ABSTRACT
As Part I o f his proposed monograph, Sylvan intended to include general survey material. The extant draft fragments o f these orientational chapters are reproduced below as individual chapter sections. Last worked on between March and July 1995-roughly a year before his death-they serve as an illuminating introduction to subsequent chapters in spite o f their unßnished state. [Editors]
3.1 Kinds and features of relevant and paraconsistent logics
Relevant logics have intimate connections with paraconsistent logics, connections important for several applications. A paraconsistent logic is nothing but a logic that can support inconsistent but nontrivial theories, that is theories such that A and ~ A both hold, for some A, but not every B obtains. Otherwise explained, a paraconsistent logic is one that does not trivialise inconsistent theories, because it does not admit the spread rule: A , ~ A / B . In more rigorous forms, paraconsistent logics can absorb logical and semantical paradoxes; it is these logics which admit dialectic applications. A central structural argument that mainstream logics are never paraconsistent exactly parallels a crucial historical argument for a negative paradox of implicationa paradox exhibiting blatant irrelevance of antecedent and consequent-a paradox that relevant logics must accordingly fault in one way or another. These parallels, shortly tabulated, expose well the close connections of relevant and paraconsistent logics. They also deliver helpful classifications of broadly relevant logics. A broadly relevant or sociative logic is one which extends conservatively a sentential part of the following sort: there are no theses of the form A —► B ( with —> a distinguished implication or conditional) where A and B fail to share a variable (or literal). Thus, in particular, no logic containing such theorems as pik ~ p — q, p — q V ~q, ~ (p p) — q, p — q —> q, p —> p — q —* p, etc. is a sociative logic. So none of the present mainstream logics, classical and intuitionist logics, are sociative. Nor are main Brasilian paraconsistent logics such as the Csystems. Relevant logics proper are sorts of sociative logics. But there are as many types of sociative logics as there are genuine alternatives for defeating arguments to the negative paradox of implication A S z ^ A —> B. Since paraconsistent logics are bound to avoid the corresponding inferential form A, ~ A / B, there is evidently a close relation between broadly relevant and
paraconsistent logics. Further, a typology for paraconsistent logics similar to that for sociative logics can accordingly be devised. The respective arguments and principles involved are now tabulated:
Ikble 3.1 Historic negative routes to logical disaster
The resulting types of sociative and paraconsistent logics are these: *A. Non-simplifying, such as very typically, connexive logics. *B. Non-additive, for example Parry logics, containment systems. *C. Non-transitive and non-cut-free, for example relational logics. *D. Non-compositive and non-adjunctive, for example confirmation logic, Jaskowski logics. *E. Non-dissembling, for example relevant logics proper, but also irrelevant ‘positive-plus’ systems such as minimal logic, C-systems, etc. In brief, relevant logics are those sociative logics which reject Disjunctive Syllogism, A & (~A V B) —»• B , and associated forms such as Antilogism, A & B -> C A k ~ C ->
So far this classification conflates sociative and paraconsistent types. For instance, C and minimal systems, are (minimally) paraconsistent but not relevant; Parry logics are sociative but not paraconsistent. However the sorts are readily separated, and will be further isolated in what follows. Nor is the classification entirely exhaustive, for in the deployment of schemata, substitution, which could be questioned, has been implicitly presumed.3
