Abstract
This paper proposes a new solution to the well-known Free Choice Permission Paradoxes (Barker 2010; Hansson 2013; Xin and Dong 2014), combining ideas from substructural logics and non-monotonic reasoning. Free choice permission is intuitively understood as “if it is permitted to do \(\alpha \) or \(\beta \) then it is permitted to do \(\alpha \) and it is permitted to do \(\beta \).” Yet, one of its logically equivalent forms allows the following inference which seems unacceptable: if it is permitted to order a vegetarian lunch then it is permitted to order a vegetarian lunch and not pay for it (Hansson 2013). The challenge for a logic of free choice permission is to exclude such counterintuitive consequences while not giving up too much deductive power. We suggest that the right way to do so is using a family of substructural logics augmented with a principle borrowed from non-monotonic reasoning. This follows up on a proposal made in Anglberger et al. (2014).
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- 1.
Subsequent literature has not always used “strong permissions” in the same sense as von Wright. Asher and Bonevac (2005) use the term in a way which is closer to what we call the “open reading” in Sect. 1.2. In Makinson and van der Torre (2003), two different senses of explicit permissions, static and dynamic, are distinguished and studied. In this paper, we will use “strong permission” in von Wright’s sense unless otherwise specified.
- 2.
We take “irrelevance” in a different sense, but still very close to relevant logic (Restall 2006): only
is not valid in our logic. Our logics validate 
, which is rejected in relevant logic. In this paper, \(\mathord {\sim }\) and \(\lnot \) are different negations, and \(\mathord {\sim }\) will be seen as the negation for actions. - 3.
By taking
and 
as classical validities, the logical equivalence between CI and FCP holds after using the substitution of logical equivalences. - 4.
- 5.
The composition operator \(\circ \) is not the sequent composition operator in propositional dynamic logic (PDL) [Harel et al. 2000, p. 168]. This operator \(\circ \), sometimes, may be understood as a non-standard concurrency operator of actions [Harel et al. 2000, p. 268, 276]. For example, \(Listen \circ WriteNote\). We suggest reading \(\alpha \circ \beta \) as “doing action \(\alpha \) and action \(\beta \) (together).”
- 6.
Later, we will see that neither the double negation introduction
nor the double negation elimination 
are valid in the (bi)-frame class. - 7.
Although after one replacement by substructure Z to Y the possibility of X[Z / Y] is not unique, it will not affect the sequent calculus introduced later.
- 8.
Notice that O-relation is not a serial relation as in standard deontic logic does. One reason is that we have not considered obligation, and to interpret it using O. Another reason is that seriality is not necessary for the characterization of (free choice) permission. The consistency is ensured by the action operators rather than the deontic one. Please refer to Theorem 9 for more details.
- 9.
As usual, when it is a double line, then the lower one is the consequent and the upper one is the conclusion.
- 10.
Recall that \(\vdash \) is a single-consequence relation.
- 11.
This model \(\mathcal {M}\) does not satisfy (cam), but it does satisfy (ram). Here is the case to invalidate (cam). We have \(Lx_1(x_3 x_1)x_4\). And for all \(z' (Lx_1x_3z' \rightarrow z' \supseteq x_1)\) because this \(z'\) must be \(x_1\) if \(Lx_1x_3z'\) exists. However, we do not have \(Lx_1x_3x_4\) in this model.
- 12.
The action negation \(\mathord {\sim }\) still satisfies the ex contradictione quodlibet rule (ECQ)
in the (bi)-frame class, which is rejected in relevant logics. - 13.
Given arbitrary frame \(\mathcal {F}\) in the open reading frame class, all states in \(\mathcal {F}\) are consistent and not complete, which can be understood in this way:
is valid in \(\mathcal {F}\), but not 
. - 14.
Removing OR and Den from the logic N, this system is a basic substructural logic for full Lambek calculus FL (Galatos et al. 2007). It is weaker than Barker’s linear logic because our fusion is neither associative nor commutative in N.
- 15.
