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Knowledge Representation: Modalities, Conditionals, and Nonmonotonic Reasoning

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Abstract

The aim of the present chapter is to overview three important tools for knowledge representation that are strongly interrelated. All three can be traced back to a fundamental limitation of classical logic: its connectives are truth-functional, which does not allow to reason about some concepts such as modalities and “if-then” relationships between propositions. To witness, most of the students in an introductory course on logic have a hard time to accept that the implication “if A then B” should be identified with “A is false or B is true”. Indeed, such an identification leads to validities that are rather counter-intuitive, such as “B implies A implies B” or “A implies B, or B implies A”. In introductory courses it is often omitted that the above interpretation of the so-called material implication was subject of much concern among scholars in the past. Their work led to the development of several families of formalisms that will be presented in this chapter: modal logics, conditional logics, and nonmonotonic formalisms. The next three sections detail the definitions of each of these: the modal logics K and S5, the conditional logics due to Stalnaker and Lewis, and the preferential and rational nonmonotonic reasoning formalisms. We then study the relationship between conditional logics and dynamic epistemic logics. The latter are a family of modal logics that got popular recently. We show that they can be viewed as particular logics of indicative conditionals: they are in the Stalnaker family and violate all of Lewis’s principles.

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Notes

  1. 1.

    We note that knowledge and belief are part of an agent’s mental attitudes. Other such attitudes exist and cannot be represented by means of truth-functional operators either. These attitudes are presented in detail in chapter “,Formalization of Cognitive-Agent Systems, Trust, and Emotions” of this volume.

  2. 2.

    Note that one might as well wish to avoid validity of the nested conditional formula \(\lnot A\Rightarrow (A\Rightarrow B) \). Such a project does not require a material implication, which amounts to studying \(\Rightarrow \) as a ‘full-fledged’ implication operator that is an alternative to \(\rightarrow \). This can e.g. be done in a logical language with operators \(\Rightarrow \), \(\lnot \), \(\wedge \) and \(\vee \). This leads us to so-called substructural logics such as intuitionistic logic or linear logic (Troelstra 1992). However, most researchers in AI tend a less radical position and study extensions of classical logic by a further logical operator \(\Rightarrow \). We consequently restrict our presentation to such approaches.

  3. 3.

    The example is Goodman’s (1947), who proposes the requirement that \(A'\) should be cotenable with \(A\) for such an inference. His paper is dedicated to the quest of a definition of such a conditional; however, having discussed several unsatisfactory proposals he ends up defining cotenability in terms of the conditional, thus resulting in a circular definition.

  4. 4.

    Precisely, for Nute’s weak conditional logic W the following holds: “Any \(\square \)-normal extension of the conditional logic W which is closed under one, is closed under all” (Nute 1980). It is however not the case that the principles are always equivalent: for example, in System C of Sect. 4.1, monotony and transitivity are equivalent, but monotony does not imply contraposition.

  5. 5.

    The terms ‘postulate’ and ‘axiom’ both designate formal properties that are desired to hold. As customary we call such a property ‘axiom’ when it is formulated in the object language and ‘postulate’ when it is formulated in the metalanguage (see e.g. the AGM revision postulates Alchourrón et al. 1985). Inference rules are therefore particular postulates having one or more object language formulas as premisses and a single object language formula as conclusion.

  6. 6.

    Actually axiom (4) is superfluous: it can be derived from (T) and (5).

  7. 7.

    We have adapted the original notation.

  8. 8.

    The link with theories of uncertainty is deepened in chapter “Representations of Uncertainty in Artificial Intelligence: Probability and Possibility” of this volume.

  9. 9.

    As noted in Makinson (1993), the condition of transitivity that was initially imposed by Burgess can be abandoned. One might as well restrict the \(\le _w\) to strict preorders.

  10. 10.

    Indeed, the first implies the second in presence of the axiom (\({\text {MOD}_{0}}\)).

  11. 11.

    Actually defaults with free variables are considered to be abbreviations to be replaced by their closed instances.

  12. 12.

    We refer to chapter “Main Issues in Belief Revision, Belief Merging and Information Fusion” of this volume for an exposition of AGM theory.

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Acknowledgements

Thanks are due to Ricardo Caferra for his careful reading of the first version of the French version of this chapter and to Henri Prade and Christos Rantsoudis for the same job on the present English version. Thanks are also due to Hans van Ditmarsch for his comments.

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  1. IRIT-CNRS, Université Paul Sabatier, Toulouse, France

    Andreas Herzig & Philippe Besnard

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  1. Andreas Herzig
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  2. Philippe Besnard
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Correspondence to Andreas Herzig .

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Editors and Affiliations

  1. CRIL-CNRS, Université d'Artois and Institut Universitaire de France, Lens, France

    Pierre Marquis

  2. Aix Marseille Université, Université de Toulon, CNRS, LIS, Marseille, France

    Odile Papini

  3. IRIT, CNRS and Université Paul Sabatier, Toulouse, France

    Henri Prade

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Herzig, A., Besnard, P. (2020). Knowledge Representation: Modalities, Conditionals, and Nonmonotonic Reasoning. In: Marquis, P., Papini, O., Prade, H. (eds) A Guided Tour of Artificial Intelligence Research. Springer, Cham. https://doi.org/10.1007/978-3-030-06164-7_2

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