On the Structure of Evidential Gluts and Gaps

by

Don Faust

Thursday, September 25, 2003
4:00 p.m.
New Science 1205

If our knowledge is absolute and consistent, then we can use Classical Logic.  If it is not absolute but remains consistent, then it is often the case that our knowledge is evidential in nature, indeed regularly involving evidential gluts and evidential gaps as well, and Evidence Logic (EL) provides an example of a suitable foundational framework.  Of course, if the conflict we are dealing with rises to the level of contradiction, then EL is also inadequate and one must use a paraconsistent logic. 

                So EL provides a framework for this middle ground where evidential conflict commonly occurs yet no contradictions arise.  EL also provides logical machinery helpful in studying further the concept of negation, which is certainly of foundational importance to paraconsistency.  In EL, for a predication P, P_c:e asserts that there is confirmatory evidence at the value e for P while P_r:e asserts refutatory evidence at value e.  See “The Concept of Evidence”, INT. J. INTELL. SYSTEMS 15 (2000), 477-493 for precise definitions and theorems delineating EL and its Boolean Sentence Algebras, and “Conflict without Contradiction: paraconsistency and axiomatizable conflict toleration hierarchies in Evidence Logic”, LOGIC AND LOGICAL PHILOSOPHY 9 (2001), 137-151 for initial examples of families of extensions of EL whose axioms reach out in a variety of ways toward domain-specific properties regarding evidential conflict.

                In this paper we will look, mostly rather informally, at how evidential gluts and gaps are each structured in EL and also how they interact.  In Section 6 we will formally examine three families of axiomatized extensions of EL:  a previously introduced family of logics each allowing no conflict at some evidence value e, a new family each allowing no paucity at value e, and a new “conjunctive” family each both allowing no conflict at value e₁ and allowing no paucity at value e₂.  The Boolean Sentence Algebras are analyzed in each case.

                Throughout we keep in mind, and indeed interpret into EL, Aristotle’s work on privatives which even 2000 years ago helped to further elucidation of the concept of negation.  His opinion that non-P IMPLIES (NOT P) is closely related to the evidential gluts we define and study in EL, while the converse of this opinion connects with our evidential gaps in EL.  By looking at some of the EL-theories as described above, which in fact contain a gradational version of the concept of privation, analysis of the interplay between privation and classical negation is achieved, and further insight into the general concept of negation is hopefully gained.