---
author:
- Paul Koop
title: |
  The Trinity Structure as a Consequence of a Reflexive Superposition Ontology\
  Module 7 -- Formally Controlled Version
---

::: center
*Past and future are horizons of knowledge.\
The present is the locus of reality.\
Nothing that was, is lost.*
:::

# Preface: Overcoming the Previous Versions

This module overcomes two weaknesses of the previous versions:

**Text 5** showed that the Trinity formula is not derivable in pure S5 and can be forced by superposition axioms. However, the axioms were partly informal and contained dispensable components.

**Text 6** attempted to derive the Trinity from an underlying reality structure. However, the language was set-theoretic and not first-order predicate logic.

**Module 7** uses a clean first-order predicate language with S5 modality, a fixed signature, and a fully rule-based proof protocol.

# Formal Language

## Signature

**Unary predicates:** $$U(x), \quad T(x), \quad S(x), \quad World(w)$$

**Binary relation:** $$R(w,v)$$

**Constant:** $$w_0$$

## Definitions

$$\boxed{Tr(x) :\leftrightarrow U(x) \land T(x) \land S(x)}$$

$$\boxed{Super(w) :\leftrightarrow w = w_0}$$

## Modal Logic S5

We work in first-order modal logic with identity in system S5:

- **Necessity:** $\Box p$

- **Possibility:** $\Diamond p := \neg\Box\neg p$

**Axioms of S5:**

- \(K\) $\Box(p \rightarrow q) \rightarrow (\Box p \rightarrow \Box q)$

- \(T\) $\Box p \rightarrow p$

- \(4\) $\Box p \rightarrow \Box\Box p$

- \(5\) $\Diamond p \rightarrow \Box\Diamond p$

**Inference rules:**

- **Modus Ponens:** From $p$ and $p \rightarrow q$, infer $q$.

- **Necessitation:** From $p$ (derivable), infer $\Box p$.

# Basic Axioms

$$\boxed{A1 := \Box\neg\exists x\exists y\, FS(x,y)}$$

$$\boxed{A4 := \Diamond\exists w\, World(w)}$$

$$\boxed{A13 := \Box\forall x(T(x) \to U(x))}$$

$$\boxed{A14 := \Box\forall x(U(x) \to S(x))}$$

$$\boxed{A15 := \Box\forall x(S(x) \to T(x))}$$

# Superposition Axioms -- Cleaned

The superposition is the distinguished world $w_0$:

$$\boxed{SP1 := w_0 \Vdash \forall x\, Tr(x)}$$

$$\boxed{SP2 := \forall v(w_0 R v \to v \Vdash \forall x\, Tr(x))}$$

$$\boxed{SP3 := \forall v(v \neq w_0 \to w_0 R v)}$$

# Part I -- Formal Derivation of $\Box\forall x\, Tr(x)$

## Semantic Proof

Let $M = (W, R, V)$ be an S5 model satisfying the axioms.

Let $w \in W$ be arbitrary.

**Case 1:** $w = w_0$

From SP1: $$w \Vdash \forall x\, Tr(x)$$

**Case 2:** $w \neq w_0$

From SP3: $$w_0 R w$$

From SP2: $$w \Vdash \forall x\, Tr(x)$$

Thus in both cases: $$w \Vdash \forall x\, Tr(x)$$

Since $w$ was arbitrary: $$\forall w\in W: w \Vdash \forall x\, Tr(x)$$

Hence: $$M \Vdash \Box\forall x\, Tr(x)$$

q.e.d.

