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A formalization of the Protagoras court paradox in a temporal logic of epistemic and normative reasons

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Abstract

We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent’s knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics for the logic: one based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to a notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz’s solution.

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Notes

  1. \({\,{\mathcal {O}}}_i \varphi \) can be alternatively read “it is obligatory for the agent i to do \(\varphi \).” In fact, in the context of deontic logic, there is a distinction between two kinds of ought: ought-to-do (the agent ought to do an action) and ought-to-be (it ought to be the case that). The former represents a normative ought while the latter represents an ideal ought. It is common to express the normative ought in terms of the ideal ought as follows: the agent ought to do \(\varphi \) iff it ought to be the case that the agent does \(\varphi \) (cf. e.g. Hansson 2013a, p. 455). While this distinction might be significant, in the present paper we shall frequently not make a distinction between these two meanings of “ought” and we will use them interchangeably.

  2. In fact, by accepting the view that strong justified permission is distinct from permission in the weak sense of not being forbidden, we reject what is called by Moore “the reflex thesis” according to which “all legal norms of permission are merely assertions of the absence (non-existence, negation, or failure) of norms of prohibition.” (cf. Moore 1973).

  3. In the rest of the paper, we will not explicitly specify the set of agents, justification variables, and atomic propositions, whenever it is clear from the context.

  4. A constant specification \(\textsf {CS}\) for \(\textsf {JTO}\) is axiomatically appropriate provided, for every axiom instance \(\varphi \) of \(\textsf {JTO}\) and for every \(i \in \textsf {Ag}\), there are constants c, d such that \(\left[ c\right] \!_i \varphi , \left[ d\right] ^{\,O}_i\! \varphi \in \textsf {CS}\), and in addition \(\textsf {CS}\) is upward closed: if \(\left[ c_{j_n}\right] \!_{i_n}\ldots \left[ c_{j_1}\right] \!_{i_1} \varphi \in \textsf {CS}\) (or \(\left[ c_{j_n}\right] ^{\,O}_{i_n}\! \ldots \left[ c_{j_1}\right] ^{\,O}_{i_1}\! \varphi \in \textsf {CS}\)), then for every \(i \in \textsf {Ag}\) there is a constant c such that \(\left[ c\right] \!_i \left[ c_{j_n}\right] \!_{i_n}\ldots \left[ c_{j_1}\right] \!_{i_1} \varphi \in \textsf {CS}\) (or \(\left[ c\right] ^{\,O}_i\! \left[ c_{j_n}\right] ^{\,O}_{i_n}\! \ldots \left[ c_{j_1}\right] ^{\,O}_{i_1}\! \varphi \in \textsf {CS}\)).

  5. See Sobel (1987) for a discussion on “first win a case” condition and “win first case” condition.

  6. If it is stipulated in the contract that Euathlus himself should plead before jurors and win his court-case, then one trivial way for Euathlus to avoid paying the fee is that he would hire a lawyer to plead before jurors. Because, if he wins with a lawyer, his victory would be non-paradoxical, and if he loses with a lawyer, he would not yet have won his first case.

  7. We suppose here that cases are numbered according to their commencement dates, rather than their conclusion dates. In addition, the court resolves the case by a yea or nay verdict, and not by dismissal or suspension. Otherwise, we need a four-valued logic for formalization. In fact, in an ancient rendition of the case (e.g. Aulus Gellius, The Attic Nights, (f.c. 150 A.D.)), the court decides to postpone its decision (deferral), and in another version (e.g. Hermogenes) the case is dismissed as not really fitting a courtroom. For more details see Jankowski (2015).

  8. In fact, in the deontic justification logic presented in Faroldi et al. (2022), the connecting principle con (see Sect. 2) implies that .

  9. See Goossens (1977, p. 72) for a discussion on “the content of the ruling” and “the ruling as an in-the-world event.”

  10. Euathlus might then sue Protagoras for malicious prosecution and ask compensation.

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  1. Department of Philosophy, Faculty of Literature and Humanities, University of Isfahan, Isfahan, Iran

    Meghdad Ghari

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran

    Meghdad Ghari

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Meghdad Ghari: A preliminary version of this paper was presented in the “Annual Seminar on Mathematical Logic and its Applications”, Arak University of Technology, September 2019. The author would like to thank the organizers for their invitation. I would also like to thank Daniela Glavanic̆ová and Federico Faroldi, as well as the referees of this journal for their comments and suggestions. This research was in part supported by a grant from IPM (No. 1401030416) and carried out in IPM-Isfahan Branch.

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Ghari, M. A formalization of the Protagoras court paradox in a temporal logic of epistemic and normative reasons. Artif Intell Law (2023). https://doi.org/10.1007/s10506-023-09351-0

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