The roots of measurement

In addition to roots for familiar classes like verb, noun, and adjective, Mayan languages have a class of roots traditionally called “positional”. Positional roots are distinct from other roots most prominently in terms of requiring derivation into stems of one of the more familiar categories to be used. The goal of this work is to show that the behavior of positionals follows from semantic facts, in particular, the fact that they denote measure functions of type ⟨e,d⟩. This conclusion is supported through a series of novel arguments from the Mayan language Kaqchikel that positional roots have a scalar semantics. It then argues for the type ⟨e,d⟩ analysis by contrasting them with gradable root adjectives, which similarly make reference to ordered degrees on a scale, but which have a relational type—namely, ⟨d,et⟩. I then show that a core function of positional morphology, and the morpheme that derives positional stative predicates in particular, is to take positional roots into stems of type ⟨d,et⟩, which will account for the fact that derived positionals behave semantically like root adjectives. In this way, this work not only presents a novel account of the Mayan data, but provide additional evidence for the proposal that even within languages there can be differences in the fine-grained compositional structure of degree-denoting expressions.


Introduction
Two fruitful strands of research have recently come to prominence in the degree semantics literature. First, there has been a push to discover degree-denoting expressions across morphosyntacticcategoriesandtoaccountfortheminaunifiedtheory.Thatis,while adjectives present the canonical case, even a small sample of recent work shows that certain verbs (Hay, Kennedy & Levin 1999;Kennedy & Levin 2008), adverbs (Rett 2011), nouns(Schwarzschild2005;Morzycki2009;Champollion2010),quantifiers(Hackl2000; Rett 2008;Solt 2014;Wellwood 2015), and even modals (Yalcin 2010;Lassiter 2011) must make reference to ordered degrees on a scale. The second strand has focused on crosslinguistic morphological variation in degree constructions. The aim is to uncover the basic ingredients for a theory of degree-denoting expressions and to understand how they compose (Beck et al. 2009;Schwarzschild 2010;Grano 2011;Bochnak 2013; Bogal-Allbritten 2013: among others).
Thepresentworkfitssquarelywithinbothoftheseresearchprograms.Itcentersonan enigmatic class of roots in Mayan languages called positionals. A few examples of positionals from the Mayan language Kaqchikel are presented in (1). 1 (1) Positionals a. √jot 'elevated' b. √ch'eq 'wet' c. √set 'circular' d. √tun 'adjacent' e. √tik' 'facing a reference point' Positionalrootsstandoutfromallotherrootsclassesinthelanguage,primarilyinterms of morphology, but also in terms of lexical semantics. The goal of this work is to show that positional roots, and their behavior in both derived and underived forms follows if they denote measure functions of type ⟨e,d⟩. Crucially, with this analysis it is possible to explain the ways they contrast with adjectives in the language, which similarly make reference to ordered degrees on a scale, but which have a relational type-namely, ⟨d,et⟩. Inthisway,InotonlypresentanovelaccountoftheMayandata,butprovideadditional evidencefortheproposalthatevenwithinlanguagestherecanbedifferencesinthefinegrained compositional structure of degree-denoting expressions.
To build this argument, though, and to see how it sheds light on the grammars of Mayan languages, it is first necessary to consider some basic Mayan morphosyntax. Mayan language morphology is built around a privileged class of roots of form CVC. 2 In the canonical case, these roots all correspond to stems of some familiar category, which isdiagnosedthroughacombinationofderivationandinflection.Forinstance,theroots in(2)aretransitiveverbsrootsbecausetheycanbeimmediatelyinflectedwithergative and absolutive agreement in a transitive clause like (3a) or derived with the passive like in(3b),whilenoneoftherootsin(4),(6),or(8)canbesimilarlyinflectedorderived.The patternoffactshasbeenreplicatedacrossexamples(4-8).Foreachclassofroots,Ihave illustratedexamplesofinflectionalandderivationalmorphologythatnoneoftheother roots present can immediately take.
(12) a. *At ch'eq. a2s wet 'You're wet.' b. *X-a-ch'eq. cp-a2s-wet 'You got wet.' c. *ri ch'eq-äq ak'wal-a' the wet-pl child-pl 'the very wet children' (13) a. At nïm. a2s big 'You're big.' b. X-a-nüm. cp-a2s-get.hungry 'You got hungry.' c. ri nim-äq ak'wal-a' the big-pl child-pl 'the big children' Two major questions follow from the fact that positional roots stand as a separate root classonparwithrootsformorefamiliarcategorieslikeverbsoradjectives,yetdifferfrom these roots in needing derivational morphology for inflection. First, what kind of syntactic and semantic objects are positional roots and how do they relate to those of more familiar categories like adjective, noun, verb? Second, why do only positional roots need derivation? While these question arise from rather parochial facts about Mayan language morphology, the answers have much wider consequences. For instance, positional roots have been used to argue against the universality of lexical categories. Evans & Levinson (2009: 435) take Mayan positionals to argue that languages can go beyond the "big four" word classes (noun, verb, adjective, adverb) to have boutique lexical categories. The type-drivensemanticaccountofthepeculiarmorphologicalpropertiesofpositionalrootsthatI developunderminestheirproposalbyprovidinganalternativeexplanation.Ithenargue, in the final section, that positional roots actually have the syntactic category of verb, which is incompatible with Evans & Levinson (2009: 435), and thus removes an argument against the universality of lexical categories. Additionally,theparticulartype-theoreticaccountIdevelopallowsbothpositionalroots and adjective roots to make reference to scales, but to do so in a way that accounts for theirmorphologicalandcompositionaldifferences.Theanalysisthusaddstothegrowingbodyofevidencethatnotonlydolanguagesdifferinwhethertheymakereferenceto degrees (Bochnak 2013), but that there are a variety of compositionally viable ways to do so (Bogal-Allbritten 2013). The situation in Kaqchikel is of interest precisely because, as Iargue,thiscompositionalvariationcanbetransparentlyreadoffofthemorphology.In particular,thecruxoftheanalysisIproposeisthatdegree-denotingadjectiverootsare relations of type ⟨d,et⟩, and so can take arguments directly, while positional roots denote measure functions of type ⟨e,d⟩, and thus need to be derived into a relational expression beforebeinginflected.Positional-specificderivationalmorphologyexists,inpart,todo precisely this.

