1.1 Terminology and a Caveat

My previous discussion has already suggested that the term ‘ontology’ can be used to refer both to a scientific discipline and to its topic of study, that is the semantic ontology of natural language. (The latter, but not the former, consists of different ontological categories like ‘individual’ and ‘property.’) Since the noun ontology has a plural form (i.e. ontologies), it in principle allows for the possibility that the subject matter of natural language ontology differs from language to language, or from semantic theory to semantic theory. (Both are indeed the case, as I show in the remainder of this subsection.) To avoid having the term ‘natural language ontology’ do double duty, this Element follows Moltmann (Reference Moltmann and Zalta2022) in using ‘natural language ontology’ for the discipline and ‘semantic ontology of natural language’ (shortened ‘semantic ontology’ or, simply, ‘ontology’) for the subject matter of this discipline.

That the subject matter of natural language ontology differs from language to language is illustrated by contemporary semantics for the pair of sentences in (4). This pair comprises a German sentence with an explicit comparative (ist größer als [‘is taller than’]; in (4a)) and its closest possible translation in Motu, an Austronesian language of Papua New Guinea (in (4b); based on Beck et al., Reference Beck, Krasikova, Fleischer, Gergel, Hofstetter, Savelsberg, Vanderelst and Villalta2009).

Expectedly, the occurrence of proper names (i.e. Matti, Cleo) in the two sentences incurs a common commitment – in German and in Motu – to individuals. Many contemporary semantics argue that the intuitive truth-conditions of constructions with the comparative morpheme -er further require a commitment to degrees, $d$ (see the interpretation of (4a) in (5), where ‘ ${\mu }_{\text{height}}$(matti)’ is the degree to which Matti is tall; see e.g. Lassiter, Reference Lassiter2012; Rett, Reference Rett and Altshuler2022).

(5)	$\left(\exists d\right)\left[\right({\mu }_{\text{height}}\left(matti\right)\ge d)\wedge \neg ({\mu }_{\text{height}}\left(cleo\right)\ge d\left)\right]$

In contrast to German, Motu lacks a dedicated degree morphology and does not allow difference comparatives. To compensate for the nonavailability of comparative morphology, Motu speakers use a conjunctive strategy that juxtaposes two full clauses with antonymous predicates. Beck et al. (Reference Beck, Krasikova, Fleischer, Gergel, Hofstetter, Savelsberg, Vanderelst and Villalta2009) have shown that a suitable semantics for the resulting constructions (e.g. (4b)) requires neither a measure function nor a degree variable (see also Bochnak, Reference Bochnak2015). Instead, they interpret gradable predicates like tall as context-sensitive vague predicates (viz. ‘counts as tall in $c$’; see (6)).

(6)	$tall(c,matti)\wedge short(c,cleo)$

Beck’s semantics suggests that, in contrast to the semantic ontology of (4a), the ontology of (4a)’s Motu counterpart, (4b), does not contain degrees. The local [= German] ontology of (4a) (in (7b)) hence properly includes the local [= Motu] ontology of (4b) (in (7a)):

(7)		a.	Motu-[(4b), Beck et al., Reference Beck, Krasikova, Fleischer, Gergel, Hofstetter, Savelsberg, Vanderelst and Villalta2009]: $\{$ individuals, properties $\}$
	$\subset$	b.	German-[(4a), Lassiter, Reference Lassiter2012]: $\{$ individuals, properties, degrees$\}$

To avoid leaving ontologies underspecified with respect to the interpreted language or fragment, this Element will be explicit about the language for which it provides a semantic ontology. Whenever no language is specified, ontological claims will be relative to a reasonably representative fragment of contemporary American English (that extends Montague’s PTQ-fragment). Since there is a large overlap between the semantic ontology of this extended fragment of English and representative fragments of other languages (incl. Motu), I expect that the findings from this Element will also be relevant for the local ontologies of other languages, and for the global semantic ontology (which is shared by all natural languages).

I have mentioned at the beginning of this subsection that language-specific (i.e. ‘local’) semantic ontologies still vary with the particular semantic theory that is adopted for the respective language. This is so since different semantic theories may – at least in part – assume objects from different semantic categories. A key example of this observation can be found in the difference between contemporary degree-based semantics for comparatives (e.g. Cresswell, Reference Cresswell and Partee1976; Lassiter, Reference Lassiter2012; von Stechow, Reference von Stechow1984) and earlier degree-free approaches (e.g. Klein, Reference Klein1980; McConnell-Ginet, Reference McConnell-Ginet1973; Neeleman, van de Koot, & Doetjes, Reference Neeleman, van de Koot and Doetjes2004): Since degree-free approaches analyze degrees as equivalence classes of individuals (and, hence, drop degrees from their stock of basic entities), they ascribe English and German a Motu-style semantic ontology:

(8)	a.	Motu-[(4b), Beck et al., Reference Beck, Krasikova, Fleischer, Gergel, Hofstetter, Savelsberg, Vanderelst and Villalta2009]: $\{$ individuals, properties $\}$
$\stackrel{!}{=}$	b.	German-[(4a), Klein, Reference Klein1980]: $\{$ individuals, properties $\}$

The effect of semantic theory on local ontology is further illustrated by the difference between Beck’s (Reference Beck, Krasikova, Fleischer, Gergel, Hofstetter, Savelsberg, Vanderelst and Villalta2009) and a Keenan (Reference Keenan and Bianchi2015)-style interpretation of (7a)/(8a). Since Keenan’s semantics interprets proper names as sets of primitive [= non-decomposable] properties (see also Keenan, Reference Keenan2018; Keenan & Faltz, Reference Keenan and Faltz1985), its application to (7a) would waive Motu’s commitment to individuals (see (9a)).

(9)		a.	German-[(4a), Keenan, Reference Keenan and Bianchi2015]: {properties, degrees}
	$\supset$	b.	Motu-[(4b), Keenan, Reference Keenan and Bianchi2015]: {properties}
	$\subset$	c.	Motu-[(4b), Beck et al., Reference Beck, Krasikova, Fleischer, Gergel, Hofstetter, Savelsberg, Vanderelst and Villalta2009]: {individuals, properties}

The difference between (7b), (7a)/(8)/(9c), (9a), and (9b) illustrates that specifying a local semantic ontology requires identifying both (i) the interpreted language (or fragment) and (ii) the particular semantic theory that is adopted for its interpretation.

Ritchie (Reference Ritchie2016) has proposed to remedy the dependence on a semantic theory by adopting a Principle of Carrying Commitments (PCC). This principle assumes that, in determining a language’s ontological commitments, semantics with equal empirical adequacy should be given an equal voice (see Ritchie, Reference Ritchie2016, 20). In virtue of this assumption, it holds that a (language or) linguistic phenomenon only determinately carries a commitment to a certain ontological category if all competing semantic theories for this phenomenon carry this commitment. For example, since the ontologies in (7a)/(9c) and (9b) only agree with respect to properties, PCC’s application to the Motu sentence (4b) only determines an ontological commitment to properties. Since degree-free semantics for explicit comparatives fail to satisfy Ritchie’s adequacy requirement,Footnote ³ PCC determines an additional commitment to degrees when applied to the German sentence (4a) (see the larger intersection of (7b) and (9a)).

I will return to PCC in the later parts of this Element, where I apply this criterion to different intensional ontologies (in Section 5.3) and where I modify PCC to a criterion that respects the difference between ontologies of the semantic object- and the metatheory (in Section 8). In the meantime, I will exploit an ontology’s variance with (ii) in Sections 6 and 7. Since this exploitation will yield insights into the minimal requirements on a given language’s semantic ontology, it will inform the modification from Section 8.

2.1 Identifying a Language’s Semantic Commitments

In what follows, I review three strategies for identifying a language’s descriptive ontological commitments. These strategies are based on the semantic selection properties of natural language predicates (see Section 2.1.1), on the study of dedicated proforms and quantifiers (see Section 2.1.2), and on the identification of implicit semantic arguments (see Section 2.1.3).

2.1.1 The Lexical Strategy: Selection Properties

A first strategy for identifying a language’s semantic commitments investigates differences in the semantic selection properties of predicates. Following Vendler (Reference Vendler1967a, Reference Vendler1967b), this strategy assumes that selectional differences track ontological differences in the kinds of arguments that these predicates combine with. According to this strategy, a predicate’s ability to accept only one, but not the other of two expressions as a semantic argument suggests that these expressions belong to different ontological categories.

A simple instance of this strategy is given in (10). This instance investigates the possibility of combining the predicate occur with different syntactic arguments – in particular: with the nominal gerund John’s singing (see (10b)), with the referential DP John (or the boy; see (10a)), and with the declarative sentence (or clause) (that) John was singing (see (10c)). Since the result of this combination is deviant for the two last-mentioned arguments (as marked by superscript question marks in (10a/c)), the strategy concludes that the semantic values of John/the boy and of (that) John was singing must belong to a different ontological category than the semantic value of John’s singing. Since occur expresses a temporal property (which requires its semantic arguments to be temporal or temporally located objects), Vendler (Reference Vendler1967b) identifies semantic arguments like John’s singing in (10b) with an event (see also Grimm & McNally, Reference Grimm and McNally2022).

(10)	$\left(\begin{array}{cc}\text{a.}& {}^{??}\text{John/the boy}\\ \text{b.}& \text{John's singing}\\ \text{c.}& {}^{??}\text{(that) John was singing}\end{array}\right)\text{occurred at noon.}$

That the value of John (or the boy) (see (10a)), in turn, belongs to a different ontological category than the value of (that) John was singing (see (10c)) is shown by the possibility of taking John/the boy – but not (that) John was singing – as the object argument of the transitive verb kick (see (11)).

(11)	$\text{Mia kicked}\left(\begin{array}{cc}\text{a.}& \text{John/the boy}\\ \text{b.}& {}^{??}\text{that John was singing}\end{array}\right).$

The observation that kick only takes concrete physical arguments (which are located in a certain point in space; see Vendler, Reference Vendler1967b) then motivates the classification of the semantic values of John and the boy as an individual. The strategy from (10) and (11) has also been used to argue for a semantic category of facts (e.g. Ginzburg, Reference Ginzburg2005; Kastner, Reference Kastner2015; Vendler, Reference Vendler1967b), of propositions (Asher, Reference Asher1993; Vendler, Reference Vendler1967a), of questions (Ginzburg, Reference Ginzburg1995; Grimshaw, Reference Grimshaw1979; Lahiri, Reference Lahiri2002), and of kinds (Carlson,Reference Carlson1977; Chierchia, Reference Chierchia1998; Landman, Reference Landman2006).

