Abstract
David Kaplan observed in Kaplan (1995) that the principle \(\forall p \Diamond \forall q (Qq \leftrightarrow q = p)\) cannot be verified at a world in a standard possible worlds model for a quantified bimodal propositional language. This raises a puzzle for certain interpretations of the operator Q: it seems that some proposition p is such that is not possible to query p, and p alone. On the other hand, Arthur Prior had observed in Prior (1961) that on pain of contradiction, ∀p(Qp →¬p) is Q only if one true proposition is Q and one false proposition is Q. The two observations are related: ∀p(Qp →¬p) is elusive in that it is not possible for the proposition to be uniquely Q. Kaplan based his model-theoretic observation on Cantor’s theorem, but there is a less well-known link between this simple set-theoretic observation and Prior’s remark. We generalize the link to develop a heuristic designed to move from Cantor’s theorem to the observation that a variety of sentences of the bimodal language express propositions that cannot be Q uniquely. We highlight the analogy between some of these results and some set-theoretic antinomies and suggest that the phenomenon is richer than one may have anticipated.
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Acknowledgments
I would like to thank Andrew Bacon for discussion and two reviewers for this journal for helpful comments and suggestions. I’m grateful to audiences at Chapman, Oxford, Oslo, and St Andrews, where I presented earlier versions of the paper.
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Appendix: Syntactic Proofs
Appendix: Syntactic Proofs
We provide proof outlines for some of the lemmas introduced in Section 4. In what follows, we only explicitly annotate the use of certain modal axioms and definitions but we generally leave quantificational steps implicit, including appeals to instances of the Converse Barcan schema, \(\Box \forall p \varphi \to \forall p \Box \varphi \), which are provable in the system.
Lemma 3
\(\vdash \forall p \ (Qp \to \Box (q \to \gamma )) \to \gamma \)
We outline a derivation of ¬γ →∃q(Qq ∧♢(q ∧¬γ)):
1 | \(\neg \gamma \to \exists p(p \wedge \Box (p \to \exists q(Qq \wedge \Diamond (p \wedge q)))\) | Def γ |
2 | \(p \wedge \Box (p \to \exists q(Qq \wedge \Diamond (p \wedge q))) \to \Box \Box (p \to \exists q(Qq \wedge \Diamond (p \wedge q)))\) | 4 (\(\Box \varphi \to \Box \Box \varphi \)) |
3 | \(p \wedge \Box (p \to \exists q(Qq \wedge \Diamond (p \wedge q))) \to \exists q(Qq \wedge \Diamond (p \wedge q))\) | T |
4 | \(\neg \gamma \to \exists q(Qq \wedge \exists p (\Diamond (p \wedge q) \wedge \Box \Box (p \to \exists q(Qq \wedge \Diamond (p \wedge q)))))\) | 1, 2, 3 |
5 | \(\neg \gamma \to \exists q(Qq \wedge \exists p \ \Diamond ((p \wedge q) \wedge \Box (p \to \exists q(Qq \wedge \Diamond (p \wedge q))))) \) | 4 |
6 | \(\neg \gamma \to \exists q(Qq \wedge \Diamond (q \wedge \exists p \ (p \wedge \Box (p \to \exists q(Qq \wedge \Diamond (p \wedge q))))) \) | 5 |
7 | ¬γ →∃q(Qq ∧♢(q ∧¬γ)) | 6, Def γ |
Lemma 4
⊩ closedχ(her(χ)).
1 | \(\forall q(Qq \to \Box (q \to her(\chi ))) \to \) | Def her(χ) |
\(\forall q(Qq \to \Box (q \to \forall p (closed_{\chi }(p) \to p)))\) | ||
2 | \(\Box (q \to \forall p (closed_{\chi }(p) \to p)) \to \) | K |
\(\Box \forall p (closed_{\chi }(p) \to (q \to p))\) | ||
3 | \(\Box \forall p (closed_{\chi }(p) \to (q \to p)) \to \) | |
\(\forall p \ \Box (closed_{\chi }(p) \to (q \to p))\) | ||
4 | \(\forall p \ \Box (closed_{\chi }(p) \to (q \to p)) \to \) | K |
\(\forall p (\Box \ closed_{\chi }(p) \to \Box (q \to p))\) | ||
5 | \(\forall p (closed_{\chi }(p) \to \Box \ closed_{\chi }(p))\) | 4 (\(\Box \varphi \to \Box \Box \varphi \)) |
6 | \(\forall p \ \Box (closed_{\chi }(p) \to (q \to p)) \to \) | 4, 5 |
\(\forall p \ (closed_{\chi }(p) \to \Box (q \to p))\) | ||
7 | \(\forall q(Qq \to \Box (q \to her(\chi )) \to \) | 1, 2, 6 |
\(\forall p (closed_{\chi }(p) \to \forall q (Qq \to \Box (q \to p)))\) | ||
8 | \(\forall q(Qq \to \Box (q \to her(\chi )) \to \forall p (closed_{\chi }(p) \to (\chi \to p))\) | 7, Def closedχ(p) |
9 | ∀p(closedχ(p) →(χ →p)) →(χ →∀p(closedχ(p) →p)) | |
10 | \((\forall q(Qq \to \Box (q \to her(\chi )) \to (\chi \to her(\chi ))\) | 8, 9, Def her(χ) |
11 | \((\forall q(Qq \to \Box (q \to her(\chi )) \wedge \chi ) \to her(\chi )\) | 10 |
11 | \(\Box (\forall q(Qq \to \Box (q \to her(\chi )) \wedge \chi ) \to her(\chi ))\) | 11, RN |
Lemma 5
\(\vdash closed_{\chi }(\forall p(Qp \to \Box (p \to her(\chi )) \wedge \chi ))\).
