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New Directions in Paraconsistent Logic pp 3–38Cite as

Tutorial on Inconsistency-Adaptive Logics

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 152)

Abstract

This paper contains a concise introduction to a few central features of inconsistency-adaptive logics. The focus is on the aim of the program, on logics that may be useful with respect to applications, and on insights that are central for judging the importance of the research goals and the adequacy of results. Given the nature of adaptive logics, the paper may be read as a peculiar introduction to defeasible reasoning.

Keywords

  • Paraconsistent logic
  • Inconsistency-adaptive logic

Mathematics Subject Classication (2000)

  • 03-01
  • 03B53
  • 03B60
  • 03A05

I am indebted to Mathieu Beirlaen for careful comments on a previous draft.

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Notes

  1. 1.

    See, for example, [17] for many other real-life examples of reasoning forms for which there is no positive test. The import of a positive test is discussed further in the text.

  2. 2.

    Uniform Substitution is rule of propositional logic. Predicative classical logic is traditionally axiomatized in terms of a finite set of rules and axiom schemata, rather than axioms. So no substitution rule is then required. Substitution rules in predicate logic have been studied [56] and the outcome is very instructive.

  3. 3.

    The \(\mathbf {L}\)-consequence set of \(\Gamma \) is defined as \(\textit{Cn}_{\mathbf {L}}(\Gamma ) =_{\textit{df}}\{ A \mid \Gamma \vdash _\mathbf {L} A \}\).

  4. 4.

    Just think about usual proofs. Every formula in the proof is a consequence of the premise set and every proof may be extended into a longer proof by applications of the rules.

  5. 5.

    A logic \(\mathbf {L}\) is paracomplete (with respect to a negation \(\lnot \)) iff some A may false together with its negation \(\lnot A\); syntactically: iff there are \(\Gamma \), A and B such that \(\Gamma , A \vdash _\mathbf {L} B\) and \(\Gamma , \lnot A \vdash _\mathbf {L} B\), but \(\Gamma \nvdash _\mathbf {L} B\).

  6. 6.

    In the context of \(\mathbf {CL}^+\), Excluded Middle together with Ex Falso Quodlibet define the classical negation.

  7. 7.

    So \(p\wedge \lnot q \vDash _{\mathbf {CL}^+} \lnot q\), \(\forall x\lnot Px \vDash _\mathbf {CL^+} \lnot Pa\), and \(a=b, Px \vDash _\mathbf {CL^+} Pb\), but \(a=b, \lnot Px \nvDash _\mathbf {CL^+} \lnot Pb\).

  8. 8.

    Names and notation may obviously be different and the model may be more complex.

  9. 9.

    Take conjunction as an example. The clause allowing for gluts: \(v_M(A\wedge B)=1\) iff (\(v_M(A)=1\) and \(v_M(B)=1\)) or \(v(A\wedge B)=1\); the one allowing for gaps: \(v_M(A\wedge B)=1\) iff (\(v_M(A)=1\) and \(v_M(B)=1\)) and \(v(A\wedge B)=1\); the one allowing for both: \(v_M(A\wedge B)=v(A\wedge B)\).

  10. 10.

    I heard the claim that restricting the formation rules of natural language so as to classify “this sentence is false” as non-grammatical is illegitimate because the sentence is ‘perfect English’. I also heard the claim that invalidating Disjunctive Syllogism is illegitimate because this reasoning form is ‘perfectly sound’.

  11. 11.

    This formula is \(\mathbf {CL}\)-equivalent to A but not \(\mathbf {CLuN}\)-equivalent to it.

  12. 12.

    As q is \(\mathbf {CLuN}\)-derivable from the premises, so is \(\lnot p\vee q\). However, relying on p to repeat the move described in the text delivers a formula that was already derivable, viz. q. The same story may be retold for every \(\mathbf {CLuN}\)-consequence of \(\Gamma _{1}\) and each time the move will be harmless because nothing new will come out of it.

  13. 13.

    Do not read the “not derived” as “not derivable”. Indeed, a formula may be derivable in several ways from the same premise set.

  14. 14.

    A more accurate wording requires that one adds: in a proof from \(\Gamma _{1}\) that extends the present stage 8. Indeed, the logic we are considering is non-monotonic. So extending the premise set may result in line 6 being marked.

  15. 15.

    The reader might think that, as p is also a \(\mathbf {CLuN}\)-consequence of \(\Gamma _{1}\), \((p\wedge q)\wedge \lnot (p\wedge q)\) is also a \(\mathbf {CLuN}\)-consequence of \(\Gamma _{1}\). This however is mistaken. \(\lnot q \nvdash _\mathbf {CLuN} \lnot (p\wedge q)\).

