The Knowability Paradox and Unsuccessful Updates

In this paper we undertake an analysis of the knowability paradox in the light of modal epistemic logics and of the phenomena of unsuccessful updates. The knowability paradox stems from the Church-Fitch observation that the plausible knowability principle, according to which all truths are knowable, yields the unacceptable conclusion that all truths are known. We show that the phenomenon of an unsuccessful update is the reason for the paradox arising. Based on this diagnosis, we propose a restriction on the knowability principle which resolves the paradox.

Paradoxes, epistemic

The four primary epistemic paradoxes are the lottery, preface, knowability, and surprise examination paradoxes. The lottery paradox begins by imagining a fair lottery with a thousand tickets in it. Each ticket is so unlikely to win that we are justified in believing that it will lose. So we can infer that no ticket will win. Yet we know that some ticket will win. In the preface paradox, authors are justified in believing everything in their books. Some preface their book by claiming that, given human frailty, they are sure that errors remain. But then they justifiably believe both that everything in the book is true, and that something in it is false. The knowability paradox results from accepting that some truths are not known, and that any truth is knowable. Since the first claim is a truth, it must be knowable. From these claims it follows that it is possible that there is some particular truth that is known to be true and known not to be true. The final paradox concerns an announcement of a surprise test next week. A Friday test, since it can be predicted on Thursday evening, will not be a surprise yet, if the test cannot be on Friday, it cannot be on Thursday either. For if it has not been given by Wednesday night, and it cannot be a surprise on Friday, it will not be a surprise on Thursday. Similar reasoning rules out all other days of the week as well; hence, no surprise test can occur next week. On Wednesday, the teacher gives a test, and the students are taken completely by surprise.

INTUITIONISTIC EPISTEMIC LOGIC

We outline an intuitionistic view of knowledge which maintains the original Brouwer–Heyting–Kolmogorov semantics for intuitionism and is consistent with the well-known approach that intuitionistic knowledge be regarded as the result of verification. We argue that on this view coreflection A → KA is valid and the factivity of knowledge holds in the form KA → ¬¬A ‘known propositions cannot be false’.We show that the traditional form of factivity KA → A is a distinctly classical principle which, like tertium non datur A ∨ ¬A, does not hold intuitionistically, but, along with the whole of classical epistemic logic, is intuitionistically valid in its double negation form ¬¬(KA ¬ A).Within the intuitionistic epistemic framework the knowability paradox is resolved in a constructive manner. We argue that this paradox is the result of an unwarranted classical reading of constructive principles and as such does not have the consequences for constructive foundations traditionally attributed it.