Barker’s system does not contain the rules (OR) and (Den), which are required in our system N. Even so, N cannot be seen as an extension of Barker’s logic, as the case, we argued here. So our system N has a substantive difference from Barker’s linear logic.
- 16.
As we discussed earlier, strong permissions are not the dual of obligations.
is not valid in our logic. Our logics validate 
, which is rejected in relevant logic. In this paper, 
and 
as classical validities, the logical equivalence between CI and FCP holds after using the substitution of logical equivalences.
nor the double negation elimination 
are valid in the (bi)-frame class.
in the (bi)-frame class, which is rejected in relevant logics.
is valid in 
.References
Anglberger, A.J., Dong, H., Roy, O.: Open reading without free choice. In: Cariani, F., Grossi, D., Meheus, J., Parent, X. (eds.) Deontic Logic and Normative Systems. Lecture Notes in Computer Science, vol. 8554, pp. 19–32. Springer International Publishing (2014). https://doi.org/10.1007/978-3-319-08615-6_3
Anglberger, A.J., Gratzl, N., Roy, O.: Obligation, free choice, and the logic of weakest permissions. Rev. Symb. Log. 8, 807–827 (2015). https://doi.org/10.1017/S1755020315000209
Asher, N., Bonevac, D.: Free choice permission is strong permission. Synthese 145(3), 303–323 (2005)
Asudeh, A.: Linear logic, linguistic resource sensitivity and resumption (2006)
Barker, C.: Free choice permission as resource-sensitive reasoning. Semant. Pragmat. 3(10), 1–38 (2010). https://doi.org/10.3765/sp.3.10
Belnap, N.D., Perloff, M., Xu, M.: Facing the Future: Agents and Choices in our Indeterminist World. Oxford University Press on Demand (2001)
Blackburn, P., De Rijke, M., Venema, Y.: Modal Logic, vol. 53. Cambridge University Press (2002)
Broersen, J.: Action negation and alternative reductions for dynamic deontic logics. J. Appl. Log. 2(1), 153–168 (2004)
Di Cosmo, R., Miller, D.: Linear logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Winter 2016 edn (2016)
Dignum, F., Meyer, J.J.C., Wieringa, R.J.: Free choice and contextually permitted actions. Stud. Logica 57(1), 193–220 (1996)
Dong, H., Roy, O.: Three deontic logics for rational agency in games. Stud. Log. 8(4), 7–31 (2015)
Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, vol. 151. Elsevier (2007)
Governatori, G., Olivieri, F., Rotolo, A., Scannapieco, S.: Computing strong and weak permissions in defeasible logic. J. Philos. Log. 42(6), 799–829 (2013)
Hansson, S.O.: The varieties of permissions. In: Gabbay, D., Horty, J., Parent, X., van der Meyden, R., van der Torre, L. (eds.) Handbook of Deontic Logic and Normative Systems, vol. 1. College Publication (2013)
Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press (2000)
Hilpinen, R.: Disjunctive permissions and conditionals with disjunctive antecedents. Acta Philos. Fenn. 35, 175–194 (1982)
Horty, J.F.: Agency and Deontic Logic. Oxford University Press (2000)
Kamp, H.: Free choice permission. In: Proceedings of the Aristotelian Society, vol. 74, pp. 57–74 (1973) (JSTOR)
Kulicki, P., Trypuz, R.: On deontic action logics based on boolean algebra. J. Log. Comput. 25(5) (2015)
Kurtonina, N.: Frames and labels. A modal analysis of categorial deduction. Ph.D. thesis, PhD Thesis and University of Amsterdam (1995)
Makinson, D.: Stenius’ approach to disjunctive permission. Theoria 50(2–3), 138–147 (1984)
Makinson, D.: Bridges from Classical to Nonmonotonic Logic. King’s College (2005)
Makinson, D., van der Torre, L.: Permission from an input/output perspective. J. Philos. Log. 32(4), 391–416 (2003)
McCarthy, J.: Epistemological problems of artificial intelligence. In: Proceedings of the 5th International Joint Conference on Artificial Intelligence, vol. 2, pp. 1038–1044. Morgan Kaufmann Publishers Inc. (1977)