Positionals make reference to degrees
This section presents a series of distributional arguments that positional roots are scalar roots, that is, their denotations make reference to degrees on a scale. 4

Positional lexicalization patterns
Thefirstargumentcomesfromfactsaboutlexicalization.TheclaimisthatforKaqchikel, a language with otherwise few adjectives, positional roots lexicalize what adjectives do in English. 5 Since adjectives in languages like English are the prototypical category with scalar semantics, a natural hypothesis is that positionals lexicalize these same scalar notions that such adjectives do.
First, though, consider the domain of adjectives in Kaqchikel and Mayan languages in general. Languages of the family tend to have small to medium-sized adjective inventories (England2004).Kaqchikelisnodifferent.CountsfromPatalMajzul's(2010)Kaqchikel dictionary show that the language has around 80 morphologically simplex adjectives and around 40 root (CVC) adjectives from these categories, which is similar to what has been reported in other Mayan languages, like Mam (England 2004). 6 Moreover, the adjectives they do have are those that are typologically common in languages with small to medium adjective inventories (Dixon & Aikhenvald 2004), namely colors, valuations, ages, dimensions, tastes, and physcal properties. Contrast this to languages with large adjective inventories, like English, which can have well over 500 items (Dixon & Aikhenvald 2004). While Kaqchikel does not have many hundreds of adjectives, it does have many hundreds of positionals. A count of Kaqchikel dictionaries gives just over 300 (Tummons 2010), while sources for other Mayan languages consistently give numbers in the 300-500 range (Knowles 1984;Kaufman 1990;Haviland 1994;KaanPixab'aj&SisIb'oy2004). Mostofthese,followingEngland(2004),lexicalize physical properties and dimensions.
If these large counts of positionals round out the space of adjectives in Mayan languages like Kaqchikel with otherwise few adjectives, then we expect them to lexicalize gradable notions. This is borne out in a corpus of 304 positional roots collected byTummons(2010).Icategorizedthecorpusbyaskingofeachpositionalwhetherit lexicalized a prima facie gradable property, determined by its translation into English. 7 Wefindthatpositionalsfallintofourclasseswithrespecttoprimafaciegradability, though those positionals with some sort of gradable semantics outnumber those without 9 to 1.
Theexamplesin(17)areatfirstpassproblematicforatheorythatwantstoaccountfor the properties of positionals in general by giving them a type ⟨e,d⟩ denotation. The prob-lems are not as bad as they seem, though. First, it may be possible to assimilate many of these expressions to those in the last category, namely positionals associated with a two-pointscale.Infact,thisiswhatIwillbeargueattheendofthissectionaftertreating the more common gradable positionals. Second, the exception arguably prove the rule. As we will see below, prima facie non-gradable positionals are less productive, which is consistent with them having a non-standard semantics, namely one that is scalar, but not gradable (i.e., involving only two points). The last group of positionals in (18), which only number in the tens, stand between the non-gradable positionals like those in (17) and those with a gradable semantics like (15)(16). They are non-gradable, but still have a scalar semantics, though one that makes use of two-point scales. As noted by Beavers (2008), who defends the idea of two point scalar change predicates, gradable predicates are just a subset of the scalar ones, namely thosewithmultipointscales.Ithusdonotcountthesepositionalsasgradable,buttheformal account proposed in section 3 can cover these expressions. They can denote measure functions that map individuals to points on a two point scale.
Whatthesedatashow,summarizedinthefigurebelow,isthatthemostcommonkindof positional root lexicalizes physical properties or dimensions that are gradable adjectives in English. This supports the proposal that the class of positionals, at its heart, consists of expressions that make reference to degrees on a scale. Thisconclusionisbolsteredbyasecondargumentconcerningproductivity.Ifthecanonicalpositionalisdegree-denoting,weshouldexpectthatpositional-specificmorphology to also implicate a degree argument, predicting the prima facie gradable classes of positionals to be more regular. To get a rough measure of morphological stereotypicality, we can count how many of the four core positional derivations a root takes, where the four core derivations are the positional stative predicate derivation (e.g., (11a)), the positional adjective derivation (e.g., (11c)), the positional intransitive derivation (e.g., (11b)), and the positional transitive derivation (e.g., (11d)). The Figure 1 shows, for each root semantic type, how many derivations it takes, that is, how productive it is. 9 What we see is that the prediction is borne out. The gradable positional roots are in general more productive.
Once again, the data show that those positionals that lexicalize gradable notions are not just the most common, but also the most morphologically stereotypical. This suggests again that we should take scalarity to be the lens through which we should look at positional roots and their derivational potential. While looking at the class of positionals as a whole through translations is illuminating, translations only hint at truth conditions. Inthenextsectionwezoominonparticularpositionalsrootstoshow,throughlanguage internal tests, that they pattern like scalar expressions in more familiar languages. We will begin by focusing on the most common type of positional, those two-hundred or so with prima facie gradable semantics. At the of end of this section we will return to the other three classes and show that they too can be folded into an account that takes positionals to uniformly have a scalar semantics.

Positionals in degree constructions
The second class of arguments, involving the first language-internal tests in favor of a degree semantics account of positionals, concern their widespread acceptability in degree constructions.Ifocusonthecomparativebecauseitisarobusttestfordegree-denoting expressions and its properties are well understood from a crosslinguistic perspective.
Kaqchikel comparatives look fairly standard from this crosslinguistic perspective. They consist of a gradable predicate (in (19) a root adjective), an optional degree morpheme borrowed from Spanish, and a locative morpheme introducing the comparative standard.
Ri a Xwan (más) nïm ch-u-wäch xta Mariy. the clf Juan (more) big prep-e3s-face clf Maria 'Juan is bigger than Maria.' lit. 'Juan is big(ger) in front of Maria' While Kaqchikel comparatives are morphologically unsurprising, we must check whether theyhavethesemanticpropertiesrequiredtodiagnosegradablepredicates.Inparticular, we must show that they make use of explicit comparison (Sapir 1944;Kennedy 2010: among others). Explicit comparison, as in (20), involves the direct comparison of the degrees individuals hold on a scale, making no claim as to whether the gradable predicate holds of the individuals in question.  Explicit comparisons should thus be acceptable: (i) in crisp judgment contexts, where the topic of comparison does not stand out relative to the comparative standard, and (ii) with gradable predicates that have absolute minimum standards, which arguably do not make use of a comparison class (Kennedy 2007), though see Burnett (2012) for important recent arguments against this view. Examples (21-22) illustrate these properties for the English comparative. Example (21) shows that the standard comparative can be used even whentheindividualdoesnotstandoutrelativetoitscomparativestandard.Incontrast, the compared to comparative, which is not an explicit comparative, is infelicitous in such contexts.
(21) John and Mary are basically the same size, but John weighs a couple of pounds more and is a centimeter taller. a. John is bigger than Mary. b. #John is big compared to Mary.
(22) Rod A is bent at a 30° angle and Rod B is bent at a 10° angle. a. Rod A is more bent than Rod B. b. #Rod A is bent compared to Rod B.
The Kaqchikel comparative can be used both in crisp judgment contexts as well as with minimumstandardpredicates.Thisshowsthatwearedealingwithabonafideexplicit comparative that should only compose with scalar expressions. Moreover, we can make this point using positionals, which shows that they are such scalar expressions. Example (23) shows the felicity of a positional (in stative predicate form, which we return later), in a crisp judgment comparative. Example (24) shows the same for an absolute positional, namely one associated with a bounded scale. 10 (23) Suppose you're trying to help a friend decide which pencil to buy. The red one is a few cents more than the blue one. Can you say: a. Ri käq jot-öl r-ajil ch-u-wäch ri xar. the red elevated-p.stat E3s-price P-E3s-face the blue 'The red is more expensive than the blue.' lit. 'The red one's price is higher than the blue one's.' Suppose you have two sticks for getting things down from the high shelves in your store. Neither is straight, but one is more bent than the other. Can you tell me which is which by pointing and saying: 10 a. La xat'at' la' kot-öl ch-u-wäch ri jun chïk. that staff there bent-p.stat P-E3s-face the one other 'Thatforkedstaffthereismorebentthantheotherone.' These examples provide a strong argument that the Kaqchikel comparative expects scalar expressions,whichpositionalsprovide(alongwithadjectiveslikein20).Puttingasidethe non-gradable and two-point-scale positional roots, in general we can use positionals in the explicit comparative construction, as we see with the following additional examples.

