Non-Monotonic Reasoning

The foundational property where a previously justified conclusion can be defeated or invalidated by new evidence. In legal contexts, a contract clause may be presumed valid until a fraud exception is introduced. This contrasts with monotonic logic, where the set of entailed theorems only grows. Defeasibility is formalized through defeaters—rules that block an inference without supporting the opposite conclusion—and is essential for modeling rebuttable presumptions in statutory interpretation.

Inference rules that apply typically or by default unless an exception is triggered. Represented as A : B / C in default logic, meaning: if A is known and B is consistent with what is known, conclude C. In legal knowledge graphs, default rules encode prima facie obligations. For example:

• Default: A signed contract is enforceable
• Exception: The signatory lacked capacity
• Exception to the exception: The incapacity was self-induced This layered structure maps directly to LegalRuleML representations.

The AGM paradigm (Alchourrón, Gärdenfors, Makinson) defines rational postulates for contracting a knowledge base when contradictions arise. When a new statute conflicts with existing case law, the system must perform contraction (removing the contradiction) followed by expansion (adding the new rule). Key operations include:

• Expansion: Adding a formula without consistency checks
• Contraction: Removing a formula to eliminate inconsistency
• Revision: Adding while maintaining consistency Legal systems use partial meet contraction to preserve maximal subsets of existing precedent.

A formalization by John McCarthy that minimizes the extension of predicates to assume only those objects known to satisfy them. In legal reasoning, circumscription closes off the set of exceptions: a contract is valid unless explicitly stated otherwise. This implements the closed-world assumption locally rather than globally. Circumscription is particularly useful for modeling statutory silence—when a statute does not address a scenario, the system assumes the scenario does not trigger the rule rather than concluding uncertainty.

Formalized by Dung as directed graphs where arguments are nodes and attacks are edges. A set of arguments is admissible if it is conflict-free and defends all its members against attackers. Legal reasoning maps naturally to this structure:

• Arguments = legal interpretations
• Attacks = counter-arguments citing contrary authority
• Grounded semantics yield the most skeptical (safest) conclusions
• Preferred semantics yield maximally defensible positions This framework underpins legal argument mining systems that extract rhetorical structures from briefs and opinions.

A semantic approach where models are ordered by plausibility or specificity, and entailment is defined over the most preferred models. In legal knowledge graphs, this resolves conflicts through lex specialis (specific law overrides general law) and lex posterior (newer law overrides older law). The preference relation ≤ on interpretations encodes:

• Specificity: More detailed rules preferred over general ones
• Recency: Later enactments preferred over earlier ones
• Authority: Higher courts preferred over lower courts This provides a formal semantics for normative conflict resolution in multi-jurisdictional systems.

A formal non-monotonic logic designed to represent defeasible reasoning, where conclusions can be defeated by contrary evidence. It structures arguments using strict rules (always true), defeasible rules (typically true), and defeaters (prevent conclusions without supporting the opposite).

• Key mechanism: A priority ordering on rules resolves conflicts
• Legal application: Modeling burdens of proof where presumptions shift
• Distinction: Unlike classical logic, a conclusion can be justified today and retracted tomorrow without contradiction

A non-monotonic formalism introduced by Ray Reiter that extends classical logic with default rules of the form 'In the absence of information to the contrary, assume X.' These defaults generate extensions—maximal, internally consistent sets of beliefs.

• Structure: Defaults have prerequisites, justifications, and consequents
• Legal example: 'In the absence of evidence of incapacity, assume a contract is valid'
• Challenge: Multiple conflicting extensions can arise, requiring preference mechanisms

A declarative logic programming paradigm based on the stable model semantics, designed for non-monotonic reasoning and combinatorial search. ASP solves problems by describing what constitutes an acceptable solution rather than specifying step-by-step algorithms.

• Core concept: Rules with negation-as-failure generate stable models (answer sets)
• Legal use case: Encoding statutory regulations to compute compliant scenarios
• Tooling: Solvers like Clingo and DLV efficiently compute answer sets from logic programs

A non-monotonic reasoning technique developed by John McCarthy that minimizes the extension of specified predicates. It formalizes the commonsense assumption that 'only those objects that must have a property do have it.'

• Mechanism: Second-order logical sentences are minimized to exclude abnormal cases
• Legal relevance: Modeling the closed-world assumption in regulatory domains where unlisted items are prohibited
• Limitation: Does not handle multiple extensions or preference ordering natively

A formal approach to non-monotonic reasoning pioneered by Phan Minh Dung, where knowledge is structured as a set of arguments and an attack relation between them. Acceptable arguments are determined through admissibility semantics.

• Semantics: Grounded, preferred, stable, and complete extensions define acceptance
• Legal application: Modeling adversarial legal disputes where arguments attack and defend
• Advantage: Provides explicit dialectical structure absent in other non-monotonic formalisms

A non-monotonic inference rule stating that any atomic sentence not known to be true is assumed false. It contrasts with the Open-World Assumption (OWA) where unknown facts are simply uncommitted.

• Formalization: Reiter's CWA adds negations of ground atoms not entailed by the theory
• Legal context: Criminal law operates under a presumption of innocence (OWA), while regulatory licensing often uses CWA
• Risk: CWA can lead to inconsistency if the knowledge base is incomplete