It is important to point out that the success of this strategy stands and falls with the choice of the embedding predicate. Thus, selectionally super-flexible verbs like remember (which accept a wide range of different syntactic constructions; see Liefke, Reference Liefke2021) will indicate many fewer ontological distinctions (as is apparent from (12), which is acceptable for all of the object arguments in a–d).

(12)	$\text{Mia remembers}\left(\begin{array}{cc}\text{a.}& \text{John/the boy}\\ \text{b.}& \text{John's singing}\\ \text{c.}& \text{that John was singing}\\ \text{d.}& \text{whether John was singing}\end{array}\right).$

That the success of this strategy also depends on the choice of the syntactic argument(!) is shown by the intuitive oddness of sentences like Mia kicked a fly.

2.1.2 The Morphological Strategy: Proforms

An alternative – or complementary – strategy to the above capitalizes on so-called ‘morphological category-specific items’. These items include quantifiers as well as proforms (paradigmatically, anaphoric pronouns and, on some accounts, wh-words). The morphological strategy assumes that natural languages lexicalize reference to different types of entities, such that different quantifiers and proforms refer to entities from distinct ontological categories. A selection of English pronouns, wh-words, and quantifiers is given in Table 1 (taken from Liefke, Reference Liefke, Ginzburg and Lassiter2024b, 23; based on Rett, Reference Rett2018, 5). This selection includes individual proforms (e.g. he, she, it), temporal proforms (e.g. tense markers), modal proforms (e.g. will, would ), and propositional/sentential proforms (e.g. that).

Table 1 Proforms for entities from different semantic categories

Category	Pronoun	wh-word	Quantifier
individuals	he, she, it	who, which, what	all, everything/-one
events	it	which, when, what	always, if
worlds/situations	will, would, then	? when	must, might, if
times	then, -ed [past	when	always, daily,
	tense morpheme]		when(ever)
locations	there, it	where	everywhere,
			where(ever)
propositions	that, it	what	everything,
			what(ever)
degrees	yea, so	how (many/much)	more, -er
manners	(like) so	how	? like
kinds	so, such	how	all

Some of the items from Table 1 straightforwardly ‘reveal’ their referent’s semantic category, as intended. This holds for those items that are exclusively associated with a single semantic category, like the English third-person pronouns he and she, and the degree demonstrative yea (see (13); Rett, Reference Rett2015, 8). In particular, since she exclusively picks out individuals, it syntactically disambiguates (14) in favor of a ‘DP + adjunct’-construction. (The complement of see on the alternative, gerundive analysis, viz. a woman swimming, can only be picked up by the pronoun it; cf. (1b).)

(13)	Anna is yea tall.	[accompanied by a gesture]
(14)	Zeno saw $[$ a woman ${]}_i$ swimming. She${}_i$ was wearing a wetsuit.

The success of these examples notwithstanding, proforms and quantifiers are, in many cases, too unspecific to allow inference to a particular semantic category. In particular, many proforms refer to objects from intuitively distinct semantic categories (see the numerous occurrences of some proforms in Table 1). An example of such ‘multi-categorial’ morphological items is the singular neutral pronoun it. As is evidenced by (15), it can be used – among others – to refer to individuals (in (15a)), events (in (15b)), and propositions (see (15c)) (see Asher, Reference Asher1993; Bach, Reference Bach1986a; Krifka, Reference Krifka1990).

(15)	a.	Berta baked [a cake]${}_i$. Anna ate it ${}_i$.	(individual)
	b.	$[$ The squeaking of the door]${}_i$ caused Mia to cringe.	(event)
		It ${}_i$ made Noel’s ears hurt.
	c.	Ben believes [that figs are fruit]${}_i$. Dana doubts it ${}_i$.	(proposition)

In order to answer the co-classification challenge from (15) (multiple categorial reference) and (12) (selectional super-flexibility), Rett (Reference Rett and Altshuler2022) has proposed to distinguish semantic categories by considering the selection and reference behavior of all proforms, quantifiers, and predicates that are associated with a given semantic category, and by drawing category-distinctions on the basis of the most selective of these proforms, quantifiers, and predicates. Her proposed account would thus focus on the predicates occur and kick (see (10), (11)) and on the proforms yea and she (see (13), (14)), rather than on the expressions remember and it.

2.1.3 The Logical-Semantic Strategy: Quantificational Domains

To answer the semantic categorization problems that remain even in Rett’s combined strategy, many researchers complement the strategies from Sections 2.1.1 and 2.1.2 with a strategy that focuses on the formal semantic modeling of the investigated language. This strategy is based on the observation that certain phenomena (paradigmatically: sentential entailments) can only be modeled if we assume an extra semantic domain whose elements are denoted by implicit (logical or semantic) arguments. The strategy is based on Quine’s (Reference Quine1948) criterion of ontological commitment, according to which “to be is $\dots$ to be the value of a variable” (32). It uses Davidson’s (Reference Davidson1977) observation that “ontology is forced into the open only where the theory finds quantificational structure” (251).

The logical-semantic strategy presupposes Montague’s method of indirect interpretation, which obtains the semantic values of natural language expressions by interpreting these expressions’ logical ‘translations’ in set-theoretic models (Montague, Reference Montague and Hintikka1973; see Zimmermann, Reference Zimmermann, McNally and Szabo2022).Footnote ⁴ The strategy starts from the logical formulas that serve as these translations. It identifies semantic commitments with the domains of existential quantifiers in these formulas.

An example of the logical-semantic strategy (due to Davidson Reference Davidson1977) is given in (16)/(17). This example is based on the observation that capturing the intuitive entailments from (16) requires introducing an event variable, $e$ (see (17)).Footnote ⁵

(16)		a.	Brutus brutally stabbed Caesar on the forum.
	$\Rightarrow$	b.	Brutus brutally stabbed Caesar.	(locative-PP drop)
	$\Rightarrow$	c.	Brutus stabbed Caesar.	(adverb drop)

(17)	a.	$\left(\exists e\right)\left[stab\right(e,brutus,caesar)\wedge brutal(e)\wedge loc(e)=forum]$
	b.	$\left(\exists e\right)\left[stab\right(e,brutus,caesar)\wedge brutal(e\left)\right]$
	c.	$\left(\exists e\right)\left[stab\right(e,brutus,caesar\left)\right]$

The logical-semantic strategy has also been used to argue for a semantic category of possible worlds $w$ (see (18a); Kratzer, Reference Kratzer, von Stechow and Wunderlich1991; Montague, Reference Montague1970), times $t$ (see (18b); Cariani, in press; Matthewson, Reference Matthewson2006), kinds $k$ (see (18c); Carlson, Reference Carlson1977; Landman, Reference Landman2006), manners $m$ (see (18d); Piñón, Reference Piñón2008; Schäfer, Reference Schäfer, Bonami and Hofherr2008), and degrees $d$ (see (4a)/(5); Heim, Reference Heim2000; Lassiter, Reference Lassiter2012) – among others. In (18), $⇝$ is the translation function that sends natural language expressions (e.g. (16a)) to logical terms (here: (17a)). In (18c), $\le$ is the instantiation relation between individuals and kinds.

(18)	a.	It may be raining in Bochum.
		$⇝\left(\exists w\right)\left[rain\right(w,bochum\left)\right]$
	b.	Ben called Mia (in $c$ at $@$).
		$⇝\left(\exists t\right)[t\prec t_c\wedge call(@,t,ben,mia\left)\right]$
	c.	Fred owns a rare dog.
		$⇝\left(\exists x\right)\left[\right(dog\left(x\right)\wedge own(fred,x))\wedge (\exists k\left)\right[x\le k\wedge rare\left(k\right)\left]\right]$
	d.	John writes illegibly.
		$⇝\left(\exists e\right)[write(e,john)\wedge \left(\exists m\right)\left[manner\right(m,e)\wedge illegible(m\left)\right]]$

We will see in the next section that Montague’s semantic ontology relies heavily on the logical-semantic strategy.

2.2 Montague’s Semantic Ontology

Since Montague’s semantics targets a small, well-defined subset of 1970s English (i.e. the PTQ-fragment) that has a fully specified ontology (see Montague, Reference Montague and Hintikka1973), it is the perfect starting point for our investigation of semantic ontologies. Note that the PTQ-fragment excludes measure phrases, degree modifiers, and explicit comparatives alongside modifiers for kinds and interrogative expressions. As a result of the former, PTQ’s ontology is similar to the ontology of ‘degree-less’ languages like Motu (see Section 1.1). As a result of the latter, PTQ’s ontology lacks kinds and questions. Since some entailments of the form of (16) can be captured without reference to – or quantification over – events (see the account, in (19b), of the entailment from (19a)), Montague (Reference Montague1969) rejects a category of events (as does Montague, Reference Montague1970, Reference Montague and Hintikka1973).

(19)	a.	The sun rose at eight. $\Rightarrow$ The sun rose.
	b.	$rise(⟨@,08:00⟩,sun)\Rightarrow \left(\exists t\right)\left[rise\right(⟨@,t⟩,sun\left)\right]$

2.2.1 Montague’s Extensional Ontology

To facilitate the type-theoretic analysis of the PTQ ontology (in Sections 4–5), I divide the PTQ-fragment into an extensional part (in Table 2) and an intensional part (in Table 3), following (Liefke, Reference Liefke, Ginzburg and Lassiter2024b, section 3). These parts differ with respect to the substitutibility of their elements: While expressions from the extensional part allow the truth-preserving substitution of co-referential expressions (e.g. Bill, the man) and of truth-conditionally equivalent expressions (e.g. Bill walks, The man walks; see (20)), expressions from the intensional part are often taken to block this substitution (see (21)).

