### 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

#### Abstract

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 $\aleph_0$-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.

### Papers

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

by C.Pulley. Unpublished

#### Abstract

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 $\theta$:

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!

References:

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