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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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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 ∆. 5 If Γ CL A, then Γ respect to its semantics)46 CL A.

3 The logic CLuN is like CL, except that it allows for gluts with respect to N egation. 2. A REGULAR PARACONSISTENT LOGIC 41 may be associated with the same model if the semantics is indeterministic. 4 I refer to [Avr05, ABNK07, AK05] for some interesting technical studies of indeterministic semantics. In this book, indeterministic semantic systems will reappear in Chapter 8. CLuN also has a deterministic semantics. What I mean by this is that the logic CLuN as defined by the above indeterministic semantics is also defined by a deterministic semantics.

Similarly for the rule R∃. 32 R∃ CHAPTER 1. INTRODUCTION To derive ∃αA(α) ⊃ B from in either A(α) or B. 5, this axiom system defines Γ all Γ ⊆ Ws and A ∈ Ws . 1 CL has static proofs. Proof. 6. 1 CL is reflexive, transitive, monotonic, compact, and there is a positive test for it. Actually, CL is not decidable, there is only a positive test for it—see any good handbook, for example [BBJ02]. 2 If A1 , . . , An duction Theorem for CL) CL B, then A1 , . . , An−1 CL An ⊃ B. (De- Proof outline. Let L = C1 , .

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Adaptive Logics and Dynamic Proofs. Mastering the Dynamics of Reasoning, with Special Attention to Handling Inconsistency by Diderik Batens

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