Table 1 summarizes the properties of unmarked expressions. The values are understood as follows: ‘possible’ means that the relevant property is sometimes the case (e.g., in English, the ‘possible’ under ‘bound entailed’ means that, with an unmarked form, a bound may/may not be entailed, which is the case when comparing expressions with, e.g., straighten vs. widen when uttered out-of-the-blue); ‘possible’ also indicates that there are no lexical means (i.e., unmarked forms) requiring that the relevant property be the case—pragmatic considerations may be involved. The ‘no’ value means that the relevant property is never the case (e.g., in Aymara, the ‘no’ under ‘bound entailed’ means that, when there is an unmarked form, a bound is never entailed). ‘N/A’ means that that property cannot be tested (e.g., in Aymara, since a bound is not entailed, whether the bound is lexical cannot be tested).

The first property regards the boundedness of the scale—tested via the entailment test. In English, there is a contrast in that a bound in the scale is reached only if the relevant scale is top-closed—the sentences are thought of as uttered out-of-the-blue. (20) (=(7)) shows this contrast (see Kennedy & Levin 2008; Beavers 2011). The scale associated with the degree achievement in (20a) is top-closed; the one in (20b) is top-open.

In contrast to English, Aymara, Polish and Hungarian do not show this alternation. Regardless of the degree achievement having a top-closed/top-open scale, the scale is unbounded—the entailment test for Polish is the one Filip (2008) uses, with the continuation but not completely.

Whether the bound in the scale must be lexical with unmarked forms can only be tested for English (see Table 1)—this is the case if the degree achievement has a top-closed scale. The degree may/may not be lexical. Consider (24) (=(5a)). If it is uttered out-of-the-blue, the theme ends up fully straight. If it is made explicit in context that what counts as maximally straight does not correspond to complete straightness, the degree the theme reaches would be this contextual maximum. This would be the case in a context where the theme needs to be straightened, but it needs to be straightened in such way that it ends up a little bent (suppose that the rod needs to fit in a particular mold, so it needs to be almost fully straight).

The literature (see Kennedy & Levin 2008) indicates that English is subject to pragmatic considerations in this regard. This is the case in (24). In addition to the contextual manipulation to determine whether the maximum is lexical/contextual, it can also be made explicit that the theme does not end up straight. Instead, it may only end up straighter. Consider a case where Mary straightened the rod for a while; she will continue straightening it afterwards. In this scenario, the predicate straighten behaves more similarly to widen in that it describes an event where no maximal degree is reached.

Being subject to pragmatic considerations is also the case with sentences like (5b), repeated below. Although when uttered out-of-the-blue this sentence means that the theme ends up wider, context may make explicit that some degree actually counts as maximal in that context. Suppose that the road has steep hills on each side. Suppose that it has been widened almost reaching the points where the really steep parts begin. In this context, the road cannot be widened any more; the contextual maximal degree of width is reached.

To capture this flexibility, the literature (Kennedy & Levin 2008; Pedersen 2015) proposes that English degree achievements are subject to the pragmatic principle of Interpretive Economy. It captures that there is a default (out-of-the-blue) case, but if there are cues that indicate otherwise (e.g., contextual cues), these cues are the ones taken into consideration. This principle is stated in (26) (Kennedy & Levin 2008: 169). It guarantees that whenever the degree achievement considered has a top-closed scale, the scale is bounded (unless indicated otherwise), since there is a lexical maximum (the ‘possible’ value in Table 1 captures that the lexical maximum may be reached). In the absence of such a degree, i.e., with a top-open scale, the scale is unbounded (unless indicated otherwise). As a result, degree maximalization is pragmatically determined.

The third property in Table 1 is the quantization of the theme. That the theme must be quantized is not the case in any of the languages: it can be quantized/cumulative—the shift-of-the-theme test is applied below. This shows that unmarked sentences remain underspecified regarding the nature of the theme cross-linguistically. In Aymara, the plural marker -ra needs to be included for plural themes, indicated in parentheses. In Polish (Filip 2008), the bare nominal can thus receive a quantized/cumulative interpretation.

