Worlds, Events, and Inertia

Abstract

The semantics of progressive sentences presents a challenge to linguists and philosophers alike. According to a widely accepted view, the truth-conditions of progressive sentences rely essentially on a notion of inertia. Dowty (Word meaning and Montague grammar: the semantics of verbs and times in generative grammar and in Montague’s PTQ, D. Reidel Publishing Company, Dordrecht, 1979) suggested inertia worlds to implement this “inertia idea” in a formal semantic theory of the progressive. The main thesis of the paper is that the notion of inertia went through a subtle, but crucial change when worlds were replaced by events in Landman (Nat Lang Semant 1:1–32, 1992) and Portner (Language 74(4):760–787, 1998), and that this new, event-related concept of inertia results in a possibility-based theory of the progressive. An important case in point in the paper is a proof that, despite its surface structure, the theory presented in Portner (1998) does not implement the notion of inertia in Dowty (1979); rather, it belongs together with Dowty’s earlier, 1977 theory according to which the progressive is a possibility operator.

Appendices

Appendix A: The Language ${\mathcal {L}_{\text {Prog}}}$

1.1 A. 1 Syntax

We assume a type-theoretic language ${\mathcal {L}_{\text {Prog}}}$ with following basic types (corresponding to events, intervals, and truth-values, respectively):

$$\begin{aligned} \mathbf {BaseType} \mathop {=}\limits ^{{\mathrm{def}}}\{{\mathbf {e}},{\mathbf {i}},{\mathbf {t}}\} \end{aligned}$$

(8)

Types are generated from basic types in the usual way as the smallest set satisfying the following conditions:

$$\begin{aligned}&\mathbf {BaseType} \subseteq \mathbf {Type}\\&\sigma ,\tau \in \mathbf {Type} \mathrel {\Rightarrow }\langle \sigma , \tau \rangle \in \mathbf {Type}\\&\sigma \in \mathbf {Type} \mathrel {\Rightarrow }\langle s,\sigma \rangle \in \mathbf {Type} \end{aligned}$$

The language has nonlogical constants $\mathbf {Con}_{\sigma }$ and variables $\mathbf {Var}_{\sigma }$ in every type $\sigma$; these together constitute the set of $\sigma$-type terms $\mathbf {Term}_{\sigma }$:

$$\begin{aligned} \mathbf {Term}_{\sigma } \mathop {=}\limits ^{{\mathrm{def}}}\mathbf {Var}_{\sigma } \cup \mathbf {Con}_{\sigma } \end{aligned}$$

(9)

The set of variables $\mathbf {Var}$, constants $\mathbf {Con}$ and terms $\mathbf {Term}$ are the collections of variables, constants and terms for every possible type:

$$\begin{aligned}&\mathbf {Var} \mathop {=}\limits ^{{\mathrm{def}}}\mathop {\bigcup }\limits _{\sigma \in \mathbf {Type}} \mathbf {Var}_{\sigma } \\&\mathbf {Con} \mathop {=}\limits ^{{\mathrm{def}}}\mathop {\bigcup }\limits _{\sigma \in \mathbf {Type}} \mathbf {Con}_{\sigma } \\&\mathbf {Term} \mathop {=}\limits ^{{\mathrm{def}}}\mathop {\bigcup }\limits _{\sigma \in \mathbf {Type}} \mathbf {Term}_{\sigma } \end{aligned}$$

We define the set of well-formed expressions recursively as follows. Terms are well-formed expressions:

$$\begin{aligned} \mathbf {Term}_{\sigma }\subseteq \mathbf {Exp}_{\sigma }. \end{aligned}$$

(10)

The following clauses describe the ways in which complex expressions are made out of simpler ones in ${\mathcal {L}_{\text {Prog}}}$:

(11a)

(11b)

$$\begin{aligned} \text {If }\alpha \in {\mathbf {Exp}_{{\mathbf {t}}}} ,&\text { then }\lnot \alpha \in {\mathbf {Exp}_{{\mathbf {t}}}} \end{aligned}$$

(11c)

$$\begin{aligned} \text {If }\alpha ,\beta \in {\mathbf {Exp}_{{\mathbf {t}}}} ,&\text { then }(\alpha \wedge \beta )\in {\mathbf {Exp}_{{\mathbf {t}}}} \end{aligned}$$

