Skip to main content
SpringerLink
Log in

Uncertainty and Persistence: a Bayesian Update Semantics for Probabilistic Expressions

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

Abstract

This paper presents a general-purpose update semantics for expressions of subjective uncertainty in natural language. First, a set of desiderata are established for how expressions of subjective uncertainty should behave in dynamic, update-based semantic systems; then extant implementations of expressions of subjective uncertainty in such models are evaluated and found wanting; finally, a new update semantics is proposed. The desiderata at the heart of this paper center around the contention that expressions of subjective uncertainty express beliefs which are not persistent (i.e. beliefs that won’t necessarily survive the addition of new information that is compatible with all previous information), whereas propositions express beliefs that are persistent. I argue that if we make the move of treating updates in a dynamic semantics as Bayesian updates, i.e. as conditionalization, then expressions of subjective uncertainty will behave the way we want them to without altering the way propositions behave.

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

Similar content being viewed by others

Notes

  1. I also ignore in this paper interactions between credal operators and negation, and embedded credal operators.

  2. Example due to Dan Lassiter, p.c.

  3. Veltman defines this as ‘acceptance’. I’ve diverged in terminology here because I want to capture the fact that an information state with this relationship to φ licenses an assertion of φ.

  4. Willer’s model excludes the empty set from information states. This is of no crucial importance, and I’ve omitted that restriction here.

  5. This may be too strong: there are cases in which might φ is followed by an explicit dismissal of φ as irrelevant:

    1. (i)

      Running the Large Hadron Collider might destroyer the universe, but the chance is so astronomically low that we don’t need to worry about it.

    This certainly doesn’t seem contradictory (see [36] for further data and commentary). Whether or not this is a problem for the meaning that Willer ascribes to this structural configuration depends on whether you model the effect of ‘astronomically low chance’ or ‘we don’t need to worry about it’ as altering this characteristic structure. I think it might be better to identify this structural configuration with something like ‘acknowledged’ possibility.

  6. An anonymous reviewer has pointed out to me that neither of Yalcin’s systems is capable of capturing the difference between true epistemic contradictions (¬φ followed by might φ) and pseudo epistemic contradictions (might φ followed by ¬φ). His original system predicts that both sequences result inevitably in contradiction; his modified system predicts that neither will.

  7. Note that the concept of (un)certainty in terms of credence structures is fundamentally separate from the notion of an expression of subjective uncertainty.

  8. This is compatible with the traditional notion that a proposition denotes the set of worlds in which it is true. We could say that the semantic meaning of a proposition is exactly that set, and that the association between a proposition and probability 1 is a fact about the pragmatics of assertion, not about the semantics of propositions. This follows, for instance, from a knowledge norm of assertion (as argued for extensively by [47]): given a knowledge norm of assertion, to assert a proposition is to present yourself as though you know it to be true—as though you associate it with probability 1. Engaging thoroughly with the literature on norms of assertion is far outside the scope of this paper. For our purposes here, it’s enough to note that the difference between a semantic view (in which the semantics for propositions associates them with probability 1) and a pragmatic view (in which the semantics for propositions is classical, but asserting them has the pragmatic effect of proposing that they be associated with probability 1) is more or less notational: the Bayesian Update Semantics that I outline in this paper is isomorphic to a split system that treats propositional assertions in the manner of [51] and employs my new machinery only for expressions of subjective uncertainty.

    What is more problematic is trying to understand the difference between these two sentences:

    1. (i)

      a. Paul came to the party.

      b. Paul definitely came to the party.

    If we make the intuitive move of saying that definitely is a credal operator that associates the proposition it scopes over with the probability range 1, and we accept my proposal that asserting a proposition (either semantically or pragmatically) comprises a request for interlocutors to associate that proposition with probability 1, then we would expect these sentences to be identical. Are they? If so, then we have an inroads to cashing out the default pragmatics of assertion: we could say that the default pragmatics of asserting a proposition associates that proposition with a covert definitely operator; i.e. [[φ]] = [[definitely φ]] = 〈φ, [1,1] 〉. On this view, the sentences in (i) are semantically identical; they differ only with respect to the covertness of the definitely operator. If there are substantial empirically attested differences in the behavior or effects of the sentences in (i), we could say that precisely because asserting a proposition already associates it with probability 1, adding the definitely operator is strictly redundant, and therefore serves some other pragmatic purpose, a la Barker & Taranto’s ([3]) analysis of clear. Further empirical work is clearly required here.

