Relevant Logics Obeying Component Homogeneity

This paper discusses three relevant logics that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde, dS*fde, crossS*fde. Second, the paper establishes complete sequent calculi for S*fde, dS*fde, crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define containment logics, we explore the single-premise/single-conclusion fragment of S*fde, dS*fde, crossS*fdeand the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues

Spoiled for choice?

The transition from a theory that turned out trivial to a consistent replacement need not proceed in terms of inconsistencies, which are negation gluts. Logics that tolerate gluts or gaps (or both) with respect to any logical symbol may serve as the lower limit for adaptive logics that assign a minimally abnormal consequence set to a given premise set. The same obtains for logics that tolerate a combination of kinds of gluts and gaps. This result runs counter to the obsession with inconsistency that classical logicians and paraconsistent logicians share.\\ All such basic logics will be systematically reviewed, some variants will be outlined, and the claim will be argued for. While those logics tolerate gluts and gaps with respect to logical symbols, ambiguity logic tolerates ambiguities in non-logical symbols. Moreover, forms of tolerance may be combined, with zero logic as an extreme.\\ In the baffling plethora of corrective adaptive logics (roads from trivial theories to consistent replacements), adaptive zero logic turns out theoretically interesting as well as practically useful. On the one hand all meaning becomes contingent, depending on the premise set. On the other hand, precisely adaptive zero logic provides one with an excellent analyzing instrument. For example, it enables one to figure out which corrective adaptive logics lead, for a specific trivial theory, to a suitable and interesting minimally abnormal consequence set

Conflict without contradiction: paraconsistency and axiomatizable conflict toleration hierarchies in Evidence Logic

Evidence Logic (EL) goes beyond Classical Logic (CL) in its primitive expressivity by including both confirmatory and refutatory predications, additionally equipped with evidence level annotations. Previous work has characterized the Boolean Sentence Algebras (BSAs) of the monadic, functional, and undecidable varieties of EL [4], [5]. From the perspective that our knowledge of the world is often less-than-certain, that is to say “evidential”, application-wise EL is conceptually antecedent to CL and provides a broad foundational framework wherein axiomatizable extensions reach out to a number of the more domain-specific recent constructions of logics for the representation and processing of uncertainty in Artificial Intelligence (AI). In this paper we analyze EL from this point of view in sections 1 and 2. In Section 3 the relationship between this work and issues in paraconsistency is briefly explored

Paraconsistent Multivalued Logic and Coincidentia Oppositorum: Evaluation with Complex Numbers

Paraconsistent logic admits that the contradiction can be true. Let p be the truth values and P be a proposition. In paraconsistent logic the truth values of contradiction is . This equation has no real roots but admits complex roots . This is the result which leads to develop a multivalued logic to complex truth values. The sum of truth values being isomorphic to the vector of the plane, it is natural to relate the function V to the metric of the vector space R2. We will adopt as valuations the norms of vectors. The main objective of this paper is to establish a theory of truth-value evaluation for paraconsistent logics with the goal of using in analyzing ideological, mythical, religious and mystic belief systems

Fixed-Point Models for Theories of Properties and Classes

There is a vibrant (but minority) community among philosophical logicians seeking to resolve the paradoxes of classes, properties and truth by way of adopting some non-classical logic in which trivialising paradoxical arguments are not valid. There is also a long tradition in theoretical computer science|going back to Dana Scott's fixed point model construction for the untyped lambda-calculus of models allowing for fixed points. In this paper, I will bring these traditions closer together, to show how these model constructions can shed light on what we could hope for in a non-trivial model of a theory for classes, properties or truth featuring fixed points