Skip to main content
SpringerLink
Log in

A Uniform Theory of Conditionals

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

A uniform theory of conditionals is one which compositionally captures the behavior of both indicative and subjunctive conditionals without positing ambiguities. This paper raises new problems for the closest thing to a uniform analysis in the literature (Stalnaker, Philosophia, 5, 269–286 (1975)) and develops a new theory which solves them. I also show that this new analysis provides an improved treatment of three phenomena (the import-export equivalence, reverse Sobel-sequences and disjunctive antecedents). While these results concern central issues in the study of conditionals, broader themes in the philosophy of language and formal semantics are also engaged here. This new analysis exploits a dynamic conception of meaning where the meaning of a symbol is its potential to change an agent’s mental state (or the state of a conversation) rather than being the symbol’s content (e.g. the proposition it expresses). The analysis of conditionals is also built on the idea that the contrast between subjunctive and indicative conditionals parallels a contrast between revising and consistently extending some body of information.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. The indicative and subjunctive labels are flawed, but ubiquitous. (1) and (2) are not meant to be exhaustive, but rather paradigmatic lists. I will not propose an analysis of habitual (Bob dances if Leland dances) or future (Bob will dance if Leland dances) indicatives. These require an independent analysis of habituals and future tense, which are more complex than simple past and present tenses, with a plausible analysis of conditionals. [48] takes up this project.

  2. The cross-linguistic distribution of conditional particles indicates that they are not homonymous. Another obvious but unappreciated mark against homonymy was pointed out to me by Tom Donaldson: there isn’t a subjunctive reading of (1) or an indicative reading of (2).

  3. Many authors have explored this idea. See [76, Section 5.6].

  4. The latter work draws a distinction between imaging and conditionalization (e.g. [33, 56]) or belief revision and belief update (e.g. [46]).

  5. These constraints are detailed in Definition 16, Appendix A.1.

  6. e.g. [85, Ch.7] and [57].

  7. Consider a robot with a model of its environment who can perform actions which change that environment. How does the robot decide what has changed and what hasn’t? [63] Striking the match usually leads to fire. But sometimes not. The match could be wet, the striking surface could be too smooth or there could be too little oxygen or too much wind, or the robot could have confused a twig for a match. Where does this list end? The robot will also need a list of the things that don’t change! Goodman observed: it seems to be true that if the match had been struck, it would have lit. But it doesn’t light under any number of conditions and listing them without circularity is non-trivial. See [79].

  8. Not only are the agents taking this information for granted, but they are all taking for granted that they are taking it for granted, and taking that for granted, etc. [16, 23, 54, 58, 87, 88, Ch. 4].

  9. I’ll be a monkey’s uncle if Bob dances more than suggests that Bob won’t dance.

  10. von Fintel [28] proposes a similar view where subjunctives presuppose that the space of worlds relevant for evaluating the antecedent are not a subset of c. As he notes, none of his data distinguishes this from the Stalnakerian view. Leahy [52] proposes that indicatives presuppose the epistemic possibility of their antecedent, that subjunctives have no presuppositions and derives the Subjunctive Suggestion as an ‘antipresupposition’ using an epistemic logic. There seem to be counterexamples to this claim about indicatives: “I know that Bob danced. But I’m happy to humor you and consider the possibility that he didn’t. If he didn’t dance, Leland didn’t dance either.” Since the [52] approach truly requires this assumption, I’m inclined to the abductive approach sketched above.

  11. More specifically, the auxiliaries in the antecedents of (8a) and (8b) are marked with what looks like past tense, and the same goes for the main verb in the antecedent of (8c). When multiple tenses are layered as in (8a) and (8b), the tense on the auxiliary determines the overall tense of the sentence. The acceptability of perfective sentences like By tomorrow, Bob will have danced makes this clear. Otherwise, such a sentence would be as unacceptable as #Bob danced tomorrow.

  12. Parallel examples for (8a) and (8b) are available [42]. With subjunctives like (8b) there is the additional difference that even with singular subjects the plural conjugation is used. In isolation Bob were to dance is ungrammatical, though The men were (supposed) to dance is acceptable. The latter sentence exhibits the same ‘fake past’ pattern in that it allows tomorrow, yesterday and today.

  13. See also [97, Section 4] who suggest that even tensed languages commonly use non-temporal morphology to mark subjunctive antecedents. Murray (p.c.) also informs me that Cheyenne uses identical tense/aspect morphology in indicative and subjunctive antecedents, but a distinct hypothetical mood for the two.

  14. Kratzer [50, 51] suggests a view on which indicative and subjunctives differ only in the modal base/ordering sources used. This counts as a simple kind of uniform theory. But it has never been investigated how precisely this difference can be derived from the meaning of an operator within the scope of the restrictor clause in subjunctives. Kratzer’s account of modal variation is more at home with bare modals where often the same exact form has multiple interpretations. Further, problems identical to those discussed below arise in the Kratzerian setting. For Kratzer, the modal domain is generated from the modal base f: \(\bigcup f(w)\). To capture the idea that indicative antecedents concern only live possibilities, one could require that \(\bigcup f(w)=c\) for all w. But then, just as with Stalnaker’s approach, one ends up allowing exceptions to modus ponens. At worlds outside c, the conditional can be true while the antecedent true and consequent false. Weakening the claim that \(\bigcup f(w)=c\) for all w, leads to the same problem Stalnaker faces: it does not embody Stalnaker’s Distinction.

  15. Stalnaker makes the uniqueness assumption: for all w and p there is at most one p-world most similar to w. The above remains neutral on this matter by universally quantifying over a set of the most similar worlds, be it a singleton or more populous set. Throughout, I will translate Stalnaker’s discussion of f into a setting that does not assume uniqueness.

  16. Nolan [69, 222-3] suggests requiring c to be knowledge helps, but at least within Stalnaker’s view where the context set is a set of worlds rather than an accessibility relation, this does not help. All this guarantees is that the actual world won’t provide the counterexample to modus ponens, which is no consolation. Further this conversationally unrealistic requirement would vitiate nearly all of the explanatory applications of Stalnaker’s framework. Lycan [61] chooses to give up and reject modus ponens. In the theory proposed below, neither concession is necessary.

  17. Stalnaker’s ‘pragmatic’ constraint would have to apply sub-sententially: If Bob is dancing, then Laura would be dancing if she were here.

  18. [ϕ] is a function which applies to c and delivers c′. Although the notation [ϕ] (c) seems more appropriate, I will follow precedent [100]

  19. The present model does not distinguish implicit and explicit information, i.e. information agents have immediate access to versus information they could access by communication or thought. Models that make this distinction exist but introduce unnecessary complications here [23, Ch. 9].

  20. This way of stating the dialectic uses the properties of eliminativity and continuity often discussed in the context of dynamic semantics [10–12, 38]. Rothschild and Yalcin [77] have shown that idempotence and commutativity provide a more general characterization of the updates that don’t obey the equation above. This development leads to the same basic debate with different properties guiding it. It is therefore equally orthogonal to the kind of argument for dynamic semantics provided below.

