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Some Results on the Limits of Thought

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Abstract

Generalizing on some arguments due to Arthur Prior and Dmitry Mirimanoff, we provide some further limitative results on what can be thought.

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Notes

  1. This argument, attributed to Kaplan, is discussed by Martin Davies in [2] Appendix 9, and by many others subsequently (Kaplan’s own discussion didn’t appear until [4]). Kaplan’s puzzle begins with the observation that if the set of all worlds has cardinality κ, then the set of all propositions, determined by sets of worlds, must have cardinality 2κ. This is incompatible with the further assumption that for each proposition p, and a given agent and time, there is a world in which the agent thinks p, and only p, at the given time. Otherwise, we would have a map f from a set of worlds onto the set of all propositions, i.e., the power set of the set of all worlds, which is ruled out by Cantor’s theorem.

  2. See Lewis’s discussion in §2.3 [5].

  3. Prior’s observation predates the literature on Kaplan’s paradox, but its significance to that discussion has for the most part been overlooked. See [8].

  4. Indeed the assumption that Prior’s proposition is thought uniquely is not only metaphysically impossible but logically incoherent.

  5. The resources needed to prove Prior’s theorem are surprisingly minimal: you only need classical propositional logic and universal instantiation for the propositional quantifiers. Weakening universal instantiation allows one to block the theorem, but as argued in Bacon et al [1] this move leaves us exposed to other problems.

  6. Prior’s proposition is related to a familiar diagonal argument for Cantor’s theorem that there is no function from the set of worlds W onto the set of sets of worlds \(\mathcal {P}W\): suppose \(f: W\rightarrow \mathcal {P}W\) maps W onto \(\mathcal {P}W\) and consider Λ = {w ∈ W : w∉f(w)}. If w is such that f(w) = Λ, then w ∈Λ iff w∉Λ. If we interpret f as mapping w to the set of worlds at which belong to some proposition thought at w, then Λ just becomes the set of worlds at which Prior’s proposition is true. It follows that at no world is one thinking propositions whose disjunction is necessarily equivalent to Prior’s proposition. Thus, in particular, it follows that Prior’s proposition is not thought uniquely at any world. A similar connection is explored in [7] (above we exploited a proof that no function from W to \(\mathcal {P}(W)\) is onto, whereas Moore exploits a proof that no function from \(\mathcal {P}(W)\) to W is one-one. Since there many worlds at which any given proposition is thought uniquely, Moore’s analysis requires the axiom of choice, whereas ours does not).

  7. Unlike the liar, however, Prior’s theorem is provable in quantified propositional logic without appeal to any controversial disquotational premises.

  8. This model is not uncontroversial — indeed Deutsch [3] has questioned whether a possible world is the sort of thing that can be a member of a set or class. However, as we shall see later, nothing hangs on this modeling assumption. This result can be formulated, as Prior’s is, in the language of modal quantified propositional logic.

  9. In epistemic logic we say that a world y is epistemically accessible to x iff every proposition known at x is true at y. Parallel arguments can be made in the present context with an analogous universal definition of accessibility, but the existential definition makes some proofs more straightforward.

  10. More formally, in quantified propositional logic, we can argue as follows:

    1. 1.

      ((p1∨...∨pn) ⇔ ∀p(Qp → ¬p)) → ((Qp1∧...∧Qpn) → ¬(p1∨...∨pn)∧ ¬ ∀p(Qp → ¬p))

    2. 2.

      ((p1 ∨ ... ∨ pn) ⇔∀p(Qp → ¬p)) → ((Qp1 ∧ ... ∧ Qpn) → (¬p1 ∧ ... ∧¬pn) ∧∃p(Qp ∧ p))

    3. 3.

      ((p1 ∨ ... ∨ pn) ⇔ ∀p(Qp → ¬p)) → ((Qp1 ∧ ... ∧ Qpn) →∃p(Qp ∧ (p ≠ p1 ∧ ... ∧ p ≠ pn)))

  11. See [6].

  12. The above reasoning assumes that every world is possible relative to every other world, but the formal derivation of the paradox will only require the principles of S4. To show that the displayed sentence expresses the set of non-well-founded, without the assumption that every world is possible relative to every other world, a more subtle argument is required.

  13. In symbols:

    1. 1.

      Qγ → ¬γ from lemma 3 by logic

    2. 2.

      ∀p(Qp → p = γ) → γ from lemma 2 by logic and T

    3. 3.

      ∀p(p = γ → Qp) → Qγ logic

    4. 4.

      ¬∀p(Qp ⇔ p = γ) from 1, 2, and 3 by logic

  14. It is interesting to note that in some sense, λ is just the demodalisation of γ. If one deletes the modal operators appearing in γ on gets the formula ¬∃p(p ∧ (p → ∃q(Qq ∧ (p ∧ q)))), which by inspection is logically equivalent to ∀p(Qp → ¬p). This suggests that in the special case where there is only one possible world, the two paradoxes coincide.

References

  1. Bacon, A., Hawthorne, J., Uzquiano, G. (2016). Higher-Order Free logic and the Prior-Kaplan paradox. Canadian Journal of Philosophy, 46(4-5), 493–541.

