Logic is traditionally understood to be the study of `what follows from what' and probabilistic logic is no exception. But whereas the entailment problem typically asks whether a conclusion C is entailed by some premises P1, ... , Pn, with respect to some specified entailment relation , written as ⊨

(1) P1, ... ,Pn ⊨ C?

on our approach (Haenni et al. 2009) to probabilistic logic poses a slightly different question. In a probabilistic setting, attached to each premise Pi is a probability or set of probabilities, Xi. But it is rare to ask whether a set of probabilistic premises entails that a particular probability (set of probabilities) Y is assigned to C. Instead, the entailment problem in probabilistic logic typically concerns what probability (set of probabilities) are assigned to a particular conclusion, C, written as

(2) P1 (X1), ... ,Pn(Xn) ⊨ C?

A surprisingly wide variety of probabilistic semantics can be plugged nto (2), thereby providing different semantics for the entailment relation = and allowing those different systems to be studied as bona fide logics. In this respect we view (2) as the fundamental question of probabilistic logics and view it as the lynchpin to out proposal for unifying probabilistic logics. As for how to answer the fundamental question, we propose a unifying approximate proof procedure utilizing credal networks, which are probabilistic graphical models analogous to Bayesian networks but configured to handle sets of probability functions representing interval valued probabilities.,

This course introduces the progicnet framework for unifying probabilistic logic in three parts:

Day 1. An introduction to the fundamental question for probabilistic logic and an introduction to the most basic semantics, our generalization of the standard semantics of `Bayesian probabilistic logic', as put forward in Ramsey (Ramsey 1926) and De Finetti (de Finetti 1937), and explicitly advocated by Howson, (Howson 2001, 2003), Morgan
(Morgan 2000), and Halpern (Halpern 2003), to handle interval-valued probability assignments via sets of probabilities.

Day 2. An introduction to a semantics for handling relative frequency information, `Evidential Probability' (EP) (Kyburg 1961, Kyburg and Teng 2001, Kyburg et al. 2007, Wheeler and Williamson 2009). EP is traditionally thought of as a logic of probability rather than a probabilistic logic (Levi 2007), and our framework helps to explain why this is so. Time permitting we will introduce some extensions to EP that utilize different features of the progicnet framework.

Day 3. An introduction to credal networks (Levi 1980, Cozman 2000)and approximate proof theory we develop with this machinery. Severa open problems will be presented.

References:

Cozman, F. (2000). Credal networks. Arti?cial Intelligence, 120(2):199{233.

de Finetti, B. (1937). La prévision: ses lois logiques, ses sources subjectives. Annales de l'Institut Henri Poincaré, 7(1):1-68.

Haenni, R., Williamson, J., Romeyn, J.-W., and Wheeler, G. (2009). Probabilistic Logic and Probabilistic Networks. Synthese Library. Springer, Dordrecht.

Halpern, J. Y. (2003). Reasoning about Uncertainty. MIT Press, Cambridge, MA.

Howson, C. (2001). The logic of Bayesian probability. In Corfield, D. and Williamson, J., editors, Foundations of Bayesianism, pages 137-159. Kluwer, Dordrecht.

Howson, C. (2003). Probability and logic. Journal of Applied Logic, 1(3-4):151-165.

Kyburg, Jr., H. E. (1961). Probability and the Logic of Rational Belief. Wesleyan University Press, Middletown, CT.

Kyburg, Jr., H. E. and Teng, C. M. (2001). Uncertain Inference. Cambridge University Press, Cambridge.

Kyburg, Jr., H. E., Teng, C. M., and Wheeler, G. (2007). Conditionals and consequences. Journal of Applied Logic, 5(4):638{650.

Levi, I. (1980). The Enterprise of Knowledge. MIT Press, Cambridge, MA.

Levi, I. (2007). Probability logic and logical probability. In Harper, W. and Wheeler, G., editors, Probability and Inference: Essays in Honor of Henry E. Kyburg, Jr., pages 255-266. College Publications.

Morgan, C. G. (2000). The nature of nonmonotonic reasoning. Minds and Machines, 10:321{360.

Ramsey, F. P. (1926). Truth and probability. In Kyburg, H. E. and Smokler, H. E., editors, Studies in subjective probability, pages 23-52. Robert E. Krieger Publishing Company, Huntington, New York, second (1980) edition.

Wheeler, G. and Williamson, J. (2009). Evidential probability and objective bayesian epistemology. In Bandyopadhyay, P. and Forster, M., editors, Handbook of the Philosophy of Statistics. Elsevier Science.