  9. There are cases in which φ will not be the proposition selected by max to conditionalize a ? φ- μ on; however, in all such cases the result of conditionalizing on the proposition(s) chosen by max will be equivalent to conditionalizing on φ. Consider the following example:

    1. (i)

      W = { w 1,w 2,w 3,w 4} φ = { w 2,w 3} ψ = { w 2,w 3,w 4} \(\mu = \left [ \begin {array}{ccc} w_1 & \rightarrow & .3 \\ w_2 & \rightarrow & .5 \\ w_3 & \rightarrow & .2 \\ w_4 & \rightarrow & 0 \\ \end {array} \right ]\)

    If this μ is in an I updated with 〈φ,[1,1] 〉, the proposition chosen by max to conditionalize it on given the definition in (78) will not be φ, but ψ. However, the result will be the same:

    1. (ii)

      \(\mu \!\!\upharpoonright _{\psi } = \mu \!\!\upharpoonright _{\varphi } = \left [ \begin {array}{ccc} w_{1} & \rightarrow & 0 \\ w_{2} & \rightarrow & \frac {5}{7} \approx .714 \\ w_{3} & \rightarrow & \frac {2}{7} \approx .286 \\ w_{4} & \rightarrow & 0 \\ \end {array} \right ]\)

  10. This could potentially be cashed out by way of formal tools encoding relations of plausible causality, such as probabilistic graphical models.

  11. An anonymous reviewer points out that one might intuitively expect the opposite. Wouldn’t people be more likely to inquire after their interlocutor’s grounds for making stronger assertions, rather than weaker ones? After all, stronger claims require stronger evidence. However, it’s important to stress that the model developed here is a model of what happens after a listener has decided to accept the content of their interlocutor’s assertion at face value—we’re restricting our attention to cases in which listeners assume their interlocutors to have told the truth. I share the intuition that the stronger a claim is, the more likely a listener will be to ask questions about its grounds—because they’ll be less likely to accept it as true. Seeking grounds to justify the assertion of a proposition comes from skepticism or suspicion. However, I believe that if we assume that the listener has total faith in the veracity of the speaker’s utterance, the relevant generalization holds. Asking someone for their grounds for an expression of subjective uncertainty doesn’t necessarily sound suspicious—it’s compatible with taking their statement for granted but seeking further clarifying information. A perfectly trusting listener, when told that Paul was probably at the party last night, can good-naturedly inquire after what the source of uncertainty is: which pieces of information point toward him having been there, and which pieces of information make it seem less than certain. The same perfectly trusting listener, when told that Paul was at the party last night, will simply accept that that is a fact.

References

  1. Anand, P., & Brasoveanu, A. (2009). Modal concord as modal modification. In Prinzhor, M., Schmitt, V., & Zobel, S. (Eds.), Proceedings of Sinn und Bedeutung 14 (pp. 1936). Vienna. Prinzhor, M., Schmitt, V., & Zobel, S. (Eds.)

  2. Baltag, A., & Renne, B. (2016). Dynamic epistemic logic. In Zalta, E.N. (Ed.), , The stanford encyclopedia of philosophy. https://plato.stanford.edu/archives/win2016/ entries/dynamic-epistemic.

    Google Scholar 

  3. Barker, C., & Taranto, G. (2003). The paradox of asserting clarity Koskinen, P. (Ed.) , Proceedings of the western conference on linguistics (WECOL) 2002 (vol. 14, pp. 10–21) Fresno, CA: California State University.

  4. Beaver, D. (2001). Presupposition and assertion in dynamic semantics (Studies in Logic, Language and Information 16). Stanford, CA: Center for the Study of Language & Information (CSLI), Stanford University.

