An Update Semantics for Defeasible Obligations

Abstract: The deontic logic DUS is a Deontic Update Semantics for prescriptive obligations based on the update semantics of Veltman. In DUS the definition of logical validity of obligations is not based on static truth values but on dynamic action transitions. In this paper prescriptive defeasible obligations are formalized in update semantics and the diagnostic problem of defeasible deontic logic is discussed. Assume a defeasible obligation normally A ought to be (done)' together withthe factA is not (done).' Is this an exception of the normality claim, or is it a violation of the obligation? In this paper we formalize the heuristic principle that it is a violation, unless there is a more specific overriding obligation. The underlying motivation from legal reasoning is that criminals should have as little opportunities as possible to excuse themselves by claiming that their behavior was exceptional rather than criminal.

• The paper introduces Deontic Update Semantics (DUS), a formal framework using update semantics to model and reason about defeasible obligations dynamically.
• It addresses the diagnostic problem of distinguishing exceptions from violations by proposing a heuristic principle and using dynamic levels of ideality and normality.
• The formalization aims to minimize opportunities for excusing non-compliance, utilizing state transitions to handle specificity and irrelevance in deontic reasoning.

The paper "An Update Semantics for Defeasible Obligations" introduces a dynamic perspective to deontic logic, specifically addressing defeasible obligations within the framework of update semantics. The central theme revolves around the formalization of prescriptive defeasible obligations using update semantics, drawing from Veltman's work, and tackles the diagnostic problem inherent in defeasible deontic logic.

The authors use the cottage housing regulations scenario to illustrate the complexities of reasoning with defeasible obligations. The core problem lies in differentiating between exceptions and violations when a defeasible obligation 'normally $\alpha$ ought to be (done)' is juxtaposed with the fact '-$\alpha$ is (done).' The paper posits a heuristic principle: such a scenario represents a violation unless a more specific, overriding obligation exists. This principle is motivated by a legal reasoning perspective that seeks to minimize opportunities for individuals to excuse criminal behavior by claiming exceptional circumstances.

The approach is formalized within a Deontic Update Semantics (DUS), where the validity of obligations is based on dynamic action transitions rather than static truth values. The logic aims to capture the intuition that criminals should have as few opportunities as possible to excuse themselves by claiming that their behavior was exceptional rather than criminal. The paper emphasizes that in the absence of a cliff, a penalty should be imposed for having a fence, as the initial obligation is violated. The difference between the antecedent of the second and third obligation is represented in the deontic states of the update semantics by two different orderings: the second gives rise to levels of exceptionality (inspired by preference-based approaches to defeasible reasoning) and the third gives rise to levels of ideality (inspired by preference-based approaches to deontic reasoning).

Key aspects of the formalization include:

• Deontic States: These are possible-worlds models represented as tuples $\sigma = (W, W*, \leq_I, \leq_N, V)$, where:
  • $W$ is a set of possible worlds.
  • $W* \subseteq W$ represents an explicit sub-state used for the context of deliberation.
  • $\leq_I$ is a reflexive binary relation on $W$ representing ideality.
  • $\leq_N$ is a transitive, reflexive, and totally connected binary relation on $W$ representing normality.
  • $V$ is a valuation function for propositions at the worlds.
• Deontic Language: This is a propositional language, $L_A$, extended with a dyadic operator oblige($\alpha$ | $\beta$), interpreted as "normally $\alpha$ ought to be (done), if $\beta$ is (done)."
• Updates: The update function $\sigma[+ \phi]$ defines how a deontic state $\sigma$ changes upon receiving a sentence $\phi$. Updates can either zoom in on the deontic state (for facts) or create ideality and normality levels (for obligations).
• Reduction: The reduction of $\sigma$ by oblige($\alpha$), denoted by $\sigma$ - oblige($\alpha$ | $\beta$), modifies the ideality relation $\leq_I$ by removing pairs of worlds $(w_1, w_2)$ where $w_1$ is a most normal -$\alpha \land \beta$ world and $w_2$ is a most normal $\alpha \land \beta$ world.
• Exception: The introduction of an exceptionality level in $\sigma$ by oblige($\alpha$ | $\beta$), denoted by $\sigma$ -$_N$ oblige($\alpha$ | $\beta$), modifies both the ideality relation $\leq_I$ and the normality relation $\leq_N$ to account for conflicts arising from the obligation.
• Preference (pref): The paper uses the transitive closure of the ideality relation, $\leq_\beta$, to define a preference relation pref($\alpha$, $\beta$) = $\alpha$. It holds if and only if for all worlds $w_1 \in W_1$ there is a world $w_2 \in W_2$ such that $w_2 \leq_\beta w_1$ and there is no $w_3 \in W_1$ such that $w_3 \leq_\beta w_2$, where $W_1$ and $W_2$ are the sets of the most normal -$\alpha \land \beta$ and $\alpha \land \beta$ worlds of $W$, respectively.
• Acceptance: A formula $\phi$ is accepted in a deontic state $\sigma$, written as $\sigma \Vdash \phi$, if updating $\sigma$ with $\phi$ results in the same state, i.e., $\sigma[+ \phi] = \sigma$.
• Validity: An argument from premises $\phi_1, \dots, \phi_n$ to a conclusion $\phi$ is valid, denoted as $\phi_1, \dots, \phi_n \vdash_1 \phi$, if updating the minimal state 0 with the premises in that order yields a deontic state in which the conclusion is accepted. Nonmonotonic validity, denoted as $\phi_1, \dots, \phi_n \Vdash \phi$, requires that for all permutations $\pi$ of $1 \dots n$ such that $$\{\phi_{\pi(1)}, \dots, \phi_{\pi(n)}[+]\} \Vdash \phi_i$$ for $1 \leq i \leq n$, we also have $\phi_{\pi(1)}, \dots, \phi_{\pi(n)} \vdash_1 \phi$.

The paper illustrates how the logic formalizes the specificity principle, ensuring that more specific and conflicting obligations override more general ones. This is achieved through the introduction of exceptionality levels that dynamically re-evaluate hierarchical obligations. The logic also addresses the irrelevance problem, ensuring that irrelevant factors do not affect the acceptance of obligations.

Further, the paper introduces test operators ideal($\alpha$ | $\beta$) and ideal*($\alpha$ | $\beta$), analogous to Veltman's might and presumably operators, to evaluate norms within a specific deontic state. These operators test whether, ideally, $\alpha$ is (done) given $\beta$ is (done), considering both the context of justification and the context of deliberation.

The authors discuss Chisholm's paradox to illustrate how the operators oblige and ideal are combined, and how strengthening of the antecedent and weakening of the consequent are combined.

In summary, the paper offers a formal framework for reasoning about defeasible obligations, highlighting the importance of dynamic interpretation and context-sensitivity in deontic reasoning.