Both logic and probability provide powerful tools for reasoning about uncertainty in a dynamic environment. Our goal in this course is to examine logical frameworks that incorporate probabilistic modeling of multiagent uncertainty. We will then see how merging these two perspectives on uncertainty can help clarify various conceptual issues and puzzles (such as the Monty Hall puzzle or the sleeping beauty problem). The primary objective is to explore the formal tools used by logicians, computer scientists, philosophers and game theorists for modeling uncertainty. We will focus on both the important conceptual issues (eg., Dutch book arguments, updating with probability zero events and higher-order probabilities) and the main technical results (eg., completeness and decidability of probabilistic modal logics).

Here is an extended outline of the course.

This is an advanced but self-contained course. Students will be expected to have had some exposure to (dynamic) epistemic logic and probabilistic logic. In particular, it will be assumed that students have already been introduced to epistemic logic and some of its dynamic extensions (i.e., public announcement logic); and although we will introduce many basic concepts of probabilistic theory (e.g., measure spaces), it will be expected that students have had previous exposure to probabilistic models of uncertainty.

Below is a schedule for the course (which is subject to change). There are also brief synopses of each of the lectures and slides.

Date	Topic	Slides
Day 1 July 27, 2009	Introduction and Motivation, Informational attitudes (brief synopsis)	Lecture 1
Day 2 July 28, 2009	Probabilistic Epistemic Logic (brief synopsis)	Lecture 2
Day 3 July 29, 2009	Dynamic Epistemic Probabilistic Logic (brief synopsis)	Lecture 3
Day 4 July 30, 2009	Harsanyi Type Spaces (brief synopsis)	Lecture 4
Day 5 July 31, 2009	Dutch Book Theorems, Puzzles (brief synopsis)	Lecture 5

Day 1: Introduction & Motivation, Informational Attitudes

Todays lecture started with some general discussion about what we will cover in the course. The first part half of the lecture discussed some motivating examples. Specifically, we discusses Robert Aumann's seminal result that (assuming a common prior) there cannot be common knowledge that the posteriors are different. The second half of the lecture covered a number of basic results in measure theory. We will return to these results tomorrow when going through the completeness proofs.

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Day 2:

We discussed two different logical systems: one for reasoning about probability and the other for reasoning about knowledge and probability. The main focus of the lecture today was on proving completeness for these logical systems.

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Day 3:

Dynamic epistemic logic (DEL) adds a PDL-style dynamic operator to the basic epistemic language. Event models describe abstract epistemic events (eg., public or private announcements, misperceptions, etc.) and the product update rule describes how these events change an initial epistemic model. This lecture described how to define product update in the probabilistic epistemic logics we have been discussing.

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Day 4:

The dominate model used in the game-theory literature to describe the players (probabilistic) beliefs is the Harsanyi Type Space. We explained how type spaces work and how they are used to analyze solution concepts. We then discussed logical systems that have been proposed to reason about type spaces.

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Day 5:

We concluded with some discussion about compactness of the logical systems we discussed in this course. We then turned to Dutch book theorems. A proof of both the synchronic and diachronic Dutch book theorems was presented. Finally we discussed a number of puzzles that have been discussed in the literature (the Cable Guy Paradox, Sleeping Beauty Problem and the Two Envelope Problem).

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