Skip to main content

Independent alternatives

Ross’s puzzle and free choice

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    By ‘orthodox’, I mean any theory that makes necessity and possibility modals upward monotonic, e.g. the standard Kripke semantics that uses accessibility relations, and the context-sensitive semantics of Kratzer (2012b). ‘Underspecific’ has many different uses in the philosophy of language; the one I employ here has precedent in Zimmermann (2006) and Fara (2013).

  2. 2.

    By ‘standard theory of disjunction’, I mean especially theories like that of Partee and Rooth (1983), where disjunction is treated as the Boolean dual of conjunction.

  3. 3.

    In keeping with much of the recent literature, I use declarative ‘ought’ claims to illustrate Ross’s puzzle. Ross’s original example was presented using imperatives: where ‘Post the letter!’ entails ‘Post the letter or burn it!’ Although I do not discuss imperatives in this paper, my account can be straightforwardly extended to them.

  4. 4.

    See Kripke (1963). In Kratzer (1977, 1991), the dominant orthodox theory of natural language modals, there are several worthwhile complications of the basic quantificational idea, but they do not matter for our purposes. See Portner (2009) for a textbook presentation of various semantic theories of natural language modals.

  5. 5.

    In the Kratzerian dialect (Kratzer 1977, 1991), where modals are relative to a modal base f and an ordering source g, we define R(w) as follows:

    $$\begin{aligned} R(w) = \textsf {max}_{g(w)}(\bigwedge f(w)) \end{aligned}$$

    where \(\textsf {max}_{g(w)}\) is a function that takes a proposition and returns the subset of worlds that are maximal with respect to the order determined by g.

    In the Kripkean dialect, where modals are sensitive to an accessibility relation R between worlds in a set W, we define R(w) as follows:

    $$\begin{aligned} R(w) = \{v \in W\ |\ ( w, v) \in \textsf {R} \} \end{aligned}$$
  6. 6.

    The puzzles were discovered in early work on deontic logic and the logic of imperatives (von Wright 1968; Kamp 1973; Ross 1941). Some recent research has continued this focus on deontic flavors of modality, including Barker (2010), Cariani (2013), Fusco (2015), and Starr (2016). Of course, as some of these authors mention, there are likely ways to extend these accounts, tailored to the deontic case, to other flavors of modality.

  7. 7.

    See Zimmermann (2006), Yablo (2014), Abreu Zavaleta (2019) for some examples of invalid, non-deontic Ross inferences, and Zimmermann (2000), Nickel (2010), Romoli and Santorio (2017), Willer (2021) for some discussion of valid, non-deontic free choice inferences.

  8. 8.

    It is standardly assumed that modal auxiliaries like ‘must’ are context sensitive, and in various contexts can express different flavors of necessity, such as metaphysical, epistemic, deontic, and so on.

  9. 9.

    Throughout, by ‘modal’ or ‘modal operator’, I mean a sentential operator that is standardly analyzed as shifting the world of evaluation for its complement proposition. ‘Modal’ as I am using it thus includes not just modal auxiliaries like ‘must’ and ‘may’, but also attitude verbs like ‘want’ and ‘believe’.

  10. 10.

    This is not to claim that there are no modals for which the Ross inference is valid—perhaps there are. My claim is a weaker one: given these examples, the solution we propose to the puzzles should be flexible enough to handle any flavor of modality.

  11. 11.

    See, for example, von Fintel (2012) for a pragmatic version of this thesis and Simons (2005) for a semantic one. As I mention below, Menéndez Benito (2005), Menéndez Benito (2010) has dissented in the analogous case of free choice ‘any’ under possibility modals, and building on this work, Aloni and Ciardelli (2013) dissent in the analogous case of imperatives.

  12. 12.

    See, for example, Wedgwood (2006), von Fintel (2012).

  13. 13.

    I am making the standard assumption that modals, like other quantifiers, presuppose that their domains are non-empty. Thus, \(\Box M\) entails that there is a relevant world where M is true. This means that whenever \(\Box M\) is true, \(\Diamond M\) is also true.