Table 2 The extensional part of Montague’s PTQ-fragment

pronouns:	$h{e}_0$, $h{e}_1$, $h{e}_2,\dots$	(individuals)
proper names:	John, Mary, Bill	(G[eneralized] Q’s)
decl. sentences:	John walks, $\dots$	(truth-values)
common nouns:	man, woman, park, fish, pen	(properties of individuals)
intrans. verbs:	run, walk, talk	(properties of individuals)
transitive verbs:	find, lose, eat, date, be	(relations betw. individ’ls)
adverbs:	rapidly, slowly	(relations betw. properties)
determiners:	a, the	(property/GQ-relations)
quantifiers:	every	(property/GQ-relations)
prepositions:	in	($\dots$)

(20)		a.	$[$ The man $]$ walks.
		b.	The man is Bill.
	$\Rightarrow$	c.	$[$ Bill $]$ walks.
(21)		a.	Mary believes $[$ that the man walks $]$.
		b.	The man walks $\iff$ Bill walks
	$\Rightarrow$	c.	Mary believes $[$ that Bill walks $]$.

In virtue of its elements’ substitutivity, the extensional part of the PTQ-fragment allows proper names to be interpreted as individuals (represented by properties of properties of individuals [= ‘generalized quantifiers’]) – rather than as individual concepts (or their representing properties of properties of individual concepts).Footnote ⁶ Table 2 includes the expressions’ semantic categories in the right-most column. In this column, ‘GQ’ is short for ‘generalized quantifier’.

The fact that Table 2 contains individual proforms (viz. indexed versions of the pronoun he) already suggests that the PTQ-fragment carries a semantic commitment to individuals. While Montague’s strategy of ‘generalizing to the worst case’Footnote ⁷ (Partee, Reference Partee1983, 34) prevents him from assuming individuals as the semantic values of referential determiner phrases (DPs), individuals play a central role in his interpretation of determiners and quantifiers (viz. as objects in the domain of existential and universal quantification; see (22)).

(22)	a.	$\text{A man walks.}⇝\left(\exists x\right)\left[\text{man}\right(x)\wedge \text{walk}(x\left)\right]$
	b.	$\text{Every man walks.}⇝\left(\forall x\right)\left[\text{man}\right(x)\to \text{walk}(x\left)\right]$
	c.	$\text{The man walks.}⇝\left(\exists x\right)\left(\forall y\right)\left[\right(\text{man}\left(y\right)\leftrightarrow y=x)\wedge \text{walk}(x\left)\right]$

Interestingly, in addition to individuals, Montague’s ontology also contains points (or ‘moments’) of time. Since Montague further assumes that this domain is structured by a simple ordering, $\le$ (Montague, Reference Montague and Hintikka1973, 257–258; see Reference Bach, Marcus, Dorn and Weingartner1986b, 577), PTQ’s semantic ontology allows for a straightforward extension to tense and temporal expressions (along the lines described in [18b] and in Liefke, Reference Liefke, Ginzburg and Lassiter2024b, 4.2.3).

2.2.2 Montague’s Intensional Ontology

To capture the substitution-resistance of predicates like temperature and rise (see (23)) and of the clausal complements of verbs like believe (see (21)), Montague’s semantics interprets these expressions in terms of propositions and/or individual concepts (i.e. as properties of individual concepts, or as relations to a proposition; see the rightmost column in Table 3).

Table 3 The intensional part of Montague’s PTQ-fragment

decl. complements:	that$\dots$, to $\dots$ [inf.]	(propositions)
sentence adverbs:	necessarily	(proposition/truth-value rel’s)
intensional nouns:	unicorn, price,	(p’ties of individual concepts)
	temperature
intransitive verbs:	rise, change	(p’ties of individual concepts)
transitive verbs:	seek [= try to find]	(rel’s to a centered proposition)
clause-taking Vs:	believe, assert	(rel’s to a proposition)
control verbs:	try, wish	(rel’s to a centered proposition)
adverbs:	allegedly	(relations between properties)
prepositions:	about	($\dots$)

For reasons that will be detailed in Section 5, Montague analyzes propositions as characteristic functions of sets of ordered world/time pairs, or of sets of ‘indices’ (i.e. those indices at which the proposition is true). Individual concepts are identified with functions from indices to individuals. The interpretation of intensional nouns like temperature as properties of individual concepts blocks intuitively invalid inferences like (23) (Montague, Reference Montague and Hintikka1973, 267–268; attributed to Barbara Partee).

(23)		a.	The temperature is ninety.
		b.	The temperature rises.
	$\Rightarrow$	c.	${}^{??}$ Ninety rises.

Since Montague’s semantics prima facie does not quantify over indices, it might look like this semantics does not carry a commitment to indices (or to possible worlds). However, such quantification is covert in the modal box operator $\text{□}$ and in Montague’s analysis of the intensional operators ${}^{\vee }$ (read: ‘cup’) and ${}^{\wedge }$ (read: ‘cap’). The box operator, which is central to Montague’s interpretation of the sentence adverb necessarily, is analyzed in terms of universal quantification over worlds (see (24); Muskens, Reference Muskens1995, 37).

(24)	$\text{Necessarily }p⇝\text{□}p(\equiv \forall w.p\left(w\right))$

The intensional operators, which are needed to switch from intensions to their actual-world extensions and vice versa, correspond to application to and abstraction from the implicit index parameter (see [25], where $\alpha$ is a well-formed expression and $i$ is a fixed variable for the evaluation index).

(25)	a.	${}^{\vee }\alpha =\alpha \left(i\right)$
	b.	${}^{\wedge }\alpha =\lambda i.\alpha$

In view of the above, Montague’s semantics for the PTQ-fragment assumes a rich descriptive ontology that includes individuals and individual concepts (see (22)/(23)), possible worlds and times (see (24)), propositions (see (21)), properties of individuals, properties of individual concepts, and generalized quantifiers, alongside objects of many other semantic categories.

I will show in Sections 4 and 5 how Montague reduces this ontology to a small subset of basic semantic categories (which has the same modeling scope as the descriptive ontology). However, before I do so, I sketch the most important extensions of Montague’s ontology (in Section 2.3) and review the merits of typing the objects in these ontologies (in Section 3).

2.3 Larger Semantic Ontologies

I have already suggested (in Section 2.1) that most state-of-the-art semantics for extensions of the PTQ-fragment (e.g. Champollion, Reference Champollion2017; von Fintel & Heim, Reference von Fintel and Heim2021) assume a substantially larger ontology than Montague’s semantic ontology from Section 2.2. This is so since these extensions contain expressions (e.g. manner adverbs, measure phrases, plurals, mass nouns, and kind terms) whose behavior cannot be adequately modeled in Montague’s ontology. We have already seen an instance of this strategy in (17) (which introduces events to capture entailments like (16)). Similar observations have motivated the assumption of events, manners, degrees, pluralities, substances, and kinds (see Figure 1 and the penultimate paragraph of Section 2.1.3). I will, at this point, not delve deeper into the arguments for each of these categories (see instead Liefke, Reference Liefke, Ginzburg and Lassiter2024b, section 4). Rather, I will content myself with listing the most frequently discussed categories and (some of ) their more prominent proponents (in Table 4).

Table 4 Extensions of Montague’s ontology (overview)

events	(Davidson, 2001; Parsons, 1990; Vendler, 1967b)
manners	(Dik, 1975; Piñón, 2008; Schäfer, 2008)
situations	(Barwise & Perry, 1983; Ginzburg, 2005; Kratzer, 1989)
questions	(Ginzburg, 1995; Hamblin, 1976; Groenendijk &
	Stokhof, 1984)
intervals	(Champollion, 2017; Partee, 1973; Rooij & Schulz, 2014)
degrees	(Heim, 2000; Lassiter, 2012; von Stechow, 1984)
kinds	(Carlson, 1977; Chierchia, 1998; Landman, 2006)
pluralities	(Champollion & Brasoveanu, 2022; Link, 1983)
vectors	(Winter, 2005; Zwarts, 1997; Zwarts & Winter, 2000)
content individuals	(Bondarenko, 2020; Kratzer, 2006; Moulton, 2009)
$⋮$	$⋮$

Admittedly, the different focus of the semantic theories whose objects are listed in Table 4 gives rise to partly different semantic ontologies. For example, since Champollion’s (Reference Champollion2017) semantics of distributivity, aspect, and measurement is not concerned with modeling interrogatives or attitude reports, its ontology – plausiblyFootnote ⁸ – does not carry a commitment to questions or content individuals (in contrast to, e.g., Hamblin, Reference Hamblin and Partee1976 or Kratzer, Reference Kratzer2006).

The resulting pluralism notwithstanding, the ontologies of contemporary semantic theories converge to a surprising extent. This holds at least as soon as these theories consider larger – phenomenally more diverse – fragments of the target language (that cover a wide[r] range of phenomena), or as soon as these theories are identified with the sum of different semantics for different classes of phenomena. Independently of these considerations, many semantic theories already share a substantial part of their ontological commitments (e.g. to individuals, worlds/situations, properties, propositions, and generalized quantifiers). Quite a few contemporary semantic theories – including semantics for manner adverbs (see (18d); Schäfer, Reference Schäfer, Bonami and Hofherr2008), semantics for distributivity (see Figure 1; Champollion, Reference Champollion2017), and semantics for attitude reports (see Kratzer, Reference Kratzer2006) – further share the assumption of events.

My intermittent sketch of semantics for these phenomena has already suggested that the members of different ontological semantic categories are related in interesting ways (see also my introduction to this Element). This holds, for example, for propositions and possible worlds (at which propositions are true or false), for properties and individuals (see (22)), for questions and propositions (which serve as answers to questions), and for manners and events (see [18d]).

The following sections capture these relations with the help of the Simple Theory of Types (see Church, Reference Church1940). The latter is a bookkeeping system that allows one to keep track of different semantic categories (see Muskens, Reference Muskens and Craig2011). A key merit of type systems lies in their providing a hierarchical structure: Rather than merely assigning objects to semantic categories (as was already done in Section 2), type systems distinguish between basic (or ‘primitive’) and complex (or ‘derived’) objects. Of these objects, the former resist a decomposition into simpler objects (and, as a result, carry ontological commitment). Objects of a complex type are constructed from basic – or less complex – objects through special type-forming rules (e.g. function space formation).

