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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

Example text

The second clarification concerns notation. The lines of a standard (explicit) proof consist of three elements: (i) a line number, (ii) a formula derived, (iii) a justification: the line numbers of the sentences from which (ii) is derived, and the rule of inference that justifies the derivation. In inconsistency-adaptive logic proofs, it is preferable to add a fourth element to each line: (iv) the set of formulas that have to behave consistently in order for (ii) to be derivable by the rule mentioned in (iii) and from the lines enumerated in (iii).

Let us proceed in terms of the basic paraconsistent logic CLuN, leaving variants for later. On the one hand, CLuN determines a set of unconditional rules of inference, viz. the usual ones: if A1 , . . , An CLuN B, then one may infer B from A1 , . . , An . 6: if A1 , . . , An CLuN B ∨((C1 ∧¬C1 )∨. ∨(Cn ∧¬Cn )), one may 14 Some paraconsistent logics validate Disjunctive Syllogism but avoid Ex Falso Quodlibet by invalidating some other CL-rule. 2. 15 Explicating this kind of reasoning was at the origin of the adaptive logic programme—see [Bat89], [Bat99b], and many other papers.

An duction Theorem for CL) CL B, then A1 , . . , An−1 CL An ⊃ B. (De- Proof outline. Let L = C1 , . . , Cm be a proof of B (= Cm ) from {A1 , . . , An }. The list L = An ⊃ C1 , . . , An ⊃ Cm can be transformed to a proof of An ⊃ B from {A1 , . . , An−1 } by replacing every An ⊃ Ci in L by: (i) a proof of An ⊃ An using A⊃1 and A⊃2 if Ci is An , (ii) a proof of An ⊃ Ci from Ci using A⊃1 if Ci is an axiom or a member of {A1 , . . , An−1 }, and (iii) a proof of An ⊃ Ci from An ⊃ (D ⊃ Ci ) and An ⊃ D using A⊃2 if Ci is obtained in L from D ⊃ Ci and D by MP.

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