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An Axiomatic System and a Tableau Calculus for STIT Imagination Logic

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Abstract

We formulate a Hilbert-style axiomatic system and a tableau calculus for the STIT-based logic of imagination recently proposed in Wansing (2015). Completeness of the axiom system is shown by the method of canonical models; completeness of the tableau system is also shown by using standard methods.

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Notes

  1. Imagination differs from knowledge not just in being agentive. Knowledge is factive, imagination clearly is not. Whilst justification is generally seen as a necessary condition of possessing knowledge, this is clearly not the case for imagination. Epistemic extensions of STIT logic have been presented, for example, in [3, 16], for a survey see [14].

  2. The other Boolean connectives are defined as usual.

  3. In [16] (notation adjusted), [d] a B a A is suggested as a formalization of “agent a forms the implicit belief that A”, where B a is a KD45 modal operator. In this case, it is desired not to exclude the option of understanding [d] a B a A as a statement of indirect belief formation.

  4. Σ is satisfiable and thus consistent. Indeed, consider a model consisting of a single moment, where every agent has a vacuous choice, every set of imagination neighbourhoods is empty and every variable valuation is empty as well.

  5. We also assume, in view of the definition of ≤ below, that 0 is not an element of any element of Ξ ∪ W.

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Acknowledgments

We would like to thank the two anonymous referees for their valuable comments, and we would like to acknowledge financial support from the DFG, project WA 936/11-1, during the preparation of the revised version of this paper. Grigory Olkhovikov would like to acknowledge financial support from the Alexander von Humboldt Foundation which made it possible for him to take part in obtaining the results reported above.

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Authors and Affiliations

  1. Department of Philosophy II, Ruhr-University Bochum, Bochum, Germany

    Grigory K. Olkhovikov & Heinrich Wansing

  2. Department of Philosophy, Ural Federal University, Ekaterinburg, Russia

    Grigory K. Olkhovikov

Authors
  1. Grigory K. Olkhovikov
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  2. Heinrich Wansing
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Correspondence to Heinrich Wansing.

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Olkhovikov, G.K., Wansing, H. An Axiomatic System and a Tableau Calculus for STIT Imagination Logic. J Philos Logic 47, 259–279 (2018). https://doi.org/10.1007/s10992-017-9426-1

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  • DOI: https://doi.org/10.1007/s10992-017-9426-1

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