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**

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If a theory T was intended (and believed) to be consistent and hence was given CL as its underlying logic, but turns out inconsistent, then replacing CL by monotonic paraconsistent logics offers a theory that is much weaker than ‘what T was intended to be’. Consider the premise set {¬p ∨ q, p, r ∨ s, ¬r, ¬p}. Obviously, this is a toy example. My aim is to illustrate the point, not to offer a historical case study. As the premise set is inconsistent, it requires a formula to behave inconsistently, viz.

4 is proved by showing (i) that every axiom is a valid formula, (ii) that MP holds true in every model (if M A and M A ⊃ B, then M B), and (iii) that R∀ and R∃ hold for valid formulas (for example, for R∀, if A ⊃ B(β) and β does not occur in either A or B(α), then A ⊃ ∀αB(α)). This is safely left to the reader. In preparation of the proof of the strong completeness of CL with respect to its semantics, we need some definitions and lemmas. As I have to consider several languages and logics, the definitions are a trifle more complex than the usual ones.

Cn such that Γ CLuN A ∨ (∃(C1 ∧ ¬C1 ) ∨ . . ∨ ∃(Cn ∧ ¬Cn )). (Derivability Adjustment Theorem)13 Proof outline. ⇒ Suppose that Γ CL A. 1), there are B1 , . . , Bm ∈ Γ for which B1 , . . , Bm CL A, whence CL B1 ⊃ (. . ⊃ (Bm ⊃ A) . 2). Let X abbreviate (B1 ⊃ (. . ⊃ (Bm ⊃ A) . )). Suppose that a CLuN-model M = D, v falsifies X∨ {∃(C ∧ ¬C) | C ∈ sub(X)} and hence falsifies all members of {∃(C ∧ ¬C) | C ∈ sub(X)}. Let M = D, v be a CL-model, with D and v as for M . I show, by an induction on the complexity of formulas, that M and M verify the same closed subformulas of X and the same instances of open subformulas of X.