  5. Demey, L., Kooi, B., & Sack, J. (2016). Logic and probability. In Zalta, E.N. (Ed.), , The stanford encyclopedia of philosophy. https://plato.stanford.edu/ archives/win2016/entries/logic-probability.

    Google Scholar 

  6. DeRose, K. (1991). Epistemic possibilities. Philosophical Review, 100(4), 581–605.

    Article  Google Scholar 

  7. Dorr, C., & Hawthorne, J. (2013). Embedding epistemic modals. Mind, 122, 867–913.

    Article  Google Scholar 

  8. Farkas, D., & Bruce, K. (2010). On reacting to assertions and polar questions. Journal of Semantics, 27, 81–118.

    Article  Google Scholar 

  9. von Fintel, K., & Gillies, A. (2010). Muststaystrong!. Natural Language Semantics, 18(4), 351–383.

    Article  Google Scholar 

  10. von Fintel, K., & Gillies, A. (2011). Might made right. In Egan, A., & Weatherson, B. (Eds.) , In Epistemic modality (pp. 108130). New York: Oxford University Press.

  11. Gillies, A. (2004). Epistemic conditionals and conditional epistemics. Noû,s, 38(4), 585–616.

    Article  Google Scholar 

  12. Groenendijk, J., Stokhof, M., & Veltman, F. (1996). Coreference and modality. In Lappin, S. (Ed.) The handbook of contemporary semantic theory, blackwell (pp. 179–213).

  13. Grosz, P. (2009). Grading modality: a new approach to modal concord and its relatives. In Proceedings of sinn und bedeutung, (Vol. 14 pp. 185–201).

  14. Hacking, I. (1967). Possibility. The Philosophical Review, 76(2), 143–168.

    Article  Google Scholar 

  15. Halpern, J. (2003). Reasoning about uncertainty. Cambridge, MA: MIT Press.

    Google Scholar 

  16. Heim, I. (1982). The semantics of definite and indefinite noun phrases PhD thesis. Amherst: University of Massachusetts.

    Google Scholar 

  17. Heim, I. (1983). On the projection problem for presuppositions. In Proceedings of the second West Coast Conference on Formal Linguistics (pp. 114–125).

  18. Heim, I. (1992). Presupposition projection and the semantics of attitude verbs. Journal of Semantics, 9, 183–221.

    Article  Google Scholar 

  19. Jeffrey, R. (1992). Probability and the art of judgment. New York: Cambridge University Press.

  20. Kamp, H. (1981). A theory of truth and semantic representation. In Groenendijk, J., Janssen, T., & Stokhof, M (Eds.) Formal Methods in the Study of Language.

  21. Karttunen, L. (1972). Possible and must. In Kimball, J. (Ed.) Syntax and semantics, vol 1, (pp. 1–20): Academic Press.

  22. Kaufmann, S. (2004). Conditioning against the grain: Abduction and indicative conditionals. Journal of Philosophical Logic, 33, 583–606.

    Article  Google Scholar 

  23. Kaufmann, S. (2005). Conditional predictions: a probabilistic account. Linguistics and Philosophy, 28, 181–231.

    Article  Google Scholar 

  24. Kaufmann, S. (2009). Conditionals right and left: Probabilities for the whole family. Journal of Philosophical Logic, 38, 1–53.

    Article  Google Scholar 

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

    Article  Google Scholar 

  26. Kratzer, A. (1981). The notional category of modality. In Words, worlds and contexts: New approaches in world semantics (pp. 38–74): De Gruyter.

  27. Kratzer, A. (1991). Modality. In Von stechow, A., & Wunderlich, D. (Eds.) Semantics: An International Handbook of Contemporary Research (pp. 639–650): de Gruyter.

