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Natural Deduction for Diagonal Operators

Part of the Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques book series (PCSHPM)


We present a sound and complete Fitch-style natural deduction system for an S5 modal logic containing an actuality operator, a diagonal necessity operator, and a diagonal possibility operator. The logic is two-dimensional, where we evaluate sentences with respect to both an actual world (first dimension) and a world of evaluation (second dimension). The diagonal necessity operator behaves as a quantifier over every point on the diagonal between actual worlds and worlds of evaluation, while the diagonal possibility quantifies over some point on the diagonal. Thus, they are just like the epistemic operators for apriority and its dual. We take this extension of Fitch’s familiar derivation system to be a very natural one, since the new rules and labeled lines hereby introduced preserve the structure of Fitch’s own rules for the modal case.


  • Modal logic
  • Natural deduction
  • Two-dimensional modal logic
  • Two-dimensional semantics
  • A priori
  • Actuality
  • Necessity

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  1. 1.

    If we can say that these constitute a single research program at all. In any case, Chalmers (2004) provides a nice account of how exactly most of these differ.

  2. 2.

    The exception is Restall (2012), as mentioned below.

  3. 3.

    Thereby relegating □ to superficial necessity. Strictly speaking, the axiomatic system in Davies and Humberstone (1980) does not contain a diagonal operator, but a ‘fixedly’ operator, \(\mathcal{F}\), which alongside an actuality operator, \(\mathcal{A}\), generates truth on the diagonal. It is with respect to \(\mathcal{FA}\) that deep necessity is defined. For more on the notions of deep and superficial necessity, see Evans (1979).

  4. 4.

    Note that (S) holds at every world considered as actual, since in these worlds ‘Julius’ always picks out the actual inventor of the zip. If, by contrast, the actual world is held fixed, and we only consider different possible worlds relative to this world as the actual one, then there will be pairs at which (S) is false—precisely those in which the inventor of the zip differs from the actual one.

  5. 5.

    We purposefully conflate use/mention when the context is clear to avoid cluttering the paper.

  6. 6.

    In Lampert (forthcoming) we present several prefixed tableau systems for different two-dimensional modal logics.

  7. 7.

    For a discussion on different accounts of validity in two-dimensional logics, see Lampert (forthcoming).

  8. 8.

    Strictly, axiom-schemata.

  9. 9.

    Fritz does not list \(\mathcal{A}1\) between the axioms, but this appears in Crossley and Humberstone’s original formulation (1977, p. 14).

  10. 10.

    These are the rules where no new scope line is opened in the elimination of \(\mathcal{D}\) and the introduction of \(\mathcal{C}\), respectively.


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Thanks to Alexander Kocurek, Andrew Parisi, Shawn Standefer, the UC Davis Llemma group, and two anonymous reviewers for comments on earlier drafts that greatly improved this paper.

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Correspondence to Fabio Lampert .

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Lampert, F. (2017). Natural Deduction for Diagonal Operators. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. CSHPM 2016. Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques. Birkhäuser, Cham.

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