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.
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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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