  14. 14.

    See also Fusco (2015) for this data in the case of ‘ought’, and an alternative account of it.

  15. 15.

    See Kratzer (2012a, 2012b) for the classic theory of this phenomenon in the case of non-attitude modals, and Blumberg and Holguín (2019) for recent work on restriction effects in the case of attitude verbs.

  16. 16.

    Again, in the case of (10b) and (10c), this assumes that natural language necessity modals like ‘ought’ presuppose that their domains are non-empty.

  17. 17.

    I will not discuss free choice ‘any’ in this paper, but I think it will be clear enough how my account could be extended to that case in order to capture the data Menéndez Benito puts forward.

    See also Aloni and Ciardelli (2013), which applies Menéndez Benito’s insight to the case of imperatives.

  18. 18.

    The term used by Menéndez Benito (2005), Menéndez Benito (2010)) is not independence but exclusivity. I have opted for ‘independence’ here since I do not want to suggest any erroneous connections to ‘exclusive’ disjunction.

  19. 19.

    In fact, on the theory I go on to develop in this paper, (12) will be neither true nor false in the described situation. I discuss the status of (12) further in §8.

  20. 20.

    For example, take:

    1. (13)

      Alicia must have either some or all of the ice cream.

    The surface grammar of the embedded disjunction, ‘Alicia had either some or all of the ice cream,’ flouts Hurford’s constraint, since having all of the ice cream entails having some of it. This means that the modal claim (13) could not possibly license both of the Independence inferences, since one of them would be:

    1. (14)

      Alicia may have all but not some of the ice cream.

    (14) cannot be true, since the embedded conjunction, ‘Alicia has all but not some of the ice cream,’ is a contradiction. The assumption I will make is that in recognizing this, a hearer assigns to (13) a logical form roughly equivalent to, ‘Alicia must have either merely some, or all of the ice cream” (where the exhaustification operator merely has the semantic function of denying that Alicia has all of the ice cream. See Simons (2001), Katzir and Singh (2013), Meyer (2013, 2014), and Ciardelli and Roelofsen (2017) for recent discussion.

  21. 21.

    An anonymous reviewer suggests that the independence inferences might be computed by the combination of the Diversity inferences together and an exclusive reading of the embedded disjunction. On this hypothesis, the Independence inferences would only be licensed in cases where it is reasonable to suppose that there is no relevant world that makes the conjunction of the disjuncts true. While I cannot provide a conclusive argument against this hypothesis here, I think that the apparent invalidity of examples like (8) and (9) provide evidence against it. In those cases, the premises can be true together, and when they are both true, there is a relevant world that makes the conjunction of the disjuncts true—where Alicia mails the letter and uses the phone. In such a case, it should be natural for speakers to opt for an inclusive reading of the disjunction embedded in the conclusion, and recognize that it follows on the orthodox semantics (even supplemented with the Diversity inferences). But both inferences seem just as invalid as the original (1). This suggests to me that the Independence inferences do not arise only on an exclusive reading of the disjunction involved, and thus that the Independence inferences are not computed in the way this hypothesis claims.

  22. 22.

    Simons (2005) also uses a covering relation as a helpful way of summarizing the modal/disjunction interaction. For Simons, modals with disjunctive complements truth conditionally require that the disjuncts form a supercover of the relevant set of worlds. C is a supercover of S iff it is a cover of S and every member of C has a non-empty intersection with S. Note that every minimal cover is a super cover but not vice versa; and that a supercover semantics validates the Diversity, but not Independence, inferences. See also Nygren (2019), which systematically explores the logic of a supercover semantics.

    At the end of her paper (§6), Simons briefly considers various pragmatic ‘add-on’ requirements that she thinks may govern the felicity of disjunctions in certain contexts. One of the three requirements she outlines resembles the minimal covering relation I define here. However, she does not explore this pragmatic constraint in much detail, and clearly does not think, as I argue here, that it is part of the literal, truth conditional semantics of modals with embedded disjunctions.

  23. 23.

