality.

2.3 The Primacy of Words

Often, to say that one thing determines another is to attribute causalor explanatory priority. Although the principle of compositionality isusually not understood in this way, sometimes philosophers read it asa principle that asserts the priority of word meaning over sentencemeaning, or more generally, the priority of the meanings of lexicalitems over the meanings of complex expressions:

(P)

Complex expressions have their meanings in virtue of theirstructure and the meanings of their constituents.

(P) is often thought to be in tension with the idea that eachexpression has the meaning it does in virtue of the way it is usedwithin some linguistic community. The conflict is supposed to arisebecause (i) the use of an expression is exhausted by its employment inspeech acts, and (ii) it is sentences, not words, that can be employedto make speech acts. Against this, it can be argued thatreferring is among the speech acts speakers routinely performand that this speech act is done with words, not sentences. One mighttry to replace (i) with a stronger claim, for example, that the use ofan expression is exhausted by its employment inasserting,asking,commanding, and a few of other speech actsnot includingreferring. But even if true the stronger claimmay not save the argument against (P) because, at leastprimafacie, one can make assertions uttering isolated words; cf.Stainton (2006). Davis (2003) develops a detailed theory of meaningthat combines (P) with a version of the use theory of meaning. (Forsome arguments against(P), see Szabó (2019).)

2.4 The Context Principle

Speakers do not learn the meanings of the constituent expressions inisolation. Rather, speakers learn the meanings of many basicexpressions by seeing them used in the context of a complete sentence.For example, one might learn the meaning of a number word such as theword ‘four’ by seeing it used in sentences such as‘Jupiter has four moons’ rather than by confronting themeaning of the word on its own.

Considerations such as these led Frege to famously declare in section60 of theFoundations of Arithmetic that only within acomplete sentence do words have meaning. This has come to be referredto in the literature as Frege’scontext principle.Frege writes that “it is enough if the sentence as whole hasmeaning; thereby also its parts obtain their meanings” (Frege[1884] 1950: section 60).^[10] This context principle has many interpretations. On someinterpretations, the context principle conflicts with the principle ofcompositionality. Theprima facie tension between the contextprinciple and compositionality has been noted across times andcultures. It was, for instance, raised and addressed in the traditionof Prabhākara Mīmāṃsā in classical Indianphilosophy; cf. Ollett 2020.

Read literally, Frege’s statement asserts that words have theirmeanings in virtue of the meaning of sentences in which they occur asconstituents. This is incompatible with(P), but not with(C). Even if words are meaningful only because they occur as constituentswithin sentences, there could still be a function that maps thestructure of a sentence and the meanings of its constituent words tothe meaning of that sentence. There would remain a sense in which themeanings of the words determine the meanings of the sentences.

There is an alternative way to construe Frege’s contextprinciple, a way that makes it a determination claim, not a primacyclaim. To state it in a form that matches the generality of(C), it must be generalized to apply to all complex expressions and theirconstituents. On one interpretation (F\(_{\textit{all}}\)), themeaning of an expression such as ‘conquered’ is determinedby the meanings of the totality of expressions in which it occurs in aconstituent: ‘Caesar conquered Gaul’, ‘Ciceroconquered Gaul’, ‘Caesar conquered Rome’, and soon.

(F\(_{\textit{all}}\))

The meaning of an expression is determined by the meanings of allcomplex expressions in which it occurs as a constituent.

According to (F\(_{\textit{all}}\)), for the word‘conquered’ to have a different meaning one of theexpressions it occurs in must have a different meaning. Like theprinciple of compositionality, (F\(_{\textit{all}}\)) can beinterpreted as a claim about reference or meaning, locally orglobally, collectively or distributively, in a language-bound manneror cross-linguistically. Compositionality is about bottom-upmeaning-determination, while the context principle about top-downmeaning-determination. As long as it is not understood as a causal orexplanatory relation determination can be symmetric, so any version of(C) is compatible with the corresponding version of(F\(_{\textit{all}}\)).

