A Map of Common Knowledge Logics

In order to capture the concept of common knowledge, various extensions of multi-modal epistemic logics, such as fixed-point ones and infinitary ones, have been proposed. Although we have now a good list of such proposed extensions, the relationships among them are still unclear. The purpose of this paper is to draw a map showing the relationships among them. In the propositional case, these extensions turn out to be all Kripke complete and can be comparable in a meaningful manner. F. Wolter showed that the predicate extension of the Halpern-Moses fixed-point type common knowledge logic is Kripke incomplete. However, if we go further to an infinitary extension, Kripke completeness would be recovered. Thus there is some gap in the predicate case. In drawing the map, we focus on what is happening around the gap in the predicate case. The map enables us to better understand the common knowledge logics as a whole.     

Motivational internalism

Cases involving amoralists who no longer care about the institution of morality, together with cases of depression, listlessness, and exhaustion, have posed trouble in recent years for standard formulations of motivational internalism. In response, though, internalists have been willing to adopt narrower versions of the thesis which restrict it just to the motivational lives of those agents who are said to be in some way normal, practically rational, or virtuous. My goal in this paper is to offer a new set of counterexamples to motivational internalism, examples which are effective both against traditional formulations of the thesis as well as against many of these more recent restricted proposals.

A new formulation of the Principle of Indifference

The idea of a probabilistic logic of inductive inference based on some form of the principle of indifference has always retained a powerful appeal. However, up to now all modifications of the principle failed. In this paper, a new formulation of such a principle is provided that avoids generating paradoxes and inconsistencies. Because of these results, the thesis that probabilities cannot be logical quantities, determined in an objective way through some form of the principle of indifference, is no longer supportable. Later, the paper investigates some implications of the new principle of indifference. To conclude, a re-examination of the foundations of the so-called objective Bayesian inference is called for.     

What Are Degrees of Belief?

Probabilism is committed to two theses:

The (impossible) Future of Hermeneutics

ABSTRACT

This paper argues that the negativity of hermeneutic experience is revelatory for the following reasons. Hermeneutic failure is not the equivalent of making an erroneous step in a closed circuit of reasoning. Neither is it a refutation. It concerns becoming conscious of an omission, an oversight, an unjustifiable claim to completeness and even the displacement of one interpretation by another more suggestive. The negative dimension of hermeneutic failure is incontrovertibly connected with becoming progressively aware of how, contrary to expectations, a different way of seeing is possible: something comes to light which displaces one’s former judgement. Consciousness of failure is, then, indissociable from an emergent awareness of overlooked and unremarked ways of thinking: “I should have been alert to this” or “I failed to take account of that”. Consciousness of failure is revelatory precisely because something else and something other than my expectation has shown itself to be decisive and in so doing has displaced my former understanding. This is the basis of the claim that the educative and spiritual importance of hermeneutics lies precisely in the practical pursuit of the impossible. It is a key contention of the paper that hermeneutic understanding expands and extends itself as a consequence of its impossible quest for completion.     

Improving Accuracy on Bayesian Inference Problems Using a Brief Tutorial

Research suggests that most people struggle when asked to interpret the outcomes of diagnostic tests such as those presented as Bayesian inference problems. To help people interpret these difficult problems, we created a brief tutorial, requiring less than 10 minutes, that guided participants through the creation of an aid (either graph or table) based on an example inference problem and then showed the correct way to calculate the positive predictive value of the problem (i.e., likelihood that positive tests correctly indicate presence of condition). Approximately 70% of those in each training condition found the correct response on at least one problem in the format for which they were trained. Just under 55% of those in the control condition (i.e., no training) were able to find the correct response on at least one table or graph problem. We demonstrated a relationship between numeracy and performance on both problem formats, although we did not find evidence for a relationship between graph literacy and performance for either problem format. Potential improvements to and applications of the tutorial are discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

Löb's Theorem in a Set Theoretical Setting

We present a semantic proof of Löb's theorem for theories T containing ZF. Without using the diagonalization lemma, we construct a sentence AUT _T, which says intuitively that the predicate autological with respect to T (i.e. applying to itself in every model of T) is itself autological with respect to T. In effect, the sentence AUT _T states I follow semantically from T. Then we show that this sentence indeed follows from T and therefore is true.