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Negation in metacomplete relevant logics. (English) Zbl 1233.03030

The main goal of this article is to explain the meaning of negation in Brady’s logic \({\mathrm {DJ}}^{\mathrm{d}}\) and some surrounding relevant logics which are M1- or M2-metacomplete (varieties of metacomplete logics defined by Slaney).
The Law of Excluded Middle (\(A\lor \sim A\)) (LEM) and the Disjunctive Syllogism (\(\sim A, A\lor B\Rightarrow B\)) (DS) are defining properties of Boolean negation that respectively capture its mutual exhaustiveness and mutual exclusiveness. Since the logics in this paper are metacomplete, they do not have the LEM as a theorem; and since they are paraconsistent, they do not have the DS as a derived rule (if DS is derivable then so is Ex Falso Quodlibet (\(A, \sim A\Rightarrow B\))). We are before a De Morgan negation, which is at base four-valued.
The author represents metavaluational structures by means of tree structures and provides a common metavaluational structure for the theorems of the logics studied here to support that negations essentially come in pairs. So, negation is interconnected to itself in a circular (and useful) kind of way.
It is concluded that negation is at base a non-logical concept that by interacting with the connectives and quantifiers becomes logical in part, through a doubling up of negation. The concept of negation is bifurcated between the intensional concept of the De Morgan negation and the classical extensional concept of the Boolean negation formally captured by the LEM and DS. This is said to be the reason why this concept is so hard to capture. Boolean negation is one of a number of single negation concepts that is suitable for a range of concepts and situations. De Morgan negation is a frame into which single negations that apply for particular concepts can fit.

MSC:

03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03B53 Paraconsistent logics
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