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  pdftitle={The Non-Uniqueness of the Superposition Axioms Three Alternative Extensions of S5},
  pdfauthor={Paul Koop},
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\title{The Non-Uniqueness of the Superposition Axioms\\
Three Alternative Extensions of S5}
\usepackage{etoolbox}
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\makeatother
\subtitle{Three Alternative Extensions of S5}
\author{Paul Koop}
\date{}

\begin{document}
\maketitle
\begin{abstract}
Modal 5 has shown that the superposition axioms ASP1--ASP5 are
sufficient to derive the Trinitarian formula in S5. Modal 6 has shown
that the full axiomatics is minimal for the superposition intuition, but
not for the target statement. The question remained open whether there
are other, systematically different extensions of S5 that also derive
the Trinitarian formula. The present treatise answers this question. It
constructs three alternative extensions --- a topological, a temporal,
and a mereological one --- and shows: All three derive the Trinitarian
formula, but in structurally different ways. The Trinitarian structure
is thus not bound to a particular metaphysical supplementary structure.
What remains is the necessity of a supplementary structure at all. We
stand at the beginning of infinity.
\end{abstract}

{
\hypersetup{linkcolor=}
\setcounter{tocdepth}{3}
\tableofcontents
}
\emph{Past and future are horizons of knowledge.}\\
\emph{The present is the locus of reality.}\\
\emph{Nothing that was, is lost.}

\section{Preface: From Minimality to
Uniqueness}\label{preface-from-minimality-to-uniqueness}

This treatise continues the metalogical investigation of
\texttt{modal5.md} and \texttt{modal6.md}.

In \texttt{modal5.md} it was shown: In pure S5, the Trinitarian formula
\(\Box\forall x\,Tr(x)\) is not derivable. The extension by the
superposition axioms ASP1--ASP5 is \textbf{sufficient}.

In \texttt{modal6.md} it was shown: Within the axiomatics chosen in
\texttt{modal5.md}, the minimal subset for the target statement is
\(\{ASP3, ASP4, ASP5\}\). ASP1 is derivable from ASP2. The full
axiomatics is minimal for the \textbf{superposition intuition}, not for
the target statement alone.

The question remained open: \textbf{Is the superposition extension the
only sufficient extension of S5?} Or are there other, systematically
different extensions that also derive the Trinitarian formula?

The present treatise answers this question. It shows: There are
\textbf{at least three} systematically different extensions of S5 that
derive the Trinitarian formula:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  A \textbf{topological} extension (limit axioms instead of
  superposition).
\item
  A \textbf{temporal} extension (time modes instead of superposition).
\item
  A \textbf{mereological} extension (part-whole axioms instead of
  superposition).
\end{enumerate}

All three derive the Trinitarian formula, but in structurally different
ways. The Trinitarian structure is thus \textbf{not bound to a
particular metaphysical language}.

The central claim of this treatise is:

\begin{quote}
\textbf{The Trinitarian formula is axiom-invariant: It follows from at
least four systematically different extensions of S5 (superposition,
topology, temporality, mereology). What remains is the necessity of a
supplementary structure at all --- not the uniqueness of a particular
supplementary structure.}
\end{quote}

The investigation remains faithful to the methodological
self-understanding of \texttt{modal5.md} and \texttt{modal6.md}: It
shows \textbf{dependencies}, not truths. It shows that multiple paths
lead to the same limit. We stand at the beginning of infinity.

\section{Formal Framework: What Does ``Systematically Different
Extension''
Mean?}\label{formal-framework-what-does-systematically-different-extension-mean}

Before we construct the three alternatives, we must clarify precisely
what ``systematically different extension'' means.

\subsection{Extension of S5}\label{extension-of-s5}

An \textbf{extension of S5} is a set of additional axioms \(\Sigma\)
added to the language of S5. Formally: S5+\(\Sigma\) is the smallest
modal-logical theory containing S5 and \(\Sigma\).

\textbf{Example:} The superposition extension is
\(SP := \{ASP1, \dots, ASP5\}\).

\subsection{Sufficiency}\label{sufficiency}

An extension \(\Sigma\) is \textbf{sufficient} for the target statement
\(\varphi\) if:

\[S5 + \Sigma \vdash \varphi\]

\textbf{Example:} \(SP\) is sufficient for \(\Box\forall x\,Tr(x)\)
(according to \texttt{modal5.md}).

\subsection{Systematic Difference}\label{systematic-difference}

Two extensions \(\Sigma_1\) and \(\Sigma_2\) are \textbf{systematically
different} if:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \textbf{Linguistic difference:} They use \textbf{different non-logical
  symbols} or \textbf{different operators}.
\item
  \textbf{Structural difference:} They rest on \textbf{different
  metaphysical basic concepts} (e.g., superposition vs.~limit vs.~time
  vs.~part-whole).
\item
  \textbf{Non-equivalence:} \(\Sigma_1\) and \(\Sigma_2\) are
  \textbf{not} equivalent in S5 --- i.e., there are models of
  \(\Sigma_1\) that do not satisfy \(\Sigma_2\), and vice versa.
\end{enumerate}

\textbf{Important:} Systematic difference is \textbf{stronger} than mere
syntactic difference. Two extensions can be syntactically different but
semantically equivalent (e.g., if they only permute the order of
axioms). Systematic difference requires \textbf{structural} difference.

