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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 Unpublished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 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!

References:

  • [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]