Automated Reasoning in Normative Detachment Structures with Ideal Conditions (1810.09993v1)

Abstract: Systems of deontic logic suffer either from being too expressive and therefore hard to mechanize, or from being too simple to capture relevant aspects of normative reasoning. In this article we look for a suitable way in between: the automation of a simple logic of normative ideality and sub-ideality that is not affected by many deontic paradoxes and that is expressive enough to capture contrary-to-duty reason- ing. We show that this logic is very useful to reason on normative scenarios from which one can extract a certain kind of argumentative structure, called a Normative Detachment Structure with Ideal Conditions. The theoretical analysis of the logic is accompanied by examples of automated reasoning on a concrete legal text.

• The paper introduces the deontic logic DL* and Normative Detachment Structures with Ideal Conditions (NDSIC) to formally model and automate reasoning about normative ideality and contrary-to-duty scenarios in texts like legal documents.
• DL* extends Standard Deontic Logic with operators (O, O') distinguishing ideal from awful worlds and defines obligations (Ought*) and permissions (Perm*) based on these distinctions, providing a richer framework for complex norms.
• An automated procedure using the MleanCoP theorem prover and translating programs is presented for answering normatively relevant queries about formalized legal texts, with potential applications in legal assistance tools.

The paper "Automated Reasoning in Normative Detachment Structures with Ideal Conditions" by Libal and Pascucci discusses the automation of a deontic logic, called DL$^*$, for reasoning about normative ideality and sub-ideality, focusing on contrary-to-duty reasoning and its application to legal texts. The paper introduces the concept of a Normative Detachment Structure with Ideal Conditions (NDSIC) and demonstrates how DL$^*$ can be used to formulate and answer normatively relevant queries extracted from legal texts.

The authors begin by criticizing Standard Deontic Logic (SDL), highlighting its limitations in capturing normative scenarios, particularly those involving contrary-to-duty norms, exemplified by Chisholm's paradox. They then introduce DL$^*$, a variation of the deontic logic DL proposed by Jones and P\"{o}rn, which distinguishes between normative ideality and sub-ideality using two operators: $O$ and $O'$. The formula $OA$ is interpreted as "$A$ is true in all normatively ideal worlds," while $O'A$ means "$A$ is true in all normatively awful worlds." An operator $Ought^*$ is defined as $Ought^*(A) =_{def} OA \wedge O'\neg A$, which is used to represent obligations.

The semantics of DL$^*$ are defined using frames $\frak{F}=\langle W, R_O, R_{O'}\rangle$, where $W$ is a domain of worlds, and $R_O$ and $R_{O'}$ are binary relations over $W$ satisfying the properties:

• For all $w\in W$, there are $v,u\in W$ such that $wR_O v$ and $wR_{O'}u$.
  • $w$ is a world in $W$.
  • $v$ and $u$ are also worlds in $W$
  • $R_O$ is a binary relation between worlds in $W$, relating $w$ to $v$.
  • $R_{O'}$ is a binary relation between worlds in $W$, relating $w$ to $u$.
• For all $w\in W$, $R_O(w)\cap R_{O'}(w)=\emptyset$.
  • $w$ is a world in $W$.
  • $R_O(w)$ is the set of worlds accessible from $w$ via the relation $R_O$.
  • $R_{O'}(w)$ is the set of worlds accessible from $w$ via the relation $R_{O'}$.

The authors address the "pragmatic oddity" arising from formalizing Chisholm's example in DL by introducing the concept of ideal obligations, formalized as $O(Ought^*(P))$. They also modify the formalization of conditional obligations to allow for the construction of chains of statements that describe what ideally ought to be the case.

The paper introduces the argumentative structure NDSIC, which includes:

• A list of ideal normative statements ($Cid_1$ to $Cid_n$), expressed as $A^*_i$ ideally ought to be the case.
• A list of normative conditionals ($Ccon_1$ to $Ccon_m$), expressed as if $A_i$ then $B_i$ ought to be the case.
• Factual relations among the statements ($Crel_1$ to $Crel_k$).
• Actual circumstances that trigger the antecedents of conditionals ($Cant$).

The paper provides an example of an NDSIC extracted from the travel guidelines of passengers of the Sociedade de Transportes Colectivos do Porto. The authors also define a permission operator $Perm^*(A)=_{def} PA \wedge P'\neg A$, where $P$ and $P'$ are the duals of $O$ and $O'$ respectively.

The authors illustrate the use of DL$^*$ by formalizing excerpts from The United Nations Convention on Contracts for the International Sale of Goods. They define a language of the Convention, codifying statements as propositional symbols and introducing deontic operators and normative conditionals. They then present a formal description of statements from the Convention, creating a set of formulas $UN$.

The paper presents several normatively relevant queries concerning the Convention, which are formalized as derivability problems in DL$^*$. For example, one query asks whether there is a violation of the Convention if the contract requires the seller to take care of the carriage of goods and the goods are placed at the seller's address of business. This is expressed as the derivability of the conditional $$(\bigwedge UN \wedge D0.1 \wedge D3)\rightarrow(\neg D\vee\neg P\vee\neg G)$$ in DL$^*$.

• $\bigwedge UN$ represents the conjunction of all formulas in the set $UN$, which formalize the relevant articles of the Convention.
• $D0.1$ represents the proposition "the contract requires the seller to take care of the carriage of goods."
• $D3$ represents the proposition "the goods are placed at the seller's address of business."
• $D$ represents the proposition "the seller delivers the goods, hands over the documents, and transfers the property according to the procedure described in the contract."
• $P$ represents the proposition "the buyer pays for the goods."
• $G$ represents the proposition "the buyer takes delivery of the goods."

To automate the answering of these queries, the authors use MleanCoP, a theorem prover for multimodal logic. They developed a program to translate queries written in their encoding into MleanCoP's syntax. The program, along with MleanCoP, is available for download on Zenodo. The paper describes the syntax of both MleanCoP and the program, and provides an example of how to formulate Query 1 using the program.

The authors discuss two potential applications of their framework: a tool for helping subjects navigate legal texts and a courtroom decision support system. The first application aims to make legal texts more accessible to non-experts by allowing them to formulate queries and receive automated answers. The second application assists judges in making informed decisions by displaying all possible normative outcomes for a given set of facts, represented as a graph. A second program was developed for generating these graphs, taking as input a set of known facts, unknown parameters, and possible outcomes.

In conclusion, the paper presents a formalization of contrary-to-duty reasoning using the deontic logic DL$^*$ and demonstrates its application to legal texts through the concept of NDSIC. The authors provide an automated procedure for answering normatively relevant queries using MleanCoP and discuss potential applications of their framework in legal contexts. They address the criteria for logical representations of contrary-to-duty scenarios outlined by Carmo and Jones, demonstrating that their formalization meets these requirements. The paper argues that their approach strikes a balance between the expressivity of logical languages and the efficiency of theorem provers, making it a valuable tool for automated legal reasoning.