Hartmann : Learning Conditionals and the Problem of Old Evidence

UNIVERSITY SEMINAR ON LOGIC, PROBABILITY, AND GAMES
Learning Conditionals and the Problem of Old Evidence
Stephan Hartmann (Ludwig Maximilians-Universität München)
4:10 pm, February 13, 2015
Faculty House, Columbia University

Abstract. The following are abstracts of two papers on which this talk is based.

The Problem of Old Evidence has troubled Bayesians ever since Clark Glymour first presented it in 1980. Several solutions have been proposed, but all of them have drawbacks and none of them is considered to be the definite solution. In this article, I propose a new solution which combines several old ideas with a new one. It circumvents the crucial omniscience problem in an elegant way and leads to a considerable confirmation of the hypothesis in question.

Modeling how to learn an indicative conditional has been a major challenge for formal epistemologists. One proposal to meet this challenge is to construct the posterior probability distribution by minimizing the Kullback-Leibler divergence between the posterior probability distribution and the prior probability distribution, taking the learned information as a constraint (expressed as a conditional probability statement) into account. This proposal has been criticized in the literature based on several clever examples. In this article, we revisit four of these examples and show that one obtains intuitively correct results for the posterior probability distribution if the underlying probabilistic models reflect the causal structure of the scenarios in question.

Leitgeb: The Humean Thesis on Belief

UNIVERSITY SEMINAR ON LOGIC, PROBABILITY, AND GAMES
The Humean Thesis on Belief
Hannes Leitgeb (Ludwig Maximilian University of Munich)
4:15 pm, May 2nd, 2014
716 Philosophy Hall, Columbia University

Abstract.  I am going to make precise, and assess, the following thesis on (all-or-nothing) belief and degrees of belief: It is rational to believe a proposition just in case it is rational to have a stably high degree of belief in it.I will start with some historical remarks, which are going to motivate calling this postulate the “Humean thesis on belief”. Once the thesis has been formulated in formal terms, it is possible to derive conclusions from it. Three of its consequences I will highlight in particular: doxastic logic; an instance of what is sometimes called the Lockean thesis on belief; and a simple qualitative decision theory.

Hájek: Staying Regular?

COLUMBIA FORMAL PHILOSOPHY WORKSHOP
Staying Regular?
Alan Hájek (Australian National University)
Thursday, April 4, 2013, 4:00 – 5:30 PM
716 Philosophy Hall, Columbia University

Abstract.
‘Regularity’ conditions provide nice bridges between the various ‘box’/‘diamond’ modalities and various notions of probability. Schematically, they have the form:

If X is possible, then the probability of X is positive

(or equivalents). Of special interest are the conditions we get when ‘possible’ is understood doxastically (i.e. in terms of binary belief), and ‘probability’ is understood subjectively (i.e. in terms of degrees of belief). I characterize these senses of ‘regularity’—one for each agent—in terms of a certain internal harmony of the agent’s probability space  <Ω, F, P>. I distinguish three grades of probabilistic involvement. A set of possibilities may be recognized by such a probability space by being a subset of Ω; by being an element of F; and by receiving positive probability from P. These are non-decreasingly committal ways in which the agent may countenance a proposition. An agent’s space is regular if these three grades collapse into one.

I briefly review several of the main arguments for regularity as a rationality norm, due especially to Lewis and Skyrms. There are two ways an agent could violate this norm: by assigning probability zero to some doxastic possibility, and by not assigning probability at all to some doxastic possibility (a probability gap). Authors such as Williamson have argued for the rationality of the former kind of violation, and I give an argument of my own. So I think that the second and third grades of probabilistic involvement may come apart for a rational agent. I then argue for the latter kind of violation: the first and second grades may also come apart for such an agent.

Both kinds of violations of regularity have serious consequences for traditional Bayesian epistemology. I consider especially their ramifications for:

  • conditional probability
  • conditionalization
  • probabilistic independence
  • decision theory

Formal Philosophy Reading Group SP12 #10

The next meeting of the formal philosophy reading group will take place on Monday (April 16) 7:10 – 9:00 PM in room 720 Philosophy Hall, Columbia University. Prof. Gaifman will continue with his presentation on dilation.

Light refreshments will be served. Hope to see you all there!

Formal Philosophy Reading Group SP12 #9

The next meeting of the formal philosophy reading group will take place on Monday (April 2) 7:10 – 9:00 PM in room 720 Philosophy Hall, Columbia University. Prof. Gaifman will continue his presentation on

  • STP (sure thing principle) and its weaker and stronger versions.
  • Violations of STP (the strong version) arising in the framework of IP (indeterminate or “imprecise” probabilities).
  • Objective probabilities
  • When “ignorance is bliss” and when it is not
  • Resolving the violations of STP by appeal to basic features of objective probabilities.

Light refreshments will be served. Hope to see you all there!

Formal Philosophy Reading Group SP12 #8

The next meeting of the formal philosophy reading group will take place on Monday (March 26) 7:10 – 9:00 PM in room 720 Philosophy Hall, Columbia University.

We will be discussing Ellsberg’s Paradox, Objective chances and the problems they give rise to in the framework of indeterminate probabilities.

Light refreshments will be served. Hope to see you all there!