(25)
Ru-chi ri jay jaq-äl ch-u-wäch ru-chi ri ch'ich'. e3s-door the house open-p.stat prep-e3s-face e3s-door the car 'The door to the house is more open than the door to the car.' Ri ala' ch'eq-ël ch-u-wäch ri xtän. the boy wet-p.stat prep-e3s-face the girl 'The boy is more wet than the girl.' Ri pa's pach'-äl ch-u-wäch ri po't. the belt rip-p.stat prep-e3s-face the blouse 'The belt is more ripped than the blouse.' These facts show that the vast majority of positionals, which were prima facie gradable under translation, are in fact so. They freely occur in the comparative construction, which is a standard diagnostic for gradability.

Positional-specific degree morphology
The third class of arguments for a scalar account of positionals is even more languagespecific, and even positional-specific. In particular, the the argument in this section comesfrompositional-specificmorphology,whichIwillargueisdegreemorphology. Inparticular,Iwillbeinterestedincomparisonsbetweenmorphologythatonlyapplies topositionalrootsanddegreemodifierslikevery, slightly, completely, etc. in languages like English.
One of the most beautiful results in the literature on scalar expressions, like gradable adjectives, is that scales themselves are not uniform. They can have their own particular structures, for example being closed on either (or both) ends, and that this structure can bediagnosedthroughtheirinteractionwithscalarmodifiers.Forexample,completely in Englishtakesameasureandderivesapredicateofindividualssatisfiablebyonlythose entities that are mapped by the measure to the maximal degree on its associated scale (Kennedy & McNally 2005).
This correctly predicts that only those gradable predicates that denote functions from entities to degrees on an upper-closed scale should be grammatical. An adjective like fast, which isn't associated with a scale with the appropriate structure, is infelicitous when modified by an expression like completely, which makes reference to scales with upper bounds.
(29) a. The car is completely full. b. #The car is completely fast.
These observations set up an empirical prediction about positional roots. If positionals are scalar expressions, we should find evidence that positionals fall into different classesbasedonscalestructure.Moreover,positional-specificmorphologyispotentially sensitive to the scale structure of the positional it derives. We will see that both of these are true, which bolsters the argument that positionals are scalar expressions. It also allows us to solve a puzzle. Recall from Figure 2 that while prima facie gradable positionals are more productive, there are still those that cannot combine with certain corepositional derivations.Ishowherethatwecanmakesenseofthesegapsifcertain positional derivations are sensitive to scale structure, and not all positional have the same scale structure. Our focus will be the the positional adjective derivation. Recall from example (11c), repeatedin(30),thatthereduplicativesuffix-VC 1 derives positional roots into adjectives. We know (30) is an adjective because: (i) it occurs between a determiner and its noun complement,and(ii)itbearstheadjectivalpluralagreementsuffix-äq.
Supporting the translation is the fact that the positional adjective form asymmetrically entails the positional stative form. This is expected if -VC 1 is intensifying, that is, if it raises the contextual standard for the gradable expression, as its translation suggests. 11 In doing elicitation of large numbers -VC 1 positionals across 8 Kaqchikel speakers, I have noticed that whilestandard-raisingisthemostcommoneffect,somepositionalsunderthe-VC 1 instead indicate permanence of the condition. This is even true for the same positional across speakers. For instance, jechejïk in (35b) for some speakers means that object of predication is not greatly twisted, but permanently twisted. My current hypothesis is that -VC 1 is ambiguous between a stage-level and individual-level predicate derivation, and only the former case is it standard-raising degree morphology. In contrast, the stative derivations seems to always have the stage-interpretation. This suggests that the stativizing morphology, in addition altering the type of positional roots additionally adds a state argument, which does not occur withthepositionaladjectivederivation.Thatsaid,Iwillsavetheexplorationofindividual-levelreadings of positional adjectives for future work, and for now focus on the degree reading, which is prominent, and deserves its own account. b. #∅ jech-ej-ïk, po man ∅ jech'-ël ta. A3s soft-p.adj-sg, but neg A3s soft-p.stat irr 'It'sextremelytwisted,butit'snottwisted.' This is enough to establish that the positional adjective derivation is standard raising relativetothecurrentcontext.Inthenextsection,afterintroducingtheformalaccount of positionalroots,Iwill provide a formalaccountofthe particular flavor ofstandard raisingweseewiththepositionaladjectivederivation.Instead,nowwewillconsiderhow the particular kind of standard-raising instantiated by the positional adjective derivation interacts with scale structure, which will establish that the positional adjective derivation is degree morphology.
First, consider those positionals that are prima facie non-scalar, that is, non-gradable andnotassociatedwithatwopointscale.Invirtueofbeingrelatedtonoscale,weexpect they should be infelicitous with the positional adjective derivation. This is true for many such positionals. They simply cannot be derived into adjective via -VC 1 .
Having shown that a positional must be scalar to smoothly take the positional adjective derivation, we can now consider whether some scales are more appropriate than others. What we will see is that it is generally felicitous with positionals that have open or lower closed scales, while it is infelicitous with positionals that have upper-closed scales.
First, consider positionals associate with open scales, namely scales for which there are no natural upper or lower bound. Such positionals are felicitous with the positional adjective derivation.
Once again, we can show that such positionals are lower-closed through language internal tests. The positive form of such expressions should entail that an individual is mapped to non-zero degree on the scale. This means that its negation should contradict the assertion that the individual has the property to any degree. This is true for those positionals above, asexemplifiedin(46-47).