There is a strengthening of(F\(_{\textit{all}}\)), according to which the meaning of an expression is determined notonly by the meanings ofall expressions in which it occurs asa constituent, but by the meaning ofany one of theseexpressions:

(F\(_{\textit{any}}\))

The meaning of an expression is determined by the meaning of anycomplex expression in which it occurs as a constituent.

If (F\(_{\textit{any}}\)) is true, then the meaning of‘conquered’ is wholly fixed by the fact that ‘Caesarconquered Gaul’ means what it does. (F\(_{\textit{any}}\)) is animmediate consequence ofreverse compositionality.(F\(_{\textit{any}}\)) is incompatible with many standard semantic theories. For example, asimple intensional semantics assigns to each sentence the set ofpossible worlds where it is true. Consider a language that containsthe standard logical operators, and so any sentence is a constituentof a necessarily true sentence. Since the meaning of a necessary truthis the set of all possible worlds, this set would have to determinethe meanings of all sentences in the language, which is implausible.There are principles of intermediate strength between(F\(_{\textit{all}}\)) and(F\(_{\textit{any}}\)). These principles allow that the meaning of an expression may be fixedby the meanings of some set or sets of complex expressions in which it occurs.^[11]

3. Arguments for Compositionality

The simplest argument for compositionality is that it accords with theordinary experience of language. Human beings learn simple words andgrammatical structures. They often use their understanding of thewords and structures in order to understand more complex structures.Although there are interesting putative counterexamples (seesection 3.2), these can often be explained away through revisions of syntacticand/or semantic theories. This argument is a reasonable startingpoint, but its evidential value is limited. It does not fullydistinguish compositionality from related principles or show that itapplies in all cases. The pre-theoretic appeal of compositionalitywill not convince those who find some of the proposed revisionsunacceptable. Moreover, this argument does not reveal the overall rolecompositionality should play in semantic theory. This section willexplore two kinds of argument for compositionality. Empiricalarguments propose that compositionality best explains some linguisticphenomenon. Methodological arguments propose that compositionality isjustified by the role it plays in semantic theories.

3.1 Empirical Arguments

The empirical arguments for compositionality identify some particularobserved fact in need of explanation. They offer an explanation forthis fact in which compositionality is a central component. Theseempirical arguments are not deductively valid arguments. The argumentstake the form of inferences to the best explanation. It is argued thatcompositionality is part of the best explanation of some observedfact. The arguments fail if a better explanation can be found for theobserved fact.

3.1.1 Productivity

The most common argument for compositionality is based onproductivity. The argument begins with the observation that speakerscan understand a large number of complex expressions they have neverbefore encountered. A speaker of English will understand the sentence(8) even though they have never explicitly encountered that sentencebefore.

(8) A book thief will steal a short book before a long book becauseshort books better fit in large coats.

This ability to understand novel complex expressions requiresexplanation.

To bolster the claim that speakers do, in fact, understand complexexpressions they have never heard before, philosophers often appeal tounboundedness: although humans are finite beings, they havethe capacity to understand each of an infinitely large set of complexexpressions. Although there are dissenters—e.g., Ziff (1974) andPullum & Scholz (2010)—the claim that natural languagescontain infinitely many complex expressions is plausible.^[12] But it is equally plausible that nobody who reads this entry thefirst time has ever encountered this very sentence before, andconsequently, the detour through cardinality considerations seemssuperfluous.

