Dyadic Morality Algebra

Context

Varshney2026 - An Algebraic Exposition of the Theory of Dyadic Morality mathematizes the theory of dyadic morality using structural-causal notation, then adds psychological constraints that are meant to model human moral appraisal rather than physical causality. The reusable technical payload is the distinction between the SCM-like base graph and the extra moral-cognition operators.

Formal Statement

Let A denote perceived intentionality of an agent, P perceived vulnerability of a patient, H perceived causal harm, S observed suffering, and W perceived moral wrongness. The base dyadic template is:

$$A\stackrel{H}{\to }P.$$

The SCM-style base equations can be summarized as:

$$A=f_A(U_A),P=f_P(U_P),H=f_H(U_H),$$ $$S=f_S(P,H,U_S),W=f_W(A,S,U_W).$$

Varshney’s wrongness factor is:

$$W=k(A\times P\times H{)}^{\alpha },$$

where k is scenario-specific moral weight and alpha is a sensitivity parameter.

The TDM extensions are:

$$T(A,P)\Rightarrow A\propto \frac{1}{P},$$

for typecasting,

$$C(O)\to \{A,P\},$$

for completion from an observation O, and

$$A_{\mathrm{inferred}}=g(\mathrm{valence}(S),A)$$

for valence-dependent inference. The evidential update can be written:

$$\mathrm{Pr}(A\mid S,H)\propto \mathrm{Pr}(S\mid A,H)\mathrm{Pr}(A).$$

Derivation / Construction

The construction starts with ordinary SCM notation, but the moral-cognition model is not an ordinary causal DAG. Typecasting adds an inverse psychological coupling between agency and vulnerability, so making an entity more agent-like suppresses its patient-like vulnerability and vice versa. Completion enforces dyadic closure by supplying a missing agent or patient when an observation is morally salient but structurally incomplete. Valence-dependent inference makes bad outcomes evidence for intentionality, which helps explain why similar actions can be judged differently when one produces harm and the other produces benefit.

Scalability is handled by compression rather than graph expansion. For many agents:

$$\{A_1,\dots ,A_n\}\stackrel{N}{\to }{A}_{\mathrm{group}},A_{\mathrm{individual}}=\frac{A_{\mathrm{group}}}{n}.$$

For many patients:

$$\{P_1,\dots ,P_n\}\stackrel{N}{\to }{P}_{\mathrm{group}},P_{\mathrm{individual}}=\frac{P_{\mathrm{group}}}{n}.$$

For indirect harm, the model uses sequential dyads rather than one full multi-node moral graph:

$$W_{\mathrm{total}}=W_1+W_2.$$

Policy conflicts are then treated as shared-node conflicts across dyads. A helpfulness-safety conflict, for example, can be represented as one AI agent with outgoing edges toward immediate user satisfaction and toward protection of future or third-party patients. Varshney’s proposed graph edits include sequentializing obligations, introducing a human-review or audit node, and communicating the priority rule before action so the AI appears bounded rather than malicious.

Implications

• TDM formalizes moral appraisal as a symbolic compression over perceived agency, vulnerability, harm, and suffering.
• The framework supports patient-centric AI safety: define protected patients and operationalize suffering rather than relying only on enumerated prohibitions.
• Post-failure explanation can be modeled as an intervention on perceived agency, shifting an event from intentional wrongness toward bounded-system error when that is accurate.
• The framework is risky as a normative guide because descriptive human appraisal can encode bias, out-group agency inflation, and in-group vulnerability inflation.
• Scoped measurement matters: estimates of A, P, H, k, and alpha should be tied to communities or stakeholders rather than averaged into a supposedly universal moral prior.

Links

• Dyadic Morality
• Varshney2026 - An Algebraic Exposition of the Theory of Dyadic Morality
• Structural Equation Models
• Actual Causality
• Safe Reinforcement Learning