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  2. Davies, M. (1981). Meaning quantification, Necessity: Themes in Philosophical Logic. Routledge.

  3. Deutsch, H. (2013). Resolution of some paradoxes of propositions. Analysis, 74(1), 26–34.

    Article  Google Scholar 

  4. Kaplan, D., & Kaplan, D. (1995) In Sinnnott-Armstrong, W., Raffman, D., & Asher, N. (Eds.), A problem in possible-world semantics, (pp. 41–52). Cambridge: Cambridge University Press.

  5. Lewis, D.K. (1986). On the Plurality of Worlds. Blackwell Publishers.

  6. Mirimanoff, D. (1917). Les antinomies de Russell et de Burali-Forti: et le problème fondamental de la théorie des ensembles.

  7. Moore, A.W. (1984). Possible worlds and diagonalization. Analysis, 44(1), 21–22.

    Article  Google Scholar 

  8. Prior, A. (1961). On a family of paradoxes. Notre Dame Journal of Formal Logic, 2(1), 16–32.

    Article  Google Scholar 

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Authors and Affiliations

  1. University of Southern California, Los Angeles, CA, USA

    Andrew Bacon & Gabriel Uzquiano

  2. University of St. Andrews, Scotland, UK

    Gabriel Uzquiano

Authors
  1. Andrew Bacon
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  2. Gabriel Uzquiano
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Corresponding author

Correspondence to Andrew Bacon.

Additional information

Thanks to Jeremy Goodman for feedback on an earlier version of the paper, and thanks to audiences at the University of St. Andrews.

Appendix

Appendix

Here we sketch the outline of a proof of Lemmas 2 and 3 in modal quantified propositional logic with S4.

We begin with Lemma 2, which reads:

  • □(p → γ) →□(∀q(Qq → q = p) → γ).

Steps that can be filled in using ordinary quantificational reasoning are omitted. To show our claim it suffices to prove:

  • (∀q(Qq ⇔ p = q) ∧¬γ) → ♢(p ∧¬γ)

For if this conditional is provable then it is necessary: □((∀q(Qq ⇔ p = q) ∧¬γ) → ♢(p ∧¬γ)), and so one can conclude ♢(∀q(Qq ⇔ p = q) ∧¬γ) → ♢♢(p ∧¬γ), which implies ♢(∀q(Qq ⇔ p = q) ∧¬γ) →♢(p ∧¬γ) in S4. The lemma follows by contraposition and applying duality and the de Morgan laws.

The antecedent amounts to the following two claims:

  1. 1.

    ∀q(Qq ⇔ p = q)

  2. 2.

    ∃r(r ∧□(r → ∃q(Qq ∧♢(r ∧ q))))

Claim 2 says that some truth, r, necessitates the claim that something Q is compossible with r. It follows that in fact, something Q is compossible with r. Since, by 1, p is the only Q proposition, it follows that p in particular is compossible with r.

  1. 3.

    ♢(p ∧ r)

Moreover, given that r necessitates the claim that something Q is compossible with r, it’s necessary that it necessitates this claim, by S4:

  • 4. □□(r → ∃q(Qq ∧♢(r ∧ q)))

Given 3 and 4 we can infer:

  • 5. ♢(p ∧ r ∧□(r → ∃q(Qq ∧♢(r ∧ q))))

using the inference from ♢A and □B to ♢(A ∧ B). Now 5 entails

  • 6. ♢(p ∧∃r(r ∧□(r → ∃q(Qq ∧♢(r ∧ q)))))

using existential generalization, and the fact that we can apply logic inside the scope of ♢. 6 is just the required ♢(p ∧¬γ).

On to now a sketch of a proof of Lemma 2:

  • γ → ∀p(Qp → ¬p).

Firstly note that one can prove, using the T axiom:

  1. 1.

    □((Qp ∧ p) → (Qp ∧♢((Qp ∧ p) ∧ p)))

Applying existential generalization to p in the consequent gives us:

  1. 2.

    □((Qp ∧ p) → ∃q(Qq ∧♢((Qp ∧ p) ∧ q)))

So by propositional logic:

  1. 3.

    (Qp ∧ p) → ((Qp ∧ p) ∧□((Qp ∧ p) → ∃q(Qq ∧♢((Qp ∧ p) ∧ q))))

Applying existential generalization in the consequent to (Qp ∧ p) gives us:

  1. 4.

    (Qp ∧ p) → ∃r(r ∧□(r → ∃q(Qq ∧♢(r ∧ q))))

Or, equivalently, (Qp ∧ p) → ¬γ. Contraposing and generalizing in p we get 5 from which 6 and 7 follow:

  1. 5.

    ∀p(γ → ¬(Qp ∧ p))

  2. 6.

    γ → ∀p ¬ (Qp ∧ p))

  3. 7.

    γ → ∀p(Qp → ¬p))

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Bacon, A., Uzquiano, G. Some Results on the Limits of Thought. J Philos Logic 47, 991–999 (2018). https://doi.org/10.1007/s10992-018-9458-1

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