80 lines
3.7 KiB
Markdown
80 lines
3.7 KiB
Markdown
# Formal Proof of Extended Modal Realism (EMR) by Necessity
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---
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## Definitions
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- Let **U** be the universe, the totality of all empirical facts.
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- Let **M(U)** be a formal model of the universe: a set of propositions or axioms that, in principle, accounts for all truths about **U**.
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- Let **PSR** be the Principle of Sufficient Reason: *Every fact or truth has a sufficient explanation.*
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- Let **EMR** be **Extended Modal Realism**: *All logically possible worlds exist — the actual world is just one among them.*
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---
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## Step 1: The Universe as a Formal System of Itself
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**P1.1** Any theory that purports to explain all truths about the universe must include a model **M(U)** that expresses those truths formally.
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**P1.2** If **M(U)** is sufficiently expressive (e.g., includes arithmetic), consistent, and recursively enumerable, then Gödel’s First Incompleteness Theorem applies.
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**C1** Therefore, **M(U)** is incomplete: there exist true statements about **U** that are unprovable within **M(U)**.
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---
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## Step 2: Rejection of Strong Empiricism
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**P2.1** Strong empiricism asserts that all truths are derivable from empirical observation (i.e., from within **M(U)**).
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**P2.2** But from **C1**, some truths about **U** are not derivable from within any empirical model.
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**C2** Therefore, strong empiricism is false.
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---
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## Step 3: Refutation of Brutalism
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**P3.1** A brutalist response to **C1** is to treat Gödelian truths as brute — true but without explanation.
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**P3.2** Suppose we add such a brute proposition to **M(U)** to form **M′(U)**.
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**P3.3** By Gödel, **M′(U)** is also incomplete — a new unprovable truth **G(M′)** emerges.
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**P3.4** This process iterates infinitely. There is no final complete model reachable by brute supplementation.
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**C3** Brutalism fails to halt explanatory incompleteness and degenerates into an infinite series of ungrounded assumptions.
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---
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## Step 4: PSR is the Only Stable Principle of Explanation
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**P4.1** PSR demands that *all* truths, including foundational ones, have sufficient reason.
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**P4.2** Empiricism fails to explain all truths (**C2**); brutalism fails to explain any truths coherently (**C3**).
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**C4** Therefore, PSR is the only principle that preserves rational explanation in metaphysics.
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---
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## Step 5: Axiom-Based Theories Violate PSR
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**P5.1** All finite metaphysical systems (e.g., theism, materialism, idealism, structural realism) begin with a finite set of axioms.
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**P5.2** Axioms are, by definition, unexplained assumptions within a system.
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**P5.3** By **P4.1**, PSR rejects any unexplained assumptions.
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**C5** Therefore, all axiom-based theories of reality violate PSR and are necessarily false if PSR is true.
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---
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## Step 6: Only EMR Contains No Axioms
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**P6.1** Extended Modal Realism (EMR) asserts that all logically possible worlds exist.
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**P6.2** EMR assumes no privileged world, no arbitrary structure, no selected laws — **every possibility is realized**.
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**P6.3** Therefore, EMR posits no unexplained exclusions or brute facts — it has no axioms.
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**C6** EMR satisfies the PSR fully.
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---
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## Step 7: EMR is the Only Metaphysical Theory Consistent with PSR
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**P7.1** From **C4**, PSR is the only stable principle of explanation.
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**P7.2** From **C5**, all axiom-based theories violate PSR.
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**P7.3** From **C6**, EMR is the only theory that satisfies PSR.
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**C7** Therefore, **EMR is the only metaphysical theory that satisfies the PSR**.
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---
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## Final Conclusion
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> **If the Principle of Sufficient Reason is true, then Extended Modal Realism is necessarily true.**
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> **All other theories of reality are either incomplete, arbitrary, or incoherent.**
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∎
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