(11d)

(11e)

$$\begin{aligned} \text {If }\alpha \in {\mathbf {Exp}_{{\mathbf {e}}}} ,&\text { then }{{\mathrm{\tau }}}(\alpha )\in {\mathbf {Exp}_{{\mathbf {i}}}} \end{aligned}$$

(11f)

$$\begin{aligned} \text {If }\alpha ,\beta \in {\mathbf {Exp}_{{\mathbf {i}}}} ,&\text { then }\alpha \sqsubset \beta \in {\mathbf {Exp}_{{\mathbf {t}}}} \end{aligned}$$

(11g)

(11h)

$$\begin{aligned} \text {If }\alpha \in {\mathbf {Exp}_{\sigma }} ,&\text { then }{{\mathrm{{}^{{\!}{\wedge }}}}}\alpha \in {\mathbf {Exp}_{\langle s, \sigma \rangle }} \end{aligned}$$

(11i)

The set of well-formed expressions of the language consists of the union of the well-formed expressions in all types:

$$\begin{aligned} {\mathbf {Exp}_{}} \mathop {=}\limits ^{{\mathrm{def}}}\mathop {\bigcup }\limits _{\sigma \in \mathbf {Type}} {\mathbf {Exp}_{\sigma }}. \end{aligned}$$

(12)

1.2 A. 2 Semantics

In order to interpret the well-formed expressions of ${\mathcal {L}_{\text {Prog}}}$, we assume the following comprehensive frame,

$$\begin{aligned} {\mathcal {F}}\mathop {=}\limits ^{{\mathrm{def}}}\langle {\mathcal {W}},\langle {\mathcal {T}},\le \rangle ,\langle {\mathcal {E}},\trianglelefteq \rangle ,{{\mathrm{{\mathbf {O}}}}},f_{{{\mathrm{\tau }}}}\rangle , \end{aligned}$$

(13)

where the respective components are explained in Sect. 3. Based on the set of moments $\langle {\mathcal {T}},\le \rangle$, the set of intervals are defined as follows:

The domains of the typed expressions of ${\mathcal {L}_{\text {Prog}}}$ are given as follows.

For basic types:

$$\begin{aligned}&\mathbb {D}_{{\mathbf {e}}} \mathop {=}\limits ^{{\mathrm{def}}}{\mathcal {E}}, \\&\mathbb {D}_{{\mathbf {i}}} \mathop {=}\limits ^{{\mathrm{def}}}{\mathcal {I}}, \\&\mathbb {D}_{{\mathbf {t}}} \mathop {=}\limits ^{{\mathrm{def}}}\{{\mathbf {0}},{\mathbf {1}}\}, \end{aligned}$$

and for complex types:

By a model of ${\mathcal {L}_{\text {Prog}}}$ we understand the following sequence of components:

$$\begin{aligned} {\mathcal {M}}= \langle {\mathcal {F}},[ \! [ \cdot ] \! ]^{\mathcal {{\mathcal {M}}}}_{},{\mathcal {H}}, {\mathsf {Inr}},{{\mathrm{{\mathsf {Circ}}}}}, {{\mathrm{{\mathsf {NI}}}}}\rangle , \end{aligned}$$

where ${\mathcal {F}}$ is the frame above; and $[ \! [ \cdot ] \! ]^{\mathcal {{\mathcal {M}}}}_{}:\mathbf {Con}_{\sigma }\rightarrow ({\mathcal {W}}\rightarrow \mathbb {D}_{\sigma })$ is the interpretation function assigning intensions to the non-logical constants in every type $\sigma \in \mathbf {Type}$. The other components are explained in the text [${\mathcal {H}}$ and ${\mathsf {Inr}}$ in Sect. 4 on Dowty (1977) and Sect. 5 on Dowty (1979), and ${{\mathrm{{\mathsf {Circ}}}}}$, ${{\mathrm{{\mathsf {NI}}}}}$ in Sect. 7 on Portner (1998)].

As is usual, we use variable assignments to extend the interpretation function $[ \! [ \cdot ] \! ]^{\mathcal {{\mathcal {M}}}}_{}$ to arbitrary terms, where an assignment function $\theta :\mathbf {Var}_{\sigma }\rightarrow \mathbb {D}_{\sigma }$ assigns values to variables in every type $\sigma$.

