### 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

#### 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.

### 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 Unpublished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 $\theta$:

$\bigvee \{ a \in [0,1] | \mbox{from X we may derive$\theta$to degree a} \} = \bigvee \{ a \in [0,1] | \forall valuation W \geq X \cdot W(\theta) \geq a \}$
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!

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.