By Diderik Batens

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**Additional resources for Adaptive Logics and Dynamic Proofs. Mastering the Dynamics of Reasoning, with Special Attention to Handling Inconsistency**

**Sample text**

Ad (ii): A ∈ / ∆ in view of the supposition and the construction. Ad (iii): Suppose that C ∈ Wo and ∆ CL C. 1), there is a finite ∆ ⊆ ∆ for which ∆ CL C. It follows that there is an i for which ∆ ⊆ ∆i ,43 whence C ∈ ∆i in view of the construction. But then C ∈ ∆ by the construction. So ∆ is CL-deductively closed in Lo . Ad (iv): As A ∈ / ∆, (A ⊃ C) ⊃ A ∈ / ∆ in view of A⊃3 and (iii). So, in view of (iii), ∆ CL (A ⊃ C) ⊃ A and, where Bi is A ⊃ C, 43 There is a finite initial segment of L that contains all members of ∆ .

We all agree that A ∨ B and ¬A, the two (local) premises of Disjunctive Syllogism, are jointly equivalent to (A ∧ ¬A) ∨ (B ∧ ¬A). If the underlying logic is CL, then A ∧ ¬A is bound to be false. So if (A ∧ ¬A) ∨ (B ∧ ¬A) is true, then so is B. 1 that one sometimes needs to reason from inconsistent premises and that this requires that one considers some inconsistencies as true. This is why we need paraconsistent logics. Thus, according to CLuN, A ∧ ¬A may be true. But if A ∧ ¬A is true, then (A ∧ ¬A) ∨ (B ∧ ¬A) is true, even if B is false.

Many people have been baffled by the claim that Disjunctive Syllogism is incorrect according to some logics; some were even riled by the fact that such logics 8 The propositional fragment of CLuN is as decidable as that of CL. the literature, “positive logic” is sometimes used for positive intuitionistic logic. This is obviously weaker than positive CL. 10 RoE is valid provided A does not occur within the scope of a negation. RoI is valid provided α does not occur within the scope of a negation in A.