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 Unpublished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:
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.