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Agential Free Choice

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Abstract

The Free Choice effect—whereby \(\lozenge (p {\textsc {or}} q)\) seems to entail both \(\lozenge p\) and \(\lozenge q\)—has traditionally been characterized as a phenomenon affecting the deontic modal ‘may’. This paper presents an extension of the semantic account of free choice defended by Fusco (Philosophers’ Imprint, 15, 1–27, 2015) to the agentive modal ‘can’, the ‘can’ which, intuitively, describes an agent’s powers. On this account, free choice is a nonspecific de re phenomenon (Bäuerle 1983; Fodor 1970) that—unlike typical cases—affects disjunction. I begin by sketching a model of inexact ability, which grounds a modal approach to agency (Belnap Theoria, 54, 175–199, 1998; Perloff 2001) in a Williamson (Mind, 101, 217–242, 1992; Erkenntnis, 79, 971–999, 2014)-style margin of error. A classical propositional semantics combined with this framework can reflect the intuitions highlighted by Kenny (1976)’s dartboard cases, as well as the counterexamples to simple conditional views recently discussed by Mandelkern et al. (Philosophical Review, 126, 301–343, 2017). In Section 3, I turn to an independently motivated actual-world-sensitive account of disjunction, and show how it extends free choice inferences into an object language for propositional modal logic.

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Notes

  1. C.f. [81, pg. 985].

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Correspondence to Melissa Fusco.

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This paper benefitted from much commentary and feedback during its gestation. Warm thanks to Arc Kocurek, Karen Lewis, Una Stonić, Jeff Horty, Chris Barker, Jarek Macnar, Michael Neilsen, Ignacio Ojea Quintana, Tomasz Placek, Dan Harris, Nate Charlow, Simon Charlow, Yimei Xiang, Shawn Standefer, David Boylan, Ginger Schultheis, Matt Mandelkern, Carlotta Pavese, Malte Willer, Maria Aloni, Floris Roelofsen, Eric Swanson, and audiences at the Australia National University, NYU, Amsterdam, the 2019 Iceland Meaning Workshop, and the 2019 PhLiP Conference.

Appendix

Appendix

In this appendix, I work with a univocal lexical entry for ‘or’, and hence omit the subscript ‘w’. To simplify proof by induction, I will assume that the parse of a disjunction \(\ulcorner a_{1} \textsc {or} a_{2} \textsc {or} a_{3} \textsc {or} {\dots } \textsc {or} a_{n}\urcorner \) has the LF \((({\dots } (a_{1} \textsc {or} a_{2})\) \( \textsc {or} a_{3}) \textsc {or} {\dots } \textsc {or} a_{n})\). Hence any n-ary disjunction is at most disjunctive in its left argument.

Syntax

Let At be a set of propositional letters a1,a2.... We define three languages, \({\mathscr{L}}_{bl}\) (the Boolean fragment of \({\mathscr{L}}\)), \({\mathscr{L}}_{nonm}\) (the nonmodal fragment of \({\mathscr{L}}\)), and \({\mathscr{L}}\).

$$ \begin{array}{@{}rcl@{}} \mathcal{L}_{bl} \phi &::=& a_{i} ~|~ \neg \phi ~|~ (\phi \wedge \phi)\\ \mathcal{L}_{nonm} \phi &::=& a_{i} ~|~ \neg \phi ~|~ (\phi \wedge \phi) ~|~ (\phi \textsc{or} \phi)\\ \mathcal{L} \phi &::=& a_{i} ~|~ \neg \phi ~|~ (\phi \wedge \phi) ~|~ (\phi \textsc{or} \phi) ~|~ \blacklozenge \phi ~|~ \boxdot \phi ~|~ C \phi \end{array} $$

Semantics

A model M is a triple \(\langle W, R, \mathcal {I} \rangle \) where W a nonempty set of possible worlds, R is a reflexive binary relation on W, and \(\mathcal {I}\) is a function from the elements of At to \(\mathcal {P}(W)\) (“the interpretation function”).