  21. Adams [2, Section 1.8] doubts Import-Export. But even his sympathizers are unmoved: [65] and [9, Section 40] who calls this pattern if-and.

  22. Veltman [99] and [103, 105] make this observation but propose pragmatic views about the kind of infelicity that results.

  23. In this case: {w}[ϕ] = ∅ and since ∅[ψ] = ∅, ∅ ⊨ ψ. Hence {w}[ϕ] ⊨ ψ.

  24. There is a long history of authors who have endorsed these truth-conditions but failed combine it with a plausible logic. Starr [92] argues that this is an additional interest of the present semantics.

  25. For a similar view see [31, Section 4].

  26. This formulation makes the limit assumption which Lewis [55, Section 1.4] officially rejected: there is a set of α-worlds most similar to w. Though some reasons for adopting it exist, it’s a mere convenience here [40, 74, 85, 140–6; 104].

  27. Again, I am putting aside the limit assumption. Stalnaker formulates the selection function as returning a single world rather than a singleton set. That implementation requires contortions (undefinedness or an impossible world) when evaluating f (w, ∅), while the set selection formulation [55, Section 3.4] can just return ∅.

  28. Weak centering would do as well: w ∈ f (w, p) if w ∈ p.

  29. The [64] ‘counterexamples’ to modus ponens are easily diagnosed and diffused on the dynamic semantics for conditionals described in Section 3.2 [34, 592-5]. The similar subjunctive case from [15, Section 3.1] only arises in a structural equations framework, e.g. [73, Ch.7]. Since that is my preferred analysis of f , it is fortunate that my version of the structural equations semantics circumvents this problem by rendering the complex subjunctive false [91].

  30. Some strict accounts of subjunctive conditionals invalidate these patterns, but in a very specific case (e.g. [29, 35]). They rely on a case where the presuppositions of the premise are met but the conclusion’s are not and cannot be accommodated. Thus, the truth of the premise does not guarantee the truth of the conclusion (and hence isn’t dynamically valid either). However, the Strawsonian definition of entailment motivated in Section 3.2 does not count cases like this for or against an inference pattern. On this definition any strict analysis will validate Antecedent Strengthening and Transitivity. By contrast, on my semantics these patterns aren’t even valid in the Strawsonian sense.

  31. When restricted to persistent consequents. See Definition 25 in Appendix.

  32. This approach is particularly problematic for mixed consequents: if Cooper had eaten a donut, he would have dunked it in coffee and might have given a bite of it to Audrey. On Stalnaker’s (re)analysis this implausibly equivalent to if Cooper had eaten a donut, he would have dunked it in coffee and would have given a bite of it to Audrey.

  33. e.g. [22; 24, 453-4; 60; 70].

  34. More generally, if ϕ 1 and ϕ 2 are formulas built up from literals embedded under ◃ by ∨ and ∧, and ψ is persistent (Definition 25), then (if(ϕ 1 ∨ ϕ 2)) ψ ⊨ ((if ϕ 1) ψ) ∧ ((if ϕ 2) ψ).

  35. Another raised by Fine [24, 453-4] is that SDA leads to Antecedent Strengthening by the substitution of equivalents in the antecedent. p and (p ∧ q) ∨ (p ∧ ¬q) are equivalent, so from p > r, ((p ∧ q) ∨ (p ∧ ¬q)) > r follows by substitution of equivalents. Then by SDA (p ∧ q) > r. But ◃p and ◃((p ∧ q) ∨ (p ∧ ¬q)) are not equivalent.

  36. One might instead have the hunch that the difference between (33) and (31a) traces to the distinction between (if ◃(α ∨ β)) ψ and (if ◃α ∨ ◃β) ψ. But recall that ◃ is coming from either the past tense or have. So one would expect If Spain had fourth for the Axis or Spain had fought for the Allies, she would have fought for the Axis to be just as bad as (33). Though it is subtle, this conditional strikes me as at least significantly better than (33). I return to ◃(α ∨ β) versus ◃α ∨ ◃β shortly.

  37. I thank James Shaw for raising this issue.

  38. Alonso-Ovalle [4, 5] also presents a (more complex) analysis using alternative semantics. This analysis does not use an operator like ◃ to apply the selection function to alternatives. It instead builds an appeal to the selection function into the basic conditional semantics. This will not lead to an adequate uniform theory of conditionals for the reasons discussed in Section 2.3. The lack of ◃ and dynamic consequence in this analysis also prevents it from having a compelling response to worries about the substitution of equivalents in antecedents (see note 35).

  39. Fine [26, 47] also voices this worry, and additionally mentions examples with existential quantification as a worry. Since many also adopt an alternative semantics for existential quantification, that additional worry is mute. Fine [25, 26] offers a different semantics for subjunctive conditionals altogether that validates SDA. Fine [26, 48] sketches an inchoate version of the Warmbrod-style [102, 284] reply to [66] detailed above. Fine’s [25] semantics shares many similarities to my preferred approach mentioned above [91], but does not integrate well into the kind of uniform theory developed here.

  40. \(\langle {c_{f}}\rangle \) is a state and if s is a state then \(\langle {c^{\prime }_{f^{\prime }},s}\rangle \) is a state. This recursive structure allows one meaning for \(\phi \)to be applied to both \(\langle {c_{f},\langle {c^{\prime }_{f^{\prime }}}}\rangle \rangle \) and it’s sub-state \(\langle {c^{\prime }_{f^{\prime }}}\rangle \). This is needed for conditionals in the consequent of a conditional.

  41. Like von Fintel’s theory, the apparatus here also allows one to explain the intuitive validity of many cases of transitivity as well as the asymmetries those inferences exhibit, but that too must be reserved for another occasion.

References

  1. Adams, E.W. (1975). Counterfactual conditionals and prior probabilities. In C.A. Hooker & W. Harper (Eds.), Proceedings of international congress on the foundations of statistics. Dordrecht: D. Reidel.

    Google Scholar 

  2. Adams, E.W. (1975). The logic of conditionals. Dordrecht: D. Reidel.

    Book  Google Scholar 

  3. Aloni, M. (2003). Free choice in modal contexts. In M. Weisgerber (Ed.), Proceedings of Sinn und Bedeutung 7 (pp. 25–37). Germany: Fachbereich Sprachwissenschaft der Universität Konstanz. http://ling.uni-konstanz.de/pages/conferences/sub7/.

    Google Scholar 

  4. Alonso-Ovalle, L. (2006). Disjunction in alternative semantics. Ph.D. thesis, Amherst: UMass Amherst.

  5. Alonso-Ovalle, L. (2009). Counterfactuals, correlatives, and disjunction. Linguistics and Philosophy, 32(2), 207–244. doi:10.1007/s10988-009-9059-0.

    Article  Google Scholar 

  6. Anderson, A.R. (1951). A note on subjunctive and counterfactual conditionals. Analysis, 12, 35–38.

    Article  Google Scholar 

  7. Arregui, A. (2007). When aspect matters: the case of would-conditionals. Natural Language Semantics, 15, 221–264. doi:10.1007/s11050-007-9019-6.