The last property in Table 1 regards telicity. Applying the in/for adverbial test to (5) (=(11)) (involving a quantized theme), shows that, in English, predicates are telic in the out-of-the-blue cases if the relevant scale is top-closed. Otherwise, they are atelic. Pragmatic considerations similar to the ones discussed above may also play a role in that if it is made explicit that the maximal degree is not reached (when the relevant scale is top-closed), the predicate is atelic (so adjoining a for adverbial becomes possible). A similar reasoning holds for cases with top-open scales: if a contextual maximal degree is made salient, the predicate is telic (so adjoining an in adverbial becomes possible)—besides, if the theme is shifted to a cumulative one, the predicates are atelic (which is as expected).

In contrast, Aymara, Polish and Hungarian do not show such flexibility: the predicates in unmarked expressions are atelic—only for adverbials are admissible. In Aymara, the contrast between in/for adverbials is indicated by the presence/absence of the ablative marker -tha, its presence indicating that the temporal expression includes a bound (since the scale is unbounded in the unmarked case (the predicates are already atelic), it does not matter whether the theme is quantized/cumulative).

The maximalization possibilities in English are as follows. Since there are only unmarked forms, there is room for flexibility. There are no requirements in that the scale does not need to be bounded; if the scale is bounded, the bound need not be lexical; the theme need not be quantized; and the predicate need not be telic. Pragmatic considerations play a role regarding maximalization. In contrast, the unmarked forms in Aymara, Polish and Hungarian do have particular properties: the scale cannot be bounded and the predicate must be atelic; the only place where some flexibility is allowed regards the theme, which can be quantized/cumulative.

Table 2 summarizes the properties of marked expressions. The values are understood as follows: ‘yes’ means that the relevant property is always the case (e.g., in Aymara, the ‘yes’ under ‘bound entailed’ means that a bound is always entailed). The ‘possible’ value means that the relevant property is sometimes the case (e.g., in Polish, the ‘possible’ under ‘lexical bound’ means that the degree bound may/may not be lexical; the lexical means do not require that the relevant property be the case; pragmatic considerations may be involved). Since English does not have marked expressions, it is not discussed in this section.

The first property regards the boundedness of the scale. Applying the entailment test shows that there is a contradiction, so the marked expressions in Aymara, Polish and Hungarian require that the scale be bounded.

The question is whether the bound must be lexical, i.e., the second property in Table 2. The relevant cases involve top-closed scales, i.e., Aymara (13b), Polish (15b) and Hungarian (17b), repeated below—if the scale is top-open, the bound is contextual. Interestingly, Aymara and Hungarian pattern together, but differently from Polish. In the former, the scale bound must be lexical: sentences with these predicates become infelicitous if it is made explicit that the maximum does not correspond to the lexical one, but to a contextual one (see Appendix 1 for contexts targeting a contextual maximum).14

In contrast, in Polish, while in the out-of-the-blue cases the lexical maximum is reached, it is possible to provide cues suggesting that the maximal degree is contextual—suppose that the maximum of cleanliness does not require that the drawer ends up spotless (e.g., all dust must be cleaned, but spots may remain); thus, the drawer need not end up completely clean (as determined by the lexical maximal degree).

I propose that the flexibility Polish displays is similar to the flexibility English displays in that the latter is subject to Interpretive Economy. It follows from this that lexical maxima are preferred in the out-of-the-blue cases, but contextual maxima are possible if additional cues are present. In section 3, I derive the Polish effect based on the same mechanism that derives the English contrasts.

The next property in Table 2 regards the quantization of the theme—tested via the shift-of-the-theme test. The contrast is between Aymara vs. Polish and Hungarian. While the Aymara marked expressions are compatible with quantized/cumulative themes (41) (-ra is used with the plural), the Polish (42) and Hungarian (43) ones are only compatible with quantized themes. This is crucial to tease apart degree and event maximalization: the latter requires that both the scale be bounded and the theme be quantized; the former only requires that the scale be bounded. Thus, the marked expressions in Aymara display degree maximalization; the Polish and Hungarian ones, event maximalization. The (im)possibility of having both quantized/cumulative themes appears below. Thus, in Polish, the theme cannot be interpreted as cumulative.