To show how the typing of semantic objects is achieved, I first introduce a number of different type systems, starting with a system for Montague’s extensional ontology, TY₁, and with Gallin’s streamlined version, TY₂, of Montague’s (Reference Montague1970) Intensional Logic IL (in Sections 4–5). Following TY₂’s extension to larger linguistic fragments (in Section 6), I will then discuss arguments for reducing – and for unifying – types (in Section 7). I precede my introduction of different type systems by reviewing the merits of typing the objects in a language’s semantic ontology:.

3.1 Compositionality

I have already noted at the beginning of Section 1 that the compositional computation of semantic values requires identifying (type-)relations between the objects from different semantic categories. To see why this is the case, consider the example in (26):

Intuitively, we want to say that (26) is true if the referent of Matti, that is, $〚\text{Matti}〛$, has the property of sleeping. However, in the absence of a deeper connection between individuals and properties, we will not be able to obtain the semantic value of (26) from $〚\text{Matti}〛$ and $〚\text{sleep}〛$ (or $〚\text{is sleeping}〛$) alone. This holds in particular so long as we have not specified the semantic linking operation $+$ in the compositional analysis, $〚\text{Matti}〛+〚\text{sleep}〛$, of $〚$ Matti is sleeping $〛$.

The type-logical analysis of properties ($〚\text{sleep}〛$) solves this problem: By analyzing properties as functions from individuals to truth-values (or to propositions), this analysis straightforwardly accounts for the compositional interpretation of (26) (in (27)).

(27)	$〚\text{sleep}〛(〚\text{Matti}〛)$

Specifically, this account analyzes the semantic value, $〚\text{sleep}〛$, of sleep as a function, $x\mapsto 〚x\text{ sleeps}〛$, from individuals $x$ to the truth-value $〚x\text{ sleeps}〛$ (or equivalently, as a characteristic function of the set, $\{x:〚x\text{ sleeps}〛=\text{1}\}$, of sleeping individuals). The application of this function to the individual $〚\text{Matti}〛$ then yields the truth-value 1 [= ‘true’] (or 0 [= ‘false’]).

3.2 Type Checking and Type Inference

The typing of semantic objects enforces a rigorous type checking on the outputs of the compositional interpretation. This checking is enabled by the type-driven nature of the semantic interpretation process (see, e.g., Heim & Kratzer, Reference Heim and Kratzer1998; chapter 3.1; Zimmermann & Sternefeld, Reference Zimmermann and Sternefeld2013; based on Klein & Sag, Reference 71Klein and Sag1985). Type-driven interpretation is a general procedure for the provision of semantic values that obtains these values from the semantic values of syntactically simple expressions (e.g. lexical elements, words) together with information about the syntactic structure of the interpreted complex expression. It assumes that the semantic values of syntactically complex expressions are obtained from the values of their immediate syntactic constituents through the familiar composition rules (paradigmatically: forward or backward functional application, FA; see Charlow & Bumford, Reference Charlow, Bumford, Ginzburg and Lassiter2025, section1.1).

Specifically, type-driven interpretation assumes an interpretation relation, $〚\cdot 〛$ (or translation relation, $⇝$), between analyzed syntactic structures (that is, LFs) and (logical terms for) the objects in our ${\text{TY}}_n$ model structures.Footnote ⁹ This relation is governed by a ‘base rule’, (T0), which specifies the interpretation of lexical elements. The remaining rules specify how the interpretation of complex syntactic structures depends on the interpretation of their constituents. The two most elementary such rules, (T1) and (T2), are given next:

(T0)	If $X$ is a word and $\text{A}$ its interpretation, then $〚X〛=\text{A}$.
(T1)	If $〚X〛=\text{A}$, then a structure, $\left[X\right]$, that contains $X$ as its only constituent
	is interpreted as follows: $〚\left[X\right]〛=\text{A}$.
(T2)	If $〚X〛=\text{A}$ and $〚Y〛=\text{B}$, then if $\text{A}\left(\text{B}\right)$ is well-typed, $〚\left[XY\right]〛=\text{A}\left(\text{B}\right)$;
	if $\text{B}\left(\text{A}\right)$ is well-typed, $〚\left[XY\right]〛=\text{B}\left(\text{A}\right)$.

For example, given that $〚\text{sleep}〛(〚\text{Matti}〛)$ is well-typed, rule (T2) yields the interpretation of Matti sleeps from the interpretations, $〚\text{Matti}〛$ and $〚\text{sleep}〛$, of the proper name Matti and the intransitive verb sleep. Since type-driven interpretation allows interpretations to be “shuffled around as long as the result is a structure in which all the parts fit together appropriately” (Rothstein, Reference Rothstein2012, 239) it has sometimes been called ‘shake-and-bake semantics’ (Bach, Reference Bach1980).

Type-driven interpretation contrasts with Montague’s (Reference Montague1970, Reference Montague and Hintikka1973) ‘rule-by-rule approach’ to semantics (the label goes back to Bach, Reference Bach1977). This approach assumes that, for each syntactic rule that describes the formation of complex linguistic expressions, there is a corresponding semantic rule. Semantic rules describe how a semantic object (i.e. a member of a given semantic domain) is obtained from the semantic values of the expression’s constituents. Type-driven interpretation differs from the rule-by-rule approach in requiring a much smaller set of semantic/interpretation rules: Since it views semantic rules as dependent on syntactic rules, the rule-by-rule approach demands that same-domain objects that interpret the results of applying different syntactic rules (e.g. walk rapidly [$\text{IAV}+\text{IV}$] and try to walk [$\text{IV//IV}+\text{IV}$]) are also obtained through different semantic rules.Footnote ¹⁰ In contrast, type-driven interpretation effectively combines multiple such rules (for walk rapidly, try to walk: in the rule (T2)).

Assuming type-driven interpretation, one can explain the meaningful- or meaninglessness of complex expressions with respect to whether the semantic value of these expressions results from a seamless application of composition rules (e.g. (T2)). Such seamless application is evidenced by the semantic values of Matti is sleeping (i.e. (26)) and All children are sleeping. These values each result from different instances of Functional Application (FA): The semantic value $〚\text{Matti is sleeping}〛$ is obtained by applying the value of the sentence’s predicate, that is, sleep, to the value of the sentence’s subject, Matti (i.e. $〚X〛+〚Y〛=〚Y〛(〚X〛)$; see Figure 2a). In contrast, since quantifier phrases like all children are typically taken to denote (a characteristic function of ) a set of properties – rather than an individual –, the semantic value $〚\text{All children are sleeping}〛$ is obtained by applying the value of the sentence’s subject, that is, all children, to the value of the sentence’s predicate, sleep (i.e. $〚X〛+〚Y〛=〚X〛(〚Y〛)$; see Figure 2b).

Figure 2 Successful cases of functional application.

These assumptions about the semantic composition of intuitively meaningful expressions also help reverse engineer the semantic category of a given expression from the categories of its mother and its sister constituents (i.e. it facilitates type inferences; see Gunter, Reference Gunter1992). Thus, from the semantic categories of $〚\text{Matti}〛$ (i.e. ‘individual’) and $〚\text{Matti is sleeping}〛$ (i.e. ‘truth-value’), we can infer the category of $〚\text{sleep}〛$. The latter is the category of functions from objects of the sister category (above: individuals) to objects of the mother category (above: truth-values) (Gutzmann, Reference Gutzmann2019, 46–49; Winter, Reference Winter2016, 59–61). I will return to this kind of type inferences in Section 4.

Note that, since type-driven interpretation starts at the level of linguistic expressions, it can also be used to provide a semantic explanation of grammaticality and syntactic well-formedness. Accordingly, a complex expression is grammatical if the semantic values of its constituents at all levels of syntactic structure (i.e. clauses, phrases, and words) combine through the familiar composition rules. The fact that this is the case in Figures 2a and 2b then explains the grammaticality of the sentences Matti is sleeping and All children are sleeping.

Inversely to the above, type-driven interpretation also accounts for the ungrammaticality and meaninglessness of certain other complex expressions. This meaninglessness results from the inability to combine the semantic values of these expressions’ constituents at some level of syntactic structure through the usual composition rules. In this way, the meaninglessness of the complex expression (28) can be explained through the impossibility to combine the individual $〚\text{Matti}〛$ with the generalized quantifier $〚\text{all children}〛$ through FA (see Figure 3). This impossibility, in turn, explains the observation that Matti all children does not form a grammatical sentence and cannot serve as a constituent in a larger, more complex expression.

Figure 3 An unsuccessful case of functional application.

Arguably, these demands on ‘matching types’ also classify some expressions as deviant that are intuitively both grammatical and meaningful. These include the sentences in (29a) and (30a), which are subject to type-clashes (see (29c), (30c)). To solve this problem, most contemporary semantic theories adopt additional composition rules (e.g. Predicate Modification [PM]; see Heim & Kratzer, Reference Heim and Kratzer1998, 65)) and introduce a small set of dedicated type-shifting operations (e.g. the Montague Rule [lift], Argument Lowering, Value Raising; see Hendriks, Reference Hendriks1993, Reference Hendriks, Gutzmann, Matthewson, Meier, Rullmann and Zimmermann2020; Partee, Reference Partee and Groenendijk1987). The need for further composition rules is exemplified by (29a) (due to Charlow & Bumford, Reference Charlow, Bumford, Ginzburg and Lassiter2025), which requires PM to combine the properties that serve as the semantic values of happy and cat (see the boldfaced material in (29b)).

(29)	a.	The happy cat is purring.
	b.	$(〚\text{the}〛(\left[\right[happy\left]\right]\cap \left[\right[cat\left]\right]))(〚\text{pur}〛)$
	c.

The need for type-shifting operations is exemplified by (30a). This sentence requires that $〚\text{Matti}〛$ is shifted to the semantic type of $〚\text{all other children}〛$, that is, the type for generalized quantifiers (see (30c)). It is achieved through Partee’s (Reference Partee and Groenendijk1987) type-shifter $\text{LIFT}:=\lambda x.\{P:P(x\left)\right\}$ (based on Montague, Reference Montague1970), which sends individuals to the set of their properties (see (30b)).