    Article  Google Scholar 

  8. Arregui, A. (2009). On similarity in counterfactuals. Linguistics and Philosophy, 32, 245–278. doi:10.1007/s10988-009-9060-7.

    Article  Google Scholar 

  9. Bennett, J. (2003). A philosophical guide to conditionals. Oxford: Oxford University Press.

    Book  Google Scholar 

  10. van Benthem, J. (1986). Essays in logical semantics. Dordrecht: D. Reidel Publishing Co.

    Book  Google Scholar 

  11. van Benthem, J. (Ed.). (1991). Language in action: categories, lambdas, and dynamic logic. Amsterdam: Elsevier Science.

    Google Scholar 

  12. van Benthem, J., Muskens, R., Visser, A. (2011). Dynamics. In J. van Benthem & A. ter Meulen (Eds.), Handbook of logic and language (pp. 607–670). Amsterdam: Elsevier Science Publishers.

    Google Scholar 

  13. Bittner, M. (2011). Time and modality without tenses or modals. In R. Musan, & M. Rathers (Eds.), Tense across languages (pp. 147–188). Tübingen: Niemeyer. http://semanticsarchive.net/Archive/zliYmQxY/bittner11_tam.pdf.

    Google Scholar 

  14. Brasoveanu, A. (2010). Decomposing modal quantification. Journal of Semantics, 27(4), 437–527. doi:10.1093/jos/ffq008. http://jos.oxfordjournals.org/content/27/4/437.abstract.

    Article  Google Scholar 

  15. Briggs, R. (2012). Interventionist counterfactuals. Philosophical Studies, 160(1), 139–166. doi:10.1007/s11098-012-9908-5.

    Article  Google Scholar 

  16. Clark, H.H. (1996). Using Language. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  17. Dekker, P. (2012). Dynamic semantics. Studies in Linguistics and Philosophy (Vol. 91). Springer Netherlands, Dordrecht.

  18. Dudman, V.C. (1981). Time and tense in English. Unpublished manuscript, Macquarie University.

  19. Dudman, V.C. (1983). Tense and time in english verb clusters of the primary pattern. Australasian Journal of Linguistics, 3, 25–44.

    Article  Google Scholar 

  20. Dudman, V.H. (1984). Parsing if-sentences. Analysis, 44(4), 145–153. doi:10.1093/analys/44.4.145. http://analysis.oxfordjournals.org/content/44/4/145.short.

    Article  Google Scholar 

  21. Dudman, V.H. (1988). Indicative and subjunctive. Analysis, 48(3), 113–122. doi:10.1093/analys/48.3.113a. http://analysis.oxfordjournals.org/content/48/3/113.2.short.

    Article  Google Scholar 

  22. Ellis, B., Jackson, F., Pargetter, R. (1977). An objection to possible-world semantics for counterfactual logics. Journal of Philosophical Logic, 6(3), 355–357.

    Article  Google Scholar 

  23. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y. (1995). Reasoning about knowledge. Cambridge: The MIT Press.

  24. Fine, K. (1975). Review of lewis’ counterfactuals. Mind, 84, 451–458.

    Article  Google Scholar 

  25. Fine, K. (2012). Counterfactuals without possible worlds. Journal of Philosophy, 109(3), 221–246.

    Google Scholar 

  26. Fine, K. (2012). A difficulty for the possible worlds analysis of counterfactuals. Synthese, 189(1), 29–57. doi:10.1007/s11229-012-0094-y.

    Article  Google Scholar 

  27. von Fintel, K. (1999). Npi licensing, strawson entailment context dependency. Journal of Semantics, 16, 97–148.

    Article  Google Scholar 

  28. von Fintel, K. (1999). The presupposition of subjunctive conditionals. In U. Sauerland & O. Percus (Eds.), The interpretive tract, vol. MIT Working Papers in Linguistics 25 (pp. 29–44). Cambridge, MA: MITWPL. http://mit.edu/fintel/www/subjunctive.pdf.

    Google Scholar 

  29. von Fintel, K. (2001). Counterfactuals in a dynamic context. In M. Kenstowicz (Ed.), Ken Hale: a life in language (pp. 123–152). Cambridge: MIT Press. http://mit.edu/fintel/www/conditional.pdf.

    Google Scholar 

  30. von Fintel, K. (2012). Subjunctive conditionals. In G. Russell & D.G. Fara (Eds.), The routledge companion to philosophy of language (pp. 466–477). New York: Routledge. http://mit.edu/fintel/fintel-2012-subjunctives.pdf.

    Google Scholar 

  31. von Fintel, K., & Gillies, A.S. (2011). Might made right. In A. Egan, & B. Weatherson (Eds.), Epistemic modality (pp. 108–130). Oxford: Oxford University Press. http://semanticsarchive.net/Archive/TAyZGY2Z/mmr.pdf.

    Chapter  Google Scholar 

  32. Frank, A., & Kamp, H. (1997). On context dependence in modal constructions. In A. Lawson (Ed.), Proceedings from semantics and linguistic theory VII (pp. 151–168). Ithaca: Cornell University.

    Google Scholar 

  33. Gärdenfors, P. (1982). Imaging and conditionalization. Journal of Philosophy, 79(12), 747–760.

    Article  Google Scholar 

  34. Gillies, A. (2004). Epistemic conditionals and conditional epistemics. Noûs, 38(4), 585–616. http://rci.rutgers.edu/thony/epi_cond_nous.pdf.

    Article  Google Scholar 

  35. Gillies, A. (2007). Counterfactual scorekeeping. Linguistics & Philosophy, 30(3), 329–360. http://rci.rutgers.edu/thony/counterfactualscorekeeping_landp.pdf.

    Article  Google Scholar 

  36. Gillies, A. (2009). On truth-conditions for If (but not quite only If). Philosophical Review, 118(3), 325–349. http://rci.rutgers.edu/thony/truth-conditions-if-phil-review-2009.pdf.

    Article  Google Scholar 

  37. Goodman, N. (1947). The problem of counterfactual conditionals. The Journal of Philosophy, 44, 113–118.

    Article  Google Scholar 

  38. Groenendijk, J., Stokhof, M., Veltman, F. (1996). Coreference and modality. In S. Lappin (Ed.), The handbook of contemporary semantic theory (pp. 179–213). Oxford: Blackwell Publishers.

    Google Scholar 

  39. Heim, I. (1982). The semantics of definite and indefinite noun phrases. Ph.D. thesis, Linguistics Department. Amherst: University of Massachusetts.

  40. Herzberger, H. (1979). Counterfactuals and consistency. Journal of Philosophy, 76(2), 83–88.

    Article  Google Scholar 

  41. Hobbs, J., Stickel, M., Appelt, D., Martin, P. (1993). Interpretation as abduction. Artificial Intelligence, 63(1–2), 69–142.

