To represent mathematically how **same conditions can lead to different outcomes** in a relational causality framework, we must consider the system as **non-linear**, **multi-dimensional**, and influenced by **partial causes** interacting through feedback loops. This avoids the strict determinism of a single causal chain while maintaining lawful structure. --- ### **1. Defining the Relational Model of Causality** #### **1.1 Partial Causation** Let the "total antecedent set" at time \( T_0 \) be represented by \( p \), composed of \( n \) partial causes: \[ p = \{C_1, C_2, \ldots, C_n\}. \] Each \( C_i \) represents a partial influence on the system, which itself is influenced by other causes (potentially infinite in number): \[ C_i = f(C_{i1}, C_{i2}, \ldots, C_{im}), \] where \( f \) is a causal function for each partial cause, dependent on an underlying web of interactions. #### **1.2 Outcome Space** The state of the system at \( T_1 \), denoted \( S(T_1) \), is determined by: \[ S(T_1) = F(p), \] where \( F \) is the global causal mapping from \( p \) to the outcome state. If \( F \) were strictly deterministic, it would yield one unique \( S(T_1) \). However, in relational causality, \( F \) incorporates **multi-directional interactions** and **feedback loops**, leading to **branching outcomes**. --- ### **2. Why Same Conditions Yield Different Outcomes** #### **2.1 Non-Linear Systems** Non-linearity introduces sensitivity to small perturbations or feedback. In such systems, the mapping \( F \) is highly dependent on initial conditions but can diverge into multiple outcomes due to complex interactions. Mathematically, consider: \[ \frac{\partial S(T_1)}{\partial C_i} \gg 0, \] where small variations in \( C_i \) (even within the same "state \( p \)") can amplify into distinct macroscopic outcomes. #### **2.2 Multi-Stable Systems** In systems with multiple equilibria or attractors, the same \( p \) can lead to different stable states. For example: \[ S(T_1) \in \{A, B, C\}, \] where \( A, B, C \) are stable attractors. The outcome depends not on randomness but on: 1. **Feedback loops** within \( p \). 2. **Synergistic interactions** between \( C_i \). #### **2.3 Feedback and Context Dependence** Feedback modifies partial causes dynamically. Let \( \Phi(C_i, t) \) represent the dynamic feedback on \( C_i \): \[ C_i(t+1) = \Phi(C_i(t), \{C_j\}_{j \neq i}), \] where the feedback depends on both \( C_i \) and other partial causes \( C_j \). This introduces **context sensitivity**, where: \[ F(p) = \text{Branch}(p, \Phi), \] yielding multiple pathways even under the same \( p \), depending on how feedback shapes the interplay. --- ### **3. Incorporating Agent-Based Freedom** #### **3.1 Internal and External Causes** Introduce **agents** \( A_k \), which can modulate \( p \) by internal processes \( \Psi_k \): \[ C_i' = C_i + \Psi_k(C_i), \] where \( \Psi_k \) is an agent's internal causal function that alters \( C_i \). #### **3.2 Divergent Outcomes from Agency** Agents create a branching structure by interacting with \( p \) internally: \[ F(p) \to F(p + \Psi_k). \] Even under identical external conditions (\( p \)), the internal dynamics \( \Psi_k \) of agents allow different trajectories for \( S(T_1) \). For example: \[ S(T_1) = \begin{cases} A & \text{if } \Psi_k \text{ aligns with feedback loop 1,} \\ B & \text{if } \Psi_k \text{ aligns with feedback loop 2.} \end{cases} \] This mechanism avoids strict determinism without invoking randomness. --- ### **4. Formalizing the Divergence** #### **4.1 Multiple Equilibria in Phase Space** Represent the system in a phase space \( \mathcal{P} \). The evolution of \( p \) is governed by: \[ \frac{dp}{dt} = G(p, t), \] where \( G \) incorporates all partial causes and feedback. In relational causality, \( G \) has: 1. **Non-unique solutions**: \( G \) maps \( p \) to multiple trajectories. 2. **Attractors**: Stable outcomes \( \mathcal{A}_1, \mathcal{A}_2, \ldots \) such that: \[ \lim_{t \to \infty} p(t) \in \mathcal{A}_i. \] #### **4.2 Agent-Driven Perturbations** Agents modify the trajectory by adding perturbations: \[ \frac{dp}{dt} = G(p, t) + \sum_k \Psi_k(p, t), \] where \( \Psi_k \) introduces controlled variations. These perturbations can steer the system toward different attractors, producing lawful but divergent outcomes. --- ### **5. Refuting Determinism Mathematically** #### **5.1 Non-Uniqueness of Solutions** If \( G(p, t) \) has multiple solutions, then: \[ S(T_1) = F(p) \to \{S_1, S_2, \ldots, S_n\}. \] This refutes strict determinism by showing \( F \) is **multi-valued**. #### **5.2 Dependence on Internal Agency** When agents introduce \( \Psi_k \), the mapping becomes: \[ S(T_1) = F(p + \Psi_k), \] which depends on \( \Psi_k \). Different \( \Psi_k \) yield distinct outcomes even under identical \( p \). --- ### **6. Conclusion** The relational causality framework avoids strict determinism by: 1. Allowing **non-linear interactions** and **feedback** to shape multiple lawful outcomes from the same antecedent state \( p \). 2. Introducing **agents** with internal causal processes \( \Psi_k \), which modulate outcomes dynamically and contextually. 3. Representing the system as evolving within a **multi-dimensional phase space** with multiple attractors. This demonstrates mathematically that the same conditions can lead to different outcomes without invoking randomness, maintaining lawful structure while transcending deterministic monotony.