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I recently reposted my favourite logic puzzle, namely the blue-eyed islander puzzle. I am fond of this puzzle because in order to properly understand the correct solution (and to properly understand why the alternative solution is incorrect), one has to think very clearly (but unintuitively) about the nature of knowledge.

There is however an additional subtlety to the puzzle that was pointed out in comments, in that the correct solution to the puzzle has two components, a (necessary) upper bound and a (possible) lower bound (I’ll explain this further below the fold, in order to avoid blatantly spoiling the puzzle here). Only the upper bound is correctly explained in the puzzle (and even then, there are some slight inaccuracies, as will be discussed below). The lower bound, however, is substantially more difficult to establish, in part because the bound is merely possible and not necessary. Ultimately, this is because to demonstrate the upper bound, one merely has to show that a certain statement is logically deducible from an islander’s state of knowledge, which can be done by presenting an appropriate chain of logical deductions. But to demonstrate the lower bound, one needs to show that certain statements are not logically deducible from an islander’s state of knowledge, which is much harder, as one has to rule out all possible chains of deductive reasoning from arriving at this particular conclusion. In fact, to rigorously establish such impossiblity statements, one ends up having to leave the “syntactic” side of logic (deductive reasoning), and move instead to the dual “semantic” side of logic (creation of models). As we shall see, semantics requires substantially more mathematical setup than syntax, and the demonstration of the lower bound will therefore be much lengthier than that of the upper bound.

To complicate things further, the particular logic that is used in the blue-eyed islander puzzle is not the same as the logics that are commonly used in mathematics, namely propositional logic and first-order logic. Because the logical reasoning here depends so crucially on the concept of knowledge, one must work instead with an epistemic logic (or more precisely, an epistemic modal logic) which can properly work with, and model, the knowledge of various agents. To add even more complication, the role of time is also important (an islander may not know a certain fact on one day, but learn it on the next day), so one also needs to incorporate the language of temporal logic in order to fully model the situation. This makes both the syntax and semantics of the logic quite intricate; to see this, one only needs to contemplate the task of programming a computer with enough epistemic and temporal deductive reasoning powers that it would be able to solve the islander puzzle (or even a smaller version thereof, say with just three or four islanders) without being deliberately “fed” the solution. (The fact, therefore, that humans can grasp the correct solution without any formal logical training is therefore quite remarkable.)

As difficult as the syntax of temporal epistemic modal logic is, though, the semantics is more intricate still. For instance, it turns out that in order to completely model the epistemic state of a finite number of agents (such as 1000 islanders), one requires an infinite model, due to the existence of arbitrarily long nested chains of knowledge (e.g. “${A}$ knows that ${B}$ knows that ${C}$ knows that ${D}$ has blue eyes”), which cannot be automatically reduced to shorter chains of knowledge. Furthermore, because each agent has only an incomplete knowledge of the world, one must take into account multiple hypothetical worlds, which differ from the real world but which are considered to be possible worlds by one or more agents, thus introducing modality into the logic. More subtly, one must also consider worlds which each agent knows to be impossible, but are not commonly known to be impossible, so that (for instance) one agent is willing to admit the possibility that another agent considers that world to be possible; it is the consideration of such worlds which is crucial to the resolution of the blue-eyed islander puzzle. And this is even before one adds the temporal aspect (e.g. “On Tuesday, ${A}$ knows that on Monday, ${B}$ knew that by Wednesday, ${C}$ will know that ${D}$ has blue eyes”).

Despite all this fearsome complexity, it is still possible to set up both the syntax and semantics of temporal epistemic modal logic in such a way that one can formulate the blue-eyed islander problem rigorously, and in such a way that one has both an upper and a lower bound in the solution. The purpose of this post is to construct such a setup and to explain the lower bound in particular. The same logic is also useful for analysing another well-known paradox, the unexpected hanging paradox, and I will do so at the end of the post. Note though that there is more than one way to set up epistemic logics, and they are not all equivalent to each other.

(On the other hand, for puzzles such as the islander puzzle in which there are only a finite number of atomic propositions and no free variables, one at least can avoid the need to admit predicate logic, in which one has to discuss quantifiers such as ${\forall}$ and ${\exists}$. A fully formed predicate temporal epistemic modal logic would indeed be of terrifying complexity.)

Our approach here will be a little different from the approach commonly found in the epistemic logic literature, in which one jumps straight to “arbitrary-order epistemic logic” in which arbitrarily long nested chains of knowledge (“${A}$ knows that ${B}$ knows that ${C}$ knows that \ldots”) are allowed. Instead, we will adopt a hierarchical approach, recursively defining for ${k=0,1,2,\ldots}$ a “${k^{th}}$-order epistemic logic” in which knowledge chains of depth up to ${k}$, but no greater, are permitted. The arbitrarily order epistemic logic is then obtained as a limit (a direct limit on the syntactic side, and an inverse limit on the semantic side, which is dual to the syntactic side) of the finite order epistemic logics.

I should warn that this is going to be a rather formal and mathematical post. Readers who simply want to know the answer to the islander puzzle would probably be better off reading the discussion at the puzzle’s own blog post instead.