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1. There are some authors who call the following weaker thesis theprinciple of compositionality: The meaning of a complex expression isdetermined bythe meaning of its structure and the meaningsof its constituents. For example, Davis (2003) claims that theambiguity of ‘aardvark lover’—it can mean either‘someone who loves aardvarks’ or ‘aardvark that is alover’ is neither lexical nor structural, but due to twodifferent conventions associated with the single structureNV-er. (For an alternative take on ‘aardvarklover’ that complies with (C), see Szabó (2008.) Since itis not clear that structureitself is the sort of thing thatcan represent (and hence can have meaning) I assume thatcompositionality should be interpreted as (C). The difference becomesimportant if proponents of the weaker principle also allow thatsyntactic rules may have context-dependent meanings. Here is anillustration. For the sake of argument assume that meaning istruth-conditional content, that ‘every man’ has a singlesyntactic structure, that ‘every’ and ‘man’each have a single determinate truth-conditional content, and that thetruth-conditional content of ‘every man’ includes somecontextual restriction on the domain of quantification. Given (C),this violates compositionality; cf. Stanley and Szabó (2000).Pelletier (2003) claims that the four assumptions are compatible withcompositionality because the syntactic rule which combines thequantifier and the noun may be associated with a context-sensitivesemantic rule. Pelletier is right given the weaker construal ofcompositionality. For a precise formulation of the weakercompositionality principle, see Pagin and Pelletier (2007). Section1.4.1 introduces a weaker compositionality principle,(C\(_{\textit{coll}}\)), that is a generalization of the weakprinciple.

2. These constituency tests are not wholly theory-neutral. Differentsyntactic theories support different constituency tests. Moreover,constituency tests are not decisive. An expression may be aconstituent even though it fails standard constituency tests. Indeed,some constituency tests are inappropriate to some categories ofexpression. Nonetheless, constituency tests impose one empiricalconstraint on the selection of a syntactic theory for naturallanguage.

3. There are a number of difficult questions about the compositionalityof these signs. Would all the words that appear on many US trafficsigns but not on their European equivalents count as constituents? Isthe size of the traffic sign a constituent? (Note that the minimumsize of the No-Left-Turn sign in the US is \(24\times 24\) inches.)What exactly does it mean to compose a shape and a color pattern? Howcan the meaning of one traffic sign can supplement the meaning ofanother? Do the meanings of traffic signs depend on contextindexicals—e.g., ‘No left turnhere’? Dothe meanings of traffic signs include the understoodrestrictions—e.g., ‘If you are driving a vehicle, make noleft turn here’?

There is also some reason to doubt the compositionality of trafficsigns. Consider the following minimal pair:

The meaning of the first is something like ‘No leftturn!’; the meaning of the second is roughly ‘Left turn;recommended speed 30mph or less’. (If you didn’t know thatthe second sign means this you can check theManual of Traffic Signs.) Since the shape and color pattern of these two signs indicatesnothing beyond the fact that the first is a regulatory signprohibiting something and the second is a warning sign recommendingsomething, there is a problem for compositionality here. Whatever thecontribution of the left arrow is, it is hard to accept that themeanings of both these traffic signs are determined compositionally.(As always, there are all sorts of maneuvers that could savecompositionality. They involve fiddling with what the meaningfulconstituents of these signs might be or with what exactly they mightmean.) [Sign images are from theManual of Traffic Signs, byRichard C. Moeur.]

4. To oversimplify a bit, in Frege’s language, the universalquantifier attaches to a monadic predicates. In order to form suitablemonadic predicates, Frege (1895, §30) proposes the followingsyntactic rule: if a sentence contains a name, then one may form amonadic predicate by removing (occurrences) the name from thesentence. For example, one may remove all occurrences of‘John’ from the sentence ‘John loves John’ toyield the monadic predicate ‘(_) loves (_)’. Thispredicate will be essential to form the sentence that corresponds to‘Everyone loves themselves’. This syntactic rule appearsto take a name and a sentence to yield a predicate. If the operationsatisfied local compositionality of reference, the referent of thepredicate should be determined by the referent of the sentence and thereferent of the name. The problem is that any two true sentences havethe same referent. Yet, removing the name ‘John’ from‘John walks’ and from ‘John talks’ seem toyield monadic predicates with different referents (‘(_)walks’ and ‘(_) talks’) even if the two sentenceshave the same truth-values. For contrasting views see Wehmeier (2018)and Pickel and Rabern (2023).

