By Diderik Batens

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

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However, there is more. The theory we are after, T ∗ , should be as close as possible to T , except that it should be non-trivial. In other words, we do not want T ∗ to comprise the CL-consequences of the inconsistencies that occur in T , but we want it to comprise all other CL-consequences of T . In order to even make sense of this intuitive idea, it is essential that one locates the inconsistencies in T . Adaptive logics are capable of doing so. Before moving on, let me make explicit a convention that was used implicitly until now.

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 ∆ . As ∆ ⊆ ∆, there is no C ∈ ∆ for which ∆ ∪ {C} CL A. 7. CLASSICAL LOGIC 35 o ∆i−1 CL (A ⊃ C) ⊃ A.

3. 1. It is easily demonstrated that the set of consequences of some set Γ is generally richer than the CLuN-consequence set of Γ. For example, q is not a CLuN-consequence of the premises 1-4. If Γ is consistent, all CL-consequences of Γ are consequences of Γ and vice versa. If Γ is inconsistent, its set of consequences does not contain all sentences and hence is weaker than its set of CL-consequences. The reader may wonder what would happen if we were to continue the proof after line 11. The answer is that nothing worth mentioning will happen.

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