12
NotethatKennedy&McNally (2005)primarilyusethedistributionofdegreemodifierstodiagnosescale structure,butKaqchikelhasasmallerinventoryofsuchmodifiers,andsoit'snotclearwhatsemanticfield theexistingmodifierscover.Forthisreason,Ihavechosentojustusethosetestsfromthatworkwhichare only dependent on features of Kaqchikel that we understand, like simple entailments, negation, conjunction, etc. 13 That is, an expression related to a lower-bounded scales like bent will always be true if an entity has positive measure on the relevant scale. Thus, its negation will require that any satisfying entity be at the lower bound-i.e., not bent. But then any such entity will be at the upper-bound for that expression's antonymi.e., straight-and so will satisfy the antonym. This kind of reasoning does not hold for expressions related to scales without bounds-i.e., being not fat does not mean that an entity as at the minimal bound of fatness, and so must be skinny, because fatness/skinniness are not reckoned on a scale with a lower-/upper-bound.
(46) #Man ∅ ch'eq-ël ta w-aq'a', po k'o b'a ya' ch-r-ij. neg a3s wet-p.stat irr e1s-hand, but exist little water prep-e3s-back 'My hands are not wet, but they got a little water on them.' (47) #Man ∅ ch'em-ël ta, po jub'a x-Ø-pax el ru-chi'. neg a3s chip-p.stat irr, po little cp-a3s-break dir e3s-mouth 'Itisn'tchipped,butitsedgecrackedabitandfelloff.' Finally, while the positional adjective derivation is felicitous with positionals with open andlower-closedscales,weseetheeffectofscalestructurewithpositionalsassociated with upper-closed scales. The positional adjective derivation is generally infelicitous with such expressions. 14 (49) UpperClosed a. chol 'straight', but # cholochïk b. jäm 'empty', but # jamajïk c. titz' 'shut and full', but # titz'itïk d. kaw 'empty', # kawakïk As with the other cases, we can show that these positionals are, in fact, upper-closed, that is, denoting on a scale with maximal degrees. This is true, even though diagnosing upper-closedscalesisknowntobedifficultduetotheeffectsofimprecision.Ingeneral, though, gradable predicates with upper-closed scales have the default interpretation that an individual possesses the maximal degree on a scale (Kennedy 2007). This means that the positive form should be contradicted by asserting that an individual has any degree other than the maximum. Examples (50-51) show this to be the case.
14 As with the non-gradable positionals, while speakers certainly reject many upper-closed positionals with -VC 1 , coercion is also possible here. The positional √yun 'shut (mouth)' is upper-closed, but under the -VC 1 derivation additionally includes the wrinkliness of the face, which is gradable.
Thepositionaladjectivesuffixrequiresthisstandardtoshiftupward.Likeotherdegree morphology, the positional adjective derivation can be infelicitous depending on the scale structureofitspositionalargument.Inparticular,positionalswithupperclosedscalesare generally infelicitous with the positional adjective derivation. This means that positional morphology is degree morphology and we should take positional roots to have a scalar semantics.

Beyond the prima facie gradable positionals
So far we have focused on those positionals that, according to their definitions in the corpus, lexicalize clearly gradable notions. This is the vast majority of positionals, but as notedatthebeginningofthissection,therearepositionalthat,atfirstpass,appearnotto fitinthisgradablecategory.Iwillproposeacategorizationinthissectionthatallowusto fold some these positionals into the gradable category, while grouping others into a class with a scalar, though non-gradable semantics.
First, consider those positionals with what I called partially gradable semantics. These are positionals like √tär 'standing (broken)' which have clearly gradable aspect to their meaning-i.e., broken-while having a second prima facie non-gradable aspect-i.e., standing. What we see is that, in general, these expressions behave like simple gradable positionals in constructions that we have argued have a gradable semantics. For instance, García Matzar & Rodríguez Guaján (1997: 335) says positional √tär in its positional stative predicate form taräl characterizes things that standing and broken, while in the positional adjective form taratïk,whichIhavearguedhasastandard-raisingsemantics,requiresthe individual be quite broken (lit. bien quebrada). This suggests that for the purposes of positional morphology, partially gradable positionals behave just like gradable positionals, but with the the degree expressions targeting the gradable aspect of their meaning.
We see a similar fact with the behavior of partially gradable positionals and comparatives. What is compared with these expressions is the gradable aspect of their meaning. For instance, given a root like √qëb 'seated (fat)', one can form comparative like (52).

(52)
Ri ak'wäl mas qeb'-äl chi-ki-waäch r-ach'alal. the child more seated(fat)-p.stat prep-e3p-front e3s-brother 'The boy is fatter (seated) than his brothers.' Fully understanding the lexical structure of these expressions with dual aspects to their meaning is still an open question, but the fact that these kinds of positionals can participate in degree constructions shows that we can treat them as scalar expressions just like those simple gradable positionals we have considered thus far.
While partially gradable positionals are fairly simple to deal with, the positionals with prima facie non-gradable semantics are more challenging. To begin, we do actually have evidence that these are not gradable. The previous section argued that the positional adjective form should be treated as standard-raising morphology. When we look at the 41 non-gradable and two-point scale positionals, the vast majority (33) do not occur in the positional adjective form. This suggests that they are, in fact, not gradable. 15 Given that these positionals are, in fact, non-gradable, if they are to have a uniform scalaraccount,Imustarguethattheyshouldinvolvereferencetotwo-pointscales.For 15 The exceptions are likely due to a combination of coercion, which we have seen allows for certain positionals with upper closed scales to be modified by the positional adjective, or the fact that translations can be misleading. For instance, the positional √pïtz' is usually translated as 'disemboweled', which would be prima facie non-gradable, but if it is semantically more like 'squished', then this would explain why it occurs in the positional adjective form. The result of these considerations is that as far as scalarity is concerned, there are two semantic classes of positions. On one hand, there are the gradable positionals and those simultaneously lexical two notions, one of which is gradable. These positionals behave as if they are related to multipoint scale of degrees that is accessible to various kind of degree operators. On the other hand, there is a small group of positionals (∼40) that do not make reference to a multipoint scale, meaning they are not gradable. That said, we can treat these positionals as making reference to a scale with two degrees because they lexicalize a notion with a clear opposite.

