Independent alternatives

Abstract

Orthodox semantics for natural language modals give rise to two puzzles for their interactions with disjunction: Ross’s puzzle and the puzzle of free choice permission. It is widely assumed that each puzzle can be explained in terms of the licensing of ‘Diversity’ inferences: from the truth of a possibility or necessity modal with an embedded disjunction, hearers infer that each disjunct is compatible with the relevant set of worlds. I argue that Diversity inferences are too weak to explain the full range of data. Instead, I argue, modals with embedded disjunctions license ‘Independence’ inferences: from the truth of a modal with an embedded disjunction, hearers infer that each disjunct is an independent alternative among the relevant set of worlds. I then develop a bilateral inquisitive semantics for modals that predicts the validity of these Independence inferences. My account vindicates common intuitions about both Ross’s puzzle and the puzzle of free choice permission, and explains the full range of data.

Appendix: Bilateral minimal covering semantics

1.1 Language

Given a countable set of atomic sentence letters, $\textsf {At}$, and $p \in \textsf {At}$, wffs are defined by the following grammar:

$$\begin{aligned} p \ |\ \lnot \phi \ |\ \phi \wedge \psi \ |\ \phi \vee \psi \ |\ !\phi \ |\ \phi \rightarrow \psi \ |\ \Diamond \phi \ |\ \Box \phi \end{aligned}$$

1.2 Models

A Model $\mathcal {M}$ is a triple $\mathcal {M}=( W_\mathcal {M}, R_\mathcal {M}, V_\mathcal {M})$ where $W_\mathcal {M}$ is a set of worlds, $R_\mathcal {M}$ is a function from worlds to sets of worlds ($R_\mathcal {M}:W \mapsto \wp (W)$), and $V_\mathcal {M}$ is a function from atomic sentences to truth sets ($V_\mathcal {M}: \textsf {At}\mapsto \wp (W)$).

1.3 Bilateral propositions

A bilateral proposition P in a model $\mathcal {M}$ is a pair $(P^+, P^-)$ of downward-closed (relative to the subset relation) sets of sets of worlds, such that their intersection is the singleton containing the empty set. In other words, where $P^\circ$ $\in \{P^+, P^-\}$, and $s, t \subseteq W_\mathcal {M}$:

$$\begin{aligned} P^\circ&\subseteq \wp (W_\mathcal {M})\\ \text {if } s \in P^\circ \text { and } t \subseteq s&\Rightarrow t \in P^\circ \\ P^+ \cap P^-&= \{\emptyset \} \end{aligned}$$

Let $\mathcal {P}_\mathcal {M}$ be the set of all bilateral propositions in $\mathcal {M}$.

1.4 Information and alternatives

For $P \in \mathcal {P}_\mathcal {M}$:

$$\begin{aligned} \textsf {info}(P^+)&= \bigcup P^+ \quad \text {(the set of worlds where } P \text { is true)}\\ \textsf {info}(P^-)&= \bigcup P^-\quad \text {(the set of worlds where } P \text { is false)}\\ \textsf {alt}(P^+)&= \{ s \in P^+\ |\ \lnot \exists t \in P^+:\ t \supset s \}\quad \text { (the positive alternatives offered by } P)\\ \textsf {alt}(P^-)&= \{ s \in P^-\ |\ \lnot \exists t \in P^-:\ t \supset s \}\quad \text { (the negative alternatives offered by } P) \end{aligned}$$

1.5 Minimal cover

$$\begin{aligned} C\text { is a cover of } S \text { iff }&\ S \subseteq \bigcup C \\ C\text { is a minimal cover of } S \text { iff }&\ C\text { is a cover of } S \text { and } \\&\ \text {there is no } C' \subset C: C'\text { is a cover of } S \end{aligned}$$

1.6 Model update

The $\phi$-update of an accessibility function $R_\mathcal {M}$, written , is defined as follows:

$$\begin{aligned} R_\mathcal {M}\upharpoonright \phi = \lambda w \in W_\mathcal {M}.\ \textsf {info}^+([\phi ]_\mathcal {M}) \cap R_\mathcal {M}(w) \end{aligned}$$

So is the function that takes w to the subset of $R_\mathcal {M}(w)$ where $\phi$ is true in $_\mathcal {M}$. We use this notion to define a model update. The $\phi$-update of a model $\mathcal {M}$, written () is defined:

$$\begin{aligned} \mathcal {M}\upharpoonright \phi = ( W_\mathcal {M}, R_\mathcal {M}\upharpoonright \phi , V_\mathcal {M} ) \end{aligned}$$

1.7 Semantics

The proposition denoted by a wff $\phi$ in a model $\mathcal {M}$ is denoted $[\phi ]_\mathcal {M} = ([\phi ]_\mathcal {M}^+, [\phi ]_\mathcal {M}^-)$ (I drop the model subscript for readability except when important). For atomic $p \in \textsf {At}$:

$$\begin{aligned} {[}p] = (\wp (V(p)), \wp (W\setminus V(p)) ) \end{aligned}$$

Non-modal complex $\phi$:

$$\begin{aligned}{}[\lnot \phi ]&= ([\phi ]^-, [\phi ]^+)\\ [\phi \wedge \psi ]&= ( [\phi ]^+ \cap [\psi ]^+, [\phi ]^- \cup [\psi ]^- )\\ [\phi \vee \psi ]&= ( [\phi ]^+ \cup [\psi ]^+, [\phi ]^- \cap [\psi ]^- )\\ [!\phi ]&= ( \wp (\textsf {info}([\phi ]]^+)), \wp (\textsf {info}([\psi ]^-)))\\ [ \phi \rightarrow \psi ]_\mathcal {M}&= ( [\psi ]^+_{\mathcal {M} \upharpoonright \phi }, [\psi ]^-_{\mathcal {M} \upharpoonright \phi } ) \end{aligned}$$