5. As stated, (C\(_{\textit{occ}}\)) is oversimplified: it evaluateseach constituent of a complex expression within the same context. Amore adequate formulation would allow for context-shift within largerexpressions. To formulate such a principle adequately one must take astand on what contexts are—a question bypassed here. SeeStojnić 2021.

6. Recanati (2012) distinguishes between thecontent and theoccasion meaning of an expression. The former is supposed tobe derived from the standing meaning via a mandatory process he callssaturation, while the latter is supposed to be arrived atemploying the full array of contextual processes, including optionalmodulation as well. (For the distinction between saturationand modulation, see Recanati 2004.) He also suggests that the contentof complex expressions is not compositionally determined, although itis determined by its structure and the occasion meanings ofits constituents.

7. Higginbotham (1985, 2007) argues that questions of grammaticalitymust be kept separate from questions of meaningfulness. If so, despiteits ungrammaticality ‘John is probable to leave’ means(roughly) the same as ‘John is likely to leave’.

8. To capture global compositionality formally is somewhat complicated.Here is an attempt. Say that the expressions \(e\) and \(e'\) arelocal equivalents just in case they are the results ofapplying the same syntactic operation to lists of expressions suchthat corresponding members of the lists are synonymous. (Moreformally: for some natural number \(k\) there is a \(k\)-ary \(F\) inE, and there are some expressions \(e_1, \ldots,e_k\), \(e_1 ', \ldots, e_k '\) in \(E\), such that \(e = F(e_1,\ldots ,e_k)\), \(e' = F(e_1 ',\ldots ,e_k ')\), and for every \(1\lei \le k\), \(m(e_i) = m(e_i ')\).) It is clear that \(m\) is locallycompositional just in case locally equivalent pairs of expressions areall synonyms. Say that the expressions \(e\) and \(e'\) areglobalequivalents just in case they are the results of applying thesame syntactic operation to lists of expressions such thatcorresponding members of the lists are either (i) simple andsynonymous or (ii) complex and globally equivalent. (Here is therecursive definition more formally. Say that the expressions \(e\) and\(e'\) are1-global equivalents just in case they aresynonymous simple expressions. Say that the expressions \(e\) and\(e'\) aren-global equivalents just in case for some naturalnumber \(k\) there is a \(k\)-ary \(F\) inE, andthere are some expressions \(e_1 ,\ldots ,e_k, e_1 ',\ldots ,e_k '\)in \(E\), such that \(e = F(e_1 ,\ldots ,e_k), e' = F(e_1 ',\ldots,e_k ')\), and for every \(1 \le i \le k\) there is a \(1 \le j \ltn\) such that \(e_i\) and \(e_i '\) are \(j\)-global equivalents.Finally, say that the expressions \(e\) and \(e'\) areglobalequivalents just in case for some natural number \(n\) they are\(n\)-global equivalents.) It is a bit hard to define in fullgenerality what it is for an expression to be a constituent ofanother. (This problem does not arise if the syntactic algebra must beaterm algebra.) Say that \(e\) is a 1-constituent of \(e'\)just in case it is in the domain of some syntactic operation whosevalue is \(e'\). Say that \(e\) is an \(n+1\)-constituent of \(e'\)just in case it is in the domain of some syntactic operation whosevalue is an \(n\)-constituent of \(e'\). Finally, say that \(e\) is aconstituent of \(e'\) just in case there is a natural number \(n\)such that \(e\) is an \(n\)-constituent of \(e'.\) Assuming that everyexpression can be generated from simple ones through a finite numberof applications of the syntactic operations, this will do. \((e\) issimple iff it is not the value of any syntactic operation.) For analternative abstract characterization of constituency, see Hodges 2012(251–2).

