An Adaptive Logic for Compassionate

In [11], Neil Tennant describes a method for restricting the consequence relation of classical logic in such a way that the First Lewis Paradox (A,∼A ` B) is avoided, while all proofs, needed for mathematics and the hypothetico-deductive method in the natural sciences, are retained. This comes down to the fact that he allows all classical consequences of a premise set to be derivable, as long as it can not be shown that those conse-quences are irrelevant in relation to the premises from which they were derived. This means that he seeks to ensure that the results we can prove ‘relevantly ’ are those whose conclusions do not follow from their premises only because of the joint inconsistency of those premises.1 The result is that the “canon of deductive reasoning has been relevantized, ” while “our powers of reasoning have been left intact.”2 In other words, in Tennant’s approach (called “Compassionate Relevantism ” in [11]), only Ex Falso Quodlibet, the only “derivation rule ” which does not “use”, but “abuse ” the premises in order to get to their conclusions, is dropped. All other classical derivation rules, such as Addition and Disjunctive Syllogism, are retained. It is important to notice that Tennant’s notion of relevance does not apply to the implication (as all classical paradoxes of implication are valid), but to the deduction process. This is why he also likes to call it “relevance at the turnstile”. In this paper, I will show that Tennant’s “relevantizing ” of classical logic can also be reached by means of the ambiguity-adaptive logic AALns,3 which also allows for all and only the “relevant ” derivations of classical logic.? Research for this paper was supported by subventions from Ghent University an