Propositional logic extended with a pedagogically useful relevant implication

First and foremost, this paper concerns the combination of classical propositional logic with a relevant implication. The proposed combination is simple and transparent from a proof theoretic point of view and at the same time extremely useful for relating formal logic to natural language sentences. A specific system will be presented and studied, also from a semantic point of view. The last sections of the paper contain more general considerations on combining classical propositional logic with a relevant logic that has all classical theorems as theorems

Logics for qualitative inductive generalization

The paper contains a survey of (mainly unpublished) adaptive logics of inductive generalization. These defeasible logics are precise formulations of certain methods. Some attention is also paid to ways of handling background knowledge, introducing mere conjectures, and the research guiding capabilities of the logics

On a Partial Decision Method for Dynamic Proofs

This paper concerns a goal directed proof procedure for the propositional fragment of the adaptive logic ACLuN1. At the propositional level, it forms an algorithm for final derivability. If extended to the predicative level, it provides a criterion for final derivability. This is essential in view of the absence of a positive test. The procedure may be generalized to all flat adaptive logics.Comment: 18 pages. Originally published in proc. PCL 2002, a FLoC workshop; eds. Hendrik Decker, Dina Goldin, Jorgen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/

Devising the set of abnormalities for a given defeasible rule

Devising adaptive logics usually starts with a set of abnormalities and a deductive logic. Where the adaptive logic is ampliative, the deductive logic is the lower limit logic, the rules of which are unconditionally valid. Where the adaptive logic is corrective, the deductive logic is the upper limit logic, the rules of which are valid in case the premises do not require any abnormalities to be true. In some cases, the idea for devising an adaptive logic does not relate to a set of abnormalities, but to one or more defeasible rules, and perhaps also to one of the deductive logics. Defeasible rules are not universally valid, but are valid in ‘normal situations’ or for unproblematic parts of premise set. Where the idea is such, the set of abnormalities has to be delineated in view of the rules. The way in which this task may be tackled is by no means obvious and is the main topic studied in the present paper. The outcome is an extremely simple and transparent recipe. It is shown that, except for very special cases, the recipe leads to an adequate result

Pluralism In scientific problem solving : why inconsistency is no big deal

Pluralism has many meanings. An assessment of the need for logical pluralism with respect to scientific knowledge requires insights in its domain of application. So first a specific form of epistemic pluralism will be defended. Knowledge turns out a patchwork of knowledge chunks. These serve descriptive as well as evaluative functions, may have competitors within the knowledge system, interact with each other, and display a characteristic dynamics caused by new information as well as by mutual readjustment. Logics play a role in the organization of the chunks, in their applications and in the exchange of information between them. Epistemic pluralism causes a specific form of logical pluralism. Against this background, the occurrence of inconsistencies will be discussed together with required reactions and systematic ways to explicate them. Finally, the place of inconsistencies in the sciences will be considered. Seven theses will be proposed and argued for. The implications of each of these for pluralism will be considered. The general tenet is that paraconsistency plays an important role, bound to become more explicit in the future, but that the occurrence of inconsistencies does not basically affect the need for pluralism

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