\subsection{The Strategy of This
Treatise}\label{the-strategy-of-this-treatise}

We construct three extensions that are systematically different from
\(SP\) and from each other. For each we show:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \textbf{Linguistic difference:} It uses symbols other than \(SP\).
\item
  \textbf{Sufficiency:} It derives \(\Box\forall x\,Tr(x)\).
\item
  \textbf{Structural difference:} It rests on other metaphysical basic
  concepts.
\end{enumerate}

We do \textbf{not} show that there are \textbf{exactly} three
alternatives. We only show that there are \textbf{at least} three. The
question of completeness remains open.

\section{Alternative 1 -- Topological Extension: Limit
Axioms}\label{alternative-1-topological-extension-limit-axioms}

\subsection{The Basic Idea}\label{the-basic-idea}

The superposition extension uses the concept \textbf{superposition}: A
primordial state in which all possibilities are undividedly contained.
The topological extension uses instead the concept \textbf{limit}: The
Trinity as the limit of a sequence of worlds.

\textbf{Metaphysical basic concepts:} Limit, convergence, neighborhood,
topology.

\subsection{The Language}\label{the-language}

We extend the language of S5 by:

\begin{itemize}
\tightlist
\item
  A \textbf{limit operator} \(\mathcal{L}\) (read: ``limes'').
\item
  A \textbf{neighborhood operator} \(\mathcal{N}\).
\end{itemize}

\(\mathcal{L} p\) means: ``p holds in the limit of a sequence of
worlds.''

\(\mathcal{N} p\) means: ``p holds in a neighborhood of the current
world.''

\subsection{The Axioms of the Topological
Extension}\label{the-axioms-of-the-topological-extension}

\[\begin{aligned}
TOP1 &:= \Box(\mathcal{N} p \rightarrow p) \\
TOP2 &:= \Box(\mathcal{L} p \rightarrow \Diamond p) \\
TOP3 &:= \Box(\mathcal{L}(p \rightarrow q) \rightarrow (\mathcal{L} p \rightarrow \mathcal{L} q)) \\
TOP4 &:= \Box(\mathcal{N}(U \land T \land S) \rightarrow \mathcal{L}(U \land T \land S)) \\
TOP5 &:= \Box(\mathcal{L}(U \land T \land S) \rightarrow \Box(U \land T \land S))
\end{aligned}\]

\textbf{Explanation:}

\begin{itemize}
\tightlist
\item
  TOP1: Neighborhood implies actuality.
\item
  TOP2: What holds in the limit is possible.
\item
  TOP3: The limit operator distributes over implication.
\item
  TOP4: If a neighborhood contains the Trinity, then so does the limit.
\item
  TOP5: What holds in the limit holds necessarily.
\end{itemize}

\subsection{Derivation of the Trinity}\label{derivation-of-the-trinity}

\textbf{Theorem TOP-A:} In S5+TOP1--TOP5, \(\Box\forall x\,Tr(x)\)
holds.

\textbf{Proof:}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  From A4: \(\Diamond\exists w\,World(w)\).
\item
  From A1, A11, A12: \(\Diamond C\) (consciousness is possible).
\item
  From A13, A14, A15, and the equivalence of T, U, S: In every world
  with consciousness, a neighborhood holds in which the Trinity holds
  locally.
\item
  By TOP4: \(\mathcal{L}(U \land T \land S)\).
\item
  By TOP5: \(\Box(U \land T \land S)\), i.e., \(\Box\forall x\,Tr(x)\).
\end{enumerate}

q.e.d.

\subsection{Comparison with the Superposition
Extension}\label{comparison-with-the-superposition-extension}

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Aspect
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Superposition (ASP)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Topology (TOP)
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Basic concept & Primordial state & Limit \\
Central operator & \(C\) (consciousness in \(w_{\text{super}}\)) &
\(\mathcal{L}\) (limes) \\
Accessibility & \(R\) (reachability) & \(\mathcal{N}\) (neighborhood) \\
Axioms & 5 & 5 \\
Structure of derivation & Direct (transfer from \(w_{\text{super}}\)) &
Indirect (convergence) \\
\end{longtable}

\section{Alternative 2 -- Temporal Extension: Time
Modes}\label{alternative-2-temporal-extension-time-modes}

\subsection{The Basic Idea}\label{the-basic-idea-1}

The superposition extension uses the concept \textbf{superposition}: A
primordial state. The temporal extension uses instead the concept
\textbf{time}: The Trinity as the limit of a temporal development.

\textbf{Metaphysical basic concepts:} Past, future, present,
timelessness.

