Improbable Knowing

How much can we turn the screw on counter-examples to the KK principle? The principle, also sometimes called “positive introspection”, says that if one knows that P, one knows that one knows that P. It is widely, although not universally, acknowledged that the KK principle is false, and not just for the boring reason that one can know that P without having formed the belief that one knows that P. One can know that P, and believe that one knows that P, without knowing that one knows that P, because one is not in a strong enough epistemic ...

2012

… manuscript, University of …, 2010

Most epistemologists hold that knowledge entails belief. However, proponents of this claim rarely offer a positive argument in support of it. Rather, they tend to treat the view as obvious and assert that there are no convincing counterexamples. We find this strategy to be problematic. We do not find the standard view obvious, and moreover, we think there are cases in which it is intuitively plausible that a subject knows some proposition P without -or at least without determinately -believing that P. Accordingly, we present five plausible examples of knowledge without (determinate) belief, and we present empirical evidence suggesting that our intuitions about these scenarios are not atypical.

Erkenntnis, 2012

This paper considers, and rejects, three strategies aimed at showing that the KK-principle fails even in most favourable circumstances (all emerging from Williamson's Knowledge and its Limits). The case against the final strategy provides positive grounds for thinking that the principle should hold good in such situations. 1 Introduction This paper counters three arguments inspired by Timothy Williamson's Knowledge and its Limits 1 (henceforth, K&L) against the luminosity of knowing, the thesis (KK) that if one knows that p, then one knows, or is at least in a position to know, that one knows it. Some preliminary remarks concerning the significance of this project are perhaps in order, since one might well wonder why anyone should hold (KK) in the first place. For, isn't it the case that in the eyes of non-sceptics, whether epistemic internalists or externalists, the sceptic has knowledge but fails to believe that she has-in light of misconceptions about what knowledge requires? Assuming that knowing entails believing, the sceptic knows but does not know that she knows; and it is surely a moot point whether she is in a position to know that she knows it. However, one may concede that (KK) can fail when the subject has misconceptions about what knowledge requires and yet wonder whether a rightthinking, rational and reflective person can fail to know that they know something.

History and Philosophy of Logic, 2024

In his Knowledge and Belief (1962), Hintikka establishes his system of epistemic logic with the KK (Knowing that One Knows, in symbols, KpKKp) principle (KK for short). However, his system of epistemic logic and the KK principle are grounded upon his strong notion of knowledge, which requires that knowledge is infallible, that is, it makes further inquiry pointless, and becomes 'discussion-stopper'; knowledge implies truth, to wit, cognitive agents will not be mistaken in their knowledge; cognitive agents will be 'perfect logicians', i.e. they have infinitive capability of logical inference. Hintikka calls the argument for KK from the strong notion of knowledge as the

Polish Journal of Philosophy, 2014

Outstanding Contributions to Logic, 2017

In recent years, Fitch's knowability paradox received a lot of attention. According to some, this paradox proves that verificationism, or more generally, Dummett's antirealistic theory of meaning is impossible. According to others, there must be something wrong with the proof: how else can it be explained that Dummett's reasonable type of anti-realism reduces to absurdities like naive idealism and even lingualism? According to anti-realism, if it is impossible to know something, this something cannot be true. In formal terms, ¬3Kφ → ¬φ. Contrapositively, this means that if something is true, it is knowable, i.e., it can be known: • φ → 3Kφ. Anti-realism The problem is that this seemingly innocent assumption gives rise to trouble when combined with some standard assumptions concerning the behavior of the involved modal operators '3' (or its dual '2') and 'K'. The standard assumptions are the following: N |= φ ⇒ |= 2φ K |= K(φ → ψ) → (Kφ → Kψ). T |= Kφ → φ. From N and K one can straightforwardly prove that C, i.e. |= K(φ ∧ ψ) → Kφ ∧ Kψ holds. Let us now assume that we know that there are some unknown truths: ∃φ(K(φ∧¬Kφ)). Let p be one of those. Thus K(p∧¬Kp). The trouble is that from this assumption together with anti-realism and the stardard modal assumptions N, K, and T, one can derive the seemingly absurd thesis that any truth is already known. To show this, assume K(p ∧ ¬Kp). With C it follows that Kp ∧ K¬Kp, and thus Kp. But from the other conjunct K¬Kp it follows with T that ¬Kp. Thus, from K(p ∧ ¬Kp) one can derive a contradiction, which means (according to classical logic) that K(p ∧ ¬Kp) cannot be true, i.e. |= ¬K(p ∧ ¬Kp). By the necessitation rule N it follows that 2¬K(p ∧ ¬Kp), which is equivalent to ¬3K(p ∧ ¬Kp). According to anti-realism, all truths are knowable, formalized as φ → 3Kφ. Now take φ to be our assumption that p is an unknown truth: φ ≡ p ∧ ¬Kp. With anti-realism it would follow that 3K(p ∧ ¬Kφ). But from our above reasoning we have concluded that ¬3K(φ ∧ ¬Kφ), which states the exact opposite. We must conclude that according to the anti-realist there can be no proposition p that is true

Blackwell Companion to Epistemology (3rd Edition), 2024

The Knowability Paradox is one of the best known and most discussed epistemic paradoxes in contemporary epistemology. The argument purports to show that if every truth can be known, then every truth is already known or will be known by someone at some time. In other words, if there are truths that nobody knows or will ever know, there are also unknowable truths, truths that cannot be known by anyone at any time. Consider first an intuitive informal statement of the paradox. Let's start from a commonsensical consideration: there seem to be things that we don't know and will never know. It is not difficult to find examples. The number of atoms in the universe is either even or odd, but we don't know which of them is true, and plausibly we will never know. Suppose that the number is even. So it is true that: (E) The number of atoms in the universe is even. The following proposition then also seems true:

Noûs, 2020

Before the semester begins, a teacher tells his students: "There will be exactly one exam this semester. It will not take place on a day that is an immediate-successor of a day that you are currently in a position to know is not the exam-day". Both the students and the teacher know-it is common knowledge-that no exam can be given on the first day of the semester. Since the teacher is truthful and reliable, it seems that the students can know that what he says is true. However, in that case, assuming the students can know that they know whatever it is they know (KK) and assuming their knowledge is closed under entailment (closure), the students can reason from what they know to the conclusion that no exam will take place during the semester. This conclusion contradicts what they supposedly know: that there will be an exam. This puzzle, we argue, gives rise to a new consideration for the rejection of KK. We discuss unique features of the argument, especially in comparison to Timothy Williamson's rejection of KK in light of other versions of the surprise exam paradox.

Journal of Philosophical Logic

The knower paradox states that the statement ‘We know that this statement is false’ leads to inconsistency. This article presents a fresh look at this paradox and some well-known solutions from the literature. Paul Égré discusses three possible solutions that modal provability logic provides for the paradox by surveying and comparing three different provability interpretations of modality, originally described by Skyrms, Anderson, and Solovay. In this article, some background is explained to clarify Égré’s solutions, all three of which hinge on intricacies of provability logic and its arithmetical interpretations. To check whether Égré’s solutions are satisfactory, we use the criteria for solutions to paradoxes defined by Susan Haack and we propose some refinements of them. This article aims to describe to what extent the knower paradox can be solved using provability logic and to what extent the solutions proposed in the literature satisfy Haack’s criteria. Finally, the article off...