The last property in Table 2 regards telicity. With a quantized theme, the Aymara marked cases are telic, which is expected (the scale is also bounded). Thus, adjoining an in adverbial is possible; adjoining a for adverbial yields ungrammaticality (Martínez Vera 2018b). This appears in (44). If the theme is cumulative, the predicate is atelic, (it does not matter that the scale is bounded). Adjoining a for adverbial is possible; adjoining an in adverbial yields ungrammaticality. This appears in (45).

In contrast to Aymara, the marked expressions in Polish (Filip 2008) and Hungarian (Kardos 2016) include telic predicates. This follows since the scale must be bounded and the theme, quantized. Thus, adjoining an in adverbial is possible; adjoining a for adverbial yields ungrammaticality. This is shown below.

The discussion shows that the marked forms in Aymara (with -su), involve degree maximalization—degree maximalization requires that the bound in the scale corresponds to the lexical maximum (if available).15 The marked forms contrast with the unmarked forms (without -su), where degree maximalization cannot occur. In Polish and Hungarian, the marked forms (with a prefix or particle respectively) involve event maximalization—the scale must be bounded and the theme, quantized. The unmarked forms are used when event maximalization does not occur. Polish and Hungarian differ in that Hungarian requires that, in the marked forms, the lexical maximum corresponds to the bound in the scale (if available) (see (39)–(40)).

The discussion of Tables 1 and 2 indicates that there are four language types regarding maximalization, as stated in (48) (see (3)). The next section proposes a formal implementation that derives these properties.

Recall the language type English represents (including the properties of its unmarked forms in Table 3):

The assumptions made above already account for English, since there is no lexical specification (determined by overt means) regarding degree/event maximalization. I focus on the denotations involving the verb and theme (tense is ignored). The LF that I assume appears in (63).

Examples (5a)–(5b) are repeated; their denotations in (64b)–(65b) indicate that the theme increases in some property along a scale in an event.

Regarding the value of index i under g, there are preferences: when uttered out-of-the-blue, in sentences with a top-closed degree achievement the maximum is reached; in sentences with a top-open degree achievement some non-maximal degree is reached (see section 2.3.1). This is captured via Interpretive Economy (Kennedy & Levin 2008). I implement Interpretive Economy in (66):

(66) indicates that conventional means are preferred over contextual cues. The lexical maximum or minimum is preferred over the contextual maximum as a value for the standard, capturing the meaning of the English sentences, where conventional means are used whenever available. Based on Kennedy & Levin (2008), I propose that the lexical maximal degree is preferred over the minimum as a value for the standard, because the former is more informative—only the former requires that the whole scale be considered (thus, the homomorphic relation makes reference to the largest interval of the scale).

This approach captures in what sense degree and event maximalization are left open in English. Regarding degree maximalization, whether the scale is bounded is subject to Interpretive Economy. Regarding event maximalization, in addition to the former, there are no restrictions on the quantization of the theme. As a result, a predicate will be telic in general in the out-of-the-blue cases if the scale associated with the predicate is top-closed, since the standard’s value will be the maximum, and if the theme’s quantity can be determined (once event maximalization has been discussed, I come back to the presence/absence of event maximalizers in English (and Aymara) in section 3.3).

Recall the language type Aymara represents (including the properties of its unmarked/marked expressions in Table 4):

The novelty here regards degree maximalization. I assume that Aymara -su_i is a degree morpheme similar to POS_i, the difference being that the former’s presupposition state that the value of the standard equals the relevant maximum (Martínez Vera 2018b). The proposal is stated in terms of Maximize Presupposition (Heim 1991). For illustration, consider a classical example involving the (in the singular) and a. These are lexical alternatives in competition that have (almost) identical denotations but differ in that the has a uniqueness presupposition (i.e., there is a unique relevant individual) that a does not have. Thus, the, but not a, has a restricted domain. The idea is that whenever there are two expressions ϕ and ψ that differ in that ϕ has the and ψ has a, and there is a unique (relevant) individual under consideration, the alternative with the, ϕ, is used instead of the alternative with a, ψ, because ϕ, with the, has a restricted domain. Only if the presupposition of the in ϕ is not satisfied, ψ, with a, is used. This is illustrated in (68). First imagine a context where the victim’s biological mother is considered; there is one unique (relevant) individual, so the alternative with the is preferred over the alternative with a (the former has a restricted domain). Now consider a context where the victim has two mothers (e.g., a lesbian couple has a child who is the victim); the presupposition of the is not met and the alternative with a is used.