Interim summary
This section has provided a set of arguments from various domains to argue that positional roots in Kaqchikel should receive a scalar semantics. We started by taking a broad look at the positional lexicon, showing that nearly all positional roots are prima facie gradable and that the gradable roots are more productive. After this, we saw that those prima facie gradable positionals do, in fact, behave like gradable expressions across languages. We saw that they freely occur in degree constructions like the comparative. Moreover, we saw that positional morphology, in this case -V C 1 , can be degree morphology that is sensitive toscalestructure.Finally,Ihavearguedthatthosepositionalsthatarenotprimafacie gradable can still be given a scalar account. Having produced evidence that positional roots require a scalar semantics, we will now provide precisely that kind of analysis. Moreover, we will show that the particular account we choose will explain the restricted distribution of positional roots in Kaqchikel, as well as shed light on the similarities and differencesbetweenpositionalsandgradablerootadjectives,whichalsorequireascalar semantics. The view we come to is that, semantically speaking, positionals are a kind of proto-adjective. They denote measure functions that positional morphology then derives into degree relations, the kind of expressions that gradable adjectives denote.

Positional roots are measure functions
In this section we extend the proposal three ways. First, we argue that the positional stativepredicatederivationin(11a)isalsodegreemorphology,thoughnotanovertPOS morphemeasitmayatfirstappear.Second,wegiveaformalaccountofthepositional stative derivation, as well as the positional adjective derivation described in the previous section. Finally, and more importantly, we argue for a formal account of positional roots themselves in which they denote measure functions-i.e., expressions of type ⟨e,d⟩. Inthisway,theKaqchikelcomestobeacasestudyinresolvingathornyissueformany languages, namely whether to treat gradable adjectives as measure functions or degree relations of type ⟨d,et⟩. For instance, Kennedy & McNally (2005) note that English gradable adjectives can be treated as either type ⟨e,d⟩ or ⟨d,et⟩ tothesameeffect,andthereappear to be no good semantic arguments to distinguish them. They note that crosslinguistically these two kinds of account will most likely only be distinguishable via (morpho)syntactic facts. This is exactly what we see in Kaqchikel, where their are morphosyntax suggests that positional roots are of type ⟨e,d⟩, while gradable adjectives have the relational type ⟨d,et⟩.Tomakethisargument,though,wefirsthavetobetterunderstandthepositional stative predicate derivation.
The positional stative predicate form of positionals is their citation form. Not only is the positional stative predicate derivation the most productive, positionals thus derived have the widest syntactic distribution. The positional stative predicate derivation is illustrated in example (53).
Insimplenon-verbalpredicateconstructionslike(53),positionalsreceiveanorm-related or evaluative reading. That is, they are satisfied by individuals that have a degree of some measure that exceeds some standard of comparison. This is clearly illustrated for open-scale positionals, where the standard is clearly context dependent, just as with the positive form adjectives in English.

(54)
Suppose your friend says she bought a pencil on the bus for Q20. ∅ Jot-öl r-ajil. a3s elevated-p.stat e3s-price 'That's expensive.' (55) Suppose your friend says she bought a silk shawl on the bus for Q20. #∅ Jot-öl r-ajil. a3s elevated-p.stat e3s-price 'That's expensive.' Ifpositionalstativepredicatescanhaveevaluativesemantics,could-V l be the morphological instantiation of pos in Kaqchikel? Consider what that would entail. If we take position roots to denote measure functions, as we will argue for in detail below, then -V l wouldhavethetranslationin(57).Itwouldtakeameasureandpredicateofindividuals that is true of an individual, just in case the individual's degree on the measure in question exceeds the contextual standard. First, it incorrectly predicates that positional stative predicates should not be able tobethetargetoffurtherdegreemodification.Example(58-59)showthatpositional stative predicates do allow further degree modification, which should not be possible if -V l were pos, which derives predicates of individuals with no exposed degree argument. 16 (58) a. Yalan ∅ jot-öl r-ajil. very a3s elevated-p.stat e3s-price 'It'sveryexpensive.' b. Yalan ∅ tew. very a3s cold 'It'sverycold.' A second related problem is that positional stative predicates, like underived adjectives, can appear in the comparative. More importantly, these comparatives are acceptable in crisp-judgment contexts, which is not expected if they have a pos-like semantics.
(60) Suppose you're trying to help a friend decide which pencil to buy. The red one is a few cents more than the blue one. Can you say: a. Ri käq (mas) jot-öl r-ajil ch-u-wäch ri xar. the red (mas) elevated-p.stat e3s-price p-e3s-face the blue 'The red is more expensive than the blue.' lit. 'The red one's price is higher than the blue one's.' (61) Suppose there two boys, twins, but one is one centimeter taller than the other. Can you say: a. Ri jun ala' (mas) nïm r-aqan ch-u-wäch ri jun chïk. the one boy (mas) big e3s-leg p-e3s-face the one other 'The one boy is taller than the other.' lit. 'The one boy's leg is bigger than the other one's.' Instead,thedatashowthatpositionalsderivedby-V l do not denote simple predicates of individuals, but still have an exposed degree argument that can be targeted by conditionals and other degree morphology, including positself.If-V l is not pos, then, what is its function?Iarguethatitisdegreemorphology,justlikethepositionaladjectivederivation.Itssemanticfunctionisdifferent,though.Iwillarguethatitmerelytakes measure functions of type ⟨e,d⟩ into degree relations of type ⟨d,et⟩, which I argue is the type of 16 While we have assumed that gradable expressions require a degree-based analysis, a Kleinian account of gradable is still an open possibility (Klein 1982). The goal of this paper is not argue for or against degreebased analyses in general, but it is important to note that while a Kleinian account could deal with degree modifiers like in (58-59), the fact that Kaqchikel has bounded gradable expressions that support crisp judgments, like (60-61), is more problematic for an alternative Kleinian analysis which would treat such expressions as vague predicates.
gradable adjectives in Kaqchikel. In essence, the -V l derivation will give positionals adjective denotations, but with non-adjectival morphosyntax. This account, developed formally in the following section, will explain why positionals require derivation, but once derived have a similar distribution to gradable adjective, modulo the fact that they have a verb-like syntactic distribution.

