# Nomological Deductive Reasoning for Trustworthy, Human-Readable, and Actionable AI Outputs
## 5 Nomological Deductive Reasoning (NDR): A Proposed Solution
### Mathematical Model of the NDR Framework  
1. **Laws (L)** Let \(\mathbf{L}\) represent the set of laws or rules governing a certain real-world domain (healthcare diagnosis, bank credit score, traffic code for mobility applications, criminal justice, etc.). These laws are formalized as logical statements or principles that provide the foundation for reasoning in the system. Each law \(\mathbf{L}_{i}\in\mathbf{L}\) corresponds to a specific rule or law within the system. Example (in credit assessment settings): * \(L_{1}\): \"If an applicent has multiple financial obligations and a credit purpose that are considered risky, then they are at high risk of default\". * \(L_{2}\): \"If the loan duration is \(>\)12 months and the applicent has stable employment, then the credit is classified as 'Good'\". 2. **Conditions (C)** Let \(\mathbf{C}\) denote the set of antecedents or conditions that must hold true in order for a law to be applicable to a particular data instance. This means that each condition \(\mathbf{C}_{j}\in\mathbf{C}\) is a prerequisite or condition that must be satisfied for the corresponding law to be activated or relevant. Example (in medical settings): * \(C_{1}\): \"Applicent has stable employment\". * \(C_{2}\): \"The loan duration is \(>\)12 months\". 3. **Data instances (D)** Let \(\mathbf{D}=\{\mathbf{d}_{1},\ \mathbf{d}_{2},\ \ldots,\mathbf{d}_{k}\}\) represent the set of input data fed into the AI system. Each \(\mathbf{d}_{i}\in\mathbf{D}\) represents a specific instance or data sample. Example (in credit assessment): * \(d_{1}\): A data sample where the applicent has stable employment and is applying for a loan payable in more than 12 months. * \(d_{2}\): A data sample where the applicent does not have any job and is asking for a loan payable in 3 months. 4. **Hypothesis or Prediction (H)** Let \(\mathbf{H}=\big{\{}\mathbf{h}_{1},\ \mathbf{h}_{2},\ \ldots,\mathbf{h}_{p}\big{\}}\) represent the set of predictions or outcomes generated by the AI model. Each \(\mathbf{h}_{i}\in\mathbf{H}\) corresponds to a specific prediction or outcome generated for the instance \(\mathbf{d}_{i}\). Example (in credit assessment): * \(h_{1}\): \"The applicent is at high risk for loan payment default\" \begin{table}
\begin{tabular}{c c c} \hline
**Explanation Constituents** & **Description** & **Example** \\ \hline \multirow{2}{*}{Explanandum} & Sentence describing the phenomenon to & \multirow{2}{*}{The leaves of the tea plant are yellowing.} \\  & be explained (not the phenomenon itself) & \\ \hline \multirow{8}{*}{Explanans} & Class of those sentences which are & \multirow{8}{*}{While the weather and soil conditions seem to be normal, the leaves of tea plant are yellowing} \\  & Reduced to account for the phenomenon. & \\ \cline{1-1}  & These sentences fall into two subclasses: & \\ \cline{1-1}  & Certain sentences \(C_{1},\ C_{2},\ \ldots,\ C_{n}\), which & \\ \cline{1-1}  & state specific antecedent conditions, & \\ \cline{1-1}  & and & \\ \cline{1-1}  & a set of sentences \(L_{1},\ L_{2},\ \ldots,\ L_{m}\), which & \\ \cline{1-1}  & represent general laws in the domain of & \\ \cline{1-1}  & the problem at hand. & \\ \hline \end{tabular}
\end{table} 
Table 2: Premises of explanation arguments. \(h_{2}\): \"The applicant is not at high risk for loan payment default\". 5. **Formalized Deductive Inference** The key goal of the NDR framework is to use deductive reasoning to formalize the process of how the AI model generates a prediction \(\mathbf{h_{i}}\) based on the combination of conditions \(C\) and laws \(L\) applied to the input data \(\mathbf{d_{i}}\).