The denotation $[ \! [ \cdot ] \! ]^{\mathcal {{\mathcal {M}}}}_{\theta ,w}$ of a well-formed expression of the language is computed relative to a model $\mathcal {{\mathcal {M}}}$, variable assignment $\theta$ and possible world w according to the following clauses.

The denotation of terms (nonlogical constants and variables) are given by clauses (14) and (15):

$$\begin{aligned}&\text { if } c\in {\mathbf {Con}_{}}, \text { then } [ \! [ c ] \! ]^{\mathcal {{\mathcal {M}}}}_{\theta ,w} = [ \! [ c ] \! ]^{\mathcal {{\mathcal {M}}}}_{}(w), \end{aligned}$$

(14)

$$\begin{aligned}&\text { if } v\in {\mathbf {Var}_{}}, \text { then }[ \! [ v ] \! ]^{\mathcal {{\mathcal {M}}}}_{\theta ,w} = \theta (v). \end{aligned}$$

(15)

The denotation of the complex expressions of ${\mathcal {L}_{\text {Prog}}}$ are given by the following clauses (16a)–(16i):

(16a)

(16b)

$$\begin{aligned}&\text {If }\alpha \in {\mathbf {Exp}_{{\mathbf {t}}}}, \text { then } [ \! [ \lnot \alpha ] \! ]^{\mathcal {{\mathcal {M}}}}_{\theta ,w} = {\mathbf {1}}\mathop {\iff }\limits ^{\mathrm{def}}[ \! [ \alpha ] \! ]^{\mathcal {\mathcal {{{\mathcal {M}}}}}}_{\theta ,w}={\mathbf {0}}. \end{aligned}$$

(16c)

(16d)

(16e)

$$\begin{aligned}&\text {If }\alpha \in {\mathbf {Exp}_{{\mathbf {e}}}}, \text { then } [ \! [ {{\mathrm{\tau }}}(\alpha ) ] \! ]^{\mathcal {{\mathcal {M}}}}_{\theta ,w}\mathop {=}\limits ^{{\mathrm{def}}}f_{{{\mathrm{\tau }}}}\left( [ \! [ \alpha ] \! ]^{\mathcal {{\mathcal {M}}}}_{\theta ,w},w\right) . \end{aligned}$$

(16f)

(16g)

(16h)

(16i)

Finally, for every expression $\alpha$ of type ${{\mathbf {t}}}$, the proposition expressed by $\alpha$ is given by the following definition:

(17)

Appendix B: Kratzer’s Theory of Modality

Since Portner (1998) uses Kratzer’s modal framework (Kratzer 1977, 1981, 1991), I summarize its main points very briefly below. The point of this section is not that of making a historically faithful survey of Kratzer’s theory but, rather, the description of the theoretical means and terminology used in Sect. 7.

For expository purposes in this section, I define a simple propositional language ${\mathcal {L}_K}$ containing two modal operators ${{\mathrm{{\Box }}}}$ and ${{\mathrm{{\Diamond }}}}$ as follows. Let $\mathbf {Atom}$ be a denumerable set of atomic formulas $\psi , \psi ', \psi '',\dots$ The set of formulas $\mathbf {Form}$ is the smallest set satisfying the following clauses:

$$\begin{aligned} \text {if } \alpha \in \mathbf {Atom},&\text { then } \alpha \in \mathbf {Form};\\ \text {if } \alpha ,\beta \in \mathbf {Form},&\text { then } (\alpha \wedge \beta ), \lnot \alpha , {{\mathrm{{\Box }}}}(\alpha ), {{\mathrm{{\Diamond }}}}(\alpha )\in \mathbf {Form}. \end{aligned}$$