    Article  Google Scholar 

  42. Iatridou, S. (2000). The grammatical ingredients of counterfactuality. Linguistic Inquiry, 31(2), 231–270.

    Article  Google Scholar 

  43. Ippolito, M. (2003). Presuppositions and implicatures in counterfactuals. Natural Language Semantics, 11(2), 145–186. doi:10.1023/A:1024411924818.

    Article  Google Scholar 

  44. Isard, S. (1974). What would you have done if..Theoretical Linguistics, 1, 233–255.

    Article  Google Scholar 

  45. Kamp, H. (1981). A theory of truth and semantic representation. In J.A. Groenendijk, T. Janssen, M. Stokhof (Eds.), Formal methods in the study of language (pp. 277–322). Dordrecht: Foris.

    Google Scholar 

  46. Katsuno, H., & Mendelzon, A. (1992). On the difference between updating a knowledge base and revising it. In P. Gärdenfors (Ed.), Belief revision (pp. 183–203). Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  47. Kaufmann, S. (2000). Dynamic context management. In M. Faller, S. Kaufmann, M. Pauly (Eds.), Formalizing the dynamics of conversation (pp. 171–188). Stanford: CSLI Publications.

    Google Scholar 

  48. Kaufmann, S. (2005). Conditional truth and future reference. Journal of Semantics, 22(3), 231–280. doi:10.1093/jos/ffh025. http://jos.oxfordjournals.org/content/22/3/231.abstract.

    Article  Google Scholar 

  49. Kratzer, A. (1989). An investigation of the lumps of thought. Linguistics and Philosophy, 12(5), 607–653.

    Article  Google Scholar 

  50. Kratzer, A. (1991). Modality. In A. von Stechow & D. Wunderlich (Eds.), Semantics: an international handbook of contemporary research (pp. 639–650). Berlin: de Gruyter.

    Google Scholar 

  51. Kratzer, A. (2012). Modals and conditionals: new and revised perspectives. New York: Oxford University Press.

    Book  Google Scholar 

  52. Leahy, B. (2011). Presuppositions and antipresuppositions in conditionals. In N. Ashton, A. Chereches, D. Lutz (Eds.), Proceedings from semantics and linguistic theory (SALT) 21 (pp. 257–274).

  53. Lewis, C.I. (1914). The calculus of strict implication. Mind, 23(90), 240–247. http://www.jstor.org/stable/2248841.

    Article  Google Scholar 

  54. Lewis, D.K. (1969). Convention: a philosophical study. Cambridge: Harvard University Press.

    Google Scholar 

  55. Lewis, D.K. (1973). Counterfactuals. Cambridge: Harvard University Press.

    Google Scholar 

  56. Lewis, D.K. (1975) In K. Gunderson (Ed.), Language, mind, and knowledge. minnesota studies in the philosophy of science (Vol. 7, pp. 3–35). Minneapolis: University of Minnesota Press.

    Google Scholar 

  57. Lewis, D.K. (1979). Counterfactual dependence and time’s arrow. Noûs, 13, 455–476.

    Article  Google Scholar 

  58. Lewis, D.K. (1979). Scorekeeping in a language game. Journal of Philosophical Logic, 8(1), 339–359. http://www.jstor.org/stable/30227173.

    Article  Google Scholar 

  59. Lewis, K.S. (2012). Discourse dynamics, pragmatics, and indefinites. Philosophical Studies, 158, 313–342. doi:10.1007/s11098-012-9882-y.

    Article  Google Scholar 

  60. Loewer, B. (1976). Counterfactuals with disjunctive antecedents. Journal of Philosophy, 73(16), 531–537.

    Article  Google Scholar 

  61. Lycan, W.G. (2001). Real conditionals. Oxford: Oxford University Press.

    Google Scholar 

  62. Lyons, J. (1977). Semantics (Vol. 2). Cambridge: Cambridge University Press.

    Book  Google Scholar 

  63. McCarthy, J., & Hayes, P.J. (1969). Some philosophical problems from the standpoint of artificial intelligence. In B. Meltzer & D. Michie (Eds.), Machine intelligence 4 (pp. 463–502). Edinburgh: Edinburgh University Press.

    Google Scholar 

  64. McGee, V. (1985). A counterexample to modus ponens. Journal of Philosophy, 82, 462–471.

    Article  Google Scholar 

  65. McGee, V. (1989). Conditional probabilities and compounds of conditionals. Philosophical Review, 98(4), 485–541.

    Article  Google Scholar 

  66. McKay, T.J., & van Inwagen, P. (1977). Counterfactuals with disjunctive antecedents. Philosophical Studies, 31, 353–356.

    Article  Google Scholar 

  67. Montague, R. (1970). Pragmatics and intensional logic. Synthese, 22(1/2), 68–94. http://www.jstor.org/stable/20114750.

    Article  Google Scholar 

  68. Moss, S. (2012) On the pragmatics of counterfactuals. Noûs, 46(3), 561–586. doi:10.1111/j.1468-0068.2010.00798.x. http://www-personal.umich.edu/ssmoss/counterfactuals.pdf.

    Article  Google Scholar 

  69. Nolan, D. (2003). Defending a possible-worlds account of indicative conditionals. Philosophical Studies, 116(3), 215–269.

    Article  Google Scholar 

  70. Nute, D. (1975). Counterfactuals and the similarity of worlds. The Journal of Philosophy, 72(21), 773–8.

    Article  Google Scholar 

  71. Nute, D. (1980). Conversational scorekeeping and conditionals. Journal of Philosophical Logic, 9, 153–166.

    Article  Google Scholar 

  72. Pearl, J. (1996). Causation, action, and counterfactuals. In Y. Shoham (Ed.), Theoretical aspects of rationality and knowledge: proceedings of the sixth conference (TARK 1996) (pp. 51–73). San Francisco: Morgan Kaufmann.

    Google Scholar 

  73. Pearl, J. (2000). Causality: models, reasoning, and inference. Cambridge: Cambridge University Press.

    Google Scholar 

  74. Pollock, J. (1976). The possible worlds analysis of counterfactuals. Philosophical Studies, 29, 469–476.

    Article  Google Scholar 

  75. Roberts, C. (1989). Modal subordination and pronominal anaphora in discourse. Linguistics and Philosophy, 12(6), 683–721.

    Article  Google Scholar 

  76. van Rooy, R. (2006). Attitudes and changing contexts. Dordrecht: Springer.

    Google Scholar 

  77. Rothschild, D., & Yalcin, S. (2013). On the dynamics of conversation. Ms. Oxford University and UC Berkeley. http://philpapers.org/rec/ROTOTD.

  78. Schulz, K. (2007). Minimal models in semantics and pragmatics: free choice, exhaustivity, and conditionals. Ph.D. thesis, University of Amsterdam: Institute for Logic, Language and Information, Amsterdam. http://www.illc.uva.nl/Publications/Dissertations/DS-2007-04.text.pdf.

  79. Shanahan, M. (2009). The frame problem. In E.N. Zalta (Ed.), The stanford encyclopedia of philosophy, winter 2009 edn. http://plato.stanford.edu/archives/win2009/entries/frame-problem/.

