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  pdftitle={The Trinity Structure as a Consequence of a Reflexive Superposition Ontology – Module 7},
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\title{The Trinity Structure as a Consequence of a Reflexive Superposition Ontology \\ \large Module 7 – Formally Controlled Version}
\author{Paul Koop}
\date{}

\begin{document}
\maketitle

\begin{center}
\emph{Past and future are horizons of knowledge. \\ The present is the locus of reality. \\ Nothing that was, is lost.}
\end{center}

\newpage
{
\setcounter{tocdepth}{3}
\tableofcontents
}
\newpage

\section{Preface: Overcoming the Previous Versions}

This module overcomes two weaknesses of the previous versions:

\textbf{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.

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

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

\section{Formal Language}

\subsection{Signature}

\textbf{Unary predicates:}
\[
U(x), \quad T(x), \quad S(x), \quad World(w)
\]

\textbf{Binary relation:}
\[
R(w,v)
\]

\textbf{Constant:}
\[
w_0
\]

\subsection{Definitions}

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

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

\subsection{Modal Logic S5}

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

\begin{itemize}
\item \textbf{Necessity:} \(\Box p\)
\item \textbf{Possibility:} \(\Diamond p := \neg\Box\neg p\)
\end{itemize}

\textbf{Axioms of S5:}
\begin{itemize}
\item (K) \(\Box(p \rightarrow q) \rightarrow (\Box p \rightarrow \Box q)\)
\item (T) \(\Box p \rightarrow p\)
\item (4) \(\Box p \rightarrow \Box\Box p\)
\item (5) \(\Diamond p \rightarrow \Box\Diamond p\)
\end{itemize}

\textbf{Inference rules:}
\begin{itemize}
\item \textbf{Modus Ponens:} From \(p\) and \(p \rightarrow q\), infer \(q\).
\item \textbf{Necessitation:} From \(p\) (derivable), infer \(\Box p\).
\end{itemize}

\section{Basic Axioms}

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

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

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

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

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

\section{Superposition Axioms – Cleaned}

The superposition is the distinguished world \(w_0\):

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

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

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

\section{Part I – Formal Derivation of \(\Box\forall x\, Tr(x)\)}

\subsection{Semantic Proof}

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

Let \(w \in W\) be arbitrary.

\textbf{Case 1:} \(w = w_0\)

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

\textbf{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.

\subsection{Syntactic Proof in Natural Deduction}

\begin{table}[h]
\centering
\begin{tabular}{llll}
\toprule
\textbf{Line} & \textbf{Formula} & \textbf{Rule} & \textbf{Premises} \\
\midrule
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 \\
\bottomrule
\end{tabular}
\end{table}

\section{Part II – Eschatological Extension}

\subsection{Additional Axioms}

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

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

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

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

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

\subsection{Definition of Resurrection}

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

\subsection{Proof}

\begin{table}[h]
\centering
\begin{tabular}{llll}
\toprule
\textbf{Line} & \textbf{Formula} & \textbf{Rule} & \textbf{Premises} \\
\midrule
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 \\
\bottomrule
\end{tabular}
\end{table}

\section{Result}

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

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

\section{Metalogical Classification}

\begin{table}[h]
\centering
\begin{tabular}{ll}
\toprule
\textbf{Statement} & \textbf{Status} \\
\midrule
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 \\
\bottomrule
\end{tabular}
\end{table}

\begin{center}
\emph{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.}
\end{center}

\end{document}