  28. Lassiter, D. (2011). Measurement and modality: The scalar basis of modal semantics. PhD thesis: NYU.

  29. Lassiter, D. (2014). The weakness of must: In defense of a Mantra. Proceedings of SALT, 24, 597–618.

    Article  Google Scholar 

  30. Lassiter, D. (2016). Must, knowledge, and (in)directness. Natural Language Semantics, 24(2), 117–163.

    Article  Google Scholar 

  31. MacFarlane, J. (2011). Epistemic modals are assessment-sensitive. In Egan, A., & Weatherson, B. (Eds.), , Epistemic modality (pp. 144–178). New York: Oxford University Press.

    Chapter  Google Scholar 

  32. McCready, E., & Ogata, N. (2007). Evidentiality, modality and probability. Linguistics and Philosophy, 30, 147–206.

    Article  Google Scholar 

  33. Moss, S. (2015). On the semantics and pragmatics of epistemic vocabulary. Semantics and Pragmatics, 8, 1–81.

    Article  Google Scholar 

  34. Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Morgan Kaufmann.

  35. Rothschild, D. (2012). Expressing credences. In Proceedings of the Aristotelian Society.

  36. Rudin, D. (2016). Deriving a variable-strength might. In Proceedings of Sinn und Bedeutung 20.

  37. Stalnaker, R. (1978). Assertion. In Cole, P. (Ed.) Syntax and semantics 9, academic press (pp. 315–32).

  38. Stephenson, T. (2007). Judge dependence, epistemic modals, and predicates of personal taste. Linguistics and Philosophy, 30, 487–525.

    Article  Google Scholar 

  39. Stone, M. (1994). The reference argument of epistemic Must. In Proceedings of IWCS 1 (pp. 181–190).

  40. Swanson, E. (2006). Interactions with context. PhD thesis: MIT.

  41. Swanson, E. (2011). How not to theorize about the language of subjective uncertainty. In Egan, A., & Weatherson, B. (Eds.), Epistemic Modality: Oxford University Press.

  42. Swanson, E. (2016). The application of constraint semantics to the language of subjective uncertainty. Journal of Philosophical Logic, 45(2), 121–146.

    Article  Google Scholar 

  43. Teller, P. (1972). Epistemic possibility. Philosophia, 2(4), 303–320.

    Article  Google Scholar 

  44. Veltman, F. (1996). Defaults in upyear semantics. Journal of Philosophical Logic, 25(3), 221–261.

    Article  Google Scholar 

  45. Willer, M. (2010). Modality in flux PhD thesis. Austin: University of Texas.

    Google Scholar 

  46. Willer, M. (2013). Dynamics of epistemic modality. Philosophical Review, 122 (1), 45–92.

    Article  Google Scholar 

  47. Williamson, T. (2000). Knowledge and its limits. Oxford: Oxford University Press.

    Google Scholar 

  48. Yalcin, S. (2007). Epistemic modals. Mind, 116, 983–1026.

    Article  Google Scholar 

  49. Yalcin, S. (2010). Probability operators. Philosophy Compass, 5(11), 916–937.

    Article  Google Scholar 

  50. Yalcin, S. (2011). Nonfactualism about epistemic modals. In Egan, A., & Weatherson, B. (Eds.) Epistemic Modality: Oxford University Press.

  51. Yalcin, S. (2012). Context probabilism. In Proceedings of the 18th Amsterdam Colloquium (pp. 12–21).

  52. Yanovich, I. (2014). Standard contextualism strikes back. Journal of Semantics, 31, 67–114.

    Article  Google Scholar 

Download references

Acknowledgments

Thanks above all to Pranav Anand for countless hours of productive and enjoyable conversation, without which this paper would not exist. For reading and commenting on this paper in various incarnations, thanks to Adrian Brasoveanu, Donka Farkas, Dan Lassiter and Margaret Kroll. Thanks also to Floris Roelofsen, for helpful discussion and a pleasant walk, and especially grateful thanks to Chris Barker & Dylan Bumford, who were this paper’s saviors at one point in its history. Finally, thanks to Frank Veltman and an anonymous reviewer for JPL for efficiently shepherding this paper to publication, and improving it substantially in the process. All errors are mine.