Speakers cannot so easily understand novel simple expressions. Aspeaker who has never encountered the word ‘lugubrious’before will need its meaning explained. The argument from productivityis that compositionality is part of the best explanation ofspeakers’ ability to understand novel complex expressions. Theargument can be found in Frege, who claimed that

the possibility of our understanding sentences which we have neverheard before rests evidently on this, that we can construct the senseof a sentence out of parts that correspond to words. (Frege [c. 1914]1980: 79)

The details of the argument can be expanded without assuming thatmeanings are Fregean senses.^[13]

How can a speaker’s ability to understand novel complexexpressions such as (8) be explained? It can be explained by assuming(Prod1) that (8) has constituents that are put together according tocertain rules; (Prod2) that speakers know the meanings of theseconstituent expressions and know the rules for putting them together;and (Prod3) that if anyone can understand some expressions and thesyntactic rules for deriving a complex expression out of these simplerexpressions, then they can understand the complex expression. Given(Prod1), (Prod2), and (Prod3), the observed fact that speakersunderstand (8) would be explained.

Compositionality figures in this explanation because (Prod3) itselfneeds to be explained. How can speakers pass from their understandingof some more basic expressions and a syntactic rule for putting themtogether to form a larger expression to understanding the meaning ofthe complex expression? The argument from productivity assumes that(Prod3) is best explained if some form of the principle ofcompositionality is true, and the meaning of the complex is determinedby its structure and the meaning of its constituents.

Occasionally, the fact that natural languages arelearnableis also used to argue for compositionality. This is not a whollyindependent argument: one reason it is remarkable that one can learn anatural language is that once one has learnt it, one’sunderstanding is productive. If one could not understand expressionsone has never encountered before without detailed empirical study,then it might be that the language was learned by rote memorization.That said, compositionality may explain the fact that naturallanguages are learnable by constraining the space of hypotheses aboutthe meanings of the complex expressions of the language; cf. Del Pinal2015.

3.1.2 Systematicity

Another argument for compositionality is based onsystematicity, the fact that there are definite andpredictable patterns among the sentences a speaker understands. Inparticular, the observed datum is that speakers can generallyunderstand grammatical recombinations of sentences they alreadyunderstand. A speaker who understands the sentences ‘the dog isasleep’ and ‘the cat is awake’ can generallyunderstand the sentences ‘the dog is awake’ and ‘thecat is asleep’.

Like the argument from productivity, the argument fromsystematicity seeks to explain how speakers can go from theirunderstanding of some portion of the language to their understandingof novel sentences. The portion of the language that a speaker isassumed to understand is a set of sentences. The novel expressions arethose that result from permuting the basic vocabulary.

The explanation is meant to run as follows: (Sys1) A speaker whounderstands some complex expressions understands the rules thatgenerate it and knows the meanings of its constituents; (Sys2)speakers who understands the rules that generate a complex expressioncan apply it to other expressions they understand; and (Sys3) if aspeaker understands the meanings of some constituent expressions andunderstands a rule for putting them together, then the speaker canunderstand the complex expression that results from putting the basicexpressions together. A speaker who understands ‘the dog isasleep’ and ‘the cat is awake’ can also understandthe constituents ‘the dog’, ‘the cat’,‘is asleep’, and ‘is awake’ and the rule forcombining noun phrases and verb phrases. The speaker who understandsthese words and the rule for combining noun phrases and verb phrasesis able to understand ‘the dog is awake’ and ‘thecat is asleep’.

Again, compositionality is meant to explain assumption (Sys3).Speakers are able to know the meanings of complex expressions fromtheir knowledge of the meanings of basic expressions and theirknowledge of the rules for putting these expressions together becausethe meanings of the complex expressions are determined by the meaningof the complex expressions.

The argument from systematicity exhibits some important differencesfrom the argument from productivity. One difference is in theplausibility of the purported fact to be explained. There are manyexamples where speakers are able to understand recombinations ofexpressions that they already understand. However, it is unclear howfar to generalize this claim. Do all who understand ‘within anhour’ and ‘without a watch’ also understand‘within a watch’ and ‘without an hour’? And doall who understand ‘halfway closed’ and ‘firmlybelieved’ also understand ‘halfway believed’ and‘firmly closed’? As Johnson (2004) argues, the claim thatnatural languages are systematic presupposes a natural non-overlappinglinguistic categorization of all the expressions. The claim that inunderstanding a sentence one must be able to understand itsconstituents is related to reverse compositionality; cf. section1.4.2; Fodor & Lepore 2001: 59; Pagin 2003: 292.^[14]