The formal account
Indevelopingtheformalaccountwefocusontwotwogeneralizations.First,positionals cannot be used underived (unlike expressions of any other root class). Second, if a (predicative) adjective can occur in a degree construction, a positional derived by -V l can appear there too. We will see that these generalizations can be accounted for if (i) positional roots denote measures of type ⟨e,d⟩, meaning they are not predicates and their degree argument is not exposed, and (ii) -V l derives measures into relations of type ⟨d,et⟩, which is the same type as root gradable adjectives. First, consider a standard account of gradable adjectives with evaluative semantics in simple predicational constructions (e.g., Cresswell 1977;von Stechow 1984;Heim 2000;Schwarzschild 2005). Adjectives denote degree relations, like (62), while pos saturates the degree argument, as in (63-64). 1718 The result of derivation by pos in (64) is a predicate of individuals that is true of an individual x just in case x's degree of whiteness is at least d and d is greater than the standard for whiteness in the context. These are precisely the truth conditions of expressions like (65) in Kaqchikel.
Here, the positional root √jot 'high', is translated as a function that takes an individual and returns a degree measuring its height.
Onefunctionofpositionalmorphology,then,istoturnmeasurefunctionsintobonafide relations,thatis,expressionsthatevaluatetotruthvalues.Inthecaseofthepositional stative predicate derivation -V l,Iproposethatitderivespositionalrootsintoexpressions that are structurally equivalent to root adjectives. This effect is shown in (68), which looks exactly like (62) modulo the measure function at its core. 17 The particular formulation of pos presented here is not critical for the analysis. Any treatment of pos that takes an expression of type ⟨d,et⟩ and sets the contextual standard based on the denotation of that expression could be consistently substituted. 18 Noteherethattosavespace,insteadofreproducingthetranslationof(62),Ipasstothestandard-setting function the translation of √säq-written as T(√säq).
Because V l-derived positionals are semantically equivalent with root adjectives, they can compose with pos, as in (69).
The result is a predict of individuals that is true of an x just in case the degree measuring x's height is at least d and d is greater than the standard for height in the context. These are the truth conditions of the positional stative predicate in (70).
(70) ∅ jot-öl. a3s high-p.stat 'It'shigh.' NotethatwhileIhaveillustratedtheinteractionofpos and -V l with a gradable positional √jot,theaccountworksjustaswellforpositionalsassociatedwithtwopointscales.In particular, such a positional will denote a measure that takes individuals to one of two degrees. The -V l morpheme will take such a measure into a relation between individuals and those to degrees that holds just in case the individual exceeds the standard. Finally, posfixesthestandardsothatonlyindividualsmappedtothedegreehigheronthescale. Essentially, these positionals behave exactly as if they were associated with upper closed scales, except that they are not gradable in virtue of having only two degrees.
Taking a broad view, the core claim of the analysis is that positional roots are a kind of proto-adjective. They denote expressions that can be explicitly derived into expressions that denote just as root adjectives do. Beyond getting the truth conditions of root adjectives and positional stative predicates in evaluative contexts, this type of account makes aseriesofcorrectpredictionsaboutthesimilaritiesanddifferencesbetweenpositionals and adjectives.
First, the analysis explains the core generalization that positionals are unique in needingderivation,asshownin(71).Positionalscannotbeusedunderivedbecausetheyare not predicates-type ⟨et⟩. The positional root in (71) maps its argument to a degree, not a truth value, and so (71) is unassertable.
a3s high e3s-price 'It'sexpensive.' Note that unlike with root adjectives, the phonologically null pos morpheme cannot help because it is of the wrong type, as shown in (72).
(72) *pos ⟨⟨d,et⟩,et⟩ (high ⟨ed⟩ ) The only way for pos to compose with a positional is for it to first compose with an expression like -V l above, which produces an degree relation of type tupled,et. A second argument in favor of this type difference works in the opposite direction. While positional roots cannot compose with degree morphology specialized for root gradable adjectives, positional degree morphology cannot compose with adjectives. This is surprising given that it is simple degree morphology. Example (73) illustrates that the derivationaldegreemodifier-VC 1 we encountered in the previous section cannot target adjectives. 19 (73) *saq-as-ïk white-p.adj-sg (74) *-VC ⟨ed,⟨d,et⟩⟩ (saq ⟨d,et⟩ ) At the level of the root, then, the analysis makes correct predictions about the differences between adjectives and positionals, even though both have a scalar semantics. The analysis also makes correct predictions about similarities between root adjectives and derived positionals, in particular, positionals in the positional stative predicate form. The analysisclaimsthatwhilepositionalstativepredicatesaremorphosyntacticallydifferent thanbonafideadjectives,theyhavethesamedenotations.Wecorrectlypredictthatthey should occur in the same degree constructions. We see in examples (58-59) that both positionals and adjectives alike accept the same degree modifiers once the former has been derived.
very a3s cold 'It'sverycold.' (76) a. Jub'a ∅ jot-öl r-ajil. little a3s elevated-p.stat e3s-price 'It'salittleexpensive.' b. Jub'a ∅ k'äy. little a3s bitter 'It'salittlebitter.' Also note that both adjectives and derived positionals freely occur in the comparative (e.g., 60-61). The fact that derived positionals can appear in a similar range of degree constructions as root adjectives supports the proposal that positionals have a scalar semantics and are derived into expressions of the same type as root gradable adjectives.
Afinalargumentinfavoroftheparticulartype-theoreticaccountofpositionalsdevelopedhereconcernsovertmeasurearguments.Inalltheexampleswehaveseenthusfar, the positionals have their the degree argument targeted, though not overtly satisfied. Examples like (77) show that V l-derived positionals have exactly the type structure the proposal predicts.
(77) Ju-jaj jot-öl nu-ch'akät chi kaj. one-arm.length elevated-p.stat e1s-chair p heaven 'My chair is one arm-length tall in height.' The derived positional is of type ⟨d,et⟩,expectingadegreeargumentfirst.Thisissupplied by jujaj, which denotes degrees that measure one arm's length. This saturates the degree argument of jotöl as in (78), generating a predicate of individuals who measure on the height scale is greater than an arm-length.
Example(77)thusassertsthatthespeaker'schairsatisfiesthepredicatein (78),namely it is one arm-length tall, which are the correct truth conditions. More importantly for the argument here, constituency accords with the predicted type for positional stative predicates. 20 Theyare,infact,degreerelationsthatcomposefirstwithadegreeargumentand then an individual argument to return a truth value. The behavior of positional roots and their derived counterparts support the core analysishere.Positionalrootdenotemeasurefunctions,whilepositionalmorphologyderives positional stems that denote relations. We have seen how this works for the positional stative predicate derivation -V l.Inowwanttoreturntothestandard-raisingpositional derivation -VC 1 that the previous section showed to be degree morphology. We will see that -VC 1 has the same type as -V l,butwithastandard-raisingsemanticeffect.
Recall the two major generalizations concerning the positional adjective derivation -VC 1 . First, -VC 1 is standard-raising morphology. Second, positional roots with upper-closed scales reject -VC 1 derivation.ThissuggeststhefollowinganalysisthatIwilldevelophere. Standard raising by -VC 1 requires the an individual to exceed in measure, not just the contextual standard on the scale at issue, but all contextual relevant degrees on the scale. Now, if the upper bound is always contextually salient for upper-closed positionals, then -VC 1 would derive for these positional predicates of individuals who are off the scale, which should be infelicitous.
We have already seen that the second crucial ingredient of the analysis holds in Kaqchikel. Just like in English, positionals with upper closed scales do in fact take the maximal degree to be the standard in the positive form (see examples 50-51). This shows that the upper-bound for upper-closed positionals is, in fact, by default contextually salient. Kennedy (2007), for instance, predicts this to be the case via a pragmatic principle of interpretiveeconomy, which says that by default, if setting the contextual parameters for pos can be done using only the conventional meaning of the expressions involved, it will be. Having observed this though, we can hard code it into our standard pos operator for simplicity, keeping in mind that this is merely an expository shortcut. The crucial clause for understanding -VC 1 is that upper-closed positionals in the positive form will require satisfying individuals to have the maximal degree on the scale by default.
where a. s(g) = max(g), if g has an upper-closed scale b. s(g) = min(g), if g has a lower-closed scale c. s(g) = the contextual standard for g, if g has a fully open scale 20 We know that jujaj jotöl plausibly forms a constituent and that jujaj is not some higher adverbial because usually stative subjects can prepose to a position in front of the stative predicate, but before higher adverbials.Thisiscompletelyimpossiblewithmeasures,asshownbelow.Insteadtheonlypossiblepreposing moves the subject over both measure and stative predicate, which suggests they form a stative predicate together.
Nu-ch'akät ju-jaj jot-öl chi kaj. e1s-chair one-arm.length elevated-p.stat p heaven 'My chair is one arm-length tall in height. ' We can now turn to the standard-raising effect of -VC 1 , which will interact with the default interpretation of pos for upper-closed position to generate infelicity. I borrow an idea from the account of extreme adjectives-gigantic, fantastic, gorgeous, etc.-due to Morzycki(2012).Inadditiontocontextuallyspecifiedstandarddegrees,thereisaalso a set C of salient degrees that act as domain restriction for degree quantification. For instance, for a domain-restricted version of pos above, we would simply require that d ∈ C in addition to satisfying the degree relation and being greater than the contextual standard (that would also be in C).Myproposalforthesemanticeffectof-VC 1 is that it targetsthedomainrestriction,furtherrestrictingquantificationbypickingoutjustthose degrees that are greater than every degree in C.
First, note that this analysis accounts for the fact that VC 1 -derived positionals are indeed standard-raising. We know that any contextual standard, by definition, must be in C. The -VC 1 derivation derives a relation between degrees and individuals that holds just in case the individual's degree exceeds the maximum standard allowed in the context. Essentially, the -VC 1 derivation says that an individual doesn't just exceed the standard,butexceedstherangefromwhichstandardsarechosen.Itisstandard-raising in this sense. Second, the analysis says that the type of the VC 1 -derivation is exactly the same as the -V l positional stative predicate derivation. This is important because it correctly predicts that these derived positional adjectives, in virtue of having an unsaturated degree argument, should be able to appear in a similar range of degree constructions as the positional stative predicate. This is illustrated for the comparative construction in (82).
(83) ∅ ch'eq-ech-ïk. a3s wet-p.adj-sg 'It'sverywet.' We assume that in non-verbal predicate constructions like (83), the positional adjective also composes with pos, just as with the positional stative predicate constructions above. The result is the predicate in (84), which is true of an individual x just in case x is at least d-wet and d is not only greater than the contextual standard for wetness (namely the lower bound as given by 80), but d is greater than every contextually salient degree of wetness. That is, x is very wet, which are the truth conditions of the predicate in (83).
Itistrueofanindividualx just in case it is d-empty and d is not just greater than the context standard, but greater than all salient degrees of empty. With upper-closed scales, though, the greatest degree on the scale is the contextual standard, and thus in C. But now -VC 1 will only be true of individuals who have a degree greater than any in C, which is impossible because there is no such degree. That is, x must be more empty than completely empty, which should be infelicitous.
While there is certainly more to explore, both for the positional adjective derivation andthepositionalstativepredicatederivation,wehaveseenherethatpositionalspecific morphology can be given scalar semantics that captures both their truth conditions and compositional structure. The latter has played an especially important role in explaining the distribution of positional roots and stems. The core idea throughout has been that positional roots denote measure functions, while derived positionals denote degree relations of the same type as root adjectives. This explains the fact that positional roots are extremely constrained, while the positional stems considered here have a similar distribution to root adjectives across a variety of degree constructions.

