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Modal Rules: Extending Defeasible Logic with Modal Operators

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Rules and Reasoning (RuleML+RR 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11092))

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Abstract

In this paper we present a general methodology to extend Defeasible Logic with modal operators. We motivate the reasons for this type of extension and we argue that the extension will allow for a robust knowledge framework in different application areas.

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Notes

  1. 1.

    For example the modal logic of provability where the interpretation of \(\Box \) as the provability predicate of Peano Arithmetics forces a one to one mapping.

  2. 2.

    In addition to the aspects we discuss here, we would like to point out that it has been argued [27, 29] that deontic logic is better than a predicate based representation of obligations and permissions when the possibility of norm violation is kept open. As we have argued elsewhere [20] a logic of violation is essential for the representation of contracts where rules about violations are frequent. Obviously this argument only proves that modal operators are superior to deontic modalities, but we are confident that the argument can be extended to other modal notions.

  3. 3.

    To be precise contradictions can be obtained from the monotonic part of a defeasible theory, i.e., from facts and strict rules.

  4. 4.

    Notice that a strict rule can be defeated only when its antecedent is defeasibly provable.

  5. 5.

    The language can be extended to deal with other notions. For example to model agents, we have to include a (finite) set of agents, and then the modal operators can be parameterised with the agents. For a logic of action or planning, it might be appropriate to add a set of atomic actions/plans, and so on depending on the intended applications.

  6. 6.

    A modal operator \(\Box _i\) is reflexive iff the truth of \(\Box _i\phi \) implies the truth of \(\phi \). In other words \(\Box _i\) is reflexive when we have the modal axiom \(\Box _i\phi \rightarrow \phi \).

  7. 7.

    All the coherency results stated in this paper are a consequence of using the principle of strong negation to define the proof conditions. The principle mandates that the conditions for proof tags \(+\#\) and \(-\#\) are the strong negation of each other, where the strong negation transforms conjunctions in disjunctions, disjunctions in conjunctions, existential in universal, universal in existential, conditions for a tag \(-@\) in \(+@\) and conditions for \(+@\) in \(-@\). [17] proves that if the proof conditions are defined using such a principle then the corresponding logic is coherent.

  8. 8.

    The proofs of these results as well as that of Propositions 3 and 4 are modifications and generalisation of the proofs given in [21].

  9. 9.

    Notice that the ‘seriality’ axiom \(\Box \phi \rightarrow \Diamond \phi \) more generally corresponds to the ‘consistency’ of the modal operator. Given the interpretation of the modal operators given in modal logic as derivability in DL and the consistency of DL, this reading is appropriate for the present context.

  10. 10.

    Here we will ignore all temporal aspects and we will assume that the sequence of actions is done in the correct order.

  11. 11.

    This algorithm outputs \(+\partial \); \(-\partial \) can be computed by an algorithm similar to this with the “dual actions”. For \(+\Delta \) we have just to consider similar constructions where we examine only the first parts of step 1 and 2. \(-\Delta \) follows from \(+\Delta \) by taking the dual actions.

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Author information

Authors and Affiliations

  1. Data61, CSIRO, Dutton Park, 4102, Australia

    Guido Governatori

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  1. Guido Governatori
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Correspondence to Guido Governatori .

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Editors and Affiliations

  1. Computer Science, Freie Universität Berlin, Berlin, Germany

    Christoph Benzmüller

  2. University of Calabria, Rende, Italy

    Francesco Ricca

  3. Université du Luxembourg, Esch-sur-Alzette, Luxembourg

    Xavier Parent

  4. SINTEF/University of Oslo, Oslo, Norway

    Dumitru Roman

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Governatori, G. (2018). Modal Rules: Extending Defeasible Logic with Modal Operators. In: Benzmüller, C., Ricca, F., Parent, X., Roman, D. (eds) Rules and Reasoning. RuleML+RR 2018. Lecture Notes in Computer Science(), vol 11092. Springer, Cham. https://doi.org/10.1007/978-3-319-99906-7_2

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