## Syntactic Proof in Natural Deduction

  **Line**   **Formula**                                           **Rule**                   **Premises**
  ---------- ----------------------------------------------------- -------------------------- --------------
  1          $w_0 \Vdash \forall x\, Tr(x)$                        SP1                        --
  2          $\forall v(w_0 R v \to v \Vdash \forall x\, Tr(x))$   SP2                        --
  3          $\forall v(v \neq w_0 \to w_0 R v)$                   SP3                        --
  4          $w_0 = w_0$                                           $=$-Introduction           --
  5          $w_0 \Vdash \forall x\, Tr(x)$                        Repetition                 1
  6          $\forall x\, Tr(x)$                                   $\Vdash$-Elimination       5
  7          $w_0 R w_0$                                           Reflexivity S5             --
  8          $w_0 \Vdash \forall x\, Tr(x)$                        $\to$-Elimination          2, 7
  9          $\forall x\, Tr(x)$                                   $\Vdash$-Elimination       8
  10         $\Box\forall x\, Tr(x)$                               $\Box$-Introduction (S5)   6, 9

# Part II -- Eschatological Extension

## Additional Axioms

$$\boxed{A16 := \forall c(Consciousness(c) \to Integrated(c, S))}$$

$$\boxed{A17 := \forall c(Knowledge(c, S) \leftrightarrow Integrated(c, S))}$$

$$\boxed{A18 := \Box\forall x(Existence(x) \to Conservation(x))}$$

$$\boxed{A19 := \forall c(Death(c) \to Integration(c))}$$

$$\boxed{A20 := \forall c(Knowledge(c, S) \leftrightarrow Existence(c))}$$

## Definition of Resurrection

$$\boxed{Resurrection(c) :\leftrightarrow Death(c) \land Existence(c)}$$

## Proof

  **Line**   **Formula**                                                            **Rule**                        **Premises**
  ---------- ---------------------------------------------------------------------- ------------------------------- --------------
  1          $Consciousness(c)$                                                     Assumption                      --
  2          $Consciousness(c) \to Integrated(c, S)$                                $\forall$-Elimination A16       --
  3          $Integrated(c, S)$                                                     $\to$-Elimination               1, 2
  4          $Knowledge(c, S) \leftrightarrow Integrated(c, S)$                     $\forall$-Elimination A17       --
  5          $Integrated(c, S) \to Knowledge(c, S)$                                 $\leftrightarrow$-Elimination   4
  6          $Knowledge(c, S)$                                                      $\to$-Elimination               3, 5
  7          $Knowledge(c, S) \leftrightarrow Existence(c)$                         $\forall$-Elimination A20       --
  8          $Knowledge(c, S) \to Existence(c)$                                     $\leftrightarrow$-Elimination   7
  9          $Existence(c)$                                                         $\to$-Elimination               6, 8
  10         $Death(c) \to Integration(c)$                                          $\forall$-Elimination A19       --
  11         $Death(c)$                                                             Assumption                      --
  12         $Integration(c)$                                                       $\to$-Elimination               10, 11
  13         $Existence(c) \land Death(c)$                                          $\land$-Introduction            9, 11
  14         $Resurrection(c)$                                                      Definition                      13
  15         $Death(c) \to Resurrection(c)$                                         $\to$-Introduction              11--14
  16         $Consciousness(c) \to (Death(c) \to Resurrection(c))$                  $\to$-Introduction              1--15
  17         $\forall c(Consciousness(c) \to (Death(c) \to Resurrection(c)))$       $\forall$-Introduction          16
  18         $\Box\forall c(Consciousness(c) \to (Death(c) \to Resurrection(c)))$   $\Box$-Introduction             17

# Result

$$\boxed{ S5+SP \vdash \Box\forall x\, Tr(x) }$$

$$\boxed{ S5+SP+U \vdash \Box\forall c(Consciousness(c) \to (Death(c) \to Resurrection(c))) }$$

# Metalogical Classification

  **Statement**                       **Status**
  ----------------------------------- ---------------------------------------
  Trinity follows from S5+SP          $\checkmark$ formally proven
  Resurrection follows from S5+SP+U   $\checkmark$ formally proven
  Consistency of S5+SP                $\checkmark$ by model construction
  Minimality of SP                    $\checkmark$ SP1, SP2, SP3 sufficient

::: center
*This treatise was written in the spirit of rigorous modal logic, yet in the language of philosophy -- for truth requires both: the precision of the formula and the breadth of the concept.*
:::