For modal $\phi$:

$$\begin{aligned} {[}\Box \phi ]^+&= \wp (\{ w \in W \ |\ \textsf {alt}([\phi ]^+)\text { is a minimal cover of } R(w) \})\\ [\Box \phi ]^-&= \wp (\{ w \in W \ |\ \text { there is a non-empty } R' \subseteq R(w)\text { such that }\\&\quad \quad \textsf {alt}([\phi ]^-)\text { is a minimal cover of } R' \}\\ [\Diamond \phi ]^+&= \wp (\{ w \in W \ |\ \text { there is a non-empty } R' \subseteq R(w)\text { such that }\\&\quad \quad \textsf {alt}([\phi ]^+)\text { is a minimal cover of } R' \})\\ [\Diamond \phi ]^-&= \wp (\{ w \in W \ | \textsf {alt}([\phi ]^-)\text { is a minimal cover of } R(w)\}) \end{aligned}$$

1.8 Truth and falsity

A sentence $\phi$ is true at a point M, w iff $w \in \textsf {info}([\phi ]_\mathcal {M}^+)$.

A sentence $\phi$ is false at a point M, w iff $w \in \textsf {info}([\phi ]_\mathcal {M}^-)$.

1.9 Entailment

A sentence $\phi$ entails a sentence $\psi$ in a model $\mathcal {M}$ (written $\phi \vDash _\mathcal {M} \psi$), when the following condition holds:

$$\begin{aligned} \phi \vDash _\mathcal {M} \psi \text { iff } \textsf {info}^+([\phi ]_\mathcal {M}) \subseteq \textsf {info}^+([\psi ]_\mathcal {M}) \end{aligned}$$

1.10 Summary of results

As mentioned above, since I am using this simple language to model natural language disjunction, I am interested in how modals interact with disjunctions that obey Hurford’s constraint. For this reason, I focus on the set of all models such that for $p, q \in \textsf {At}$ and neither entails the other; we will call these admissible.

Definition 1

(Admissible model) A model $\mathcal {M}$ is admissible iff for $p, q \in \textsf {At}$:

$$\begin{aligned} V_\mathcal {M}(p)&\not \subseteq V_\mathcal {M}(q)\\ V_\mathcal {M}(q)&\not \subseteq V_\mathcal {M}(p) \end{aligned}$$

Let $\mathfrak {M}$ be the set of all admissible models, and we will say that $\phi \vDash _\mathfrak {M} \psi$ iff for every $\mathcal {M} \in \mathfrak {M}$, $\phi \vDash _\mathcal {M} \psi$.

Fact 1

(Independence inferences)

$$\begin{aligned} \Box (p \vee q)&\vDash _\mathfrak {M} \Diamond (p\wedge \lnot q)\\ \Box (p \vee q)&\vDash _\mathfrak {M} \Diamond (q\wedge \lnot p)\\ \Diamond (p \vee q)&\vDash _\mathfrak {M} \Diamond (p\wedge \lnot q)\\ \Diamond (p \vee q)&\vDash _\mathfrak {M} \Diamond (q\wedge \lnot p) \end{aligned}$$

Fact 2

(Independence conditionals)

$$\begin{aligned} \Box (p \vee q)&\vDash _\mathfrak {M} \lnot p\rightarrow \Box q\\ \Box (p \vee q)&\vDash _\mathfrak {M} \lnot q\rightarrow \Box p\\ \Diamond (p \vee q)&\vDash _\mathfrak {M} \lnot p\rightarrow \Diamond q\\ \Diamond (p \vee q)&\vDash _\mathfrak {M} \lnot q\rightarrow \Diamond p \end{aligned}$$

Fact 3

(Ross inference)

$$\begin{aligned} \Box p \nvDash _\mathfrak {M} \Box (p \vee q) \end{aligned}$$

Fact 4  (Free Choice)

$$\begin{aligned} \Diamond ( p\vee q) \vDash _\mathfrak {M} \Diamond p \end{aligned}$$

Fact 5

(Modal duality) For any $\phi$ in any model $\mathcal {M}$:

$$\begin{aligned} {[}\Box \phi ]_\mathcal {M}&= [\lnot \Diamond \lnot \phi ]_\mathcal {M}\\ {[}\Diamond \phi ]_\mathcal {M}&= [\lnot \Box \lnot \phi ]_\mathcal {M} \end{aligned}$$

Fact 6

(Impossibility distribution over $\vee$)

$$\begin{aligned} \lnot \Diamond (p \vee q)&\vDash _\mathfrak {M} \lnot \Diamond p\\ \lnot \Diamond (p \vee q)&\vDash _\mathfrak {M} \lnot \Diamond q \end{aligned}$$

Fact 7

(Unnecessity distribution over$\vee$)

$$\begin{aligned} \lnot \Box (p \vee q)&\vDash _\mathfrak {M} \lnot \Box p\\ \lnot \Box (p \vee q)&\vDash _\mathfrak {M} \lnot \Box q \end{aligned}$$