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.