We define the standard intension of ϕ, V (ϕ), on \({\mathscr{L}}_{nonm}\) as follows:

$$ \begin{array}{@{}rcl@{}} V(a) &=& \mathcal{I}(a) \\ V(\neg \phi) &=& W \setminus V(\phi)\\ V(\phi \wedge \psi) &=& V(\phi) \cap V(\psi) \\ V(\phi \textsc{or} \psi) &=& V(\phi) \cup V(\psi) \end{array} $$

A point of evaluation in M is a triple 〈h, y, x〉 such that h is a serial, reflexive subset of W (∀wh, wRw), and a pair of worlds y, xh.

Truth at a Point of Evaluation

For any model M and point of evaluation 〈h, y, x〉 in M, propositional letter a, wffs ϕ, ψ:

$$ \begin{array}{@{}rcl@{}} h, y, x \vDash a \quad & \text{ iff } & \quad x \in V(a) \\ h, y, x \vDash \neg \phi \quad & \text{ iff } & \quad \text{ there is no } y^{\prime} \text{ such that } h, y^{\prime}, x \nvDash \phi \\ h, y, x \vDash (\phi \wedge \psi) \quad & \text{ iff } & \quad h, y, x \vDash \phi \text{ and } h, y, x \vDash \psi \\ h, y, x \vDash \blacklozenge \phi \quad & \text{ iff } & \quad \exists w \in h: h, y, w \vDash \phi \\ h, y, x \vDash \boxdot \phi \quad & \text{ iff } & \quad \forall x^{\prime} \in h \textup{: if } xRx^{\prime}, \text{ then } h, y, x^{\prime} \vDash \phi \\ h, y, x \vDash C \phi\quad & \text{ iff } & \quad \exists w \in h \textup{ s.t.: } \forall v \textup{s.t. } wRv: h, y, v \vDash \phi. \\ \end{array} $$

...these entries are the entries of the toy language in Table 2 (§2), with a free y parameter added.

Given a pair of sentences ϕ1 and ϕ2 in \({\mathscr{L}}\), the w-relative answer set of ϕ1 and ϕ2 is

$$ Ans_{w}(\phi_{1}, \phi_{2}) = \begin{cases} \{\phi_{1}\} & \text{if } h, w, w \vDash \phi_{1} \text{ and } h, w, w \nvDash \phi_{2} \\ \{\phi_{2}\} & \text{if } h, w, w \vDash \phi_{2} \text{ and } h, w, w \nvDash \phi_{1} \\ \{\phi_{1}, \phi_{2}\} & \text{otherwise.} \end{cases} $$

Now disjunction can be added:

$$ h, y, x \vDash (\phi \textsc{or} \psi) \quad \text{ iff } \quad \exists \beta: \beta \in Ans_{y} (\phi, \psi) \text{ and } h, y, x \vDash \beta $$

Two interdefinitions of C hold, given the reflexivity of R: (i) \(C \phi := \blacklozenge \boxdot \phi \) [39, pg. 606]; (ii) \(C \phi := \blacklozenge (\phi \wedge \boxdot \phi )\).

1.1 Consequence

There are four notions of consequence available in our system, corresponding to some choice of local or global, and diagonal or two-dimensional.

 

global

local

diagonal

\(\vDash _{1}\)

\(\vDash _{2}\)

two-dimensional

\(\vDash _{3}\)

\(\vDash _{4}\)

We are interested primarily in the preservation of diagonal acceptance, which corresponds to \(\vDash _{1}\): ϕ is accepted at h iff ∀wh: \(h, w, w \vDash \phi \). For short, we use hϕ := ∀wh: (\(h, w, w \vDash \phi \)).

Lemma 1 (Nondisjunctive Stability)

For any \(\phi \in {\mathscr{L}}_{bl}\), any \(h \subseteq W\), and \(x, y, y^{\prime } \in h\): \(h, y,x \vDash \phi \) iff \(h, y^{\prime }, x \vDash \phi \).