  80. Simons, M. (2005). Dividing things up: the semantics of or and the modal/or interaction. Natural Language Semantics, 13(3), 271–316. doi:10.1007/s11050-004-2900-7.

    Article  Google Scholar 

  81. Skyrms, B. (1974). Contraposition of the conditional. Philosophical Studies, 26(2), 145–147.

    Article  Google Scholar 

  82. Stalnaker, R.C. (1968). A theory of conditionals. In N. Rescher (Ed.), Studies in logical theory (pp. 98–112). Oxford: Basil Blackwell Publishers.

    Google Scholar 

  83. Stalnaker, R.C. (1975). Indicative conditionals. Philosophia, 5, 269–286. Page references to reprint in [87].

    Article  Google Scholar 

  84. Stalnaker, R.C. (1978). Assertion. In P. Cole (Ed.), Syntax and semantics 9: pragmatics (pp. 315–332). New York: Academic Press. References to reprint in [87].

    Google Scholar 

  85. Stalnaker, R.C. (1984). Inquiry. Cambridge: MIT Press.

    Google Scholar 

  86. Stalnaker, R.C. (1998). On the representation of context. Journal of Logic, Language, and Information, 7(1), 3–19.

    Article  Google Scholar 

  87. Stalnaker, R.C. (1999). Context and content: essays on intentionality in speech and thought. Oxford: Oxford University Press.

    Book  Google Scholar 

  88. Stalnaker, R.C. (2002). Common ground. Linguistics and Philosophy, 25(5–6), 701–721.

    Article  Google Scholar 

  89. Stalnaker, R.C. (2005). Conditional propositions and conditional assertions. In J. Gajewski, V. Hacquard, B. Nickel, S. Yalcin (Eds.), New work on modality, MIT working papers in linguistics (Vol. 51). Cambridge: MITWPL.

    Google Scholar 

  90. Starr, W.B. (2010). Conditionals, meaning and mood. Ph.D. thesis, Rutgers University: New Brunswick. http://hdl.rutgers.edu/1782.1/rucore10001.600001ETD.000056780.

  91. Starr, W.B. (2013). Structured possible worlds. Ms. Cornell University.

  92. Starr, W.B. (2013). Indicative conditionals, strictly. Ms. Cornell University.

  93. Starr, W.B. (to appear ). What ‘If’? Philosopher’s Imprint.

  94. Stone, M. (1997). The anaphoric parallel between modality and tense. Tech Rep. 97-06, University of Pennsylvania Institute for Research in Cognitive Science, Philadelphia, PA. http://www.cs.rutgers.edu/mdstone/pubs/ircs97-06.pdf.

  95. Stone, M. (1999). Reference to possible worlds. Tech. Rep. 49, Rutgers University Center for Cognitive Science. http://www.cs.rutgers.edu/mdstone/pubs/ruccs-49.pdf. Updated version of [94].

  96. Thomason, R.H. (2007). Conditionals and action logics. In Logical formalizations of commonsense reasoning: papers from the AAAI spring symposium, AAAI technical report SS-07-05. Menlo Park: AAAI Press.

  97. Van linden, A., & Verstraete, J.C. (2008). The nature and origins of counterfactuality in simple clauses: cross-linguistic evidence. Journal of Pragmatics, 40(11), 1865–1895. doi:10.1016/j.pragma.2008.03.00803.008. http://www.sciencedirect.com/science/article/pii/S0378216608000647.

    Article  Google Scholar 

  98. Veltman, F. (1985). Logics for conditionals. Ph.D. dissertation, Amsterdam: University of Amsterdam.

  99. Veltman, F. (1986). Data semantics and the pragmatics of indicative conditionals. In E. Traugott, A. ter Meulen, J. Reilly, C. Ferguson (Eds.), On conditionals. Cambridge: Cambridge University Press: Cambridge.

    Google Scholar 

  100. Veltman, F. (1996). Defaults in update semantics. Journal of Philosophical Logic, 25(3), 221–261. doi:10.1007/BF00248150.

    Article  Google Scholar 

  101. Veltman, F. (2005). Making counterfactual assumptions. Journal of Semantics, 22, 159–180. http://staff.science.uva.nl/veltman/papers/FVeltman-mca.pdf.

    Article  Google Scholar 

  102. Warmbrōd, K. (1981). Counterfactuals and substitution of equivalent antecedents. Journal of Philosophical Logic, 10(2), 267–289.

    Article  Google Scholar 

  103. Warmbrōd, K. (1981). An indexical theory of conditionals. Dialogue, Canadian Philosophical Review, 20(4), 644–664.

    Article  Google Scholar 

  104. Warmbrōd, K. (1982). A defense of the limit assumption. Philosophical Studies, 42(1), 53–66.

    Article  Google Scholar 

  105. Warmbrōd, K. (1983). Epistemic conditionals. Pacific Philosophical Quarterly, 64, 249–265. http://philpapers.org/archive/WAREC.1.pdf.

    Google Scholar 

  106. Williams, J.R.G. (2008). Conversation and conditionals. Philosophical Studies, 128, 211–233. http://www.personal.leeds.ac.uk/.

    Article  Google Scholar 

Download references

Acknowledgments

I benefitted greatly from participants in my Spring 2012 graduate seminar at Cornell and from audiences at NYU and the University of Michigan Philosophy and Linguistics Workshop. The basic ideas here come from my dissertation [90, Ch.3], so the lengthy acknowledgments there apply again here. I owe a special debt to conversations with Barbara Abbott, Ana Arregui, Josh Armstrong, Maria Bittner, Carlos Fasola, Thony Gillies, Gabe Greenberg, Michael Johnson, Jeff King, Philipp Koralus, Ernie Lepore, Karen Lewis, Barry Loewer, Salvador Mascarenhas, Sarah Murray, Carlotta Pavese, Adam Rigoni, Bob Stalnaker, Jason Stanley, James Shaw, Matthew Stone, Rich Thomason, Brian Weatherson and Malte Willer. I also wish to thank two anonymous reviewers whose insightful comments improved this paper significantly.

Author information

Authors and Affiliations

  1. Cornell University, Ithaca, NY, USA

    William B. Starr

Authors
  1. William B. Starr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to William B. Starr.