The arguments also differ because the systematicity argument aims toexplain a speaker’s ability to understand novel sentences interms of their existing ability to understand whole sentences. It maybe that in understanding the reconfigurations of the alreadyunderstood sentences, speakers appeal not only to the meanings of theparts of the sentence but also to the meanings of the whole sentences.So, the speaker may use their understanding of the whole sentences‘the dog is asleep’ and ‘the cat is awake’(and all of their constituents) to understand the novel sentence‘the dog is awake’. This is compatible with thepossibility that it is not merely the structure and meanings of theconstituents that determine the meaning of the complex expression, butalso the meanings of various other complex expressions. This is not tosay that the latter explanation is better, but only that theassumption that the speaker has more knowledge makes availableexplanations that are not available in the argument fromproductivity.

3.1.3 Challenges to the Empirical Arguments

There are many challenges to these empirical arguments in theliterature. One of the most prominent challenges purports to offer abetter explanation of the fact that speakers can know the meaning of acomplex expression by knowing the rule for constructing the expressionand knowing the meanings of its constituents.

Here is an alternative explanation: the claim that the complexexpression has a certain meaning iscomputable from the claimthat it results from applying a certain rule to certain constituentexpressions and from the claim that the constituents have certainmeanings. One claim is computable from a set of claims when there isan effective procedure deriving a claim from the set of claims. Theexplanation of systematicity would require this: the claim that‘the dog is asleep’ meansthat the dog is asleepis computable from the claims (for example) that ‘the dog isasleep’ results from combining ‘the dog’ and‘is asleep’, that ‘the dog’ meansthedog, and that ‘is asleep’ meansis asleep.This explanation does not require compositionality, because thecomputation may appeal to factors other than the structure and themeanings of ‘the dog’ and ‘is asleep’. Forexample, the computation of the claim that ‘the dog isasleep’ meansthat the dog is asleep might appeal tothe expressions ‘the dog’ and ‘is asleep’themselves and not merely to their meanings; cf. Schiffer 1987, 1991,Pagin 2012, Dever 2005.

A more recent challenge comes from large language models (LLMs). LLMsproduce coherent strings of text similar to those produced by naturalspeakers by repeatedly predicting the next word from an input stringof text. The models associate a long vector with each basic expression(or “tokens”) and derives a series of vectors for stringsof vectors in terms of the vectors of the basic expressions making upthe string. The final derived vector gives rise to a probabilitydistribution for the next word. The values of the vectors are set bytraining on a large body of text, cf. (Wolfram 2023). The trainingdata include far more sentences than any normal human being is exposedto. However, LLMs can produce and process novel sentences on the basisof this training data. The question is whether LLMs encode anythinglike a recursive syntax or compositional semantics for theselanguages. If they do not, then this poses a problem for the argumentsfor compositionality appealing to productivity and systematicity,because then LLMs would be examples of systems that display competencewith novel sentences and the ability to recombine existing sentenceswithout compositionally processing them. Because of the size of thevectors associated with words and the complexity of the computations,separating out elements of the representation is challenging. Thisproject is the subject of ongoing research. See Lake and Baroni 2018and Dankers et al. 2022.

A final challenge to the empirical arguments is that they do notconclusively establish compositionality as a universal claim. Generalconsiderations of productivity and systematicity cannot rule outisolatedexceptions to compositionality. Suppose someonesuggests that the complex expressione is a counterexample to(C). The fact that speakers tend to understand all sorts of complexexpressions they have never heard before does not mean that they wouldunderstande on the first encounter. Similarly, the fact thatspeakers can generally understand a new sentence by reconfiguringwords in a sentence they already understand does not entail that theywould understande. Even ifin general humansunderstand complex expressions they have never heard before in virtueof their knowledge of the expressions’ structures and themeanings of the constituents, speakers might understande insome other way.