Proof

A trivial induction on the complexity of \(\phi \in {\mathscr{L}}_{bl}\). □

Theorem 1 (Diagonal Classicality)

For any \(h \subseteq W, w \in h,\) and \(\phi \in {\mathscr{L}}_{nonm}\): \(h, w, w \vDash \phi \) iff wV (ϕ).

Proof

By induction. The atomic, negation, and conjunction cases are trivial.

  • Disjunction. We need to show: \(h, w, w \vDash (\phi \textsc {or} \psi )\) iff w ∈ (V (ϕ) ∪ V (ψ)). Assume for the Inductive Hypothesis that (i) \(h, w, w \vDash \phi \) iff wV (ϕ), and (ii) \(h, w, w \vDash \psi \) iff wV (ψ).

    (⇒) If \(h, w, w \vDash (\phi \textsc {or} \psi )\), then w ∈ (V (ϕ) ∪ V (ψ)).

    If \(h, w, w \vDash (\phi \textsc {or} \psi )\), then ∃β: βAnsw(ϕ, ψ) and \(h, w, w \vDash \beta \). For any such h, w, and β: β ∈{ϕ, ψ}. Hence if \(h, w, w \vDash \beta \), then \(h, w, w \vDash \phi \) or \(h, w, w \vDash \psi \). Hence (by Inductive Hypothesis) wV (ϕ) or wV (ψ). Hence w ∈ (V (ϕ) ∪ V (ψ)).

    (⇐) If w ∈ (V (ϕ) ∪ V (ψ)), then \(h, w, w \vDash (\phi \textsc {or} \psi )\).

    If w ∈ (V (ϕ) ∪ V (ψ)), then wV (ϕ) or wV (ψ).

    Case 1. wV (ϕ). Then by IH, \(h, w, w \vDash \phi \). By the definition of the Alt function, it follows that ϕAnsw(ϕ, ψ). Hence ∃β(= ϕ) ∈ Answ(ϕ, ψ) such that \(h, w, w \vDash \beta \). Hence \(h, w, w \vDash (\phi \textsc {or} \psi )\). Case 2 is similar, but with ψ/ϕ.

Lemma 2 (Classical Theoremhood)

For any \(\phi \in {\mathscr{L}}_{nonm}, \vDash _{1} \phi \) iff ϕ is a theorem of classical propositional logic.

1.2 Application: The Dartboard [49]

We identify worlds with ordered pairs 〈τ(n),m〉 consisting of a position tried for (n), and a position hit (m).Footnote 1τ(n),m〉 is globally possible—possible with respect to the modal base—if |nm|≤Δ, where Δ is the agent’s margin of error. For the local accessibility relation R on worlds, we assume \(\langle \tau (n), m \rangle R \langle \tau (n^{\prime }), m^{\prime } \rangle \) iff

  • n = n (the agent is omniscient w.r.t. her tryings); and

  • τ(n),m〉 and \(\langle \tau (n^{\prime }), m^{\prime } \rangle \) are both globally possible.

We can show that:

Fact 1 (FC)

h ⊳ (E orO) but \(h \ntriangleright C(E \textsc {or} O)\).

Suppose the agent will try to hit either 2 or 3 in the figure below, and that Δ = 1, and so the dart will fall in the range [1,4]. Our modal base is \(\{w_{1} {\dots } w_{6}\}\), where w1 = 〈τ(2), 1〉, w2 = 〈τ(2), 2〉, w4 = 〈τ(3), 2〉, and so on. \(\mathcal {I}(E)\) = {w2,w4,w6}. \(\mathcal {I}(O) = h \setminus \mathcal {I}(E)\).

figure a

Proof

By Classicality, h ⊳ (E orO) iff h ⊳ (EO). That this latter claim is true is clear by inspection of the model.