McNamara, P.: Deontic logic. In: Handbook of the History of Logic, vol. 7, pp. 197–289 (2006)
McNamara, P.: Deontic logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Fall 2010 edn (2010)
Negri, S., Von Plato, J., Ranta, A.: Structural Proof Theory. Cambridge University Press (2008)
Paoli, F.: Substructural Logics: A Primer, vol. 13. Springer Science & Business Media (2013)
Pelletier, F.J., Asher, N.: Generics and defaults. In: Handbook of Logic and Language, pp. 1125–1177. Elsevier (1997)
Restall, G.: An Introduction to Substructural Logics. Routledge (2000a)
Restall, G.: An Introduction to Substructural Logics. Psychology Press (2000b)
Restall, G.: Logic: An Introduction. McGill-Queen’s University Press (2006)
Schurz, G.: Relevant Deduction. In: Erkenntnis Orientated: A Centennial Volume for Rudolf Carnap and Hans Reichenbach, pp. 391–437. Springer (1991)
Segerberg, K., Meyer, J.J., Kracht, M.: The logic of action. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Winter, 2016 edn (2016)
Simons, M.: Dividing things up: the semantics of or and the modal/or interaction. Nat. Lang. Semant. 13(3), 271–316 (2005)
Trypuz, R., Kulicki, P.: Towards metalogical systematisation of deontic action logics based on boolean algebra. In: Deontic Logic in Computer Science, pp. 132–147. Springer (2010)
van Benthem, J.: Minimal deontic logics. Bull. Sect. Log. 8(1), 36–42 (1979)
van Benthem, J.: Language in Action: Categories, Lambdas and Dynamic Logic. MIT Press (1995)
van Benthem, J.: What one may come to know. Analysis 64(282), 95–105 (2004)
van de Putte, F.: “That will do”: logics of deontic necessity and sufficiency. Erkenntnis (2016) (in print)
von Wright, G.H.: Norm and Action-A Logical Enquiry. Routledge (1963)
von Wright, G.H.: An Essay in Deontic Logic and the General Theory of Action. North-Holland Publishing Company (1968)
Weingartner, P., Schurz, G.: Paradoxes solved by simple relevance criteria. Logique et Analyse 113, 3–40 (1986)
von Wright, G.H.: On the logic of norms and actions. In: New Studies in Deontic Logic: Norms, Actions, and the Foundations of Ethics, pp. 3–35. Springer Netherlands, Dordrecht (1981). https://doi.org/10.1007/978-94-009-8484-4_1
Xin, S., Dong, H.: The deontic dilemma of action negation, and its solution. In: The Eleventh Conference on Logic and the Foundations of Game and Decision Theory (LOFT 2014). http://hdl.handle.net/10993/19577 (2014)
Zimmermann, T.E.: Free choice disjunction and epistemic possibility. Nat. Lang. Semant. 8(4), 255–290 (2000)
Acknowledgements
We would like to thank Albert J. J. Anglberger, Sabine Frittella, Fei Liang, Johannes Korbmacher, Piotr Kulicki, Clayton Peterson, and Robert Trypuz for their comments and suggestions on the early version, and thanks to Johan van Benthem, O. Foisch, Xiaowu Li, and the anonymous reviewers for their insightful comments and suggestions on the latest version. All authors are supported by the PIOTR research project [No. RO 4548/4-1]. Huimin Dong is supported by the China Postdoctoral Science Foundation funded project [No. 2018M632494], the MOE Project of Key Research Institute of Humanities and Social Sciences in Universities [No. 17JJD720008], the National Social Science Fund of China [No. 18ZDA290], and by the Fundamental Research Funds for the Central Universities of China.
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Dong, H., Gratzl, N., Roy, O. (2019). Open Reading and Free Choice Permission: A Perspective in Substructural Logics. In: Liao, B., Ågotnes, T., Wang, Y. (eds) Dynamics, Uncertainty and Reasoning. CLAR 2018. Logic in Asia: Studia Logica Library. Springer, Singapore. https://doi.org/10.1007/978-981-13-7791-4_5
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