    One of the key differences between the two frameworks arises in cases when the set of worlds where ‘q’ is true is a subset of the worlds where ‘p’ is true (\({\llbracket }q{\rrbracket }\subseteq {\llbracket }p {\rrbracket }\)). On the traditional possible worlds analysis of propositions and disjunction, in this case the proposition denoted by ‘\(p \vee q\)’ is identical to the one denoted by just ‘p’. In inquisitive semantics, the same is true. But standard versions of alternative semantics distinguish between these propositions (Roelofsen (2013), Ciardelli et al. (2017)). This means that alternative semantics, but not inquisitive semantics, gives up the traditional explanation of Hurford’s constraint in terms of redundancy (Ciardelli and Roelofsen 2017).

  24. 24.

    A formal summary of the semantic framework and results is contained in the appendix at the end of this paper.

  25. 25.

    The basic behavior of the connectives in the bilateral approach is formally similar to the ‘radical inquisitive semantics’ of Groenendijk and Roelofsen (2010) and Aher (2012), the dual update semantics in Willer (2018), the bilateral ‘state-based’ semantics of Aloni (2018), and the bilateral truthmaker semantics of Yablo (2014) and Fine (2017a, 2017b). While these other theorists share a similar semantic framework, and some also share an interest in our two puzzles, none of these accounts offers a theory that supports the Independence inferences we are interested in in this paper.

  26. 26.

    ‘Largest sets’ here means the sets in the proposition such that there is no proper superset also in the proposition. Officially:

    $$\begin{aligned} \textsf {alt}(\phi ) = \{s \in [\phi ]\ |\ \text {for every } t \in [\phi ], \text { if } s\subseteq t \text { then } s=t\ \} \end{aligned}$$
  27. 27.

    In richer versions of inquisitive semantics, other expressions like existential quantifiers and interrogative operators also introduce multiple alternatives.

  28. 28.

    See the dynamic model update conditional in van Ditmarsch et al. (2008), or the appendix of this paper, for precise versions of such a conditional.

  29. 29.

    The official semantics for the operator is:

    $$\begin{aligned} {[}!\phi ] = \wp (\textsf {info}(\phi )) \end{aligned}$$
  30. 30.

    For example, as Alonso-Ovalle (2006) shows, the semantics of Simons (2005) suffers these problems with negation. See Aloni (2018) and Willer (2018) for alternative bilateral solutions to these problems, and Aloni (2007) for a unilateral response to these issues based on ambiguity.

  31. 31.

    See Groenendijk and Roelofsen (2010), Aloni (2018) and Willer (2018) for other versions of bilateral inquisitive semantics with similar motivations.

  32. 32.

    It is easy to see that given the semantics of negation, conjunction, and disjunction I outline here, the semantics predicts that each of the classical De Morgan equivalences holds. Some of these equivalences are controversial, especially when embedded under modals or conditionals. The theory I give here thus inherits some of this controversy. For example, it validates Dual Free Choice: when p and q are atoms that obey Hurford’s constraint, \(\lnot \Box (p\wedge q) \vDash \Diamond \lnot p \wedge \Diamond \lnot q\). For given duality between \(\Box\) and \(\Diamond\), \([\lnot \Box (p\wedge q)]=[\Diamond \lnot (p\wedge q)]\). Then, given the De Morgan equivalences, \([\Diamond \lnot (p\wedge q)] = [\Diamond (\lnot p\vee \lnot q)]\). Finally, given the validity of free choice, clearly \(\Diamond (\lnot p\vee \lnot q)\) entails \(\Diamond \lnot p\) and \(\Diamond \lnot q\). The obvious culprit appears to be the De Morgan equivalence between \(\lnot (p\wedge q)\) and \(\lnot p \vee \lnot q\), which are not equivalent on the unilateral model of the previous section or in standard inquisitive semantics. One way to modify the present system in order to invalidate dual free choice would be to change the rule for the negative part of conjunction, so that it is not equivalent to a disjunction of negations. Since this issue independent of the original puzzles and the data adduced in Sect. 4, I do not want to take a stand on it here. For simplicity and completeness, I have opted to validate all of the De Morgan equivalences. For further discussion of the De Morgan equivalences in modal and conditional contexts, see Fox (2007); Chemla (2009); Ciardelli et al. (2018); Romoli and Santorio (2019); Marty et al. (ms).