1 | \(\Box (\forall q(Qq \to \Box (q \to her(\chi )) \wedge \chi ) \to her(\chi ))\) | Lemma 4 |
2 | \(\forall q(Qq \to \Box (q \to (\forall p(Qp \to \Box (p \to her(\chi )) \wedge \chi ))) \to \) | |
\(\forall q (Qq \to \Box (q \to her(\chi ))))\) | 1 | |
3 | \((\forall q(Qq \to \Box (q \to (\forall p(Qp \to \Box (p \to her(\chi )))) \wedge \chi )) \wedge \chi )\to \) | |
\( (\forall q (Qq \to \Box (q \to her(\chi )))\wedge \chi )\) | 2 | |
4 | \(\Box (\forall q(Qq \to \Box (q \to (\forall p(Qp \to \Box (p \to her(\chi )) \wedge \chi ))) \wedge \chi )\to \) | |
\( (\forall q (Qq \to \Box (q \to her(\chi )))\wedge \chi )\) | 3, RN |
Lemma 7
\(\vdash her(\chi ) \wedge Q \ her(\chi ) \to \Diamond (her(\chi ) \wedge Q \ her(\chi ) \wedge \forall q (Qq \rightarrow \Box (q \to \\ \neg (her(\chi ) \wedge Q \ her(\chi )))))\).
1 | her(χ) →∀p(closedχ(p) →p) | Def her(χ) |
2 | (her(χ) ∧Q her(χ)) →¬ closedχ(¬(her(χ) ∧Q her(χ))) | 1 |
3 | \(\neg closed_{\chi } (\neg (her(\chi ) \wedge Q \ her(\chi ))) \to \neg \Box ((\forall q (Qq \to \) | |
\(\Box (q \to \neg (her(\chi ) \wedge Q \ her(\chi )))) \wedge \chi ) \to \neg (her(\chi ) \wedge Q \ her(\chi )))\) | Def closedχ(φ) | |
4 | her(χ) ∧Q her(χ) →♢(χ ∧her(χ) ∧Q her(χ) ∧ | |
\(\forall q (Qq \to \Box (q \to \neg (her(\chi ) \wedge Q \ her(\chi )))))\) | 2, 3 |
Lemma 8
⊩¬(her(χ) ∧ Q her(χ)).
1 | \(\forall q(Qq \to \Box (q \to \neg (her(\chi ) \wedge Q \ her(\chi )))) \to (Q \ her(\chi ) \to \) | |
\(\Box ((her(\chi ) \to \neg (her(\chi ) \wedge Q \ her(\chi )))))\) | 1 | |
2 | \(\Box (her(\chi ) \to \neg (her(\chi ) \wedge Q \ her(\chi ))) \to (her(\chi ) \to \) | T |
¬(her(χ) ∧Q her(χ))) | ||
3 | \(\forall q(Qq \to \Box (q \to \neg (her(\chi ) \wedge Q \ her(\chi )))) \to (Q \ her(\chi ) \to \) | |
(her(χ) →¬(her(χ) ∧Q her(χ)))) | 1, 2 | |
4 | (her(χ) ∧Q her(χ)) →¬(Q her(χ) →(her(χ) → | |
¬(her(χ) ∧Q her(χ)))) | ||
5 | \((her(\chi ) \wedge Q \ her(\chi )) \to \neg \forall q(Qq \to \Box (q \to \neg (her(\chi ) \wedge Q \ her(\chi ))))\) | 3, 4 |
6 | \((\chi \wedge (her(\chi ) \wedge Q \ her(\chi )) \to \neg \forall q(Qq \to \Box (q \to \) | 5 |
¬(her(χ) ∧Q her(χ)))) | ||
7 | \(\Box (\chi \wedge (her(\chi ) \wedge Q \ her(\chi )) \to \neg \forall q(Qq \to \Box (q \to \) | 6, RN |
¬(her(χ) ∧Q her(χ))))) | ||
8 | \(\neg \Diamond (\chi \wedge (her(\chi ) \wedge Q \ her(\chi )) \wedge \forall q(Qq \to \Box (q \to \) | 7 |
¬(her(χ) ∧Q her(χ))))) | ||
9 | ¬(her(χ) ∧Q her(χ)) | 8, Lemma 7 |
Lemma 10
\(\vdash ^{\prime } Q! \ her(\tau ) \to \tau \).
1 | \(closed_{\tau }(\forall p(Qp \to \Box (p \to her(\tau )))\wedge \tau )\) | Lemma 5 |
2 | \(\Box (her(\tau ) \to \forall p (Qp \to \Box (p \to her(\tau ))))\) | 1, Def her(τ) |
3 | \(Q \ her(\tau ) \to \forall p_{2}\Box ((her(\tau ) \wedge Qp_{2})\to \Box (p_{2} \to her(\tau )))\) | 2 |
4 | \(Q! \ her(\tau ) \to \forall p_{1} (Qp_{1} \to \forall p_{2}\Box ((p_{1} \wedge Qp_{2})\to \Box (p_{2} \to her(\tau ))))\) | 3 |
5 | \(\forall p (world(p) \to (Q! \ her(\tau ) \to \Box (p \to Q \ her(\tau ))))\) | Def Q!, W |
6 | \(\forall p (world(p) \to (\forall p_{1}(Qp_{1} \to \forall p_{2} \Box ((p_{1} \wedge Qp_{2}) \to \Box (p_{2} \to \) | |
\( \exists p_{3} (p_{3} \wedge \Box (p \to Qp_{3})))))))\) | 4, 5 |
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Uzquiano, G. Elusive Propositions. J Philos Logic 50, 705–725 (2021). https://doi.org/10.1007/s10992-020-09582-5
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DOI: https://doi.org/10.1007/s10992-020-09582-5