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Higher-Order Modal Logics: Automation and Applications

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Reasoning Web. Web Logic Rules (Reasoning Web 2015)

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Abstract

These are the lecture notes of a tutorial on higher-order modal logics held at the 11th Reasoning Web Summer School. After defining the syntax and (possible worlds) semantics of some higher-order modal logics, we show that they can be embedded into classical higher-order logic by systematically lifting the types of propositions, making them depend on a new atomic type for possible worlds. This approach allows several well-established automated and interactive reasoning tools for classical higher-order logic to be applied also to modal higher-order logic problems. Moreover, also meta reasoning about the embedded modal logics becomes possible. Finally, we illustrate how our approach can be useful for reasoning with web logics and expressive ontologies, and we also sketch a possible solution for handling inconsistent data.

C. Benzmüller—This work has been supported by the German Research Foundation DFG under grants BE2501/9-1,2 and BE2501/11-1.

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Notes

  1. 1.

    thf stands for typed higher-order form and it refers to a family of syntax formats for higher-order logic. So far only the fully developed thf0 format, for simple type theory, is in practical use.

  2. 2.

    In thf0, which is a concrete syntax for HOL, $i and $o represent the HOL base types i and o (Booleans). $i>$o encodes a function (predicate) type. Predicate application, as in A(X, W), is encoded as ((A@X)@W) or simply as (A@X@W), i.e., function/predicate application is represented by @; universal quantification and \(\lambda \)-abstraction as in and are represented as in

    figure a

    ; comments begin with %.

  3. 3.

    The 3480 problems for logic S4 can be download from http://christoph-benzmueller.de/papers/THF-S4-ALL.zip.

  4. 4.

    See file QML.thy available at https://github.com/FormalTheology/GoedelGod/blob/master/Formalizations/Isabelle/.

  5. 5.

    See file QML_var.thy at the github url from above.

  6. 6.

    The keyword

    figure e

    indicates a lambda abstraction:

    figure f

    (or

    figure g

    ) denotes the function \(\lambda x:t.p\), which takes an argument x (of type t) and returns p.

  7. 7.

    The underlying proof system of Coq (the Calculus of Inductive Constructions (CIC) [57]) is actually more sophisticated and minimalistic than the calculus shown in Fig. 5. But the calculus shown here suffices for the purposes of this tutorial. This calculus is classical, because of the double negation elimination rule. Although CIC is intuitionistic, it can be made classical by importing Coq’s classical library, which adds the axiom of the excluded middle and the double negation elimination lemma.

  8. 8.

    The natural deduction calculus with the rules from Figs. 5 and 6 is sound and complete relatively to the calculus of Fig. 5 extended with a necessitation rule and the modal axiom K [67]. Starting from a sound and Henkin-complete natural deduction calculus for classical higher-order logic (cf. Fig. 5), the additional modal rules in Fig. 6 make it sound and Henkin-complete for the rigid higher-order modal logic K.

  9. 9.

    More elegantly, we could employ an \(@_{cw}\)-operator; for example, (A6) would then be encoded as \(@_{cw} (likes Mary Bill)\) (see also Sect. 5.4).

  10. 10.

    Fitting [36] (pp. 83ff) actually does not use a translation to higher-order logic, where worlds become part of the syntax. But what he does, using his style of syntax (which distinguishes extensional and intensional types), is essentially analogous to the translation described here.

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Acknowledgments

We would like to thank João Marcos for consistently useful discussions about discussive logics and paraconsistency. Various persons have contributed or positively influenced this line of research in the past, including, Larry Paulson, Chad Brown, Geoff Sutcliffe, and Jasmin Blanchette.

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Authors and Affiliations

  1. Freie Universität Berlin, Berlin, Germany

    Christoph Benzmüller

  2. Vienna University of Technology, Vienna, Austria

    Bruno Woltzenlogel Paleo

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Correspondence to Christoph Benzmüller .

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  1. University of Huddersfield, Huddersfield, United Kingdom

    Wolfgang Faber

  2. FU Berlin Institut für Informatik, Berlin, Germany

    Adrian Paschke

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Benzmüller, C., Woltzenlogel Paleo, B. (2015). Higher-Order Modal Logics: Automation and Applications. In: Faber, W., Paschke, A. (eds) Reasoning Web. Web Logic Rules. Reasoning Web 2015. Lecture Notes in Computer Science(), vol 9203. Springer, Cham. https://doi.org/10.1007/978-3-319-21768-0_2

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