Deontic Logic Tensor

Deontic logic tensors encode the three classical modalities—obligation, permission, and prohibition—as distinct, potentially orthogonal dimensions within the tensor structure. Each modality occupies its own axis or slice, allowing a single tensor to simultaneously represent the truth value of all three deontic statuses for a given proposition. This architecture enables vector operations that compute normative collisions directly, such as detecting when an obligation vector and a prohibition vector are both active for the same action, forming the basis for deontic conflict detection.

The standard representation uses a rank-3 tensor with dimensions corresponding to:

• Subject (Agent): The legal person or entity bearing the obligation
• Action: The conduct being regulated
• Modality: The deontic status (obligation, permission, prohibition)

Each cell contains a truth value, often a continuous score between 0 and 1, representing the degree of normative force. This structure naturally supports normative entailment checks by performing tensor contractions that propagate obligations through chains of related actions and agents.

Unlike classical deontic logic's binary true/false assignments, deontic logic tensors employ continuous truth values in the range [0,1]. This enables the modeling of defeasible reasoning where obligations have graded strength. A general rule may carry a normative force of 0.6, while a specific exception (lex specialis) carries a force of 0.95. During conflict resolution, the higher-weighted norm naturally dominates in vector operations, implementing rule preference ordering without explicit procedural code.

To model temporal reasoning in contracts and lex posterior principles, deontic logic tensors can be extended to a rank-4 tensor by adding a time dimension. This axis encodes the effective date, repeal date, or temporal scope of each norm. Tensor slicing along the time axis retrieves the normative landscape at any given point, enabling the system to algorithmically apply the lex posterior derogat priori rule by comparing the temporal indices of conflicting norms and selecting the later-enacted provision.

A fifth dimension can encode normative hierarchy, representing the authority level of the issuing body (constitutional, statutory, regulatory, contractual). This enables the tensor to inherently model lex superior derogat inferiori by assigning higher-magnitude embedding vectors to superior norms. When a normative collision matrix is computed, the authority embeddings interact to automatically resolve conflicts in favor of higher-ranking sources, forming a normative hierarchy graph directly within the tensor's geometric structure.

The core computational operation for normative conflict resolution is tensor contraction along the modality dimension. When two norms with opposing deontic statuses (e.g., obligation vs. prohibition) apply to the same action-agent pair, their tensor product is contracted using a predefined normative collision matrix. This matrix defines the resolution outcome for each modality pair:

• Obligation ⊗ Prohibition → Prohibition (safety override)
• Permission ⊗ Prohibition → Prohibition (restrictive default)
• Obligation ⊗ Permission → Obligation (permissive strengthening) The result is a single, conflict-free deontic status.

The foundational legal maxim that a specific rule overrides a general rule. In a Deontic Logic Tensor, this is encoded as a specificity weighting in the vector space. When two norms collide, the system computes a specificity score based on the number of applicability conditions—the rule with more precise factual predicates receives higher activation priority. This principle is the algorithmic backbone of normative exception handling.

A mode of inference where conclusions are tentative and retractable. Unlike classical logic, defeasible reasoning allows a Deontic Logic Tensor to model prima facie obligations—duties that hold until a superior norm or exception defeats them. This is implemented via non-monotonic logic, where adding new premises (e.g., a contrary-to-duty obligation) can invalidate previously valid conclusions without causing system-wide contradiction.

A directed acyclic graph (DAG) that encodes precedence relationships between legal rules. Each node represents a norm, and edges define superiority based on three axes:

• Lex Superior: Authority level (constitution > statute > regulation)
• Lex Specialis: Specificity of the rule's scope
• Lex Posterior: Temporal enactment order

The Deontic Logic Tensor traverses this graph during conflict resolution to determine which norm's deontic vector dominates.

A deontic construct specifying what an agent must do after violating a primary obligation. For example: 'You ought not to breach a contract; but if you do breach, you ought to pay damages.' This creates a temporal chain of secondary norms that the Deontic Logic Tensor must model without contradiction. It is a classic stress test for any normative reasoning system, requiring the tensor to maintain coherent obligation states across time steps.

A computational method for resolving normative conflicts by identifying the largest subset of non-contradictory rules from an inconsistent rule base. When a Deontic Logic Tensor detects an obligation-prohibition collision, the MCS algorithm selects the subset that maximizes rule retention while eliminating inconsistency. This is often guided by rule preference ordering, where higher-priority norms are preferentially retained in the conflict-free output set.

A structured two-dimensional array that maps all possible pairwise interactions between deontic modalities to their predefined resolution outcomes. For example:

• Obligation vs. Prohibition: Unresolvable deadlock requiring external precedence
• Permission vs. Prohibition: Prohibition typically overrides
• Obligation vs. Permission: Obligation takes precedence

This matrix serves as a lookup table for the Deontic Logic Tensor's conflict resolution layer, enabling deterministic handling of modal collisions.