Appendix A: Dynamic Conditional Logic with ◃ (DCL ◃)

Appendix A: Dynamic Conditional Logic with ◃ (DCL ◃)

1.1 A.1 Syntax and Semantics

Definition 12 (DCL ◃ Syntax)

  • (1)    \(p\in {\mathcal {W}\mathit {ff}_{A}}\)    if \(p\in \mathcal {A}t=\mathsf {\{p_{0},p_{1},\ldots \}}\)

  •     (2)    \(\neg \phi \in \mathcal {W}\mathit {ff}_{A}\)    if \(\phi \in \mathcal {W}\mathit {ff}_{A}\)

  •     (3)    \(\Diamond \phi \in \mathcal {W}\mathit {ff}_{A}\)    if \(\phi \in \mathcal {W}\mathit {ff}_{A}\)

  •     (4)    \(\Box \phi \in \mathcal {W}\mathit {ff}_{A}\)    if \(\phi \in \mathcal {W}\mathit {ff}_{A}\)

  •     (5)    \((\phi \wedge \psi )\in \mathcal {W}\mathit {ff}_{A}\)    if \(\phi ,\psi \in \mathcal {W}\mathit {ff}_{A}\)

  •     (6)    \((\phi \vee \psi )\in \mathcal {W}\mathit {ff}_{A}\)    if \(\phi ,\psi \in \mathcal {W}\mathit {ff}_{A}\)

  •     (7)    \((\mathsf {if}\,{\phi })\,{\psi }\in \mathcal {W}\mathit {ff}_{C}\)    if \(\phi ,\psi \in \mathcal {W}\mathit {ff}_{A}\)

  •     (8)    \((\mathsf {if}\,{\phi })\,{\psi }\in \mathcal {W}\mathit {ff}_{C}\)    if \(\phi \in \mathcal {W}\mathit {ff}_{A}, \psi \in \mathcal {W}\mathit {ff}_{C}\)

  •     (9)    \(\neg \phi \in \mathcal {W}\mathit {ff}_{C}\)    if \(\phi \in {\mathcal {W}\mathit {ff}}_{C}\)

  •     (10)    \(\Diamond \phi \in \mathcal {W}\mathit {ff}_{C}\)    if \(\phi \in {\mathcal {W}\mathit {ff}}_{C}\)

  •     (11)    \(\Box \phi \in \mathcal {W}\mathit {ff}_{C}\)    if \(\phi \in {\mathcal {W}\mathit {ff}}_{C}\)

  •     (12)    \((\phi \wedge \psi )\in \mathcal {W}\mathit {ff}_{C}\)    if \(\phi ,\psi \in \mathcal {W}\mathit {ff}_{C}\cup \mathcal {W}\mathit {ff}_{A}\)

  •     (13)    \((\phi \vee \psi )\in \mathcal {W}\mathit {ff}_{C}\)    if \(\phi ,\psi \in \mathcal {W}\mathit {ff}_{C}\cup \mathcal {W}\mathit {ff}_{A}\)

  •     (14)    \(\lhd \phi \in \mathcal {W}\mathit {ff}_{A\lhd }\)    if \(\phi \in \mathcal {W}\mathit {ff}_{A}\)

  •     (15)    \(\phi_{1}\vee \phi_{2}\in \mathcal {W}\mathit {ff}_{A\lhd }\)    if \(\phi_{1},\phi_{2},\in \mathcal {W}\mathit {ff}_{A\lhd }\)

  •     (16)    \(\phi_{1}\wedge \phi_{2}\in \mathcal {W}\mathit {ff}_{A\lhd }\)    if \(\phi_{1},\phi_{2},\in \mathcal {W}\mathit {ff}_{A\lhd }\)

  •     (17)    \((\mathsf {if}\,{\phi })\,{\psi }\in \mathcal {W}\mathit {ff}_{\lhd }\)    if \(\phi \in \mathcal {W}\mathit {ff}_{A\lhd },\psi \in \mathcal {W}\mathit {ff}_{A}\)

  •     (18)    \(\Box \lhd \phi \in \mathcal {W}\mathit {ff}_{\lhd }\)    if \(\phi \in \mathcal {W}\mathit {ff}_{A}\)

  •     (19)    \(\Diamond \lhd \phi \in \mathcal {W}\mathit {ff}_{\lhd }\)    if \(\phi \in \mathcal {W}\mathit {ff}_{A}\)

  •     (20)    \((\mathsf {if}\,{\phi })\,{\psi }\in \mathcal {W}\mathit {ff}_{\lhd }\)    if \(\phi \in \mathcal {W}\mathit {ff}_{A\lhd }, \psi \in \mathcal {W}\mathit {ff}_{\lhd }\)

  •     (21)    \(\neg \phi \in \mathcal {W}\mathit {ff}_{\lhd }\)    if \(\phi \in {\mathcal {W}\mathit {ff}}_{\lhd }\)

  •     (22)    \((\phi \wedge \psi )\in \mathcal {W}\mathit {ff}_{\lhd }\)    if \(\phi ,\psi \in \mathcal {W}\mathit {ff}_{C}\cup \mathcal {W}\mathit {ff}_{A}\cup \mathcal {W}\mathit {ff}_{\lhd }\)

  •     (23)    \((\phi \vee \psi )\in \mathcal {W}\mathit {ff}_{\lhd }\)    if \(\phi ,\psi \in \mathcal {W}\mathit {ff}_{C}\cup \mathcal {W}\mathit {ff}_{A}\cup \mathcal {W}\mathit {ff}_{\lhd }\)

Remark 1

This does not permit indicative conditionals to nest in subjunctive conditionals or vice-versa, though that could be easily achieved. This also prohibits any conditional from nesting in the antecedent of a conditional. This too could be permitted, but seems unmotivated by natural language syntax.

Definition 13

(Worlds) W : 𝒜 ↦{1,0} where 𝒜t = {p 0,p 1,…}

Definition 14

(Contextual Possibilities/Information) \(c\subseteq W\)

Definition 15

(Contextual Alternatives)

  • C is a non-empty set of subsets of W

  • \(\varnothing \neq C\subseteq \mathcal {P}{(W)}\)

  • 𝒞 is the set of all such C

  • \(\bigcup C\) is the information embodied by C; the sets in C are called alternatives; overlapping and non-maximal alternatives are allowed.

Definition 16

(Selection Functions) (Let p, p′ ⊆ W and w ∈ W)

  •     (a)    f (w, p) ⊆ p    success

  •     (b)    f (w, p) = {w}, if w ∈ p    strong centering

  •     (c)    f (w, p) ⊆ p′ & f (w, p′) ⊆ p ⇒ f (w, p) = f (w, p′)    uniformity

  •     (d)   f (w, p) contains at most one world    uniqueness

  • While Stalnaker assumes (d), I will not.

Definition 17

(Contexts)

  • If C is a set of contextual alternatives and f is a selection function, then 〈C, f〉> is a context.

  • Nothing else is a context

Remark 2

(C f )

C f is an abbreviation for 〈C, f〉, and any set theoretic notation involving C f should be read as operating on the C inside C f . E.g. \(C_{f}\cap C'_{f}\) is an abbreviation for \(\langle {C\cap C'}\rangle_{f} \). This notation will never be used in a case like \(C_{f}\cap C'_{f'}\) if f ≠ f ′.

