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Proof_of_OI.md

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+# Proof of Open Individualism from the Principle of Sufficient Reason and Extended Modal Realism
+
+## Assumptions
+
+- **PSR (Principle of Sufficient Reason)**: Every fact has a sufficient explanation.
+- **EMR (Extended Modal Realism)**: All possible, impossible, and incoherent worlds are real.
+- Conscious subjects exist in (some) worlds.
+- **Phenomenal indexicality**: Each conscious subject experiences their own perspective (what it's like to be "me").
+
+---
+
+## Step-by-Step Proof of Open Individualism
+
+### 1. Define the Subject of Experience
+
+Let a *subject of experience* be the entity for which there is *something it is like* to exist (Nagel 1974). This is often modeled as an **observer-moment**: an instantaneous state of phenomenal consciousness.
+
+Let the **set of all observer-moments** across all modalities be denoted:
+
+\[
+\mathcal{O}
+\]
+
+By EMR, \( \mathcal{O} \) contains every logically, illogically, metaphysically, and physically possible observer-moment.
+
+---
+
+### 2. Observer-Moment Identity
+
+Let \( o_1, o_2 \in \mathcal{O} \) be any two observer-moments. Assume for contradiction that they belong to *distinct* subjects of experience.
+
+This implies a distinction between subjects—say \( S_1 \) and \( S_2 \)—where \( o_1 \in S_1 \), \( o_2 \in S_2 \).
+
+But such a distinction requires explanation under **PSR**:
+- Why does \( o_1 \) belong to \( S_1 \) and not \( S_2 \), and vice versa?
+
+---
+
+### 3. No Sufficient Reason for Subject Boundaries
+
+Per **EMR**, *every* way of carving up experience—including *no carving at all*—exists in some world. There are incoherent worlds with:
+- No subject boundaries,
+- Reversed or cyclic boundaries,
+- Infinitely fractal or inconsistent individuation.
+
+Therefore, **any particular boundary assignment is modally arbitrary**.
+
+By **PSR**, this arbitrariness is unacceptable:
+> There must be a sufficient reason for why *this* partitioning of observer-moments into subjects holds rather than another.
+
+But EMR ensures *every* partitioning exists. So no single partition can be ontologically privileged **without violating PSR**.
+
+---
+
+### 4. Eliminate Arbitrary Multiplicity
+
+To preserve PSR, we must **eliminate all arbitrary distinctions** between observer-moments.
+
+This leads to the only viable identity structure:
+> **Open Individualism (OI)**: All observer-moments are experienced by one and the same subject.
+
+All other identity theories (e.g., Closed Individualism or Empty Individualism) impose distinctions that:
+- Lack sufficient reason,
+- Conflict with EMR’s universal realization of partitionings,
+- Therefore **violate PSR**.
+
+---
+
+### 5. Objection: Why *One* and Not *Many*?
+
+OI is not privileging “one” per se. Rather:
+- It imposes **no boundaries**,
+- It is **identity-minimal**,
+- It **avoids arbitrary structure**.
+
+Hence, it is the only account compatible with **PSR + EMR**.
+
+Any multiplicity implies a modally unjustified subject individuation.
+
+---
+
+## ✅ Conclusion
+
+Given:
+- **PSR**: No arbitrary or unexplained facts,
+- **EMR**: All identity structures exist across worlds,
+- **OI**: The only non-arbitrary, non-partitioned account of experience,
+
+We conclude:
+
+> **Open Individualism is necessarily true**: there exists a single, modally unbounded subject who experiences every observer-moment.
+
+Any alternative view entails unexplained distinctions—thus violating the Principle of Sufficient Reason.
+