Now, we evaluate the claim that hC(E orO). Using the second paraphrase (\(C(\phi ) := \blacklozenge (\phi \wedge \boxdot \phi )\)), hC(E orO) iff ∀wh: ∃vh s.t.: (i) \(h, w, v \vDash (E \textsc {or} O)\) and (ii) \(h, w, v \vDash \boxdot (E \textsc {or} O)\). We instantiate w with w2. Hence:

$$ \begin{array}{@{}rcl@{}} \exists v \in h \text{ s.t.: } (i)~ h, w_{2}, v \vDash (E \textsc{or} O) \text{ and } (ii)~ h, w_{2}, v \vDash \boxdot (E \textsc{or} O) \text{ iff} \\ \exists v \in h \text{ s.t.: } (i)~ \exists \beta \in Ans_{w_{2}}(E, O): h, w_{2}, v \vDash \beta \text{ and } (ii)~ h, w_{2}, v \vDash \boxdot (E \textsc{or} O) \text{ iff} \\ \exists v \in h \text{ s.t.: } (i)~ \exists \beta \in \{E\}: h, w_{2}, v \vDash \beta \text{ and } (ii)~ h, w_{2}, v \vDash \boxdot (E \textsc{or} O) \text{ iff} \\ \exists v \in h \text{ s.t.: } (i)~ h, w_{2}, v \vDash E \text{ and } (ii)~ h, w_{2}, v \vDash \boxdot (E \textsc{or} O) \text{ iff} \\ \exists v \in h \text{ s.t.: } (i)~ h, w_{2}, v \vDash E \text{ and } (ii)~ \forall w^{\prime} \text{ s.t. } vRw^{\prime}: h, w_{2}, w^{\prime} \vDash E \text{ iff} \\ \exists v \in h \text{ s.t.: } (i) ((v = w_{2}) \vee (v = w_{4}) \vee (v = w_{6})) \text{ and } (ii)~ \forall w^{\prime} \text{ s.t. } vRw^{\prime}: w^{\prime} \in \mathcal{I}(E) \\ \text{...but there is no such \textit{v}: each } v \in E \text{ is s.t. } \exists w^{\prime}: vRw^{\prime} \text{ and } w^{\prime} \notin \mathcal{I}(E). \end{array} $$

Hence \(h \ntriangleright C(E \textsc {or} O)\). □

For the next Fact, it is in the interest of generality not to presume a dartboard model. (A dartboard-specific version of the proof, in terms of margins of error, appears below (Fact 3).)

Fact 2 (FC+ for historically possible and mutually exclusive disjuncts)

C(p orq), \(\blacklozenge (p \wedge \neg q)\), \(\blacklozenge (q \wedge \neg p) \vDash C(p) \wedge C(q)\).

Proof

hC(p orq) iff ∀wh: ∃vh s.t.: (i) \(h, w, v \vDash (p \textsc {or} q)\) and (ii) \(h, w, v \vDash \boxdot (p \textsc {or} q)\). By the premise \(\blacklozenge (p \wedge \neg q)\), \(\exists w^{\prime } \in h\) such that \(h, w^{\prime }, w^{\prime } \vDash (p \wedge \neg q)\) (call this world “wp”). By the premise \(\blacklozenge (q \wedge \neg p)\), \(\exists w^{\prime \prime } \in h\) such that \(h, w^{\prime \prime }, w^{\prime \prime } \vDash (q \wedge \neg p)\) (call this world “wq”).

Case 1. First, we instantiate w with wp. As above, it follows that ∃vh (call it wp) s.t. (i) \(h, w_{p}, w_{p*} \vDash (p \textsc {or} q)\) and (ii) \(h, w_{p}, w_{p*} \vDash \boxdot (p \textsc {or} q)\).

For the first conjunct: \(h, w_{p}, w_{p*} \vDash (p \textsc {or} q)\) iff \(\exists \beta \in Ans_{w_{p}}(p, q)\) s.t. \(h, w_{p}, w_{p*} \vDash \beta \). Because \(Ans_{w_{p}}(p, q)\) is the singleton {p}, it follows that \(h, w_{p}, w_{p*} \vDash p\).