(30)	a.	Matti and all other children are sleeping.
	b.	$((〚\text{and}〛(〚\text{all other children}〛))(\text{LIFT}(\left[\right[Matti\left]\right])))(〚\text{sleep}〛)$
	c.

One may worry that this remedy overshoots its aim, resulting in an unwelcome ‘rescue strategy’ for a large number of intuitively meaningless expressions. While this worry is not unfounded (see the possibility of ‘salvaging’ (28) by sending $〚\text{Matti}〛$ to a generalized quantifier through lift and combining the result with $〚\text{all children}〛$ through PM), the small number of composition rules (see Charlow, Reference Charlow2014; Section 1 of this Element) and the strong restrictions on admissible type-shifts (e.g. injectivity, lambda-definability, permutation-invariance; see van Benthem, Reference van Benthem1991; Zimmermann, Reference Zimmermann2020) impose strict limits on this strategy for the resolution of type-clashes.Footnote ¹¹

3.3 Parsimony

We have already seen that type theory distinguishes between basic (or ontologically primitive) and complex (or ontologically derived) objects. Since complex objects are constructed from basic objects through the iterated application of special type-forming rules (see below), type-theoretic ontologies need not commit – at least not in the same way – to all of their elements. Rather, their commitments are reduced to elements of a basic type, alongside the specific operations (e.g. function-space formation, set formation) that derive elements of a complex type from elements of a more basic type. The commitment to function-space and/or set formation brings with it a commitment to abstract objects like functions and sets (see Quine, Reference Quine1948).

Consider the extensional ontology of Montague’s PTQ-fragment (see Section 2.2.1). By Montague’s assignment of basic types, only individuals (i.e. the referents of proper names and pronouns) and truth-values (i.e. the extensions of declarative sentences) are entities of a basic type. On the level of type-forming operations, related observations hold for set formation or, more generally, for function-space formation. From individuals and truth-values, these operations enable the formation of (characteristic functions of ) sets of individuals or, equivalently, of functions from individuals to truth-values (see Section 4). Once these entities and operations are available, the remaining objects of Montague’s extensional ontology come ‘for free’: In particular, if we assume that properties of individuals are adequately represented (or coded) by sets of individuals, Montague’s parsimonious type system straightforwardly provides properties. The same holds for propositions (if we assume a basic type for possible worlds and the possibility of adequately representing propositions as sets of possible worlds).

Importantly, the use of type theory to reduce a semantic theory’s ontological commitments obeys the principle of ontological parsimony (more commonly known as Ockham’s razor). This principle demands that the number of entities or ontological categories that a specific theory assumes should not be increased beyond necessity – in Ockham’s original formulation, “Entiae non sunt multiplicanda praeter necessitatem” (Clauberg, Reference Clauberg2009, 320). Applied to the ontology of natural language semantics, Ockham’s principle demands that ontological categories should only be adopted if the semantic phenomena that they are intended to explain (see, e.g., the phenomena in Section 2.1) resist an explanation through those categories that are assumed anyway. I will show in Section 6 that this reasoning can – perhaps surprisingly – be used to argue for the assumption of primitive events and kinds.

In the history of analytic philosophy, Ockham’s principle has famously been used to argue against a commitment to properties (or universals). In particular, class nominalists hold that the ‘one over many’ problem – which demands an account of the multiple instantiation of properties (e.g. the exemplification of redness by Ed Sheeran’s hair, the red pen on my desk, and all British double-decker buses) – has a simple solution in terms of class membership: If properties are construed as classes, multiple instantiation is simply membership in the same class (e.g. in the class $\{$ Ed Sheeran’s hair, my red pen, UK Bus-1, UK Bus-2, $\dots \}$; see Armstrong, Reference Armstrong1980, 28–43; Lewis, Reference Lewis1983). Replacing classes by sets (or their characteristic functions) straightforwardly yields the Montagovian – extensional – picture of properties.

3.4 The Meaning/Reference-Relation

Beyond this, typing objects makes explicit the meaning/reference-relation. This holds at least so long as non-referential meanings are understood as Carnapian intensions, that is, as functions from possible worlds to the expressions’ extensions at these worlds (see Carnap, Reference Carnap1988). On Montague’s Carnapian ontology, the intensions of declarative sentences (i.e. propositions) are functions from possible worlds (or indices) to the propositions’ truth-value at these worlds. The intensions of common nouns and intransitive verbs (i.e. properties) are functions from possible worlds to the sets of individuals that exemplify the property at these worlds (see (25)).

Since Montague’s ontology identifies the extensions of an expression with the range of the function that serves as the expression’s intension (for sentences: truth-values; for common nouns: sets of individuals), it directly encodes the relation between meaning and reference. Specifically, the intension, $f$, of an expression determines the expression’s extension, that is, $f\left(w\right)$, for each evaluation world $w$. Inversely, since Montague identifies functions with their courses-of-values, the extensions, $x_w$, of an expression for each world jointly determine the expressions’ intension as that function $f$ such that, for each $w$, $f\left(w\right)=x_w$ (Zimmermann, Reference Zimmermann and Kompa2017, 193–194; see Ben-Avi & Winter, Reference Ben-Avi, Winter and Muskens2007; de Groote & Kanazawa, Reference de Groote and Kanazawa2013).

The described relation between intensions and extensions is reminiscent of the relation between Frege’s (Reference Frege and Beaney1997) notions of sense [Sinn] and reference [Bedeutung]. In particular, by determining the referent (or truth-conditional contribution) of a given expression at the relevant world, Carnapian intensions emulate the semantic function of Fregean senses (see Burge, Reference Burge2005). Montague’s type label for possible worlds, that is, $s$, is explicitly chosen to pay reverence to Frege’s notion of Sinn (Zimmermann, Reference Zimmermann, McNally and Szabo2022, 340). To capture the Fregean spirit of Montague’s ontology – as well as of its particular type-theoretic implementation (Church, Reference Church and Henle1951; see Section 5) – this ontology is sometimes called the ‘Frege-Church ontology’ (Kaplan, Reference Kaplan1976; see Liefke, Reference Liefke2024a).

Importantly, a type-theoretic encoding of the meaning/reference-relation is evidenced by all intensional ontologies that construe meanings as functions from some evaluation index to referents (as pointed out in Muskens, Reference Muskens2005). These include the ontologies of Kratzer-style situation semantics (Bondarenko, Reference Bondarenko2022; Kratzer, Reference Kratzer2002), of impossible-world semantics (Barwise, Reference Barwise1997; Rantala, Reference Rantala1982, Zalta, Reference Zalta1997), of truthmaker semantics (Liefke, Reference Liefke2020; Moltmann, Reference Moltmann2020), and of semantics with more complex evaluation indices (Plummer & Pollard, Reference Plummer and Pollard2012; Pollard, Reference Pollard2015). The meaning/reference-relation is even encoded in the semantic ontology of some hyperintensional theories (see e.g. Muskens, Reference Muskens2005; Thomason, Reference Thomason1980): While these theories treat non-referential meanings as hyper-intensions (for declarative sentences: as semantically primitive propositions; see Sections 6.1.2, 6.2.2), they still assume Carnap-style intensions and extensions. The meaning/reference-relation is then made available by a mapping from hyperintensions to intensions. For example, this mapping sends primitive propositions to sets of possible worlds. Since hyperintensional theories construct the hyperintensions of other expressions as functions to primitive propositions, this mapping generalizes to the hyperintensional meanings of all expressions.

4.1 Types and Rules of TY₁

I have already pointed out that type systems distinguish between basic and complex types. To capture the ontology of the extensional part of the PTQ-fragment, Montague (Reference Montague and Hintikka1973) assumes two basic types, that is, individuals (type $e$) and truth-values (type $t$). From these types, the types of all other extensional objects are obtained through the rule of function-space formation (i.e. Rule 1; see Church, Reference Church1940). This rule establishes that the type for functions from objects of one type to objects of another type is itself a type:Footnote ¹²

Rule 1 (Function-space formation).

If $\alpha$ and $\beta$ are types, then $⟨\alpha ,\beta ⟩$ is a type, where $⟨\alpha ,\beta ⟩$ is the type for functions from objects of type $\alpha$ to objects of type $\beta$.

A single application of Rule 1 to the types $e$ and $t$ yields (among others) the type for functions from individuals to truth-values, $⟨e,t⟩$, and the type for functions from truth-values to truth-values, $⟨t,t⟩$. Objects of type $⟨e,t⟩$ are characteristic functions of sets of individuals (i.e. extensional properties) that serve as the extensions of common nouns and intransitive verbs. Objects of type $⟨t,t⟩$ are characteristic functions of sets of truth-values that serve as the extensions of unary sentential operators (paradigmatically, of sentential negation).

A second application of function-space formation (to types $⟨e,t⟩$ and $t$) yields the type for (characteristic functions of ) sets of extensional properties, $⟨⟨e,t⟩,t⟩$. Objects of this type (i.e. extensional generalized quantifiers) serve as the extensions of determiner and quantifier phrases. The simplest types that are obtained from $e$ and $t$ through Rule 1 are given in Figure 4. In this figure, the previously discussed types are marked in boldface.

Figure 4 The hierarchy of extensional (TY₁) types.

One may expect that some expressions from Montague’s PTQ-fragment are interpreted as $n$-ary functions or relations, rather than as unary functions (as is enforced by Rule 1). This holds, for example, for sentential conjunction (i.e. and) as well as for transitive verbs like find: Intuitively, and denotes a function that sends ordered pairs of truth-values [= the extension of each of the two conjuncts] to a single truth-value [= the extension of the conjunctive sentence] (see Figure 5a). The verb find denotes a relation between individuals, or a relation between a generalized quantifier and an individual.

Figure 5 Binary versus unary semantics for conjunction.

