Abstract
In Hollis’ paradox, A and B each chose a positive integer and whisper their number to C. C then informs them, jointly, that they have chosen different numbers and, moreover, that neither of them are able to work out who has the greatest number. A then reasons as follows: B cannot have 1, otherwise he would know that my number is greater, and by the same reasoning B knows that I don’t have 1. But then B also cannot have 2, otherwise he would know that my number is greater (since he knows I don’t have 1). This line of reasoning can be repeated indefinitely, effectively forming an inductive proof, ruling out any number – an apparent paradox. In this paper we formalise Hollis’ paradox using public announcement logic, and argue that the root cause of the paradox is the wrongful assumption that A and B assume that C’s announcement necessarily is successful. This resolves the paradox without assuming that C can be untruthful, or that A and B are not perfect reasoners, like other solutions do. There are similarities to the surprise examination paradox. In addition to a semantic analysis in the tradition of epistemic logic, we provide a syntactic one, deriving conclusions from a set of premises describing the initial situation – more in the spirit of the literature on Hollis’ paradox. The latter allows us to pinpoint which assumptions are actually necessary for the conclusions resolving the paradox.
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Notes
- 1.
Hollis [9] already hints at this assumption: “...each of us has to assume that the other is not stupid...”.
- 2.
- 3.
See [12] for an overview of different variants and a discussion of historic origins.
- 4.
In muddy children, that happens in the last joint announcement by the children.
- 5.
Here and in the following we mean “truthful” in the strong sense that the announcement is in fact true (rather than only believed to be true).
- 6.
If \(\phi \) is, e.g., a primitive proposition, then “was true” is the same as “is true”. However, this is not the case in general: it could be that \(\phi \) was true in a certain state before the announcement, but became false in the same state as a result of the announcement. The canonical example of the latter is the so-called Moore sentence \(\phi = p \wedge \lnot K_a p\).
- 7.
Gerbrandy [8, pp. 27–29] discusses the same point in the context of surprise examination.
- 8.
This can be expressed elegantly by the iterated announcement operator in [13]: \(M_1 \models \langle ann2^*\rangle \lnot ann2\), which is true iff \(M_1 \models \underbrace{\langle ann2 \rangle \cdots \langle ann2 \rangle }_n\lnot ann2\) for some \(n \ge 1\). See also [20] for a further discussion of this and related operators.
- 9.
There are two schemas but actually infinitely many formulas.
- 10.
We could assume that these premises are common knowledge, writing e.g., \(C_{\{a,b\}} (1_i \rightarrow \lnot p_i)\). However, it turns out that assuming common knowledge is not needed, and it is of interest to illucidate exactly how many levels of nested knowledge are sufficient: e.g., two levels for (A4).
- 11.
Observe that \(\alpha \) expresses that (1) the two numbers are different, and (2) both agents consider each of the numbers to be the greatest (\(\alpha \) implies \(ann1 \wedge ann2\) but not the other way around).
- 12.
Nevertheless, that was no problem for \(157_a \rightarrow [ann1][ann2]\lnot K_a\lnot 2_b\), which happens to hold in those models too.
- 13.
\((we \wedge \lnot K we) \vee (th \wedge [\lnot we]\lnot K th) \vee (fr \wedge [\lnot we][\lnot th] \lnot K fr) \vee K \bot \). Note that Gerbrandy assumes that the knowledge modalities are K45 rather than S5.
- 14.
Note that this kind of self-reference is not the same as saying that “you don’t know it now and you still don’t know it after it is announced that you don’t know it now” as briefly discussed at the end of Sect. 3.
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The first author was supported by L. Meltzers Høyskolefond.
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Ågotnes, T., Sakama, C. (2023). A Formal Analysis of Hollis’ Paradox. In: Alechina, N., Herzig, A., Liang, F. (eds) Logic, Rationality, and Interaction. LORI 2023. Lecture Notes in Computer Science, vol 14329. Springer, Cham. https://doi.org/10.1007/978-3-031-45558-2_24
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