For Aymara, the proposal is that POS_i and -su_i are lexical alternatives (LEXALT) in competition (69) ((69b)=(57)).23 The difference between POS_i and -su_i is that -su_i has a presupposition specifying that the value of the standard is the maximum in the relevant scale, which is done by cases, distinguishing the lexical maximal degree case from the contextual maximal degree case. This is similar to Interpretive Economy in (66) in that there is an order, where lexical means prevail. Assuming that the option with a restricted domain is preferred, -su_i blocks POS_i whenever the maximum is reached in an event. The value of the standard in the absence of -su_i (with POS_i) is the minimum: the inference is that, whenever degree maximalization occurs, there is a marker indicating this. It follows that the scale is bounded with -su_i and unbounded without it—degree maximalization is thus captured.

The LF assumed here is (70) (see the LF for English (63))—-su_i can also occupy the slot of POS_i.

The denotations of (13a)–(13b) appear below. The former is a case without -su_i (i.e., with POS_i); the latter includes -su_i. Degree maximalization occurs only in the latter. The denotations of the verb combined with the theme indicate the difference regarding the standard: the lexical maximal degree must be used—with predicates with top-open scales (14b), the maximum is contextual, as determined by -su_i.

This captures the language type where degree maximalization is tied to lexical means, i.e., -su_i (unlike English). It has nothing to say regarding the quantization of the theme (like English). Thus, telicity contrasts do not follow from the presence/absence of -su_i only.

This section turns to the third language type, represented by Polish (including the properties of marked/unmarked expressions in Table 5):

Here I capture the maximalization/non-maximalization contrasts, which correspond to the presence/absence of a prefix. The analysis also captures the variability regarding lexical/contextual maxima.

The crucial notion is event maximizalization. Following Kardos (2016) (see Filip 2008; Beavers 2012), I assume that event maximalization is formalized via a monadic operator that applies to a predicate of events to yield a set of maximal events that is a subset of that predicate:

A maximal event constitutes the largest unique event in the denotation of predicate of events Σ. I assume that the scale provides a criterion for establishing an ordering in the event considered such that some part of the theme corresponds to a subevent in that event (as follows from the FPR). The idea is that an event of, e.g., straightening the rod to some degree may develop into a larger event of straightening the rod to some greater degree (e.g., to the maximal degree)—i.e., the latter is a “more developed version” of the former (Landman 1992: 23), with the former (and similar events) constituting stages of ‘larger’ events. The maximal event is the largest unique event in the denotation of a predicate; it includes all other smaller events.24 This is the kind of ordering that satisfies the input requirement of max_E (Filip 2008: 221). Note that a bound in the scale must be reached at the end of the event, so event maximalization can occur, since, otherwise, any two subevents of straightening the rod to some degree are indistinguishable—they would fall under the same description. If event maximalization occurs, the predicate is telic, i.e., for any event the predicate describes, it does not describe a subpart of it (see (61))—in Beavers’s (2012: 46) words, “no non-final subevent of any event related to some specific quantity of the theme is also an event related to the same specific quantity of the theme.”

A question arises regarding the locus of event maximalization. Following Filip (1999) (see Gehrke 2008), I assume that event maximalization, which is tied to a prefix, is merged low in the structure, close to the verb—I assume that event maximilizers take degree achievements as arguments (i.e., they combine with the scalar element that has combined with a degree morpheme). Assuming that the degree morpheme in Polish is POS_i, the LF I assume is (75)—EM=event maximalizer; see below for the (apparent) type mismatch between the maximalization operation EM introduces and max_E in (74).25