  16. 16.

    Names like \(\mathbf {LLL}\), \(\mathbf {AL}\), \({\mathbf {AL}^\textit{r}}\), and \(\mathbf {ULL}\) are used as generic names to define the standard format and to study its features. The names refer to arbitrary logics that stand in a certain relation to each other.

  17. 17.

    Similarly for those models together with the trivial model—the model that verifies all formulas.

  18. 18.

    The notion played a rather central role in discussions on scientific heuristics. A very clear and argued position was for example proposed by Dudley Shapere [60].

  19. 19.

    This obviously does not mean that \(\mathbin {\hat{\vee }}\) is a symbol of the language. It is a conventional name to refer to a symbol of the language that has the meaning of classical disjunction. It may even refer ambiguously: if there are several classical disjunctions, \(\mathbin {\hat{\vee }}\) need not always refer to the same one.

  20. 20.

    Axioms are suppose to be closed formulas. So \(A\in \mathcal {W}_s\). The idea is that \(\mathbf {CLuN}\)-valid rules are fully retained in the extension. One of these rules is: from \(\vdash A(a)\supset B\) to derive \(\vdash \exists xA(x)\supset B\) provided a does not occur in B.

  21. 21.

    The axiom schema may be restricted to \(A \in \mathcal {W}_s^a\), but there is no need to do so.

  22. 22.

    The exception may be caused by the logic, which is then called a flip-flop, or by the premise set—for example if the premise set comprises the formulas verified by a \(\mathbf {LLL}\)-model.

  23. 23.

    Infinite stages can be extended by inserting lines in the sequence.

  24. 24.

    It is ironic that the study of the computational complexity of adaptive logics started with a paper arguing that they are too complex [41]. The philosophical complaints and misunderstandings in that paper were answered in [26]; a mistaken theorem was corrected in [68]. Extremely interesting and more detailed studies followed [54, 55].

  25. 25.

    Stepwise: the language \(\mathcal {L}_s\) of \(\mathbf {CLuN}\) is extended with the symbol \(\mathord {\sim }\) and \(\mathbf {CLuN}\) is extended with axioms or rules that give \(\mathord {\sim }\) its classical meaning—for example the schemas \(A\supset (\mathord {\sim }A\supset B)\) and \((A\supset \mathord {\sim }A)\supset \mathord {\sim }A\).

  26. 26.

    The classical symbols were actually superimposed on \(\mathcal {L}\): in the extended language, they never occur within the scope of the original logical symbols of \(\mathcal {L}\).

  27. 27.

    The distinction warrants that the reference to a finite proof stage in Definition 1.5 is all right.

  28. 28.

    The mistake is caused by a confusion between symbols and concepts. If \(\check{\vee }\) occurs in a premise, and so in \(\mathcal {L}\), then \(\check{\vee }\) is not a new symbol of the extended language. So one needs to extend the language with another symbol, say \(\tilde{\vee }\), and call that the checked disjunction.

  29. 29.

    All that is new in the restored version is the notion of an inferred \(\textit{Dab}\)-formula.

  30. 30.

    Adaptive versions of \(\mathbf {D2}\) and other Jaśkowski logics were extensively studied [48–50].

  31. 31.

    The low computational complexity of the consequence set is rather artificial. We suppose that at least one \(\varphi \cap \Delta = \emptyset \) is given, but precisely locating a \(\varphi \) may be a very complex task.

  32. 32.

    The logic-like entity has a rather limited application field. For some \(\Gamma \), \(\Phi (\Gamma )\) is not only infinite but also uncountable.

  33. 33.

    That is (i) \(\mathcal {W} \subseteq \mathcal {W}^+\) and (ii) if \(A,B \in \mathcal {W}^+\), then \((A\mathbin {\hat{\vee }}B) \in \mathcal {W}^+\).

  34. 34.

    Classical theories, which have \(\mathbf {CL}\) as underlying logic, fail to define such a theory. Their consequence relation is much less complex. If A is not a theorem of a classical theory, humans or Turing machines may never find this out. However, if A is a theorem of the classical theory, humans or Turing machines will find that out at a finite point in time. As this point may be two million years from here, the point is slightly theoretical.

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    Diderik Batens

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Batens, D. (2015). Tutorial on Inconsistency-Adaptive Logics. In: Beziau, JY., Chakraborty, M., Dutta, S. (eds) New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics, vol 152. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2719-9_1

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