Abstract
Nearly all popular reasoning forms that handle inconsistencies in a defeasible way have been characterised in terms of inconsistency-adaptive logics in standard format. This format has great advantages, which are explained in the first two sections. This suggests that inconsistency-adaptive logics form a suitable unifying framework for handling such reasoning forms. I shall present four new arguments in favour of this suggestion. (1);Identifying equivalent premise sets proceeds along familiar lines and is much easier than for many other formats. (2);Inconsistency-adaptive logics offer maximally consistent interpretations by themselves, without requiring tinkering from their user. (3);Characterization in terms of inconsistency-adaptive logics offers easy extensions and variations (a fascinating new type of example will be given). (4);Inconsistency-adaptive logics allow for axiomatisations that identify a set of isomorphic models and enable one to describe inconsistent models in an unambiguous way.
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Notes
- 1.
- 2.
Replacement of Identicals is not derivable in \(\mathbf{CLuN}\) but can be added.
- 3.
Tarski logics that are compact and semi-recursive may be characterised as logics that have static proofs, whereas defeasible logics have dynamic proofs. A first version of the theoretical analysis of such notions is presented in Batens (2009a).
- 4.
All \({\mathbf{C}}_{n}\) logics defined below in the text are identical to da Costa’s, except that he introduces C ω as the limit. C ω is like \({\mathbf{C}}_{\overline{\omega }}\) except that the former has positive intuitionistic logic where the latter has positive classical logic. An interesting study of limits of the hierarchy is presented in Carnielli and Marcos (1999). The logic \({\mathbf{C}}_{\overline{\omega }}\) is there called C min .
- 5.
While ¬A ∧ A and \(A \wedge \neg A\) are \({\mathbf{C}}_{\overline{\omega }}\)-equivalent, ¬( ¬A ∧ A) and \(\neg (A \wedge \neg A)\) are not. Which of both is taken to express the consistency of A is a conventional matter.
- 6.
The approach is related to, but different from, the one followed in Carnielli et al. (2007), where a consistency operator, ∘ A, belongs to the standard language and is implicitly defined by, for example, ∘ A ⊃ ((A ∧ ¬A) ⊃ B).
- 7.
Just as A 1 is a \(\mathbf{CL}\)-theorem, viz. a \({\mathbf{C}}_{0}\)-theorem, A m is a \({\mathbf{C}}_{n}\)-theorem whenever m > n. So one may suppose that no formula of the form A m or A (m) is \({\mathbf{C}}_{n+1}\)-derivable from the non-logical axioms of a theory that has \({\mathbf{C}}_{n}\) as underlying logic.
- 8.
This is typical for inconsistency-adaptive logics, not for other adaptive logics.
- 9.
By “classical arithmetic” I obviously mean the set of formulas true in the standard model and not the theorems of some axiom system.
- 10.
This further clarifies the claim made in the previous paragraph. Although \({\mathit{Cn}}_{\mathbf{LP}}(\Delta ) ={ \mathit{Cn}}_{\mathbf{L{P}^{\mathit{m}}}}(\Delta )\), \(\langle {\mathbf{PA}}_{1}^{2},\mathbf{{LP}^{\mathit{m}}}\rangle\) identifies ℳ 1 2 whereas \(\langle {\mathbf{PA}}_{1}^{2},\mathbf{LP}\rangle\) does not.
- 11.
The most obvious justification for contraposition is consistency. So I always wondered why so many relevant logicians want their implications to be contraposable.
- 12.
One shouldn’t make too much of the “different only” phrase. In Priest’s view it may be true together with “the characters are the same”, for otherwise “This sentence is false and only false.” would produce triviality.
- 13.
The first quoted claim is obviously false: all formulas derivable by the lower limit logic are adaptively derivable, whether consistent or inconsistent. However, some further consequences are adaptively derivable by taking as many other inconsistencies to be false as the premises permit. So inconsistency-adaptive logics do acknowledge the full role of all the premises and do not dodge any inconsistencies in them. They presuppose that inconsistencies are false unless and until proven otherwise, from the premises that is.
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Acknowledgements
Research for this paper was supported by subventions from Ghent University and from the Fund for Scientific Research—Flanders. I am indebted to Graham Priest for comments on a former draft of this paper.
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Batens, D. (2013). New Arguments for Adaptive Logics as Unifying Frame for the Defeasible Handling of Inconsistency. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4438-7_7
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