Author information

Authors and Affiliations

  1. Stevenson 238, University of California Santa Cruz, 1156 High Street, Santa Cruz, CA, 95064, USA

    Deniz Rudin

Authors
  1. Deniz Rudin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Deniz Rudin.

Appendices

Appendix A: Bayesian Update Semantics in one page

  1. (90)

    Syntax for \(\mathcal {L}\) and \(\mathcal {CL}\):

    1. a.

      Any p \(\in \mathcal {A}\) is a formula of \(\mathcal {L}\) Where \(\mathcal {A}\) is the set of all atomic propositions

    2. b.

      If φ and ψ are formulas of \(\mathcal {L}\), then so are (φ∧ψ) and ¬(φ)

    3. c.

      Nothing else is a formula of \(\mathcal {L}\)

    4. d.

      Every formula of \(\mathcal {L}\) is a formula of \(\mathcal {CL}\)

    5. e.

      For any formula φ of \(\mathcal {L}\) and any C in \(\mathfrak {C}\), C φ is a formula of \(\mathcal {CL}\) Where \(\mathfrak {C}\) is the set of all credal operators

    6. f.

      Nothing else is a formula of \(\mathcal {CL}\)

  1. (91)

    Semantics for \(\mathcal {L}\) and \(\mathcal {CL}\): Relative to some model M with domain W,

    1. a.

      [[p]] = {w ∈ W : p is true in w}

    2. b.

      [[φ∧ψ]] = [[φ]]∩[[ψ]]

    3. c.

      [[¬φ]] = W - [[φ]]

    4. d.

      [[C φ]] = [[C]]([[φ]])

  1. (92)

    Credal Operators ∀C\(\in \mathfrak {C}\), C is a function of the form λ φ.〈φ, int 〉 Where int is a real interval and φ⊆ W Example denotations:

    1. a.

      [[might]] = λ φ.〈φ, (0,1] 〉

    2. b.

      [[probably]] = λ φ.〈φ, (.5,1] 〉

  1. (93)

    Probability Measures Given some finite set of worlds W, μ : ℘(W) → [0,1] is a probability measure iff

    1. a.

      realism: μ(W) = 1

    2. b.

      finite additivity: [ ∀φ,ψ⊆ W : φ∩ψ = ∅](μ(φ) + μ(ψ) = μ(φ∪ψ))

  1. (94)

    Information States

    1. a.

      Any I \(\subseteq \mathfrak {M}\) is an information state Where \(\mathfrak {M}\) is the set of all probability measures

    2. b.

      The maximally uninformed information state is \(\mathfrak {M}\)

  1. (95)

    The Update Function

    1. a.

      I[ Φ] = { μ : ∃μ ′∈I s.t. μ∈μ ′[ Φ]}

    2. b.

      μ[ 〈φ, int 〉] = { μ ′ : ∃ψ∈max(↑(μ, φ, int)) s.t. \(\mu \!\!\upharpoonright _{\psi }\) = μ ′} Where ↑(μ, φ, int) { ψ⊆ W : \(\mu \!\!\upharpoonright _{\psi }\)(φ) ∈int}, and For some set of propositions S, max(S) = { ψ∈ S : ¬∃ψ ′∈ S s.t. ψ⊂ψ ′}

    3. c.

      μ[ φ] = μ[ 〈φ, [1,1] 〉]

Appendix B: More on Epistemic Contradiction

One of Yalcin’s ([48]) arguments that epistemic contradictions really are contradictions is that they, unlike the Moore paradoxes that they resemble, are still bad under suppose; this turns out to be true regardless of conjunct order:

  1. (96)

    a. #Suppose it is raining and it might not be raining.

    b. #Suppose it is not raining and it might be raining.

    c. #Suppose it might not be raining, and it’s raining.

    d. #Suppose it might be raining, and it’s not raining.

Willer ([46]) has this to say:

Insofar as the point of a supposition is to adopt a state of information that supports a certain hypothesis, we then expect that supposing “It might not be raining and it is raining” is just as odd as supposing “It is raining and it might not be raining.”