Having the syntax in place, let us turn to the semantics. The formulas of ${\mathcal {L}_K}$ are interpreted in a model $\mathcal {K} = \langle {\mathcal {W}}, {{\mathrm{{\mathfrak {B}}}}}, {{\mathrm{{\mathfrak {O}}}}},[ \! [ \cdot ] \! ]^{\mathcal {K}}_{}\rangle$, where ${\mathcal {W}}$ is a nonempty set of possible worlds, ${{\mathrm{{\mathfrak {B}}}}}:{\mathcal {W}}\rightarrow \wp (\wp ({\mathcal {W}}))$ is a function called the modal base, ${{\mathrm{{\mathfrak {O}}}}}:{\mathcal {W}}\rightarrow \wp (\wp ({\mathcal {W}}))$ is a function called the ordering source, and $[ \! [ \cdot ] \! ]^{\mathcal {K}}_{}:\textsf {Atom}\rightarrow ({\mathcal {W}}\rightarrow \{{\mathbf {0}},{\mathbf {1}}\})$ is the interpretation function assigning characteristic functions of propositions—i.e., sets of worlds—to the atomic formulas; intuitively, $[ \! [ \alpha ] \! ]^{\mathcal {K}}_{}$ is the characteristic function of the set of worlds at which $\alpha$ is true in model $\mathcal {K}$. The modal base and the ordering source are called conversational backgrounds.

The denotation of a formula not containing ${{\mathrm{{\Box }}}}$ or ${{\mathrm{{\Diamond }}}}$ as main operator is given at a possible world w as follows:

$$\begin{aligned} \text {if } \alpha \in \textsf {Atom}, \text {then }&[ \! [ \alpha ] \! ]^{\mathcal {K}}_{w} = [ \! [ \alpha ] \! ]^{\mathcal {K}}_{}(w);\\ \text {if } \alpha ,\beta \in \textsf {Form}, \text {then }&[ \! [ \lnot \alpha ] \! ]^{\mathcal {K}}_{w} = {\mathbf {1}}\mathop {\iff }\limits ^{\mathrm{def}}[ \! [ \alpha ] \! ]^{\mathcal {K}}_{w} = {\mathbf {0}}, \text { and}\\&[ \! [ (\alpha \wedge \beta ) ] \! ]^{\mathcal {K}}_{w} ={\mathbf {1}}\mathop {\iff }\limits ^{\mathrm{def}}[ \! [ \alpha ] \! ]^{\mathcal {K}}_{w} = {\mathbf {1}}\text { and } [ \! [ \beta ] \! ]^{\mathcal {K}}_{w} = {\mathbf {1}}. \end{aligned}$$

Crucial to the interpretation of the two modal operators ${{\mathrm{{\Box }}}}$ and ${{\mathrm{{\Diamond }}}}$ is the ideality ordering $\preceq _{{{\mathrm{{\mathfrak {O}}}}}(w)}$ over ${\mathcal {W}}$ induced by ${{\mathrm{{\mathfrak {O}}}}}$ in the following way:

Definition 17

(Ideality preorder) Let the ordering source ${{\mathrm{{\mathfrak {O}}}}}(w)$ be assigned to possible world w. For any $w',w''\in {\mathcal {W}}$, $w'$ is said to be at least as good as $w''$ with respect to ${{\mathrm{{\mathfrak {O}}}}}(w)$, in notation: $w'\preceq _{{{\mathrm{{\mathfrak {O}}}}}(w)} w''$, if $w'$ verifies every proposition in ${{\mathrm{{\mathfrak {O}}}}}(w)$ that $w''$ verifies:

Based on $\preceq _{{{\mathrm{{\mathfrak {O}}}}}(w)}$, we can define the set of “best worlds” which contains those worlds in the modal base ${{\mathrm{{\mathfrak {B}}}}}(w)$ that verify maximal subsets of the ordering source ${{\mathrm{{\mathfrak {O}}}}}(w)$:³⁹

Definition 18

(Best worlds)

As Kratzer points out, there is in general no guarantee that the set of the best worlds is not empty, that is, in the general case she does not subscribe to the Limit Assumption:⁴⁰

Assumption 1

(Limit Assumption) For every world $w\in {\mathcal {W}}$,

$$\begin{aligned} {{\mathrm{{\textsc {Best}}}}}\bigl ({{\mathrm{{\mathfrak {B}}}}}(w),{{\mathrm{{\mathfrak {O}}}}}(w)\bigr ) \ne \emptyset . \end{aligned}$$

In applications of Kratzer’s theory, the Limit Assumption is usually assumed to hold.⁴¹ In this case the definition of the modal operators in ${\mathcal {L}_K}$ are as follows.⁴²

Definition 19

(${{\mathrm{{\Box }}}}$ and ${{\mathrm{{\Diamond }}}}$)