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A Logic for Human Actions

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Applications of Formal Philosophy

Part of the book series: Logic, Argumentation & Reasoning ((LARI,volume 14))

Abstract

The present paper introduces an action logic able to model human actions. We begin by providing an analysis of the proof-theory of action logics from the perspective of category theory. Action logics are classified as different types of monoidal deductive systems with respect to their categorical structure. This enables us to correlate the properties of the logical connectives with the type of deductive system that is used. We then provide a philosophical analysis of action connectives and, in light of our analysis, show which type of deductive system is required to model human actions. According to the usual distinction between actions and propositions in dynamic logic, we distinguish between an action logic, representing the formal structure of actions, and a propositional action logic, expressing the formal structure of the language we use to talk about actions.

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Notes

  1. 1.

    See Mac Lane [52, pp. 13, 16, 80-2] for the definitions of functors, natural transformations and adjoint functors.

  2. 2.

    See also Marquis [53] for arguments in favor of this point.

  3. 3.

    This section is built on previous work. See also [59].

  4. 4.

    It should be noted that there is a distinction between a proof of \(\psi \) from \(\varphi \), represented by the consequence relation \(\varphi \longrightarrow \psi \) (where \(\varphi \) is a formula), and the derivation of a proof, corresponding to its demonstration.

  5. 5.

    A rigorous presentation needs a different notation for deductive systems and categories. We will use lower case letters of the end of the alphabet when speaking of categories and Greek letters when speaking of deductive systems.

  6. 6.

    See Mac Lane [52, p. 16] for the definition.

  7. 7.

    See Barr [9, p. 161] for the definition of a dualizing object.

  8. 8.

    See [32] for an introduction and [7] for a historical sketch.

  9. 9.

    See [20] for an introduction to modal logics.

  10. 10.

    Note that (K1)–(K8) can all be derived using the definition of a MCcoC.

  11. 11.

    For more information on categories and dynamic logic, the reader can consult [73]. For other approaches in dynamic deontic logic, see [74] for a dynamic logic of permission, [66] for dynamic logic and agency and [14] for the analysis of action negation.

  12. 12.

    See also [12] for Kleene algebras and Pratt’s action logic.

  13. 13.

    On this subject, see also Blute and Scott [13].

  14. 14.

    Note that when a poset is seen as a category, \(\le \) must be viewed as an arrow between equivalence classes of actions rather than actions per se insofar as \(\le \) is an antisymmetric relation.

  15. 15.

    Recall that stit stands for seeing to it that.

  16. 16.

    This was reprinted in [37] and further developed in [38, 39].

  17. 17.

    Note that ‘bi-intuitionistic’ here refers to an intuitionistic logic with two conditionals. The relationship between bi-Heyting algebras and modal logic has been discussed by Reyes and Zolfaghari [65].

  18. 18.

    The abbreviations in Table 5.1 stand for associativity, commutativity, idempotence and distributivity.

  19. 19.

    Segerberg [68] also distinguishes between actions and assertions but uses a Boolean algebra of actions and Boolean propositional connectives.

  20. 20.

    At this point, one might be tempted to object that ‘logic’ is concerned with declarative sentences (i.e., with sentences that have the potential to be true or false) and, in this respect, the notion of an action ‘logic’ where actions are not understood as declarative sentences violates the current use of the term. It should be noted, however, that this view of ‘logic’ is too restrictive. Indeed, the notion of ‘truth’ is only a mere tool that can be discarded. For instance, in proof-theory, formulas are not understood as declarative sentences but are taken as having the potential to be ‘provable’ (rather than ‘true’).

  21. 21.

    By the same reasoning, if we want to be able to deal with limited resources, sequence must not be idempotent.

  22. 22.

    Walton [77, p. 320], for example, distinguished between fourteen different locutions of action negation within the natural language.

  23. 23.

    To be precise, the mens rea refers to the agent’s state of mind. Acting volontarily does not presuppose an explicit intention but only means that the agent is in a state of mind such that he has the capacity to understand what he is doing, even though he cannot anticipate every possile outcome of his actions.

  24. 24.

    Although criminal liability usually requires both an illegal act and a criminal mind, it should be noted that there are exceptions within Canadian law. First, some cases involve the notion of absolute liability, which depends on the act alone (see R. v. Canning, 2004 SKPC 13 for the definition). Second, there are specific cases where murders are straightaway considered as first degree murders (e.g., the killing of a police officer or a warden acting in the course of their duties), notwithstanding the intention of the person perpetrating the act (see for instance the Criminal Code, R.S.C., 1985, c. C-46, art. 231.4-231.6.).

  25. 25.

    This is quite straightforward assuming that \(\alpha \cong \beta \) since \(\alpha \cong \beta \cong \beta ^{**} \cong *\ominus \beta ^*\), and hence \(\alpha \otimes \beta ^* \cong *\) by the fact that \(\ominus \) is the adjoint of \(\otimes \), and by the same reasoning one obtains \(\beta ^*\cong *\ominus \alpha \cong \alpha ^*\).

  26. 26.

    It should be noted that logics that are analogues to compact deductive systems are used in quantum physics and quantum logic. See for instance Duncan’s [26] Ph.D. thesis and further work.

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Acknowledgements

I would like to thank Jean-Pierre Marquis for many discussions on this subject. I am also grateful to Andrew Irvine, François Lepage and Yvon Gauthier for helpful comments and discussions. Thanks to an anonymous referee for comments on a previous draft of this paper. This research was financially supported by the Social Sciences and Humanities Research Council of Canada.

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  1. Munich Center for Mathematical Philosophy, Munich, Germany

    Clayton Peterson

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  1. Institute of Philosophy, Sociology and Journalism, University of Gdansk, Poland and , Centre for Logic and Philosophy of Science, Ghent University,, Ghent, Belgium

    Rafał Urbaniak

  2. Dalhousie University , Halifax, Nova Scotia, Canada

    Gillman Payette

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Peterson, C. (2017). A Logic for Human Actions. In: Urbaniak, R., Payette, G. (eds) Applications of Formal Philosophy. Logic, Argumentation & Reasoning, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-58507-9_5

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