That’s all Willer says about this; I’ll make this argument at greater length and in greater detail here. The idea is that there’s something about the suppositional project that makes conjunctions under suppose behave differently than conjunctions normally behave; in a dynamic semantics, conjunction generally is translated as sequential update; however, in a suppositional project, the goal seems to be more like world-building; in other words, when a speaker instructs an addressee to suppose a list of conjuncts, the implicit directive is something like: ‘construct the largest possible information state that licenses all of these’. If conjuncts under suppose are incompatible, then this project is impossible, and anomaly results.

However, it’s possible to force the interpretation of suppositional projects to be narrative/sequential; in this case, conjunction seems to behave like sequential update:

  1. (97)

    Suppose that it might be raining, and you go outside to check, and it’s not raining.

This sounds perfectly fine! But for truly illicit update sequences, narrativizing the suppositional project does not alleviate the badness:

  1. (98)

    a. #Suppose that your dad was born in Greece and your dad wasn’t born in Greece.

    b. #Suppose that your dad was born in Greece, and you receive a new familial record, and your dad wasn’t born in Greece.

As Yalcin notes, epistemic contradictions are also bad in the antecedents of conditionals, and this is independent of conjunct order as well:

  1. (99)

    a. #If it is raining and it might not be raining, then I’ll bring an umbrella just in case.

    b. #If it is not raining and it might be raining, then I’ll bring an umbrella just in case.

    c. #If it might be raining and it’s not raining, then I’ll bring an umbrella just in case.

    d. #If it might not be raining and it’s raining, then I’ll bring an umbrella just in case.

The same patterns I noted with suppose are characteristic of conditional antecedents:

  1. (100)

    a. If it might be raining, and you go outside to check, and it’s not raining, then you can open all the windows.

    b. #If your dad was born in Greece, and you receive a new familial record, and your dad wasn’t born in Greece, then you must be very surprised.

If the antecedent is made to seem narrative, the anomaly associated with conjoined incompatible updates disappears; the anomaly associated with incoherent update sequences remains.

Finally, as [7] note, these conjunctions are bad when embedded in disjunctions:

  1. (101)
    1. a.

      #Either it’s raining and it might not be raining, or it’s a good day for a picnic.

    2. b.

      #Either it’s not raining and it might be raining, or it’s a good day for a picnic.

    3. c.

      #Either it might be raining and it’s not raining, or it’s a good day for a picnic.

    4. d.

      #Either it might not be raining and it’s raining, or it’s a good day for a picnic.

Once again, we see the same pattern:

  1. (102)
    1. a.

      Either it might have been raining and you went outside to check and it wasn’t raining, or you haven’t been doing your job.

    2. b.

      #Either your dad was born in Greece and you received a new familial record and your dad wasn’t born in Greece, or you haven’t been doing your job.

Narrativizing the conjunction again rescues the incompatible updates; the incoherent updates remain bad. It seems that we’ve discovered a fact about the pragmatics of embedded conjunction in at least some contexts: often the implicit project suggested by embedded conjunction is to build an information state compatible with all conjuncts. It follows from the definition of incompatibility that incompatible updates will be anomalous relative to this project. Developing, clarifying, formalizing and defending the proposal that embedded conjunction often gives rise to ‘world-building’ pragmatics would be material for a paper in itself; my only goal in bringing it up here is to have identified some salient differences between the behavior in these contexts of updates that I’ve classified as incoherent and updates that I’ve classified as merely incompatible, thereby discrediting or at least complicating the claim that the anomalousness of might φ plus ¬φ sequences in these contexts is due to the fact that those sequences are truly incoherent. In fact, I believe that I have found support in my brief examination of these contexts for the notion that these sequences behave exactly as we would expect incompatible but non-incoherent update sequences to behave.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rudin, D. Uncertainty and Persistence: a Bayesian Update Semantics for Probabilistic Expressions. J Philos Logic 47, 365–405 (2018). https://doi.org/10.1007/s10992-017-9431-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-017-9431-4

Keywords

Search

Navigation