By Diderik Batens

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

**Example text**

7 is derivable from 2 and 5 because p behaves consistently at stage 6 of the proof. 8 is derivable from 3 and 7 because q behaves consistently at stage 7. However, as ¬q was only derivable because p behaves consistently— see line 7—we should add p ∧ ¬p in the fourth element of line 8; and indeed RC forces us to do exactly so. If p had not behaved consistently, then we would not have been able to derive ¬r in the way we did. Finally, q is derivable from 3 and 6 because r behaves consistently at stage 8.

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. So, if the logic is paraconsistent, the joint truth of A∨B and ¬A does not warrant the truth of B. So, if the logic is paraconsistent, Disjunctive Syllogism is not truth preserving. It is instructive to realize that the situation may be described as follows.

For every α ∈ C ∪ O, let α = {β | α = β ∈ ∆}. Define a CL-model M = D, v as follows: D = { α | α ∈ C ∪ O} and (i) for all α ∈ C ∪ O: v(α) = α , (ii) v(A) = 1 iff A ∈ ∆ ∩ S,44 and (iii) where π ∈ P r , v(π) = { α1 , . . , αr | πα1 . . αr ∈ ∆}. 45 The basis of the induction is where A ∈ S, or A has the form πα1 . . αr with π ∈ P r and α1 , . . , αr ∈ C ∪ O, or A has the form α = β with α, β ∈ C ∪ O. 2) holds for these A is obvious in view of the definition of M , and so are the cases of the induction step in view of the properties of ∆.