Abstract
Legal dynamics is an important aspect of legal reasoning that inspired the area of belief revision. While formal models of belief revision have been thoroughly examined, the formalisation of legal dynamics has been mostly neglected. In this contribution we propose Temporal Defeasible Logic to model legal dynamics . We build such a logic in steps starting from basic defeasible logic , and we show how to use it to model different forms of modifications such as derogations, textual modifications, abrogation and annulment.
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Notes
- 1.
Defeasible Logic does not impose any property for \(\prec \). However, in many application is useful to assume that the transitive closure to be acyclic to prevent situations where, at the same time a rule overrules another rule and it is overridden by it.
- 2.
More correctly, we should use \(r: \{a_{1},\ldots,a_{n}\}\hookrightarrow c\). However, to improve readability, we drop the set notation for the antecedent of rule.
- 3.
The relationships between Defeasible Logic and argumentation are, in fact, deeper than the similarity of the argumentation like proof theory. Governatori et al. (2004) prove characterisation theorems for defeasible logic variants and Dung style argumentation semantics (Dung 1995). In addition Governatori (2011) proved that the Carneades argumentation framework (Gordon et al. 2007), widely discussed in the AI and Law literature, turns out to be just a syntactic variant of Defeasible Logic .
- 4.
Here we concentrate on proper defeasible derivations. In addition we notice that defeasible derivations inherit from definite derivations, thus we can assert + ∂ p if we have already established +Δ p.
- 5.
It is possible to give different definitions of support to obtain variants of the logic tailored for various intuitions of non-monotonic reasoning. Billington et al. (2010) show how to modify the notion of support to obtain variants capturing such intuitions, for example by weakening the requirements for a rule to be supported: instead of being defeasibly provable a rule is supported if it is possible to build a reasoning chain from the facts ignoring rules for the complements.
- 6.
As explained elsewhere (Governatori and Rotolo 2010), we do not add a deontic operator in the consequent of rules (i.e., \(a_{1},\ldots,a_{n} \Rightarrow \mathsf{O}b\)), but we rather differentiate the mode of conclusions by distinguishing diverse rule types. This choice has a technical motivation: (a) it considerably makes simpler and more compact the proof theory; (b) it allows us to characterise a specific logical consequence relation for \(\mathsf{O}\). However, another version of the logic (much more cumbersome) can be adopted where deontic rules have the form \(a_{1},\ldots,a_{n} \Rightarrow \mathsf{O}b\) without affecting our treatment of legal dynamics .
- 7.
In the three formulas below \(\rightarrow \) is the material implication of classical logic.
- 8.
This norm makes use of “must not”, to see that “must not” is understood as prohibition in legal documents see, the Australian National Consumer Credit Protection Act 2009, Section 29, whose heading is “Prohibition on engaging in credit activities without a licence”, recites “(1) A person must not engage in a credit activity if the person does not hold a licence authorising the person to engage in the credit activity”.
- 9.
The idea of using defeaters to introduce permissions was introduced by Governatori et al. (2005b).
- 10.
We equate arguments with rules, thus this is the same as saying that there is (defeasible) rule such that all the elements in its antecedent are provable and the conclusion is p (t′, y).
- 11.
In the following formalisation, we have used t + 2d to indicate the time when a complaint has been received, represented by the variable t, plus 2 days. We formalism we propose is neutral about the representation of time and the granularity used in such a representation. All we need is a representation of time isomorphic to the set of natural numbers. Moreover, the time variables in the rules have to be instantiated with the specific time values.
- 12.
There are many subclauses of clause 4.1.2 and it is well outside the scope of this work to formalise them. Hence, we limit ourselves to formalise a template that has to be instantiated with the appropriate content. In any case, all rules will have the same duration specifications.
- 13.
We do not need to impose that the function is injective: while each label should have only one content at any given time, we may have that different labels (rules) have the same content.
- 14.
For instance, if we have \(s \prec _{\mathit{Monday}}^{2007}r\) and \(r \prec _{\mathit{Tuesday}}^{2007}s\), it means that, according to the regulation in force in 2007, on Mondays rule s is stronger than rule r, but on Tuesdays r is stronger than s.
- 15.
Two meta-rules are conflicting, when the two meta-rules have the same rule as their head, but with a different content.
- 16.
In this way, we detach from the terminology adopted in several legal systems and accepted, e.g., by Alchourrón and Makinson (1981, 1982) and Alchourrón and Bulygin (1981). Indeed, since we clearly distinguish the dynamics of obligations and permissions from the ones of legal norms, we can identify various reasons for undoing legal effects by manipulating the same set of legal norms: when legal effects are undone via adding exceptions, we will have derogations, when they are undone via norm removal we will have annulments or abrogations, etc. See below and, for further references, cf. Governatori et al. (2005a) and Stolpe (2010).
- 17.
In the remainder of the paper, when temporal parameters are not essential we will not specify them and will just add a superscript x.
- 18.
Recall that, for any rule s, A(s) denotes the set of antecedents of s, while ∂ −(D) stands for the set of negative conclusions of the theory D. i.e., the literals occurring in conclusions of the form − ∂.
- 19.
The general procedure to block conclusions when conclusions persist over repositories can be very complex: for all details, see Governatori and Rotolo (2010).
- 20.
Hence, derogation was not meant by Alchourrón and Makinson as a process of adding exceptions. On this point, see above footnote 16.
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NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.
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Governatori, G., Rotolo, A. (2015). Logics for Legal Dynamics. In: Araszkiewicz, M., Płeszka, K. (eds) Logic in the Theory and Practice of Lawmaking. Legisprudence Library, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-19575-9_12
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