Definition 18

(Update Semantics)

Where \(C=\{c_{0},\ldots ,c_{n}\}\) and \(\overline {C}^{\phi }:=\{\,c_{0}-\bigcup (\{c_{0}\}[\phi ]),\ldots ,c_{n}-\bigcup (\{c_{n}\}[\phi ])\,\}\):

  •     (1)    C f [p]  =    \(\{\{w\in c_{0}\mid w(\mathsf {p})=1\},\ldots ,\{w\in c_{n}\mid w(\mathsf {p})=1\}\}_{f}\)

  •     (2)    C f [¬ϕ]  =    \(\overline {C}^{\phi }_{f}\)

  •     (3)    \(C_{f}[\phi \wedge \psi ]\)  =    \((C_{f}[\phi ])[\psi ]\)

  •     (4)    \(C_{f}[\phi \vee \psi ]\)  =    \(C_{f}[\phi ]\cup C_{f}[\psi ]\)

Remark 3

Above, \(\overline {C}^{\phi }\) may be pronounced the \(\phi \) complement of C. Forming this set amounts to eliminating the \(\phi \)-worlds from each alternative in C. So \(C_{f}[\neg \phi ]\) will eliminate \(\phi \)-worlds from each alternative in C.

Definition 19

(Support, Truth in w)

  1. (1)

    Support \(C_{f}\vDash \phi \Leftrightarrow \bigcup (C_{f}[\phi ])=\bigcup C_{f}\)

  2. (2)

    Truth in w,f \(w,f\vDash \phi \Leftrightarrow \{\{w\}\}_{f}[\phi ]=\{\{w\}\}_{f}\)

Definition 20

(Propositional Content) ⟦ϕ⟧f={w∣w,f⊨ϕ}

Definition 21

(Inquisitive Content) \(\lfloor \phi \rfloor_{f}=C'\iff \{W\}_{f}[\phi ]=\langle {C',f}\rangle \)

Remark 4

Since \(\lhd \) is the only operator sensitive to f , I will often omit f when writing \(\lfloor \phi \rfloor \) and ⟦ϕ⟧ for ϕ that do not contain it.

Definition 22

(Entailment) \(\phi_{1},\ldots ,\phi_{n}\vDash \psi \Leftrightarrow \forall C_{f}:C_{f}[\phi_{1}]\cdots [\phi_{n}]\vDash \psi \)

  • If \(C_{f}[\phi_{1}]\cdots [\phi_{n}][\psi ]\) is defined.

Definition 23

(Conditional Semantics) \(C_{f}[(\mathsf{if}\,{\phi})\,{\psi}]=\left\{\begin{array}{ll}\{a\in C\mid C_{f}[\phi]\vDash \psi\}_{f} &\text{if }\bigcup(C_{f}[\phi])\neq\varnothing\\\text{Undefined} &\text{otherwise}\end{array}\right.\)

Definition 24

(Alternative Counterfactual Expansion ◃) Let f be a selection function and \(c=\bigcup C\):\(\displaystyle C_{f}[\lhd \alpha ]=\{\{w'\mid \exists w\in c: w'\in f(w,a)\}\mid a\in \lfloor \alpha \rfloor \}_{f}\)

Remark 5

To illustrate the definition, consider C f [◃A]:

$$\begin{array}{rll} C_{f}[\lhd\mathsf{A}] &=& \{\{w'\mid\exists w\in c: w'\in f(w,a)\}\mid a\in\lfloor\mathsf{A}\rfloor\}_{f}\\ &=& \{\{w'\mid\exists w\in c: w'\in f(w,⟦\mathsf{A}⟧)\}\}_{f} \end{array} $$

The second line follows since \(\lfloor \mathsf {A}\rfloor =\{⟦\mathsf {A}⟧ \}\). By the meaning of ∨, C f [◃A ∨ ◃B] = C f [◃A] ∪ C f [◃B]. This results in (51).

  1. (51)

    {{w′∣ ∃w ∈ c : w′ ∈ f (w, ⟦A⟧)}, {w′ ∣ ∃w ∈ c : w′∈ f (w, ⟦B⟧)}} f

As desired, C f [◃(A ∨ B)] will yield the same result.

$$\begin{array}{rll} C_{f}[\lhd\mathsf{(A\vee B)}] &=& \{\{w'\mid\exists w\in c: w'\in f(w,a)\}\mid a\in\lfloor\mathsf{A\vee B}\rfloor\}_{f}\\ &=& \{\{w'\mid\exists w\in c: w'\in f(w,⟦\mathsf{A}⟧)\},\{w'\mid\exists w\in c: w'\in f(w,⟦\mathsf{B}⟧)\}\}_{f} \end{array} $$

In this case, the second line follows since \(\lfloor \mathsf {A\vee B}\rfloor =\{⟦ \mathsf {A}⟧ ,⟦ \mathsf {B}⟧ \}\). Thus, evaluating a disjunctive antecedent comes to the same thing whether the disjunction takes wide or narrow scope. It comes to evaluating whether or not the expanded set of alternatives supports the consequent. This raises a pertinent question: I’ve said what it takes for a set of worlds to support a sentence, but what does it take for a set of alternatives to support a sentence? Requiring that C[ϕ] = C produces an interesting but more fine-grained notion of support, requiring not only that ϕ provided no new information, but also introduced no new alternatives. To capture the old notion of support, I want to say that the underlying set of worlds is the same in C as in C[ϕ]. This is accomplished in Definition 22.

1.2 A.2 Results

Fact 3

(Simple Subjunctive Conditional Propositions)

For \(\psi \in \mathcal {W}\mathit {ff}_{A}\) and free of \(\Diamond \)and \(\Box \), where \(\lfloor \alpha \rfloor =\{a_{0},\ldots ,a_{n}\}\):

  1. 1.

    \(⟦ (\mathsf {if}\,{\lhd \alpha })\,{\psi }⟧_{f}\) is well-defined iff \(\forall w\in W\), \(\exists a\in \lfloor \alpha \rfloor :f(w,a)\neq \varnothing \)By Def’s 20, 19.2, 23 and 24, \(⟦ (\mathsf {if}\,{\lhd \alpha })\,{\psi }⟧_{f}= \{w\mid \{f(w,a_{0}),\ldots ,f(w,a_{n})\}_{f}\vDash \psi \}\) when defined. By Def 23, this is defined if and only if for all \(w\in W, \bigcup \{f(w,a_{0}),\ldots ,f(w,a_{n})\}\neq \varnothing \). This holds at an arbitrary w iff \(\exists a\in \lfloor \alpha \rfloor :f(w,a)\neq \varnothing \).

  2. 2.

