October 1989 to October 1991

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


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 aleph0-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 Unpublished


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.


  • [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: [PDF]


October 1988 to October 1989

Department of Mathematics, University of Manchester

Title: Some Completeness and Incompleteness Results for Pavelkan-Type Logics on the Unit Interval

Supervisor: Professor J.Paris

Keywords: Fuzzy Logic, Fuzzy Reasoning, Inexact Reasoning


Pavelkan-Type logics are essentially logics which deduce sentances to some fuzzy degree or truth-value from fuzzy sets of hypothesis. I will provide some axiomatizations for these logics which will improve on the work of others in the area. That the work of others may be improved stems from the often partial notions of soundness and completeness for these logics.

So as to understand what possible functions may or may not be defined in the structures used to interpret Pavelkan-type logics on the unit interval, I shall provide a generalization of the results of McNaughton concerning the aleph0-valued logic of Lukasiewicz.

Finally, some partial answers are provided to the existence of sound and complete axiomatizations for some structures first introduced by Goguen for inexact reasoning.


Pavelkan Logics on the unit Interval: Proof Boundaries and Sound and Complete Logics

by C.Pulley. Unpublished


In [Pavelka], Pavelka gives a sound and complete axiomatisation for a logic that reasons with fuzzy sets of hypothesis and conclusions rather than just sets, as in (say) classical and intuitionistic logics.

This paper presents axiomatizations that are not only simpler than those presented in [Pavelka], but which also have simpler soundness and completeness arguments. In addition, we shall also be concerned with studying Pavelka's notions of soundness and completeness.

Pavelka determines soundness and completeness using a limit concept. More precisely, for any fuzzy set X and logical sentence θ:

∨ { a ∈ [0,1] | from X we may derive θ to degree a } = ∨ { a ∈ [0,1] | ∀ valuation W ≥ X · W(θ) ≥ a }
Since we are showing that limit values are equal in our P-soundness and P-completeness results, it should be no surprise that the associated deductive system may fail to derive the limit value, but still be able to derive all non-limit values (ie. those belief values strictly less than the limit value). This observation leads us to present a number of logics for which we may characterize the provable limit values.

In particular, we present a logic in which one may assume mutually inconsistent information, but one may not derive every sentence to every belief value!


  • [Pavelka]: Pavelka, J. On Fuzzy Logic III: Semantical Completeness of some Many Valued Propositional Calculi Zeitshrift für Math. Logik und Grundlagen d. Math., Vol. 25, 1979.

Full Paper: [PDF]


October 1983 to July 1986

Nottingham University

BSc Honours in Mathematics


September 1976 to July 1983

North Cestrian Grammar School

4 A levels (including Mathematics, Physics and Chemistry to grade A), 1 A/S level and 10 O levels.


31st October 2008


Computer Hacking Forensic Investigator

18th July 2008


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