3.2 Methodology

A popular reason for believing in compositionality is that it works.According to this argument, compositionality has been a high-levelempirical hypothesis that is not directly related to observation.Rather, linguists have adopted various versions of the principle ofcompositionality as a working hypothesis and developed semantictheories on their basis.

Compositionality is a powerful working hypothesis, but it is wellknown that it can’t be an empirical hypothesis all by itself.(Partee 2004: 14)

In some cases, broad frameworks such as Montague semantics have beenmotivated, in part, by the appeal to various versions of the principleof compositionality. The resulting theories have provided intuitivelysatisfactory explanations for a wide range of linguistic data, such asthe validity or invalidity of inferences or contrasts in acceptabilitybetween minimal pairs. Moreover, when certain phenomena appeared toconflict with compositionality, it was usually shown that this is notso: reasonably elegant and comparatively natural compositionaltheories were just around the corner; cf. section 3.2. Finally,compositionality (or its specific implementations) has also beeninvoked to motivate narrower interventions in syntactic or semantictheory (e.g. Partee 1975, 511-513).

There are two components to the argument from methodology. The firstcomponent is a concession, that compositionality is not directlytestable. This concession is motivated by various trivializationarguments that have been introduced. The second component is that thesuccess of semantic theories does provide indirect evidence forcompositionality.

3.2.1 The Empirical Significance of Compositionality

Compositionality alone has few direct empirical implications. Tosubstantively constrain semantic theorizing, compositionality must bepaired with auxiliary hypotheses about syntax and semantics.

Janssen (1983) proved that one can turn any meaning assignment on arecursively enumerable set of expressions into a compositional one, ifone can replace the syntactic operations with different ones. Thisshows that for compositionality to remain substantive, the semantictheory must be paired with some auxiliary hypotheses about thesyntactic structure of complex expressions; cf. Westerståhl 1998.^[15]

From the side of semantics, Zadrozny (1994) showed that starting withan assignment of meanings to the expressions of a language, one cangenerate a new compositional assignment of meanings that encode theoriginal meanings. Given a setS of strings generated from anarbitrary alphabet via concatenation and a meaning functionmwhich assigns the members of an arbitrary setM to themembers ofS, one can construct a new meaning function\(\mu\) such that for all \(s, t \in S \mu(s{.}t) = \mu(s)(\mu(t))\)and \(\mu(s)(s) = m(s)\). Zadrozny’s argument shows thatcompositionality alone does not constrain the meanings of the complexexpressions provided that the meanings of the simpler expressions canbe replaced with new “meanings” from which the old areuniformly recoverable.^[16] (Cf. Kazmi and Pelletier (1998), Westerståhl (1998), Dever(1999).)

Horwich (1997) offers an argument that compositionality does notconstrain the meanings of the basic expressions, either. He considersthe view that the meanings of complex expressions are phrase structuretrees with the meanings of the constituent lexical items assigned totheir terminal nodes. The meanings of complexes are straightforwardlydetermined from the meanings of the simple expressions together withtheir syntactic mode of composition. This shows that if the semantictheory is lax enough with the meanings assigned to the complexexpressions, any assignment of meanings to basic expressions iscompatible with the principle of compositionality.^[17]

These results show that for compositionality to be a substantiveconstraint on natural language theorizing, it must be paired withbackground constraints on syntax, the semantics of the basic lexicalitems, and the semantics of composites (cf. Dever 2008).^[18]

3.2.2 Compositionality as Hypothesis

The principle of compositionality has been a successful workinghypothesis in linguistics. Does this give evidence to believe it? Thisdepends on the role that compositionality plays in the success oflinguistic theory. A high-level methodological hypothesis might beadopted for a subject matter for reasons of methodological simplicityor to commence an investigation. It may thereby lead to a successfultheory without playing any role in that success. If this were thecase, then there might be no reason to believe the principle ofcompositionality. Of course, this is compatible with using theprinciple as an instrumental hypothesis in semantics.

