### Combining Probability and Logic

Fabio Cozman, Rolf Haenni, Jan-Willem Romeijn, Federica Russo, Gregory Wheeler, Jon Williamson
2009 Journal of Applied Logic
This volume arose out of an international, interdisciplinary academic network on Probabilistic Logic and Probabilistic Networks involving four of us (Haenni, Romeijn, Wheeler and Williamson), called Progicnet and funded by the Leverhulme Trust from 2006-8. Many of the papers in this volume were presented at an associated conference, the Third Workshop on Combining Probability and Logic (Progic 2007), held at the University of Kent on 5-7 September 2007. The papers in this volume concern either
more » ... he special focus on the connection between probabilistic logic and probabilistic networks or the more general question of the links between probability and logic. Here we introduce probabilistic logic, probabilistic networks, current and future directions of research and also the themes of the papers that follow. What is probabilistic logic? Probabilistic logic, or progic for short, can be understood in a broad sense as referring to any formalism that combines aspects of both probability theory and logic, or in a narrow sense as a particular kind of logic, namely one that incorporates probabilities in the language or metalanguage. In the latter case, if the probabilities are incorporated directly into the logical language we have what might be called an internal progic. An example is a first-order language where one or more of the function symbols are intended to refer to probability functions. Thus one can form expressions like (P 1 (F a) = 0.2 ∧ P 2 (Rab) 0.5) → Gb. This kind of language is suitable for reasoning about probabilities and is explored by Halpern , for instance. If, on the other hand, the probabilities are incorporated into the metalanguage we have an external progic. For example, one might attach probabilities to sentences of a propositional language: (p ∧ q) → r 0.95 . This kind of language is suitable for reasoning under uncertainty, and maintains a stricter distinction between the level of logic and the level of probability [see, e.g., 10]. A logic that incorporates probabilities both within the language and the metalanguage is a mixed progic. The central question facing a progic is which conclusions to draw from given premisses. For an internal progic the question is which ψ to conclude from given premisses ϕ 1 , . . . , ϕ n , where these are sentences of a language involving probabilities. This is analogous to the question facing classical logic. But for an external (or mixed) progic, the question is rather different. Instead of asking what ψ Y to conclude from given premisses ϕ 1 X 1 , . . . , ϕ n X n one would normally ask what Y to attach to a given ψ, when also given premisses ϕ 1 X 1 , . . . , ϕ n X n . Note that, depending on the semantics in question, the X 1 , . . . , X n , Y might be probabilities or sets of probabilities. Since the fundamental question of an external probabilistic logic differs from that of a non-probabilistic logic, different techniques may be required to answer this question. In non-probabilistic logics one typically appeals to a