Lectures, European Summer School in Logic, Language and Information, 4-15 August 2008, Hamburg.

Introduction

If we wish to reason more effectively, we can draw on both probability theory and deductive logic to offer guidance. However, these are very different formalisms: deductive logic tells us how the structure of sentences can be exploited to draw conclusions while probability theory tells us how uncertainties interact. For example, deductive logic tells us that from “Jack has bronchitis” and “if Jack has bronchitis then Jack has a cough” we can conclude “Jack has a cough”. Probability theory can be used to tell us the probability that “Jack has bronchitis given that he has a cough” if we know the relative incidences of this symptom and disease in the population.

A probabilistic logic offers a richer formalism, one that combines the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure. Potential applications are numerous and are to be found in disciplines as diverse as the philosophy of science (where we need to model theory choice and theory change, and to understand statistical methodology), artificial intelligence (where computers need to combine statistics with structural knowledge in order to forecast the weather for instance), bioinformatics (where we need to combine deductive reasoning about chemical structure with statistical reasoning about observed biological characteristics) and legal argumentation (where we would like to model legal theory formation from case law). In each of these problem domains probabilistic information and structural knowledge needs to be combined and a probabilistic logic offers a framework for handling this combination.

The difficulty with probabilistic logics is that they tend to multiply the complexities of their probabilistic and logical components. Probabilistic logics can be hard to understand, and inference using probabilistic logics can be time-consuming and complex. In probability theory, probabilistic networks (including what are known as Bayesian nets and credal nets) have been developed to simplify the task of probabilistic reasoning. These nets can afford a simple representation of complex problems and can be used to dramatically speed-up the time it takes to perform calculations.

These lectures investigate the application of probabilistic networks to probabilistic logic. It is shown how probabilistic networks render probabilistic logic simpler and more perspicuous and render applications of probabilitic logic feasible at last.

Course Leaders

Jan-Willem Romeijn, Psychology, University of Amsterdam & Philosophy, University of Groningen

Jon Williamson, Philosophy, University of Kent

Course Materials

An Executive Summary

Jon Williamson: A note on probabilistic logics and probabilistic networks, The Reasoner 2(5), 2008.

Full discussion

Rolf Haenni, Jan-Willem Romeijn, Gregory Wheeler and Jon Williamson: Probabilistic logics and probabilistic networks.

While in principle probabilistic logics might be applied to solve a range of problems, in practice they are rarely applied at present. This is perhaps because they seem disparate, complicated, and computationally intractable. However, we shall argue in this programmatic paper that several approaches to probabilistic logic fit into a simple unifying framework: logically complex evidence can be used to associate probability intervals or probabilities with sentences.

Specifically, we show in Part I that there is a natural way to present a question posed in probabilistic logic, and that various inferential procedures provide semantics for that question: the standard probabilistic semantics (which takes probability functions as models), probabilistic argumentation (which considers the probability of a hypothesis being a logical consequence of the available evidence), evidential probability (which handles reference classes and frequency data), classical statistical inference (in particular the fiducial argument), Bayesian statistical inference (which ascribes probabilities to statistical hypotheses), and objective Bayesian epistemology (which determines appropriate degrees of belief on the basis of available evidence).

Further, we argue, there is the potential to develop computationally feasible methods to mesh with this framework. In particular, we show in Part I how credal and Bayesian networks can naturally be applied as a calculus for probabilistic logic. The probabilistic network itself depends upon the chosen semantics, but once the network is constructed, common machinery can be applied to generate answers to the fundamental question introduced in Part I.

Lectures

• Monday:
  • The progicnet framework.
    • Reading: sections 1 and 8 of the above paper. Slides:
• Tuesday:
  • The standard probabilistic semantics.
    • Reading: sections 2 and 9. Slides:
• Wednesday:
  • Evidential probability.
    • Reading sections 4 and 11. Slides:
  • Probabilistic argumentation; classical statistics,
    • Reading: sections 3,10,5,12. Slides:
• Thursday:
  • Bayesian statistics.
    • Reading: sections 6 and 13. Slides:
• Friday:
  • Objective Bayesian epistemology.
    • Reading: sections 7 and 14. Slides:

Background reading: See the progicnet web page