For the second conjunct: \(h, w_{p}, w_{p*} \vDash \boxdot (p \textsc {or} q)\) iff ∀v s.t. wpRv, \(\exists \beta \in Ans_{w_{p}}(p, q)\) s.t. \(h, w_{p}, v \vDash \beta \). Again, because \(Ans_{w_{p}}(p, q)\) is the singleton {p}, it follows that ∀v s.t. wpRv: \(h, w_{p}, v \vDash p\). Hence \(h, w_{p}, w_{p*} \vDash \boxdot p\).

Hence for any wh: ∃v (viz., wp) s.t. (i) \(h, w, v \vDash p\) and (ii) \(h, w, v \vDash \boxdot p\). It follows that \(h \triangleright \blacklozenge (p \wedge \boxdot p)\), and hence that hC(p). ✓

Case 2. Second, we instantiate w with wq. A symmetric argument to the argument in Case 1 with q/p will show that hC(q). ✓□

Fact 3 (I)

\(C(\phi ) \nvDash C(\phi \textsc {or} \psi )\).

For this example, we consider ϕ = K (composite) and ψ = P (prime) as in the main text, restricting for convenience to n between 46 and 54 (Fig. 3).

If the agent’s margin of error is 2 or less, she can reliably guarantee K in this range by aiming for e.g. 50. However, she cannot reliably guarantee P. But by a similar proof to the proof of Fact 3 above, \(C(K \textsc {or} P) \vDash C(K) \wedge C(P)\). Since \(h \ntriangleright C(P)\), \(h \ntriangleright C(K \textsc {or} P)\).

1.3 General (n-ary) Disjunction

Suppose \(\phi , \psi \in {\mathscr{L}}_{nonm}\). Then for any disjunctive wff (ϕ orψ), \(\psi \in {\mathscr{L}}_{bl}\). We want to show, where ⊗ is exclusive-or:

Fact 4 (Procedure for disjunction)

\(h, y, x \vDash (\phi \textsc {or} \psi )\) iff either (i) \(h, y, y \vDash (\phi \otimes \psi )\) and \(h, y, x \vDash \chi \), where χ is the α ∈{ϕ, ψ} s.t. \(h, y, y \vDash \alpha \), or (ii) \(h, y, y \nvDash (\phi \otimes \psi )\) and \(h, y, x \vDash (\phi \vee \psi )\).

Proof

This follows from inspection of the clause for “or”. (i) covers the first two cases of the Ansy(ϕ, ψ) function, while (ii) covers the third case. □

For the next theorem, we use the following

Notation. For \(\phi \in {\mathscr{L}}_{bl}\) and x, yW: \(x \sim _{\phi }y\) iff \(h, y, y \vDash \phi \) and \(h, x, x \vDash \phi \). NB that by Nondisjunctive Stability, above, this is equivalent to: \(x \sim _{\phi }y\) iff \(h, y, y \vDash \phi \) and \(h, y, x \vDash \phi \).

Theorem 2 (Characterization of 2D Disjunction.)

If, for \(\phi _{1}{\dots } \phi _{n} \in {\mathscr{L}}_{bl}\): \(h, y, x \vDash (\phi _{1} \textsc {or} {\dots } \textsc {or} \phi _{n})\) and \(\exists ! \phi _{i} \in \{\phi _{1} {\dots } \phi _{n}\}\) s.t. \(h, y, y \vDash \phi _{i}\), then \(y \sim _{\phi _{i}} x\).

Proof by induction on the length of n.

Proof

Atomic case (viz., two disjuncts in \({\mathscr{L}}_{bl}\).)

We show that for \(\phi , \psi \in {\mathscr{L}}_{bl}\), if \(h, y, x \vDash \phi \textsc {or} \psi \) and ∃!α ∈{ϕ, ψ} s.t. \(h, y, y \vDash \alpha \), then \(y \sim _{\alpha } x\).

Assume \(h, y, x \vDash (\phi \textsc {or} \psi )\) for \(\phi , \psi \in {\mathscr{L}}_{bl}\) and ∃!ϕi ∈{ϕ, ψ} s.t. \(h, y, y \vDash \phi _{i}\). Then either (i) \(h, y, y \vDash \phi \) and \(h, y, y \nvDash \psi \), or (ii) \(h, y, y \vDash \psi \) and \(h, y, y \nvDash \phi \). We show that in either case, \(y \sim _{\phi _{i}} x\).

