### Department of Mathematics, University of Manchester

#### Title: Some Results Concerning Probabilistic Logics and Inference Process Logics

Supervisor: Professor J.Paris

Keywords: Probability, Probabilistic Reasoning, Fuzzy Logic, Proof Theory, Model Theory, Cut Elimination, Inference Processes

#### Abstract

This thesis has two main aims. First, we give a general result that enables us to produce sound and complete axiomatizations for various (non-classical) propositional logics. This result is achieved using some elementary proof theory and model theory. We then illustrate how to use the result by providing sound and complete axiomatizations for the Lukasiewicz $\aleph_0$-valued logic, a logic related to the work of J.Pavelka and some logics of probability that are due to R.Fagin, J.Halpern and N.Megiddo. Our results here simplify and expand on work presented elsewhere.

Our second aim is to initiate an investigation into the search for a suitable deductive theory for modeling how inference processes reason. We present and consider a particular deductive framework. This deductive framework has the unfortunate property of sometimes identifying deductive theories that are for different inference principles. In attempting to avoid this collapsing we are led to consider adding in a type theory to our deductive framework.

### A Propositional Logic That Handles Conditional Probabilities

by C.Pulley. Unpublished

#### Abstract

In [FagHalMeg90], two logics for reasoning about probabilities were introduced. The first was unable to reason about conditional probabilities. To rectify this [FagHalMeg90] introduce their second probabilistic logic. However, unlike their first, they find it necessary to introduce quantifiers. In this paper we rectify this defect by presenting a propositional logic that can reason about conditional probabilities. Our method of proof is such that we may generalize it to encapsulate soundness and completeness for numerous other logics (eg. the first probabilistic logic of [FagHalMeg90]; fuzzy logic etc.). Our main requirement for these generalizations is that our logics should have a first-order metatheory.

References:

• [FagHalMeg90]: Fagin, R., Halpern, J.Y., and Megiddo, N. (1990), A Logic for Reasoning about Probabilities, Information and Control Vol. 87 pp.78-128.

Full paper: