Perfect validity, entailment and paraconsistency

An adaptive logic for relevant classical deduction

“…In this section, I will describe the semantics of Neil Tennant's CR, 7 and I will show that it can be used to capture relevant classical deduction, because of the fact that CR isolates EFQ from CL (see [10] and [11]). …”

“…In other words, is it possible to decide for any Δ CL A whether A 'follows' from Δ by dint of Δ's inconsistency, rather than by dint of any genuine deductive connection between Δ and A. 2 In [10] and [11], Neil Tennant proved it to be possible to split up CL-proofs into explosive classical proofs and relevant classical proofs. 3,4 The consequences of explosive classical proofs solely depend on the inconsistency of that premise set in order to be derivable, while the consequences of relevant classical proofs follow from the premises by "relevant use" of the classical derivation rules.…”

On Argumentation Logic and Propositional Logic

“…In particular, AL may be seen as following in spirit some of Tennant's works [24][25][26], sharing their aim to develop proof theories that on the one hand preserve fully classical entailment from consistent theories but on the other hand do not trivialise when the theories are inconsistent. 14 This is given by discarding the exfalso quodlibet principle as well as some forms of reasoning, e.g., in [24], the excluded middle law, while keeping others, e.g., again in [24], the disjunctive syllogism (or resolution).…”

Inference Claims

“…Rejection of the principles ex falso quodlibet and ex quolibet verum requires revising standard logic, whether classical or intuitionist. Tennant (1979Tennant ( , 1980Tennant ( , 1984 has shown one way of doing so, although he modifies the relations of deducibility and entailment rather than the relation of following logically. He treats entailment as the converse of deducibility, and then puts restrictions on deducibility that simultaneously restrict the extension of the entailment relation.…”