I further assume an abstract element that remains neutral with regard to event maximalization—intuitively, it is the POS_i equivalent for event maximalization. This assumption captures that Polish displays categorical contrasts between examples with/without a prefix regarding the requirements that the scale be bounded and the theme have a quantity. Of relevance is that, with the prefix, the scale must be bounded; without it, the scale is not bounded. This is similar to Aymara: there is a categorical contrast between sentences with/without -su_i (see sections 2.3.1-2.3.2). Therefore, I adopt a similar approach for event maximalization: I formalize event maximalization in terms of Maximize Presupposition. There are two lexical items: event maximalizer EM and an element that remains neutral in this regard, NM. EM, but not NM, presupposes event maximalization. Thus, event maximalization occurs with EM, which blocks NM (the alternative with a restricted domain is preferred (i.e., ‘accidental’ event maximalization without EM does not occur; see Table 5)—my proposal thus analyzes degree and event maximalization uniformly. Thus, when event maximalization occurs, there is a marker indicating this.

The denotations of NM and EM appear below. The former requires that the event be maximalized. The two items constitute lexical alternatives where EM blocks NM if event maximalization occurs—Δ is a variable for degree achievements (the result of combining scalar predicate δ and a degree morpheme).

The denotations of (15a)–(15b) appear below. Both denotations convey that the theme becomes cleaner. Only in the latter does event maximalization occur, i.e., the theme reaches the maximum in this case. This is relevant in connection to the standard. When event maximalization does not occur, there is no largest event, so the scale is unbounded and the standard equals the minimum of cleanliness, i.e., When event maximalization occurs (78), where the scale is top-closed, there are two options: the scalar bound could correspond to the lexical or the contextual maximal degree of cleanliness, i.e., or (if the scale is top-open, e.g., (16b), the maximal degree must be contextual). As discussed in section 2.3.2, there is a preference for the lexical maximum. This is expected if Polish is subject to Interpretive Economy (66): the lexical value is chosen in out-of-the-blue cases; the contextual maximum is a possible value when additional cues are given. Thus, POS_i in Polish is subject to pragmatic considerations (like English). The difference lies in that the minimum is excluded in the presence of event maximalization, since the scale must be bounded.

This analysis captures that event maximalization is marked overtly (via prefixes). It captures that event maximalization always involves telic predicates; without event maximalization, predicates are atelic—this is also tied to the presence/absence of themes whose quantity can be determined. There is, nonetheless, room for pragmatic considerations regarding the value of the standard (like English).

The last type is represented by Hungarian (including the properties of marked/unmarked expressions in Table 6):

My analysis builds on the analysis of Aymara and Polish (sections 3.2.2–3.2.3). I assume that Hungarian has two degree morphemes (like Aymara), and that event maximalization in this language is like Polish (with EM/NM).26

The LFs for Hungarian appear in (80). (80a) is the case without a particle; (80b) includes a particle—EM=event maximalizer (the particle), NM=EM’s non-maximalization counterpart, DM_i=degree maximalizer.27 Note that, since POS_i and DM_i are alternatives (see section 3.2.2), only having POS_i in the non-maximalization case and DM_i in the maximalization case follows: POS_i cannot assign the maximum to the standard (which is possible only in the presence of DM_i), making it semantically incompatible with EM. DM_i cannot appear in (80a), because the relevant scale would be bounded, which is not possible in the absence of event maximalization (I assume that all lexical alternatives are replaced together when this takes place (80a); the satisfaction of the presuppositions against the relevant context is then determined at the same time).

The denotations of POS_i and DM_i, and of NM and EM are repeated below (see (69), (76)).

Consider the denotations of (17a) and (17b). Both denotations convey that the theme increases in straightness, the difference lying in whether there is event maximalization, which only occurs in the latter. Additionally, when event maximalization occurs, the standard equals the lexical maximum—with top-open scales, e.g., (18b), that degree is contextual.

My analysis captures that event maximalization is tied to marked forms (involving particles) in Hungarian (like Polish), and always involves telic predicates (in contrast to cases without event maximalization, which involve atelic predicates). The analysis of Hungarian contrasts with that of Polish regarding the value of the standard in connection to maximal degrees. Due to the presence of a degree maximalizer (like Aymara -su_i), lexical maxima must be used if available.