On the other hand, it may be that investigation reveals thatcompositionality is an important part of the explanation of variousphenomena under the scope of the theory. These phenomena might includeobservations about productivity and systematicity, discussed insection 2.2.1. If this were the case, then these empirical argumentsfor compositionality would each individually provide the principlewith some indirect empirical evidence.^[19]

4. Arguments Against Compositionality

The most prominent arguments against compositionality come fromapparent counterexamples. Putative counterexamples to(C) are complex expressions whose meaning appears to depend not only onthe meanings of their constituents and on their structure but on somethird factor as well. The third factor might be the contextin which the sentence is used (§3.2.1 and §3.2.4), itslinguistic environment (§3.2.3), auxiliary conventions(§3.2.2), or someone’s beliefs about what the expressionmeans (§3.2.5).

In each case, compositionality is not the only hypothesis generatingthe counterexample. The counterexamples depend on auxiliary hypothesesabout syntactic structure and about the meanings of the constituentexpressions. In many cases, compositionality can be maintained byrevising some of these auxiliary hypotheses. The choice betweenrevising compositionality and revising an auxiliary hypothesis willdepend on complicated theoretical constraints.

4.1 How Compositionality Might Fail

Before surveying the putative counterexamples to compositionality fromthe semantics literature, consider a simple non-linguistic case wherespeakers’ understanding is productive and systematic despiteapparent lack of compositionality in the system of representations.Specifically, consider the Algebraic notation for chess.^[20] Here are the basics. The rows of the chessboard are represented bythe numerals \(\b{1},\b{2}, \ldots ,\b{8}\); the columns arerepresented by the lower-case letters \(\ba, \bb, \ldots ,\bh\). Thesquares are identified by column and row; for example \(\b{b5}\) is atthe intersection of the second column from the left and the fifth rowfrom the top. Upper-case letters represent the pieces: \(\bK\) standsfor king, \(\bQ\) for queen, \(\bR\) for rook, \(\bB\) for bishop, and\(\bN\) for knight. Moves are typically represented by a tripletconsisting of an upper-case letter standing for the piece that makesthe move and a sign standing for the square where the piece moves.Moves made by pawns lack the upper-case letter from the beginning.When more than one piece of the same type could reach the same square,the sign for the square of departure is placed immediately in front ofthe sign for the square of arrival. Finally, there are symbols forspecial moves such as castling, for check, and for making commentariesabout moves.

Someone who understands the Algebraic notation can follow descriptionsof particular chess games and, in particular, can tellwhichmove is represented by particular lines within such a description.Nonetheless, in isolation, the line \(\b{Nb5}\) is not be enough tofigure out what this move is supposed to be. It must be a move to\(\b{b5}\) made by a knight. However, the square from which it movesis unspecified. All this can be determined by following thedescription of the game from the beginning, assuming that one knowswhat the initial configurations of figures are on the chessboard, thatwhite moves first, and that afterwards black and white move one afterthe other. But staring at \(\b{Nb5}\) itself will not help.

The first moral of the example is that one can have a productive andsystematic understanding of representations even if one does notunderstand complex representationsmerely by understandingtheir simple components and the way those components are combined. Thereason this could happen is that all who understand the system knowcertain things (e.g., the initial configuration of pieces and theorder of moves) from which they can figure out the missing information(e.g., which figure is moving and from where).

The second moral is that given certain assumptions about meaning inchess notation we can have productive and systematic understanding ofrepresentations even if the system itself is not compositional. Theassumptions in question are that (a) the description in the firstparagraph of this section fully determines what the simple expressionsof chess notation mean and also how they can be combined to formcomplex expressions, and that (b) the meaning of a line within a chessnotation determines a move. These assumptions are auxiliaryhypotheses.