  33. 33.

    The bilateral, dynamic inquisitive semantics of Willer (2018) also postulates some truth value gaps for modal sentences.

  34. 34.

    See for example, Schwarzschild (1993) and Križ (2015, 2016). It is controversial whether homogeneity effects give rise to gaps, and if they do, whether these gaps should be thought of as presuppositions or not.

  35. 35.

    This may be why Ross inferences are so strongly repugnant: the premise not only fails to ensure the truth of the conclusion: it ensures the conclusion is neither true nor false.

References

  1. Abreu Zavaleta, M. (2019). Weak speech reports. Philosophical Studies 176 (8), 2139–2166.

    Article  Google Scholar 

  2. Aher, M. (2012). Free choice in deontic inquisitive semantics (dis). In M. Aloni, V. Kimmelman, F. Roelofsen, G. W. Sassoon, K. Schulz, and M. Westera (Eds.), Logic, Language and Meaning, Berlin, Heidelberg, pp. 22–31. Springer, Berlin Heidelberg.

    Chapter  Google Scholar 

  3. Aloni, M. (2007). Free choice, modals, and imperatives. Natural Language Semantics 15 (1), 65–94.

    Article  Google Scholar 

  4. Aloni, M. (2018). Fc disjunction in state-based semantics.

  5. Aloni, M. and I. Ciardelli (2013). A logical account of free choice imperatives. In M. F. M. Aloni and F. Roelofsen (Eds.), The Dynamic, Inquisitive, and Visionary Life of \(\phi\), \(?\phi\) , and \(\Diamond \phi\): a Festschrift for Jeroen Groenendijk, Martin Stokhof, and Frank Veltman, pp. 1–17.

  6. Alonso-Ovalle, L. (2006). Disjunction in Alternative Semantics. PhD Dissertation, University of Massachusetts Amherst.

  7. Barker, C. (2010). Free choice permission as resource-sensitive reasoning. Semantics and Pragmatics 3 (10), 1–38.

    Google Scholar 

  8. Blumberg, K. and B. Holguín (2019). Embedded attitudes. Journal of Semantics 36 (3), 377–406.

    Article  Google Scholar 

  9. Cariani, F. (2013). Ought and resolution semantics. Noûs 47 (3), 534–558.

    Article  Google Scholar 

  10. Chemla, E. (2009). Universal implicatures and free choice effects: experimental data. Semantics and Pragmatics 2, 1–33.

    Article  Google Scholar 

  11. Chemla, E. and L. Bott (2014). Processing inferences at the semantics/pragmatics frontier: disjunctions and free choice. Cognition 130 (3), 380–396.

    Article  Google Scholar 

  12. Ciardelli, I., J. Groenendijk, and F. Roelofsen (2018). Inquisitive Semantics. Oxford University Press.

    Book  Google Scholar 

  13. Ciardelli, I. and F. Roelofsen (2011). Inquisitive logic. Journal of Philosophical Logic 40 (1), 55–94.

    Article  Google Scholar 

  14. Ciardelli, I. and F. Roelofsen (2017). Hurford’s constraint, the semantics of disjunction, and the nature of alternatives. Natural Language Semantics 25 (3), 199–222.

    Article  Google Scholar 

  15. Ciardelli, I., F. Roelofsen, and N. Theiler (2017). Composing alternatives. Linguistics and Philosophy 40 (1), 1–36.

    Article  Google Scholar 

  16. Ciardelli, I., Zhang, L., and Champollion, L. (2018). Two switches in the theory of counterfactuals: A study of truth conditionality and minimal change. Linguistics and Philosophy  (6).

  17. Fara, D. G. (2013). Specifying desires. Noûs 47 (2), 250–272.

    Article  Google Scholar 

  18. Fine, K. (2017a). A theory of truthmaker content i: Conjunction, disjunction and negation. Journal of Philosophical Logic 46 (6), 625–674.