Case (i). In this case, Ansy(ϕ, ψ) = {ϕ}. Hence \(h, y, x \vDash (\phi \textsc {or} \psi )\) iff \(h, y, x \vDash \phi \); hence \(h, y, x \vDash \phi \). Hence ϕi = ϕ. Since \(h, y, y \vDash \phi \) and \(h, y, x \vDash \phi \), it follows that \(y \sim _{\phi _{i}} x\).

Case (ii) is symmetric, with ψ instead of ϕ. In this case, ϕi = ψ and \(y \sim _{\phi _{i}} x\).

Inductive Step (number of disjuncts > 2.)

Assume that if, for \(\phi _{1} {\dots } \phi _{(n-1)} \in {\mathscr{L}}_{bl}\), \(h, y, x \vDash (\phi _{1} \textsc {or} {\dots } \textsc {or} \phi _{(n-1)})\) and \(\exists ! \phi _{i} \in \{\phi _{1} {\dots } \phi _{(n-1)}\}\) s.t. \(h, y, y \vDash \phi _{i}\), then \(y \sim _{\phi _{i}} x\) (viz., that \(h, y, x \vDash \phi _{i}\)).

Show: if, for \(\phi _{1} {\dots } \phi _{n} \in {\mathscr{L}}_{bl}\): \(h, y, x \vDash ((\phi _{1} \textsc {or} {\dots } \textsc {or} \phi _{(n-1)}) \textsc {or} \phi _{n}\)), and \(\exists ! \phi _{i} \in \{\phi _{1} {\dots } \phi _{n}\}\) s.t. \(h, y, y \vDash \phi _{i}\), then \(y \sim _{\phi _{i}} x\). □

Proof

If, by “or”, \(h, y, x \vDash ((\phi _{1} \textsc {or} {\dots } \textsc {or} \phi _{(n-1)}) \textsc {or} \phi _{n})\) and \(\exists ! \phi _{i} \in \{\phi _{1} {\dots } \phi _{n}\}\) s.t. \(h, y, y \vDash \phi _{i}\), then either

  1. (i)

    \(\exists ! \phi _{i} \in \{\phi _{1} {\dots } \phi _{(n-1)}\}\) s.t. \(h, y, y \vDash \phi _{i}\) and \(h, y, y \nvDash \phi _{n}\); or

  2. (ii)

    ∃!ϕi ∈{ϕn} s.t. \(h, y, y \vDash \phi _{i}\) and \(h, y, y \nvDash (\phi _{1} \textsc {or} {\dots } \textsc {or} \phi _{(n-1)})\).

Case (i). Then by IH, \(y \sim _{\phi _{i}} x\) for i < n and hence \(\exists ! \phi _{i} \in \{\phi _{1} {\dots } \phi _{n}\}\) s.t. \(y \sim _{\phi _{i}} x\).

Case (ii). Then \(Ans_{y}(\phi \textsc {or} {\dots } \textsc {or} \phi _{(n-1)}, \phi _{n}) = \{\phi _{n}\}\). Hence \(h, y, x \vDash ((\phi _{1} \textsc {or} {\dots } \textsc {or} \phi _{(n-1)}) \textsc {or} \phi _{n})\) iff \(h, y, x \vDash \phi _{n}\). Hence \(x \sim _{\phi _{n}} y\); hence there is some unique ϕi ∈{ϕi} s.t. \(x \sim _{\phi _{i}} y\). □

Theorem 3 (or-elim+)

Let \(\boxplus \) be a normal modal operator. We show that, for \(\phi _{1}{\dots } \phi _{n} \in {\mathscr{L}}_{bl}\): if (Premise 1) \(h, y, y \vDash \boxplus (\phi _{1} \textsc {or} {\dots } \textsc {or} \phi _{n})\) and (Premise 2) \(h, y, y \vDash ((\bigwedge _{j \neq i} \neg \phi _{j}) \wedge (\phi _{i}))\), then (C) \(h, y, y \vDash \boxplus \phi _{i}\).