To accommodate this interpretation, one could extend Rule 1 to the formation of $n$-ary functional and/or relational types (along the line of Muskens, Reference Muskens1995). However, there is a straightforward procedure for turning multiary functions into unary functions of a higher type (Winter, Reference Winter2016, 57–59). This procedure, called ‘Currying’ (after Haskell Curry, Reference Curry and Jakobson1961) or ‘Schönfinkelization’ (after Moses Schönfinkel, Reference Schönfinkel1924; see Heim & Kratzer, Reference Heim and Kratzer1998, 30–31), works by feeding the function its arguments one by one in inverse order, beginning with the last element of the ordered $n$-tuple (see (31)). In (31), $x_1,\dots ,x_n$ are arguments of the types ${\alpha }_1,\dots ,$ ${\alpha }_n$. $f$ is a function that sends ordered $n$-tuples of objects of types ${\alpha }_1,\dots ,{\alpha }_n$ to truth-values:

(31)	$\text{CURRY}\left(f\right)=\lambda x_n\dots \lambda x_1.f(x_1,\dots ,x_n)$

Specifically, curry modifies the ‘binary’ interpretation of and from Figure 5a (see [32a]) into the unary function in (32b). This unary function is used in the interpretation in Figure 5b.

(32)	a.	$〚\text{and}{〛}^{\text{2}}=\lambda ⟨p,q⟩.p\wedge q$
	b.	$〚\text{and}{〛}^{\text{1}}=\text{CURRY}(〚\text{and}{〛}^{\text{2}})=\lambda q\lambda p.p\wedge q$

Remark that there is nothing wrong in principle with including $n$-ary functions and relations in one’s semantic ontology. In particular, the addition or deletion of $n$-ary functions does not affect the theory’s adequacy. However, since much of contemporary syntax assumes binary-branching structures of the form in (32b) (see Figure 5b) – and since the restriction to unary functional types is in line with (Montague, Reference Montague1970, Reference Montague and Hintikka1973) and with much contemporary work – I will hereafter assume only unary functional types (next to basic types). An overview of the types of objects in Montague’s extensional ontology is given in Table 5.

Table 5 Types in Montague’s extensional ontology

Expression	Object	Type
–	individuals	$e$ (‘entity’)
declarative sentences	truth-values	$t$
sentential negation	sets of truth-values	$⟨t,t⟩$
sentential conjunction	functions to sets of truth-values	$⟨t,⟨t,t⟩⟩$
nouns, intransitive verbs	(fcts of ) sets of individuals	$⟨e,t⟩$
DPs, quantifier phrases	extensional gen’zd quantifiers	$⟨⟨e,t⟩,t⟩$
transitive verbs	functions to sets of individuals	$⟨⟨⟨e,t⟩,t⟩,⟨e,t⟩⟩$
adverbs	functions to sets of individuals	$⟨⟨e,t⟩,⟨e,t⟩⟩$
determiners/quantifiers	fcts to generalized quantifiers	$⟨⟨e,t⟩,⟨⟨e,t⟩,t⟩⟩$
prepositions	$\dots$	$\dots$

Note that these objects and functions already supply many of the relations that serve to categorize objects in large, state-of-the-art ontologies (e.g. the ontology for Chemical Entities of Biological Interest, ChEBI; see Degtyarenko et al., Reference Degtyarenko, de Matos, Ennis, Hastings, Zbinden, McNaught, Alcántara, Darsow, Guedj and Ashburner2008).Footnote ¹³ These include ‘is a property of’ (type-$⟨e,t⟩$), ‘is a binary relation between’ (type-$⟨e,⟨e,t⟩⟩$), ‘is an $n$-ary relation between’ (type-$⟨e,⟨\dots ,⟨e,t⟩⟩⟩$), and ‘is a relatum of’.

I have stated above that Montague’s extensional ontology contains objects of two basic types (or sorts), that is, individuals and truth-values. Following Gallin’s (Reference Gallin1975) convention of subscripting a type theory’s name by the number of its basic types – not counting the truth-value type $t$ – this type system is called ‘TY₁’ (for one-sorted type theory).Footnote ¹⁴ For completeness, the definition of TY₁ types is given next:

Definition 1 (TY₁ types).

The set of TY₁ types is the smallest set such that

(i)	$e$ and $t$ are TY1 types;
(ii)	if $\alpha$ and $\beta$ are TY1 types, then $⟨\alpha ,\beta ⟩$ is a TY1 type. (Rule 1)

This completes my introduction of Montague’s extensional/TY₁ types. I will move to a restriction and intensional extension of these types and their associated objects in Sections 4.3 and 5. However, before I do so, I first show how TY₁ types give rise to the type checking mechanism that has been discussed in Section 3.2.

4.2 Type Checking in TY₁

As expected, TY₁ straightforwardly explains the semantic acceptability (or meaningfulness) of complex expressions. To help identify type matches or mismatches, Figure 6 enriches the semantic trees from Figure 3 with the types of each of their nodes. This enrichment is reminiscent of type judgments in modern type theories (see Martin-Löf, Reference Martin-Löf, Rose and Shepherdson1975) and has recently been adopted in type-theoretically conscious applications of formal semantics (see e.g. Elliott, Reference Elliott2017; Moulton, Reference Moulton2015; Winter, Reference Winter2016, chapter 3).

Figure 6 Successful cases of functional application, with typing.

In particular, the type-enrichment of Figure 2a (in Figure 6a) identifies the semantic value of sleep with a (type-$⟨e,t⟩$) function from individuals to truth-values. Since it identifies $〚\text{Matti}〛$ with a suitable argument of this function (i.e. a type-$e$ individual), it straightforwardly explains the semantic acceptability of (26). Similar observations hold for the sentence All children are sleeping (see Figure 2b; in Figure 6b): since Figure 6b identifies $〚\text{all children}〛$ with a generalized quantifier, that is, with a function from type-$⟨e,t⟩$ functions to truth-values (which takes as input arguments of the type of $〚\text{sleep}〛$), it straightforwardly explains the semantic acceptability of the sentence All children are sleeping.

Notably, assuming that $〚\text{all children}〛$ is a generalized quantifier, one could also obtain the type-$t$ interpretation of All children are sleeping by treating $〚\text{sleep}〛$ as a function from generalized quantifiers to truth-values (i.e. as an object of type $⟨⟨⟨e,t⟩,t⟩,t⟩$; see Figure 7). This treatment is assumed in Groenendijk and Stokhof (Reference Groenendijk, Stokhof, Chierchia, Partee and Turner1989) (see Hendriks, Reference Hendriks1993), where it is implemented through the operation of Argument Raising. In contrast to the interpretation of All children are sleeping from Figure 6b, the treatment from Figure 7 enables the sentence’s compositional interpretation through backward FA, analogous to Figure 6a. However, the typing from Figure 6b is preferred to the typing from Figure 7 on grounds of simplicity (reflected in a preference for lower-rank types).

Figure 7 Alternative to the interpretation in Figure 6b, with typing.

Considerations like these can also be used to explain the semantic deviance of constructions like Matti all children (see Figure 8): since the semantic value of Matti does not have the right type to serve as an argument of the function that is denoted by all children (and vice versa), the two expressions cannot compose in the expected way.

Figure 8 An unsuccessful case of functional application, with typing.

4.3 From TY₁ Types to Extensional Types

I have suggested in the context of Figure 4 that not all TY₁ types are associated with objects in Montague’s extensional ontology. In particular, while Montague’s semantics for the PTQ-fragment assumes type-$e$ individuals (see Table 4), it does not assume extensional functions to individuals (e.g. type $⟨e,e⟩$ or $⟨t,e⟩$). Instead, all complex objects in Montague’s extensional ontology are functions to truth-values (i.e. characteristic functions of sets).

The restriction of functional types to types that “end in $t$” (Groenendijk & Stokhof, Reference Groenendijk and Stokhof1990, 5) is owed to the semantic identification of non-referential expressions with their truth-conditional contribution (following Davidson, Reference Davidson1967). It is further motivated by the observation (due to Partee & Rooth, Reference Partee, Rooth, Partee and Portner1983) that characteristic functions facilitate the modeling of semantic phenomena like entailment and coordination (see Keenan & Faltz, Reference Keenan and Faltz1985). Following Partee and Rooth (Reference Partee, Rooth, Partee and Portner1983), entailment – or, more generally, semantic inclusion – is defined as the functional counterpart of set-theoretic inclusion: for two linguistic expressions $A$ and $B$, it thus holds that $A\Rightarrow B$ (read: ‘ $A$ includes $B$’) iff, for all $x$, if $〚A〛\left(x\right)=\text{1}$, then $〚B〛\left(x\right)=\text{1}$. In the semantics literature, types whose objects allow for this treatment are called conjoinable (Partee & Rooth, Reference Partee, Rooth, Partee and Portner1983), relational (Groenendijk & Stokhof, Reference Groenendijk and Stokhof1990), or Boolean (Kac, Reference Kac1992; Winter, Reference Winter2002). The set of these types is defined as follows:

Definition 2 (Boolean types).

A type $\tau$ is Boolean iff

(i)	$\tau =t$ or
(ii)	$\tau =⟨\alpha ,\beta ⟩$, where $\beta$ is a Boolean type.

Using the notion of Boolean type, one can restrict TY₁ types to those types that more accurately reflect the commitment of the extensional part of Montague’s PTQ-fragment:

Definition 3 (extensional types).

The set of extensional types is a proper subset of the set of TY₁ types such that

(i)	$e$ and $t$ are extensional types;
(ii)	if $\alpha$ and $\beta$ are TY1 types and $\beta$ is a Boolean type, then $⟨\alpha ,\beta ⟩$ is an extensional type.

This definition includes all types from Table 4. To capture the semantic commitments of the full PTQ-fragment, I will extend the set of extensional types to intensional types in Section 5.

5.1 Intensional Ontology without Fully-Fledged Worlds

To obtain the types and objects from Table 6, Montague’s Intensional Logic (IL) extends the set of TY₁ types via a rule for the formation of intensional types. This rule (i.e. Rule 2) assumes that the type for functions from possible worlds, or indices, to objects of some type $\alpha$ is also a type (Montague, Reference Montague1970, 227–228; Reference Montague and Hintikka1973, 256).

Rule 2 (Intensional type-formation).

If $\alpha$ is a type, then $⟨s,\alpha ⟩$ is a type.

The resulting set of types is specified immediately below:

Definition 4 (IL types).