    Article  Google Scholar 

  19. Fine, K. (2017b). A theory of truthmaker content ii: Subject-matter, common content, remainder and ground. Journal of Philosophical Logic 46 (6), 675–702.

    Article  Google Scholar 

  20. Fox, D. (2007). Free choice and the theory of scalar implicatures* mit,.

  21. Fusco, M. (2015). Deontic modality and the semantics of choice. Philosophers’ Imprint  15.

  22. Gazdar, G. (1979). Pragmatics: Implicature, presupposition, and logical form. Academic Press.

    Google Scholar 

  23. Goldstein, S. (2019). Free choice and homogeneity. Semantics and Pragmatics 12, 1–48.

    Article  Google Scholar 

  24. Gotzner, N., J. Romoli, and P. Santorio (2020). Choice and prohibition in non-monotonic contexts. Natural Language Semantics 28 (2), 141–174.

    Article  Google Scholar 

  25. Groenendijk, J. and Roelofsen, F. (2010). Radical inquisitive semantics.

  26. Hurford, J. R. (1974). Exclusive or inclusive disjunction. Foundations of Language 11 (3), 409–411.

    Google Scholar 

  27. Kamp, H. (1973). Free choice permission. Proceedings of the Aristotelian Society 74 (1), 57–74.

    Article  Google Scholar 

  28. Katzir, R. and R. Singh (2013). Hurford disjunctions: embedded exhaustification and structural economy. Sinn und Bedeutung 18, 201–216.

    Google Scholar 

  29. Kratzer, A. (1977). What ’must’ and ’can’ must and can mean. Linguistics and Philosophy 1 (3), 337–355.

    Article  Google Scholar 

  30. Kratzer, A. (1991). Modality. In: von Stechow, A., & Wunderlich, D. (Ed.), Semantics: An International Handbook of Contemporary Research, pp. 639–650. Berlin: de Gruyter.

  31. Kratzer, A. (2012a). Conditionals. In Modals and Conditionals, pp. 86–108. Oxford University Press.

  32. Kratzer, A. (2012b). The notional category of modality. In Modals and Conditionals, pp. 27–69. Oxford University Press.

  33. Kratzer, A. and Shimoyama, J. (2002). Indeterminate pronouns: The view from japanese. Proceedings of the Third Tokyo Conference on Psycholinguistics .

  34. Kripke, S. A. (1963). Semantical considerations on modal logic. Acta Philosophica Fennica 16 (1963), 83–94.

    Google Scholar 

  35. Križ, M. (2015). Aspects of Homogeneity in the Semantics of Natural Language. PhD Thesis, University of Vienna.

  36. Križ, M. (2016). Homogeneity, non-maximality, andall. Journal of Semantics 33 (3), 493–539.

    Article  Google Scholar 

  37. Marty, P., Romoli, J., Sudo, Y., and Breheny, R. (ms.). Negative free choice.

  38. Menéndez Benito, P. (2005). The Grammar of Choice. Ph.D. thesis, University of Massachusetts, Amherst.

  39. Menéndez Benito, P. (2010). On universal free choice items. Natural Language Semantics 18 (1), 33–64.

    Article  Google Scholar 

  40. Meyer, M.-C. (2013). Ignorance and Grammar. PhD Thesis, MIT.

  41. Meyer, M.-C. (2014). Deriving hurford’s constraint. Proceedings of SALT 24, 577–596.

    Article  Google Scholar 

  42. Nickel, B. (2010). Generically free choice. Linguistics and Philosophy 33 (6), 479–512.

    Article  Google Scholar 

  43. Nygren, K. (2019). Supercover semantics for deontic action logic. Journal of Logic, Language and Information 28 (3), 427–458.

    Article  Google Scholar 

  44. Partee, B. and Rooth. M. (1983). Generalized conjunction and type ambiguity. In C. S. Rainer Bauerle and A. von Stechow (Eds.), Meaning, Use, and Interpretation of Language, pp. 361–383.