    When defined, \(⟦ (\mathsf {if}\,{\lhd \alpha })\,{\psi }⟧_{f}= \{w\mid \bigcup \{f(w,a_{0}),\ldots ,f(w,a_{n})\}\subseteq ⟦ \psi ⟧_{f}\}\)

Proof

As previously noted, \(⟦ (\mathsf {if}\,{\lhd \alpha })\,{\psi }⟧_{f}= \{w\mid \{f(w,a_{0}),\ldots ,f(w,a_{n})\}_{f}\vDash \psi \}\). \(\psi \) is neither conditional nor modal, so \(\forall C_{f}: C_{f}\vDash \psi \) iff \(\bigcup C\subseteq ⟦ \psi ⟧_{f}\). □

Fact 4

(Import-Export) (if ϕ 1) ((if ϕ 2)ψ) ⫤⊨ (if ϕ 1 ∧ ϕ 2)ψ

Proof

When both forms are defined, they amount to the same update:

$$\begin{array}{@{}rcl@{}} C_{f}[(\mathsf{if}\,{\phi_{1}})\,{((\mathsf{if}\,{\phi_{2}})\,{\psi})}] &=& \{a\in C_{f}\mid C_{f}[\phi_{1}]\vDash(\mathsf{if}\,{\phi_{2}})\,{\psi}\}_{f} \\ &=& \{a\in C_{f}\mid \bigcup(C_{f}[\phi_{1}][(\mathsf{if}\,{\phi_{2}})\,{\psi}])=\bigcup(C_{f}[\phi_{1}])\}_{f}\\ &=&\{a\in C_{f}\mid \bigcup\{a'\in C_{f}[\phi_{1}]\mid C_{f}[\phi_{1}][\phi_{2}]\vDash\psi\}=\bigcup(C_{f}[\phi_{1}])\}_{f}\\ & =& \{a\in C_{f}\mid C_{f}[\phi_{1}][\phi_{2}]\vDash\psi\}_{f} \\ & =& \{a\in C_{f}\mid C_{f}[\phi_{1}\wedge\phi_{2}]\vDash\psi\}_{f} \\ & =& C_{f}[(\mathsf{if}\,{\phi_{1}\wedge\phi_{2}})\,{\psi}] \end{array} $$

Cases of undefinedness do not count towards validity and may be disregarded. □

Fact 5

(Modus Ponens)

(if ϕ) ψ, ϕ ⊨ ψ

Proof

Either \(C_{f}[(\mathsf {if}\,{\phi })\,{\psi }]=C_{f}\) or \(C_{f}[(\mathsf {if}\,{\phi })\,{\psi }]=\varnothing_{f}\). In the former case, \(C_{f}[\phi ]\vDash \psi \) and the condition needed for the entailment, \(\bigcup (C_{f}[(\mathsf {if}\,{\phi })\,{\psi }][\phi ][\psi ])=\bigcup (C_{f}[(\mathsf {if}\,{\phi })\,{\psi }][\phi ])\), is equivalent to \(\bigcup (C_{f}[\phi ][\psi ])=\bigcup (C_{f}[\phi ])\). But since \(C_{f}[\phi ]\vDash \psi \), this obtains. In the latter case \(C_{f}[(\mathsf {if}\,{\phi })\,{\psi }][\phi ][\psi ]=\varnothing_{f}=C_{f}[(\mathsf {if}\,{\phi })\,{\psi }][\phi ]\), so the entailment must also hold. □

Definition 25

(Persistence) ϕ is persistent iff \(C'_{f}\vDash \phi \) if \(C_{f}\vDash \phi \) and \(\bigcup C'\subseteq \bigcup C\). (I.e. \(\phi \)’s support persists after more information comes in.)

Fact 6

In general, \(\Diamond \phi \) is not persistent. Take a c containing many worlds but only one \(\phi \)-world w. Then \(\{c\}_{f}\vDash \Diamond \phi \), but \(c-\{w\}\subseteq c\) and \(\{c-\{w\}\}_{f}\nvDash \Diamond \phi \).

Fact 7

(Disj. Ants. 1) For persistent \(\psi \), \((\mathsf {if}\,{\phi_{1}\vee \phi_{2}})\,{\psi }\vDash (\mathsf {if}\,{\phi_{1}})\,{\psi }\wedge (\mathsf {if}\,{\phi_{2}})\,{\psi }\)

Proof

The premise tests that \(C_{f}[\phi_{1}]\cup C_{f}[\phi_{2}]\vDash \psi \). The conclusion presupposes that \(\bigcup (C_{f}[\phi_{1}])\neq \varnothing \) and \(\bigcup (C_{f}[\phi_{2}])\neq \varnothing \), and tests that \(C_{f}[\phi_{1}]\vDash \psi \) and \(C_{f}[\phi_{2}]\vDash \psi \). Since \(\bigcup (C_{f}[\phi_{1}])\subseteq (\bigcup (C_{f}[\phi_{1}])\cup \bigcup (C_{f}[\phi_{2}]))\) and \(\bigcup (C_{f}[\phi_{2}])\subseteq (\bigcup (C_{f}[\phi_{1}])\cup \bigcup (c[\phi_{2}]))\), this test must be successful when \(\psi \) is persistent but may not be successful when \(\psi \) isn’t persistent. □

Remark 6

(if p ∨ ¬p) ♢ p ⊮ ((if p) ♢ p) ∧ ((if ¬p) ♢ p). If there are both p and ¬p worlds in c all presuppositions will be met and the premise will successfully test c. The second conjunct of the conclusion won’t successfully test, despite its presuppositions being met.

Remark 7

While Fact 8 is limited to atomic \(\alpha \) and \(\beta \), it could be proven inductively for any \(\alpha ,\beta \in \mathcal {W}ff_{A}\), but I omit that lengthy proof here.

Fact 8

(Disj. Ants. 2) For persistent \(\psi \) and atomic \(\alpha , \beta \), \((\mathsf {if}\,{\lhd (\alpha \vee \beta )})\,{\psi }\vDash (\mathsf {if}\,{\lhd \alpha })\,{\psi }\wedge (\mathsf {if}\,{\lhd \beta })\,{\psi }\)

Proof

The premise tests that \(C_{f}[\lhd (\alpha \vee \beta )]\vDash \psi \). By Definition 24, \(C_{f}[\lhd (\alpha \vee \beta )]=\{\{w'\mid \exists w\in c: w'\in f(w,a)\}\mid a\in \lfloor \alpha \vee \beta \rfloor \}_{f}\). By Definitions 18.4, 18.1 and 21, \(\lfloor \alpha \vee \beta \rfloor =\{⟦ \alpha ⟧ ,⟦ \beta ⟧ \}\), so the first line below follows.

$$\begin{array}{@{}rcl@{}} C_{f}[\lhd(\alpha\vee\beta)] &=& \{\{w'\mid\exists w\in c: w'\in f(w,⟦\alpha⟧)\},\{w'\mid\exists w\in c: w'\in f(w,⟦\beta⟧)\}\}_{f} \\ &=&\{\{w'\mid\exists w\in c: w'\in f(w,⟦\alpha⟧)\}\}_{f}\cup\{\{w'\mid\exists w\in c: w'\in f(w,⟦\beta⟧)\}\}_{f} \\ &=& C_{f}[\lhd\alpha]\cup C_{f}[\lhd\beta] \\ &=& C_{f}[\lhd\alpha\vee\lhd\beta] \end{array} $$

The second line follows by set theory, and the last two by Definitions 24 and 18.4, respectively. From this equality and Fact 7, the entailment follows. □

Rights and permissions

Reprints and permissions

About this article

Cite this article

Starr, W.B. A Uniform Theory of Conditionals. J Philos Logic 43, 1019–1064 (2014). https://doi.org/10.1007/s10992-013-9300-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-013-9300-8

Keywords

Search

Navigation