The set of IL types is the smallest set such that

(i)	$e$ and $t$ are IL types; (basic TY1 types)
(ii)	if $\alpha$ and $\beta$ are IL types, then $⟨\alpha ,\beta ⟩$ is an IL type; (Rule 1)
(iii)	if $\alpha$ is an IL type, then $⟨s,\alpha ⟩$ is an IL type. (Rule 2)

Note that Definition 4 does not assume a basic type for possible worlds (or indices). Instead, it introduces possible worlds only indirectly, that is, through the new Rule 2. The introduction of worlds through an additional type-forming rule has a key merit: It reduces the number of predicted, but ontologically irrelevant types. The addition of this rule thus serves a similar purpose as the restriction of $\beta$ to a Boolean type in the definition of extensional types (see Definition 3). It has long been assumed that the lack of a basic type for possible worlds also accounts for the absence of pronouns for possible worlds (see von Fintel & Heim, Reference von Fintel and Heim2021, 13). However, following Stone (Reference Stone1997), more recent work has assumed modal or possible-world proforms (see Section 2.1.2, Table 1).

5.2 Intensional Ontology with Fully-Fledged Worlds

The lack of a fully-fledged type for possible worlds requires that quantification and abstraction over worlds be achieved through dedicated operators (i.e. through the operators $\text{□}$, $\diamond$, and ${}^{\wedge }$). Gallin’s (Reference Gallin1975) streamlined version of Montague’s type system, TY₂ (for two-sorted type theory), compensates for this challenge by adopting a fully-fledged type for possible worlds. (The name for this type system, TY₂, already suggests that this system assumes an additional basic type next to the basic TY₁ types $e$ and $t$.) The set of TY₂ types is specified next:

Definition 5 (TY₂ types).

The set of TY₂ types is the smallest set such that

(i)	$e$, $s$, and $t$ are TY2 types; (basic TY1 types + $s$)
(ii)	if $\alpha$ and $\beta$ are TY2 types, then $⟨\alpha ,\beta ⟩$ is an IL type. (Rule 1)

In Definition 5, the assumption of a basic type for possible worlds obviates the adoption of an intensional type-forming rule like Rule 2: the iterated application of Rule 1 to the basic TY₂ types already yields all the intensional types from Table 6. In fact, the set of TY₂ types contains many types that are irrelevant for the semantics of the PTQ-fragment (see Zimmermann, Reference Zimmermann2023). These include most complex types that ‘end in $s$ or $e$’ (an exception being the type for individual concepts; see Section 2.2.2). An example of such ontologically irrelevant types is the type $⟨t,s⟩$ (i.e. functions from truth-values to possible worlds).

5.3 Intensional Ontology with Propositional Functions

It is sometimes argued that even the fairly parsimonious ontology of IL already contains many more intensional objects than are required to capture the semantics of English (details in Kaplan, Reference Kaplan1976, Zimmermann, Reference Zimmermann2023; see Section 2.2.2). This is evidenced by the ontologies from Montague (Reference Montague and Thomason1974) and Cresswell (Reference Cresswell1973), which restrict intensions to Russell (Reference Russell1996)-style propositional functions. The latter are objects of type $⟨{\alpha }_1,⟨\dots ⟨{\alpha }_n,⟨s,t⟩⟩⟩⟩$, where ${\alpha }_1,\dots ,{\alpha }_n$ are Russellian typesFootnote ¹⁵ (defined in what follows). To emphasize that Russellian propositions are, in principle, neutral with respect to the analysis of intensions (s.t. they can be identified with more finely grained objects than sets of possible worlds; see Sections 3.4 and 6.1.2), I call the type for Russellian propositions ‘ $p$’ (for propositions). Russellian types are defined as follows:

Definition 6 (Russellian types).

The set of Russellian types is the smallest set such that

(i)	$e$ and $p$ are Russellian types;
(ii)	if $\alpha$ and $\beta$ are Russellian types, then $⟨\alpha ,\beta ⟩$ is a Russellian type.
	(Rule 1)

Because of their Russellian origin, propositional function-based ontologies like these are sometimes called ‘Russellian ontologies’ (Kaplan, Reference Kaplan1976; see Liefke, Reference Liefke2024a).

Items (34) and (35) contrast a Russellian ontology (b.) with a Frege–Church ontology (a.) for the compositional interpretations of (23b) and (33). To emphasize the ontological categories in this interpretation, I have annotated the compositional semantic values in (34) and (35)–(36) with their (IL or Russellian) types:

The Russellian interpretation in (34b) implements Kaplan’s proposal to replace individual concepts (type $⟨s,e⟩$) with one-place propositional functions (i.e. type-$⟨e,p⟩$; see Section 2.2.2). Kaplan’s account further suggests that intensional nouns like temperature and intensional transitive verbs like rise receive an interpretation in the type $⟨⟨e,p⟩,p⟩$ (i.e. as second-order propositional functions). The Russellian and the Frege-Church ontology of the PTQ-fragment are contrasted in Table 7.Footnote ¹⁶

Table 7 Intensional types in a Frege-Church and a Russellian ontology

Expression	Frege-Church (IL/TY2)	Russellian Type
proper names	$⟨s,⟨⟨s,⟨e,t⟩⟩,t⟩⟩$	e
declarative sentences	$⟨$s, t$⟩$	p
modal/sentence adverbs	$⟨⟨s,t⟩,⟨s,t⟩⟩$	$⟨p,p⟩$
nouns, IVs	$⟨⟨$s, e$⟩$, t$⟩$	$⟨⟨e,p⟩,p⟩$
determiner phrases	$⟨s,⟨⟨s,⟨e,t⟩⟩,t⟩⟩$	$⟨e,p⟩$
determiners/quantifiers	$⟨⟨⟨s,e⟩,t⟩,⟨s,⟨⟨s,⟨e,t⟩⟩,t⟩⟩⟩$	$⟨⟨⟨e,p⟩,p⟩$, $⟨e,p⟩⟩$
extensional transitive Vs	$⟨⟨s,⟨⟨s,⟨e,t⟩⟩,t⟩⟩,⟨s,⟨e,t⟩⟩⟩$	$⟨e,⟨e,p⟩⟩$
clause-taking verbs	$⟨⟨s,t⟩,⟨s,⟨e,t⟩⟩⟩$	$⟨p,⟨e,p⟩⟩$
control verbs, adverbs	$⟨⟨s,⟨e,t⟩⟩,⟨s,⟨e,t⟩⟩⟩$	$⟨⟨e,p⟩,⟨e,p⟩⟩$

Table 7 shows how different semantics for the same natural language fragment can share little to no ontological assumptions: Assume for the purpose of exposition that Russellian propositions (type $p$) are not analyzed as sets of possible worlds (type $⟨s,t⟩$). Then, by Ritchie’s (Reference Ritchie2016) Principle of Carrying Commitments (see Section 1.1), the Frege-Church and the Russellian ontology have nothing in common. This holds at least for the level of the object theory, which contains the semantic values of linguistic expressions.Footnote ¹⁷ In particular, while the simplest [= least-complex type] objects in the Russellian ontology are individuals (type $e$ – due to its adoption of a ‘simplest types first’Footnote ¹⁸ strategy; see Partee, Reference Partee, Rosner and Johnson1992, 115) and Russellian propositions (type $p$), the basic semantic objects in Montague’s IL-ontology are Frege-Church propositions (type $⟨s,t⟩$) and properties (type $⟨s,⟨e,t⟩⟩$). I have printed these objects/types in boldface in Table 7.

The above suggests that different ontologies for the same language or fragment have different motivations. For the ‘competition’ between Russellian and Frege-Church (or TY₂) ontologies, this is indeed the case: while the Russellian ontology contains the simplest objects – individuals (type $e$) are intuitively less complex than functions from possible worlds to sets of individuals (type $⟨s,⟨e,t⟩⟩$) –, only the ontology of TY₂ contains the referents of modal proforms. The ontology of Montague’s IL lies in between these two extremes. Since TY₂ allows for overt quantification and abstraction over possible worlds, the rest of this Element will use TY₂. However, the reader should not identify this methodological move with an indication of the author’s preferred ontology.

6.1 Strategy 1: Exploit Complex Types

6.1.1 Questions

Historically, one of the first extensions of the PTQ-fragment has been to interrogatives (e.g. (37a); see Groenendijk & Stokhof, Reference Groenendijk and Stokhof1984; Hamblin, Reference Hamblin and Partee1976b; Hausser & Zaefferer, Reference Hausser, Zaefferer, Guenthner and Schmidt1978; Karttunen, Reference Karttunen1977). In particular, Hamblin (Reference Hamblin and Partee1976) identifies the semantic values of interrogatives, that is, questions, with the set of all atomic propositions that count as possible answers to the question (see (37b)). Karttunen (Reference Karttunen1977) analyzes questions as the set of the question’s true answers relative to the evaluation index. Assuming the familiar possible worlds-analysis of propositions, both of these accounts treat questions as objects of type $⟨⟨s,t⟩,t⟩$.

(37)	Background: There are only three individuals, that is, Matti, Noah, and Oskar
	a.	Who is sleeping?
	b.	$\{$‘Matti is sleeping’, ‘Noah is sleeping’, ‘Oskar is sleeping’ $\}$

To specify what counts as a ‘possible answer’ to a question, newer semantics for interrogatives (e.g. Ciardelli et al., Reference Ciardelli, Groenendijk and Roelofsen2018; Theiler et al., Reference Theiler, Roelofsen and Aloni2018; Uegaki, Reference Uegaki2019) replace Hamblin’s notion of a possible answer with the notion of a piece of information that resolves the issue raised by the question. These semantics analyze questions as downward-closed sets of propositions. The downward closure of these sets is supported by the observation that if a piece of information $p$ resolves the issue raised by a given question $Q$, any stronger piece of information $q$ (s.t. $q\subset p$) will also resolve the issue raised by $Q$. The effect of replacing sets of propositions (see (37b)) by downward-closed sets is illustrated in (38).