9. Collective compositionality could be formalized using the trick usedto formalize global compositionality. Thus, \(m\) is collectivelycompositional just in case collectively equivalent pairs ofexpressions are all synonyms, where collective equivalence is definedexactly like global equivalence with one difference. The recursivestep demands not only that \(e_i\) and \(e_i '\) be \(j\)-collectiveequivalents but also that the very same semantic relations should holdamong \(e_1 ,\ldots ,e_k\) and among \(e_1 ',\ldots ,e_k '\). Thisleaves space for the possibility that ‘Cicero is Cicero’is not collectively equivalent to ‘Cicero is Tully’, eventhough they have the same structure and their proper constituents areall collectively equivalent.

10.The translation here is due to Szabó.

11. There are principles of intermediate strength between\((F_{\textit{all}})\) and \((F_{\textit{any}})\). These principlesallow that the meaning of an expression may be fixed by the meaningsof some set or sets of complex expressions in which it occurs. Oneclass of interest are the cofinal sets. In determining the meaning of‘conquered’, one may need to look at how this wordinteracts with every other expression of the language(‘Caesar’, ‘Cicero’, ‘Gaul’,‘someone’, and so on). So the set of expressions relevantto determining the meaning of ‘conquered’ may need toinclude ‘Caesar conquered Gaul’. However, not everyexpression containing ‘conquered’ and ‘Caesar’may be needed to determine the meaning of ‘conquered’. Acofinal set of expressions is a set such that any expression occurs asa constituent in at least one member of the set. One intermediatecontext principle would say that the meaning of‘conquered’ is determined by the meanings of theexpressions that occur in any set of cofinal expressions.

(F\(_{\textit{cofinal}}\)) The meaning of an expression is determinedby the meanings of all expressions with any cofinal set ofexpressions.

(Except for very odd languages, the set of all expressions within thelanguage in which some given expression occurs as a constituent is oneof the many cofinal sets of expressions, so\((F_{\textit{all}})\)follows from (F\(_{\textit{cofinal}}\)) but notthe other way around. That (F\(_{\textit{cofinal}}\) ) follows from\((F_{\textit{any}})\) but not the other way around is trivial. Oneinteresting feature of (F\(_{\textit{cofinal}}\)) is that it appearsto be in conflict with Quine’s thesis of the indeterminacy oftranslation (taken as a thesis that implies the indeterminacy ofmeaning). Assume that the set of all observation sentences is cofinalwithin a reasonably large fragment of a natural language and that themeaning of an observation sentence is identical to its stimulusmeaning—(F\(_{\textit{cofinal}}\)) ensures then that themeanings of all the words are determined within our fragment. Therehas been an attempt to\(_{\textit{any}}\)show that (F\(_{\textit{cofinal}}\)) follows from less controversialclaims, and perhaps even from claims that Quine himself was committedto; cf. Werning 2004. The heart of Werning’s argument is theExtension Theorem; cf. Theorem 14 in Hodges 2001. The theoremstates that a meaning assignment to a cofinal set of expressions thatsatisfies(H) and(S\(_{\textit{singular}}\)) has a unique extension to a meaning assignment to all expressionsthat satisfies(H),(S\(_{\textit{singular}}\)) as well as its converse (there is a generalized result mentioned inHodges 2012 (257). The extra assumptions needed to get from theExtension Theorem to a denial of indeterminacy remain questionable;cf. Leitgeb 2005.

12. An example from Platts (1979): ‘The horse behind Pegasus isbald’, ‘The horse behind the horse behind Pegasus isbald’, ‘The horse behind the horse behind the horse behindPegasus is bald’, …. Martin (1994, 7) appeals to numeralsin trying to illustrate the same point. This I find less convincing.One can indeed say that English contains the sentences ‘I haveone kumquat’, ‘I have two kumquats’, ‘I havethree kumquats’, … but perhaps this series cannot becontinuedad infinitum without recourse to elaboratemathematical notation. And the mathematical notion is arguably notpart of English.