  45. Portner, P. (2009). Modality. Oxford University Press.

    Google Scholar 

  46. Roelofsen, F. (2013). Algebraic foundations for the semantic treatment of inquisitive content. Synthese 190 (S1), 1–24.

    Article  Google Scholar 

  47. Romoli, J. and P. Santorio (2017). Probability and implicatures: A unified account of the scalar effects of disjunction under modals. Semantics and Pragmatics 10 (3), 1–61.

    Google Scholar 

  48. Romoli, J. and P. Santorio (2019). Filtering free choice. Semantics and Pragmatics 12 (12), 1–27.

    Article  Google Scholar 

  49. Ross, A. (1941). Imperatives and logic. Philosophy of Science 11 (1), 30–46.

    Article  Google Scholar 

  50. Sayre-McCord, G. (1986). Deontic logic and the priority of moral theory. Noûs 20 (2), 179–197.

    Article  Google Scholar 

  51. Schwarzschild, R. (1993). Plurals, presuppositions and the sources of distributivity. Natural Language Semantics 2 (3), 201–248.

    Article  Google Scholar 

  52. Simons, M. (2001). Disjunction and alternativeness. Linguistics and Philosophy 24 (5), 597–619.

    Article  Google Scholar 

  53. Simons, M. (2005). Dividing things up: The semantics of or and the modal/or interaction. Natural Language Semantics 13 (3), 271–316.

    Article  Google Scholar 

  54. Starr, W. B. (2016). Expressing permission. Semantics and Linguistic Theory 26, 325–349.

    Article  Google Scholar 

  55. van Ditmarsch, H., W. van Der Hoek, and B. Kooi (2008). Dynamic Epistemic Logic. Springer.

    Book  Google Scholar 

  56. von Fintel, K. (2012). The best we can (expect to) get? challenges to the classic semantics for deontic modals.

  57. von Wright, G. H. (1968). An Essay in Deontic Logic and the General Theory of Action. Amsterdam: North-Holland Pub. Co.

    Google Scholar 

  58. Wedgwood, R. (2006). The meaning of ’ought’. In R. Shafer-Landau (Ed.), Oxford Studies in Metaethics: Volume 1, pp. 127–160. Clarendon Press.

    Google Scholar 

  59. Willer, M. (2018). Simplifying with free choice. Topoi 37 (3), 379–392.

    Article  Google Scholar 

  60. Willer, M. (2021). Two puzzles about ability can. Linguistics and Philosophy 44 (3), 551–586.

    Article  Google Scholar 

  61. Yablo, S. (2014). Aboutness. Princeton University Press.

    Book  Google Scholar 

  62. Zimmermann, T. E. (2000). Free choice disjunction and epistemic possibility. Natural Language Semantics 8 (4), 255–290.

    Article  Google Scholar 

  63. Zimmermann, T. E. (2006). Monotonicity in opaque verbs. Linguistics and Philosophy 29 (6), 715–761.

    Article  Google Scholar 

Download references

Acknowledgements

I am very grateful for helpful comments and discussion on previous drafts of this paper to Jessica Collins, Melissa Fusco, Ben Holguín, Ezra Keshet, Tamar Lando, Karen Lewis, Janum Sethi, Achille Varzi, the participants of the 2021 Eastern APA session for this paper, the participants of the Dianoia Institute of Philosophy Language Workshop, and two anonymous referees for this journal.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Richard Jefferson Booth.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: Bilateral minimal covering semantics

Appendix: Bilateral minimal covering semantics

Language

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

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

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 (\(V_\mathcal {M}: \textsf {At}\mapsto \wp (W)\)).

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 \(\mathcal {M}\).

Information and alternatives

For \(P \in \mathcal {P}_\mathcal {M}\):

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

Minimal cover

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

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:

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

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 \(\phi\):

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

Truth and falsity

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


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

Entailment

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

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

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\(\vee\))

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Booth, R.J. Independent alternatives. Philos Stud (2021). https://doi.org/10.1007/s11098-021-01706-0

Download citation

Keywords

  • Modality
  • Disjunction
  • Ross’s puzzle
  • Free Choice
  • Monotonicity