Against a purely morphosyntactic account
IntheprevioussectionIarguedthatthedistributionofpositionalrootsandtheirderived stemswasmostdirectlyconstrainedbysemantics,inparticular,bytypes.Positionalroots denote measure functions, while positional morphology takes such functions into degree relations. There are alternatives, though, to a semantics-centric account. The skeptic might say that instead of types, positional roots are constrained due to their syntactic category. For instance, Evans & Levinson (2009), who argue that positionals roots belong to a novel lexical category p, could say that positionals require overt derivation into another lexicalcategorybecauseonlythosecategorieshaveinflection.Alternatively,aDistributed Morphology approach might say that positional roots are category-less, and with no zero derivation,requireovertcategory-definingmorphologytobeinflected.
Iwillargueagainstbothofthesesyntacticalternativesatonce.Inparticular,Iarguethat positional roots are neither category-less nor have a boutique category, but instead are verbal roots with the category V. 22 Moreover,Iarguethat-V l is not category-changing, so 22 Inrecentwork,Coon(2019)defendstheideathatverbalrootsinMayan,namelyintransitiveandtransitive roots are distinguished in being of type ⟨e,st⟩, where s ranges over eventualities. That is, verbs take an internal argument and produce a predicate of eventualities. There is some tension between this proposal andtheoneherebecauseIarguethatpositionalrootsareverbalbut,crucially,theymustbemeasurefunctions and not be relations of this type. There is reason to believe, though, that gradable adjectives might require both event-and state-type arguments (e.g., Wellwood 2015; 2016). There is not enough space here toconsiderwhetherpositionalsrequireaneventualityargument,thoughIexpectso.Ifwetakethisroute, thenpositionalrootsbecomeevenclosersemanticallytoverbroots,asproposedbyCoon(2019).Inthis case, positionals would just be defective verb roots, taking both an individual and eventuality argument, as Coon (2019) proposes, but relating it to a degree, not a truth value. V l-marked positionals are of category V as well. Thus, category membership alone can't constrain the distribution of positionals, and so the semantic explanation advocated here is better.
The primary argument that positional roots are verbal concerns those expressions they share derivations with. In particular, while there are positional-specific derivations, positional roots only share derivation with verb roots. Consider the instrumental nominalization, which is productive with verbs roots, as in (87) When we turn to positionals we see that they behave like verb roots, but not roots of other categories, in allowing instrumental nominalization.
(89) Positionalinstrumentalnominalization a. weq-b'äl adorned-instr 'adornments' b. tzuy-b'äl seated-instr 'seat' The fact that positional roots share derivations with verb roots is not a parochial fact about Kaqchikel. Across the Mayan family positionals share affinities with verbs. For instance, in Chol, positional intransitive derivation is the same as the passive for a subset oftransitiveverbs(Coon&Preminger2009).Similarly,Tzotzilhasaconstruction,called Color+PositionalCompounds,whichappliesproductively,onlytopositionalandverb roots (Laughlin 1975;Haviland 2003). These considerations suggest a deep historical connection between verb and positional roots that we see synchronically in Kaqchikel in patterns like (87-89). The best explanation is that positional roots are just verb roots, though a distinguished subclass with some of its own morphology to manage their degree denotations, as argued here. The idea is that just like transitive and intransitive roots form morphologically distinguishable root class, yet are both verbal in virtue of also sharing a number of derivations, positional roots would be a third distinct verbal root class.
Ifweaccept,then,thatpositionalrootshavethecategoryVinvirtueofsharingderivation with verbs, we can further show that -V lisnotcategorychanging.Positionalstative predicates derived by -V l share derivations with verbs, but not other categories. Consider, forinstance,gerundivenominalizationbyaffix-en.Theexamplesin(90)showbonafide verbs roots being nominalized by the -ensuffix,whiletheexamplesin (91) Crucially, expressions of other categories, even categories with an allied semantics like adjectives, reject derivation by -en. This shows that V l-derived positionals, in virtue of taking verbal derivations, have the category V.
In sum, the morphological evidence shows that positional roots are of category V. Moreover, V l-derived positionals are also of category V. Thus, a purely syntactic account of why underived positionals cannot occur in degree constructions faces obstacles. We cannot say that positional roots, unlike roots of any other type in the language, cannot appear because they have a boutique category or no category at all. This bolsters thesemanticaccountpreposedinthisprevioussections.Positionalrootsdenotemeasure functions, which simply cannot be used as predicates. They require derivation in order to denote an expression that can take both an individual and degree argument to return a truth value.