Proof

\(h, y, y \vDash \boxplus (\phi _{1} \textsc {or} {\dots } \textsc {or} \phi _{n})\) iff ∀x s.t. yRx, \(h, y, x \vDash (\phi _{1} \textsc {or} {\dots } \textsc {or} \phi _{n})\). By Theorem 2 and (Premise 2), \(h, y, x \vDash (\phi _{1} \textsc {or} {\dots } \textsc {or} \phi _{n})\) entails \(h, y, x \vDash \phi _{i}\). Hence by (Premise 1), ∀x s.t. xRy: \(h, y, x \vDash \phi _{i}\). Hence \(h, y, y \vDash \boxplus \phi _{i}\). □

1.4 (DPr) Under Quantificational Negation

Fact 5 ((DPr) for (FC+))

Here, we show that with the alternative entry for negation proposed in §5:

2) \(h, y, x \vDash \neg \phi \) iff there is no \(y^{\prime }\) s.t. \(h, y^{\prime }, x \vDash \phi \)

a form of (DPr) follows. We focus on the simple case \(\neg _{2} C(p \textsc {or} q) \vDash \neg _{2} C(p) \wedge \neg _{2} C(q)\), adding (as in the proof of Fact 2) the assumption that \(\blacklozenge (p \wedge \neg q)\) and \(\blacklozenge (q \wedge \neg p)\).

Proof

\(\neg _{2} C(p \textsc {or} q), \blacklozenge (p \wedge \neg q), \blacklozenge (q \wedge \neg p) \vDash \neg _{2} C(p) \wedge \neg _{2} C(q)\)

$$ \begin{array}{@{}rcl@{}} h \triangleright \neg_{2} C(p \textsc{or} q) &\text{ iff }& \forall w \in h: h, w, w \vDash \neg_{2} C(p \textsc{or} q)\\ &\text{ iff }& \forall w \in h: \text{ there's no } y^{\prime} \in h: h, y^{\prime}, w \vDash C(p \textsc{or} q) \\ &\text{ iff }& \forall w \in h: \text{ there's no } y^{\prime} \in h: \exists w^{\prime} \in h: h, y^{\prime}, w^{\prime} \vDash \boxdot(p \textsc{or} q)\\ &\text{ iff }& \text{ there's no } y^{\prime}, w^{\prime} \in h: h, y^{\prime}, w^{\prime} \vDash \boxdot(p \textsc{or} q)\\ &\text{ iff }& \forall y^{\prime}: \text{ there's no } w^{\prime} \in h: h, y^{\prime}, w^{\prime} \vDash \boxdot(p \textsc{or} q)\\ \end{array} $$

By the second premise, we know that ∃wh such that \(h, w, w \vDash p\) and \(h, w, w \nvDash q\) (call this world wp). Instantiating wp for \(y^{\prime }\) above, we can conclude that:

$$ \begin{array}{@{}rcl@{}} & \text{ there's no } w^{\prime} \in h: h, w_{p}, w^{\prime} \vDash \boxdot(p \textsc{or} q)\\ &\text{ iff } \text{ there's no } w^{\prime} \in h: h, w_{p}, w^{\prime} \vDash \boxdot p\\ \end{array} $$

Because the truth-conditions of p are not actuality-parameter sensitive, this is equivalent to:

$$ \begin{array}{@{}rcl@{}} &\forall y^{\prime} \in h: \text{ there's no } w^{\prime} \in h: h, y^{\prime}, w^{\prime} \vDash \boxdot p\\ &\text{ iff } \text{ there's no } y^{\prime}, w^{\prime} \in h: h, y^{\prime}, w^{\prime} \vDash \boxdot p \end{array} $$

Hence h ⊳ ¬2C(p). A similar argument, leveraging the third premise, shows that h ⊳ ¬2C(q). □

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Fusco, M. Agential Free Choice. J Philos Logic 50, 57–87 (2021). https://doi.org/10.1007/s10992-020-09561-w

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