13. Note that Frege’s conclusion appears to be the buildingprinciple. Whether Frege believed in compositionality is a matter ofmuch debate among scholars. One problem is that in the publishedwritings appeals to a substitutivity principle about reference, but noanalogous principle about sense. Another problem is that in theGrundlagen (1884) Frege announces his famouscontextprinciple, which on certain interpretations contradicts certaininterpretations of compositionality; cf.section 1.6.4 above. Finally, it is not even clear whether Fregean senses can beproperly construed as linguistic meanings. For more on Frege andcompositionality, see Janssen 2001 and Pelletier 2001. They alsoprovide good bibliographies.

14. Fodor (1998b) does offer an empirical argument in favor ofsystematicity. The idea is that if complex expressions could beunderstood without understanding their constituents, then it isunclear how exposure to a corpus made up almost entirely of complexexpressions could suffice to learn the meanings of lexical items. But,as a matter of empirical fact, children learn the meanings of words byencountering them almost exclusively within other expressions.However, as Robbins (2005) points out, this observation can at bestlead one to conclude that understanding asuitably large setof complex expressions in which a given expression occurs as aconstituent suffices for understanding the constituent itself. It doesnot show that understandingany complex expression sufficesfor understanding its constituents.

15. This assumes at least some level of autonomy for syntax. But it isoverwhelmingly plausible that even if syntax is not fully independentof semantics, notall syntactic regularities are to beexplained semantically.

16. The original meaning is uniformly recoverablein principle.There is no constraint here on how complex a calculation might berequired to determine the value \(\mu(s)(s)\).

17. That compositionality in itself does not constrain lexical meaningmight appear paradoxical at first, but the source of paradox is justinstability in how the label ‘compositionality’ is used.Sometimes compositionality is said to be that feature in a language(or non-linguistic representational system) which best explains theproductivity and systematicity of our understanding; cf. Fodor 2001:6.(C) is but one of the features such explanations use—others includethe context-invariance of most lexical meaning, the finiteness of thelexicon, the relative simplicity of syntax, and probably much else.These featurestogether put significant constraints on whatlexical meanings might be; cf. the papers collected in Fodor andLepore (2002) and Szabó (2004).

18. Wehmeier (2024) has recently argued that combining compositionalitywith the requirement that the semantics be suitable to model logicalconsequence imposes substantive requirements on meaningassignments.

19. The facts to be explained might include the fact that speakers areable to communicate in real time makes it overwhelmingly likely thatthe computational complexity of the interpretation algorithm employedis relatively low. In fact, it seems reasonable to think that semantictheories withminimal complexity are, other things beingequal, preferable. And there are certain results that show that,semantics theories that conform to certain strengthenings of(C) will be minimally complex; cf. Pagin 2012.

20. The algebraic notation is the standard system for describing chessgames. Its essentials go back to the Arabs of the 9^thcentury. The example and the discussion are a slightly modifiedversion of Szabó 2000a: 77–78.

21. Cf. Szabó 2000c. To defend compositionality along these linesit is by no means necessary to postulate a variable in the logicalform of the sentence; cf. Szabó 2010.

22. The explanation may follow the path of Salmon (1986). For a morerecent proposal, see Båve (2008).

23. For example, one could take the clausal complement of‘believes’ to be an interpreted logicalform—something which includes phonological information about thewords employed in the clause; cf. Higginbotham (1986), Segal (1989),Larson and Ludlow (1993). Such theories violate compositionalitybecause they maintain that the semantic value of a that-clauseincludes phonological information even though the semantic values oftheir constituents and their mode of combination do not. The fact thatsimple recursive semantic theories can violate compositionality shouldraise extra concerns about the strength of arguments from productivityand systematicity.