(38)	$\{$‘No one is sleeping’, ‘Someone is sleeping’,
	$\{$‘Matti is sleeping’, ‘Noah is sleeping’, ‘Oskar is sleeping’,
	$\{$‘Matti and Noah are sleeping’, ‘Matti and Oskar are sleeping’,
	$\{$‘Noah and Oskar are sleeping’, ‘Matti, Noah, and Oskar are all sleeping’ $\}$

The interpretation of interrogatives as downward-closed sets of propositions also captures the intuitive entailment relations between more and less specific questions (see (39)):

(39)	a.	Who is sleeping and snoring?
		$⇝\lambda p.p\subseteq \{w:\exists x.x\text{ snores and sleeps in }w\}$
$\Rightarrow$	b.	Who is sleeping?
		$⇝\lambda p.p\subseteq \{w:\exists x.x\text{ sleeps in }w\}$

At a compositional level, the analysis of questions as objects of type $⟨⟨s,t⟩,t⟩$ enables the interpretation of question words (e.g. who) and interrogative-embedding predicates (e.g. wonder) as objects of TY₂ types. These objects are identified with functions from properties to questions (type $⟨⟨s,⟨e,t⟩⟩,⟨⟨s,t⟩,t⟩⟩$) and with functions from questions to properties of individuals (type $⟨⟨⟨s,t⟩,t⟩,$ $⟨s,⟨e,t⟩⟩⟩$), respectively.

6.1.2 Possible Worlds and Situations

In an effort to capture the relation between propositions and truth-at-a-world (see Section 3.4), some hyperintensional theories (e.g. Fox, Lappin, & Pollard, Reference Fox, Lappin, Pollard, Alberti, Balough and Dekker2002; Pollard, Reference Pollard2008) analyze possible worlds as ultrafilters on propositions (type $⟨p,t⟩$, where $p$ is the type for primitive propositions; see Section 3.4, 6.2.2). These theories combine the introduction of a new type of entity (here: primitive propositions) with the use of a new complex type that is constructed from TY₂ types and this new type. They assume that propositions are elements of a Boolean algebra, $B_p$. The latter is an algebraic structure that is ordered by a reflexive transitive relation, $⊑$, that has a top element, $\top$, and a bottom element, $\perp$. Ultrafilters on propositions are then (i) upwards-closed and (ii) meet-closed proper subsets of $B_p$ (iii) that contain, for each proposition $p$, either $p$ or its complement (see Pollard, Reference Pollard2008, 263).

In virtue of (i), it holds that, if a proposition $p$, is in this set and $p⊏q$, then $q$ is also in this set. In virtue of (ii), it holds that, if two propositions, $p$ and $q$, are in this set, their meet, $p\sqcap q$, is also in this set. Proposition (iii) demands that, for each proposition $p$, either $p$ or its complement, $p^{\prime }$, is in this set. The concept of an ultrafilter is graphically illustrated in the Hasse diagram in Figure 9a. There, the ultrafilter is the set of all grey nodes.

Figure 9 Type-$⟨p,t⟩$ representations of possible worlds (a) and questions (b).

Interestingly, the analysis of possible worlds as sets of propositions inverses the relation between propositions and worlds. In Montague’s semantics, a proposition [= set of possible worlds] $p$ is true at a world $w$ iff $w$ is a set-theoretic element of $p$ (see (40a)). Inversely, in Pollard’s semantics, a primitive proposition $p$ is true at a possible world [= set of propositions] $w$ iff $p$ is a set-theoretic element of $w$ (see (40b), where $\chi \left(w\right)$ is the characteristic function of the ultrafilter $w$).

(40)	a.	Montague (Reference Montague and Hintikka1973): ‘ $p$ is true at $w$’ iff $w\in p$ (or $p\left(w\right)=\text{1}$)
	b.	Pollard (Reference Pollard2008): ‘ $p$ is true at $w$’ iff $p\in w$ (or $(\chi \left(w\right))\left(p\right)=\text{1}$)

Note that Pollard’s semantics analyzes possible worlds as objects of the same type as hyperintensional questions (see Figure 9b, based on (39b)). However, a Pollard-style hyperintensional semantics is still able to distinguish worlds from questions. This is due to the fact that the type-$⟨p,t⟩$ representations of worlds and questions have different properties (i.e. being an [upward-closed] ultrafilter resp. being a downward-closed set). It is further due to the possibility of defining subtypes by predicates for their properties (for details, see Pollard, Reference Pollard2008, 276–277). Analogous considerations hold for type-$⟨p,t⟩$ representations of situations (i.e. spatio-temporal world parts; see Kratzer, Reference Kratzer1989), which use prime filters of propositions (Liefke, Reference Liefke, Arai, Kojima and Mineshima2017).

Since prime filters do not require that, for each proposition $p$, $B_p$ contains either $p$ or its complement, they straightforwardly capture the partiality of situations. The possibility of representing situations by prime filters even extends to informationally partial situations (see Liefke & Werning, Reference Liefke and Werning2018). The latter are spatio-temporal world-parts that may be informationally incomplete (e.g. such that they only contain some of their inhabitants’ properties). Informationally partial situations are hence what Perry (Reference Perry and Sture1986) has called ‘aspects’ of a world.

To provide non-referential meanings of names and pronouns that avoid overt reference to possible worlds, Kaplan (Reference Kaplan1976) has proposed to represent individual concepts by (type-$⟨e,⟨s,t⟩⟩$) propositional functions (see my Sections 2.2.2, 5.3). This representation is preserved in a hyperintensional version of Kaplan’s approach that interprets proper names and pronouns as objects of type $⟨e,p⟩$.

Arguably, one could ‘de-hyperintensionalize’ Pollard’s semantics by replacing primitive propositions with sets of possible worlds. The resulting account would analyze indices (classically, possible worlds) as complex entities of type $⟨⟨s,t⟩,t⟩$ (i.e. as generalized quantifiers over primitive worlds). While this account is equivalent (up to coding/typing) to a model of Montague’s IL, the surjective relation between $p$ and $⟨s,t⟩$ (s.t. different primitive propositions may be associated with the same set of possible worlds; see Thomason, Reference Thomason1980) – and the resulting many-to-one relation between ultrafilters on propositions and generalized quantifiers over worlds – suggests that IL models of a given natural language fragment reduce to Pollard’s account. For a detailed discussion of this reduction, the reader is referred to Liefke (Reference Liefke, Arai, Kojima and Mineshima2017).

6.1.3 Manners

To avoid an excessive proliferation of basic types, Umbach, Hinterwimmer, and Gust (Reference Umbach, Hinterwimmer and Gust2022) have proposed to analyze manners as similarity classes of events (i.e. as objects of type $⟨v,t⟩$, where $v$ is the type for events; see Umbach & Gust, Reference Umbach and Gust2014, 325–329). Such objects are sets of events that are indistinguishable with respect to a contextual parameter that includes the relevant dimensions of comparison. In the analysis of the manner $m$ from (41), this parameter is called $F$; sim is a similarity relation that compares two events (i.e. its first and second argument) relative to this parameter (its third arguments).

(41)	$\left(\exists e\right)[\text{manner}(m,e)\leftrightarrow m=\{e^{\prime }:\text{SIM}(e^{\prime },e,F\left)\right\}]$

For the ‘John writing’ example from (18d)/(42a), the dimensions in this parameter include, for example, proficiency level (beginner, advanced), style (handwritten, typed), and neatness (orderly, messy; see Umbach & Gust, Reference Umbach and Gust2014). The manner $m$ in the interpretation of (18d)/(42a), i.e. (42b), is then analyzed as the set of events that resemble John’s writing in at least one of these parameters (presumably in their messiness and attendant illegibility; see Umbach et al., Reference Umbach, Hinterwimmer and Gust2022, 327–328).

(42)		a.	John writes illegibly.
	$⇝$	b.	$\left(\exists e\right)[\text{write}(e,\text{john})\wedge \left(\exists m\right)[\stackrel{\text{is a manner of}}{\overbrace{m=\{e^{\prime }:\text{SIM}(e^{\prime },e,F\left)\right\}}}\wedge \text{illegible}(m\left)\right]]$

The analysis of manners as sets of events helps capture intuitive entailments between embedded manner how-clauses and bare infinitives or gerundive small clauses (see Liefke, Reference Liefke, Umbach and Jedrzejowski2023b):

(43)		a.	Mary saw John write(ing) the letter.
			$[\equiv$ Mary (visually) witnessed the event of John’s writing the letter. $]$
	$\Rightarrow$	b.	Mary saw how $[\equiv$ in which manner $]$ John wrote the letter.

In particular, the entailment in (43) is supported by the observation that Mary cannot witness John’s writing of the letter without also witnessing (or noting) the particular way in which John is writing the letter. Attendantly, every scenario that makes (43a) true will also make (43b) true. I will argue in Section 7 that cross-categorial entailments like (43) provide important support for the unification of types/semantic categories.

For reasons of space, I refrain from providing a detailed discussion of other semantic categories whose objects have been analyzed through a complex TY₂ (or related) type. A selection of these categories is given in Table 8.

Table 8 Complex-type reductions of ‘new’ objects

Category	Type	Analysis	Representation	Source
individual	$⟨s,e⟩$	$⟨e,⟨s,t⟩⟩$	propositional fcts	(Kaplan, 1976)
concepts		$⟨e,p⟩$	hyperintensional	(Liefke, 2017)
			propositional functions
properties		$⟨e,t⟩$	sets of individuals	(Montague, 1973)
		$⟨e,p⟩$	propositional functions	(Chierchia, 1984)
kinds,	$k$	$⟨e,t⟩$	sets of individuals	(Link, 1983)
groups	$g$	$⟨e,t⟩$	sets of individuals	(Bennett, 1975)
worlds	$s$	$⟨p,t⟩$	ultra-filters on	(Pollard, 2008b)
			propositions
situations	$s$	$⟨p,t⟩$	prime filters on proposit’s	(Liefke, 2017)
questions	$q$	$⟨⟨s,t⟩,t⟩$	sets of alternatives	(Hamblin, 1976)
		$⟨p,t⟩$	sets of basic propositions	(Pollard, 2008a)
degrees	$d$	$⟨e,t⟩$	classes of individuals	(Cresswell, 1976)
[modal $\sim$]		$⟨⟨s,t⟩,t⟩$	sets of propositions	(Portner, 2016)
manners	$m$	$⟨v,t⟩$	similarity classes	(Umbach, 2022)
			of events
intervals	$i$	$⟨v,t⟩$	equivalence classes	(Rooij & Schulz,
			of events	2014)