Intuitionistic epistemic logic, Kripke models and Fitch’s paradox.

*(English)*Zbl 1280.03020Summary: The present work is motivated by two questions. (1) What should an intuitionistic epistemic logic look like? (2) How should one interpret the knowledge operator in a Kripke-model for it? In what follows we outline an answer to (2) and give a model-theoretic definition of the operator \(K\). This will shed some light also on (1), since it turns out that \(K\), defined as we do, fulfills the properties of a necessity operator for a normal modal logic. The interest of our construction also lies in a better insight into the intuitionistic solution to Fitch’s paradox, which is discussed in the third section. In particular we examine, in the light of our definition, DeVidi and Solomon’s proposal of formulating the verification thesis as \(\phi \rightarrow \neg \neg K\phi\). We show, as our main result, that this definition excapes the paradox, though it is validated only under restrictive conditions on the models.

##### MSC:

03B42 | Logics of knowledge and belief (including belief change) |

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\textit{C. Proietti}, J. Philos. Log. 41, No. 5, 877--900 (2012; Zbl 1280.03020)

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##### References:

[1] | Bozic, B., & Dosen, K. (1984). Models for normal intuitionistic modal logics. Studia Logica, 43, 217–244. · Zbl 0634.03014 |

[2] | Burgess, J. (2008). Can truth out? In J. Salerno (Ed.), New essays on the knowability paradox (à paraître). Oxford: Oxford University Press. |

[3] | DeVidi, D., & Solomon, G. (2001). Knowability and intuitionistic logic. Philosophia, 28, 319–334. |

[4] | Dosen, K. (1985). Models for stronger normal intuitionistic modal logics. Studia Logica, 44, 49–70. · Zbl 0634.03015 |

[5] | Goldblatt, R. (2003). Mathematical modal logic: A view of its evolution. Journal of Applied Logic, 1, 309–392. · Zbl 1041.03015 |

[6] | Hintikka, J. (1962). Knowledge and belief. Dordrecht: Reidel. · Zbl 1384.03102 |

[7] | Tennant, N. (1997). The taming of the true. Oxford: Clarendon Press. · Zbl 0929.03001 |

[8] | van Benthem, J. (2004). What one may come to know. Analysis, 64, 95–105. · Zbl 1073.03004 |

[9] | van Benthem, J. (2008). The information in intuitionistic logic. Synthese, to appear. URL http://www.illc.uva.nl/Publications/ResearchReports/PP-2008-37.text.pdf . |

[10] | Williamson, T. (1982). Intuitionism disproved? Analysis, 42, 203–207. |

[11] | Williamson, T. (1988). Knowability and constructivism. The Philosophical Quarterly, 38, 422–432. |

[12] | Williamson, T. (1992). On intuitionistic modal epistemic logic. Journal of Philosophical Logic, 21, 63–89. · Zbl 0746.03014 |

[13] | Wolter, F., & Zakharyaschev, M. (1999). Intuitionistic modal logics as fragments of classical bimodal logics. In E. Orlowska (Ed.), Logic at work (pp. 168–186). Dordrecht: Kluwer. · Zbl 0922.03023 |

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