The third moral is that one can maintain compositionality by rejectingthe auxiliary hypotheses such as (a) and (b). One can reject (a) byarguing that the meaning of \(\bN\) in \(\b{Nb5}\) contains anindexical component, it picks out a particular knight moving from aparticular square. One can also reject (b) by arguing that the meaningof \(\b{Nb5}\) is nothing more than the meaning of ‘some knightmoves from somewhere to square \(\b{b5}\)’—utterances of\(\b{Nb5}\) might carry extra information but that is of no concernfor the semantics of the notation. Both moves would savecompositionality at a trade-off. To make rejecting (a) worthwhile, onewould have to explain how the meaning of \(\bN\) depends on context.To rejecting (b) worthwhile, one would have to explain how the movecan be determined from the meaning of the utterance of \(\b{Nb5}\)combined with the background information; cf. Pickel and Rabern2022.

4.2 How Compositionality Allegedly Fails

This section discusses several famous putative counterexamples to thecompositionality of English from the semantics literature. The list isby no means exhaustive. (For a more systematic survey of howcompositionality problems are typically solved in formal semantics,see Zimmerman 2012.) For each putative counterexample, there arepossible responses considered below.

4.2.1 Possessives

An utterance of ‘Mary’s horse’ can refer to a horsethat belongs to Mary, a horse that she is riding, a horse that she hasplaced a bet on, and so on. ‘Mary’s portrait’ can beused to refer to a portrait that belongs to Mary or to a portrait inwhich she is the subject. Thus, the complex expressions‘Mary’s horse’ and ‘Mary’sportrait’ appear to be context sensitive. But, the contextsensitivity does not seem to trace to the constituents‘Mary’, ‘horse’, or ‘portrait’.These are a putative counterexample to the compositionality ofoccasion meaning. This counterexample can be accepted or backgroundassumptions can be revised. One way to preserve the compositionalityof occasion meaning would be to take the possessive construction ascontributing a context-sensitive constituent. See (Partee 1997).

4.2.2 Idioms

Idioms pose a more complicated challenge to compositionality. Idiomsinclude complex expressions with conventionalized meanings. Forexample, in English, the sentence ‘Mary kicked the bucket’can mean that Mary died. There is no obvious way to derive thismeaning from the meanings of the parts. Thus, idioms seem to beisolated exceptions to compositionality. Jackendoff (1997) estimatesthe number of English idioms to be around twenty-five thousand.

Defenders of compositionality sometimes argue that the syntacticcomplexity of these expressions is only apparent, and hence, they canbe viewed as lexical items. That is, the expression ‘kicked thebucket’ might be taken as a single atomic expression of Englishthat means died (in addition to being a complex expression of Englishthat means ‘kicks the bucket’). But unless there are clearnon-semantic grounds for singling out idioms, the move isquestion-begging. Such criteria have been proposed, but they tend tobe rather controversial; cf. Nunberg, Sag, and Wasow 1994. Anotherapproach to defending compositionality is to argue that the idiomaticmeaning of these expressions is metaphorical. That is, even if theidiomatic meaning is conventionalized, it remains metaphorical. It maybe that different responses are appropriate for different idioms.

4.2.3 Cross-sentential anaphora

Consider the following minimal pair from Barbara Partee:

(9)

I dropped ten marbles and found all but one of them. It isprobably under the sofa.

(10)

I dropped ten marbles and found nine of them. It is probably underthe sofa.

Sentence (9) is unproblematic, and sentence (10) is markedly odd. Thisdifference is plausibly a matter of meaning, and so (9) and (10)cannot be synonyms. Nonetheless, the first sentences are at leasttruth-conditionally equivalent. If necessary equivalence is sufficientfor synonymy, then this is an apparent counterexample tocompositionality.

One response would deny that the first sentences of (9) and (10) arefully synonymous. What is interesting about this example is that it isnot immediately clear how a more fine-grained account of meaning willaccount for the contrast between these sentences. The difference isobviously due to the fact that ‘one’ occurs in the firstsentence of (9), which is available as a proper antecedent for‘it’ and that there is nothing in the first sentence of(10) that could play a similar role. Some authors have suggested thatthe right way to approach this problem is to opt for a dynamicconception of meaning, one that can encode anaphoric possibilities forsubsequent sentences (Kamp 1981; Heim 1982; Groenendijk & Stokhof1990, 1991; Zeevat 1989; and Chierchia 1995). See entries onanaphora,dynamic semantics, anddiscourse representation theory.