Conclusions
Overthecourseofthispaperthreemajorclaimshavebeenmade.First,Ihaveargued that positionals in Kaqchikel should receive a scalar semantics. Their distribution and semantic properties closely follow gradable adjectives in English, as well as such adjectivesin Kaqchikelitself.Second,Ihavearguedpositionalrootsareofadifferenttypethan gradable root adjectives in Kaqchikel and that positional morphology serves, in part, to bestowpositionalstemswiththesametypeasgradableadjectives.Finally,Iarguedthat the reason positionals are the only root class that must be derived follows from these previous two claims. In particular, as expressions of type ⟨ed⟩, positional roots are not relational, and so must be derived into an expression of the appropriate type to be used. Inadditiontomakingthispositivesemanticargument,Ihavearguedhereagainstsyntactic alternatives in which positional roots are category-less or have a boutique syntactic category.
While this work is empirically focused on the proper analysis of positionals in Kaqchikel, there are wider theoretical points this work addresses. First, this work weighs in on the important question of whether languages can have lexical categories beyond the familiar verb,noun,adjective,adverb.Atfirstpass,positionalslooklikeagoodcaseofanovel lexicalcategory,andthishasevenbeenargued(Evans&Levinson2009).Ihaveshown here that this is not the case. Morphosynactically, positionals are just verbs, but have a unique distribution in virtue of their lexical semantics and semantic type. This result emphasizes the need for analytical depth when we want to make claims about lexical categories. Fairly abstract notions, like semantic type, can constrain surface morphosyntactic distributions in ways that may be misleading. The second major theoretical result of this workisthatitconfirmsapredictionbyKennedy (2007),namelythatlanguagesmaydiffer in whether degree expressions have type ⟨ed⟩ or type ⟨d,et⟩,andthatthisdifferenceis most likely only distinguishable (morpho)syntactically. We have shown here that even in the same language we can see such a split in degree-denoting expressions, and the split can be diagnosed syntactically.
Finally,becausethisworkprovidesthefirstdetailedformalanalysisofpositionalsin any Mayan language, it has only been possible to scratch the surface of what there is toexplore.Iseethreeareasforfutureworkthatareespeciallypressing.Tobegin,itis critical to being to explore how well this analysis presented here extends to other Mayan languages.IstronglyexpectittoholdfortheK'ichean-branchlanguages,butfrombrowsing dictionaries and grammars, it may be the case that positionals in some western-branch Mayan languages have a more verb-like lexical semantics than the adjective-like notions lexicalized in Kaqchikel. This does not necessarily preclude a scalar account; there are many scalar change verbs in languages like English, for instance, work must be done to see how well the account extends to these languages.
A second area that requires further exploration concerns the facts raised in footnote 11. For some positionals, varying by speaker, the positional adjective derivation prefers an individual-levelreadingthatdoesnothaveastandard-raisingflavor.Thisfactdoesnot negate the need for a standard-raising account of -VC 1 for many positionals, nor does it preclude a degree-based account of these non-standard-raising individual-level readings. Infact,thesereadingscanbehandledassigning-VC 1 a degree denotation similar to -V l, but restricted to individual-and not stage-level predication. The outstanding work is to understand which positionals can have this individual-level reading, and to tie it to some deeper lexical-semantic fact. A final avenue for future work should explore deeper reasons why some notions are lexicalized as positionals, while other are lexicalized as adjectives, and whether any such differencecouldexplainthetypedifferencesIhaveproposedhere.Inparticular,whenwe look at Kaqchikel root adjectives, almost all of them are so-called relative adjectives-they arerelatedtoscalesthathavenoupperorlowerbound.Itwouldbenicetobeabletosay adjectives have the type they do, in contrast to positionals, in virtue of lexicalizing this particular class of gradable notions, though the way forward to addressing this problem is not clear.