4.2.4 Context Sensitivity

Let’s call expressions whose occasion meaning sometimes deviatesfrom their standing meaningcontext-dependent. The scope ofcontext-dependent lexical items is a matter of controversy.Semantic minimalists hold that there are only a handful ofcontext-dependent expressions: the personal and demonstrativepronouns, a few adverbs (e.g., ‘here’, ‘now’,‘next’), and a few adjectives (e.g. ‘actual’,‘present‘, ‘local’); cf. Borg 2004 andCappelen and Lepore 2005 .Radical contextualists hold thatessentially all lexical items are context-dependent; e.g., Searle(1980). Many theorists are somewhere in the middle.

Radical contextualism is sometimes seen as a challenge tocompositionality, more precisely, to(C\(_{\textit{occ}}\)); cf. Cohen 1986, Lahav 1989, Fodor 2001. An effective argument fromcontext-dependence against(C\(_{\textit{occ}}\)) would need to show that there is at least one complex expression inL whose occasion meaning varies with context, while theoccasion meanings of its constituents all remain the same. Suppose aJapanese maple leaf, turned brown, has been painted green. Considersomeone pointing at this leaf uttering (11):

(11)

This leaf is green.

The utterance could be true on one occasion (when the speaker issorting leaves for decoration) and false on another (when the speakeris identifying the species of tree the leaf belongs to). The meaningsof the words are the same on both occasions and so is their syntacticcomposition. But the meaning of (11) on these two occasions—what(11)says when uttered in these occasions—is different.As Charles Travis, the inventor of this example puts it:“…words may have all the stipulated features while sayingsomething true, but also while saying something false” (Travis1994: 171–172; see also Travis 1996 and Lahav 1989).

There are many possible responses to this challenge to thecompositionality of occasion meaning. One is to deny the relevantintuition. Perhaps the leaf really is green if it is painted green and(11) is uttered truly in both situations. Nonetheless, one might besometimes reluctant to make such a true utterance for fear of beingmisleading. One might be taken to falsely suggest that the leaf isgreen under the paint or that it is not painted at all (cf. Sainsbury2001 and Berg 2002). The second option would be to hold that thecontext sensitivity of this example can be traced to the contextsensitivity of a constituent expression, ‘green’ (Dancy2003, Szabó 2010, Lasersohn 2012, Recanati 2012, and Borg 2016).^[21] A third option would be that the compositionally determined occasionmeaning falls short of determining a truth condition or proposition,but only propositional constraints (Bach 1994, Soames 2005, Buchanan2010, Harris 2020).

4.2.5 Propositional attitudes

A final and widely known objection to compositionality comes from theobservation that even if \(e\) and \(e'\) are synonyms, thetruth-values of sentences where they occur embedded within the clausalcomplement of a mental attitude verb often appear to differ. So,despite the fact that ‘biblioklepts’ and ‘bookthieves’ are synonyms, apparently (12) may be true and (13)false:

(12)

Carla believes that biblioklepts steal.

(13)

Carla believes that book thieves steal.

This pair apparently violates compositionality; cf. Pelletier(1994).

There is a sizable literature on the semantics ofpropositional attitude reports (see entry). Some think that considerations like this show that thereare no genuine synonyms in natural languages. If so, compositionalityis trivialized. Some deny the intuition that (12) and (13) may differin truth-conditions and seek explanations for the contrary appearancein terms of implicature.^[22] Still others abandon compositionality but still provide recursivesemantic clauses.^[23] Finally, some preserve compositionality by postulating a hiddenindexical associated with ‘believe’ (Richard 1990,Crimmins & Perry 1989, and Crimmins 1992).