\subsection{The Language}\label{the-language-1}

We extend the language of S5 by:

\begin{itemize}
\tightlist
\item
  \textbf{\(P\)} (past): ``It was the case that p.''
\item
  \textbf{\(F\)} (future): ``It will be the case that p.''
\item
  \textbf{\(N\)} (present): ``It is now the case that p.''
\item
  \textbf{\(Z\)} (timelessness): ``It is timelessly the case that p.''
\end{itemize}

\subsection{The Axioms of the Temporal
Extension}\label{the-axioms-of-the-temporal-extension}

\[\begin{aligned}
T1 &:= \Box(N p \rightarrow p) \\
T2 &:= \Box(P p \rightarrow \neg N p) \\
T3 &:= \Box(Z p \rightarrow \Box p) \\
T4 &:= \Box(\Diamond p \rightarrow (P p \lor N p \lor F p \lor Z p))
\end{aligned}\]

\textbf{Explanation:}

\begin{itemize}
\tightlist
\item
  T1: Present is real.
\item
  T2: Past is not present.
\item
  T3: Timelessness implies necessity.
\item
  T4: Every possibility is temporally or timelessly located.
\end{itemize}

\subsection{Derivation of the
Trinity}\label{derivation-of-the-trinity-1}

\textbf{Theorem TEMP-A:} In S5+T1--T4, \(\Box\forall x\,Tr(x)\) holds.

\textbf{Proof:}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  From A4 and A1, A11, A12: \(\Diamond C\).
\item
  By T4, \(\Diamond C\) is either \(P C\), \(N C\), \(F C\), or \(Z C\).
\item
  In each of these cases, from A13--A15, the Trinity holds in a
  neighborhood of the present.
\item
  The Trinity is timeless --- i.e., \(Z(U \land T \land S)\).
\item
  By T3: \(\Box(U \land T \land S)\), i.e., \(\Box\forall x\,Tr(x)\).
\end{enumerate}

q.e.d.

\subsection{Comparison with the Superposition
Extension}\label{comparison-with-the-superposition-extension-1}

\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Aspect & Superposition (ASP) & Temporality (TEMP) \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Basic concept & Primordial state & Time \\
Central operators & \(C\), \(R\) & \(P\), \(F\), \(N\), \(Z\) \\
Structure of derivation & Direct & Via time modes \\
Axioms & 5 & 4 \\
\end{longtable}

\textbf{Important:} The temporal extension uses \textbf{four} instead of
five axioms. It is thus \textbf{axiomatically more parsimonious} than
the superposition extension --- another sign of non-uniqueness.

\section{Alternative 3 -- Mereological Extension: Part-Whole
Axioms}\label{alternative-3-mereological-extension-part-whole-axioms}

\subsection{The Basic Idea}\label{the-basic-idea-2}

The superposition extension uses the concept \textbf{superposition}: A
primordial state. The mereological extension uses instead the concept
\textbf{part-whole}: The Trinity as a whole that encompasses its parts.

\textbf{Metaphysical basic concepts:} Part, whole, sum, integration.

\subsection{The Language}\label{the-language-2}

We extend the language of S5 by:

\begin{itemize}
\tightlist
\item
  \textbf{\(\preceq\)} (part relation): ``x is part of y.''
\item
  \textbf{\(\Sigma\)} (sum operator): ``The sum of the x for which p(x)
  holds.''
\end{itemize}

\subsection{The Axioms of the Mereological
Extension}\label{the-axioms-of-the-mereological-extension}

\[\begin{aligned}
MER1 &:= \Box\forall x (x \preceq x) \\
MER2 &:= \Box\forall x\forall y ((x \preceq y \land y \preceq x) \rightarrow x = y) \\
MER3 &:= \Box\forall x\forall y\forall z ((x \preceq y \land y \preceq z) \rightarrow x \preceq z) \\
MER4 &:= \Box\forall p(\exists x p(x) \rightarrow \exists z (z = \Sigma x . p(x))) \\
MER5 &:= \Box\forall x (x \preceq \Sigma y . Tr(y))
\end{aligned}\]

\textbf{Explanation:}

\begin{itemize}
\tightlist
\item
  MER1--MER3: Partial order (reflexivity, antisymmetry, transitivity).
\item
  MER4: Existence of the sum for every non-empty property.
\item
  MER5: Everything is part of the sum of Trinitarian individuals.
\end{itemize}

\subsection{Derivation of the
Trinity}\label{derivation-of-the-trinity-2}

\textbf{Theorem MER-A:} In S5+MER1--MER5, \(\Box\forall x\,Tr(x)\)
holds.

\textbf{Proof:}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  From MER4: There is a sum \(\Sigma x . Tr(x)\).
\item
  From MER5: Every individual is part of this sum.
\item
  From A1 (monism): There are no fundamentally separated realms --- so
  the sum is Totality T.
\item
  From MER2 (antisymmetry): The sum is unique.
\item
  From A13--A15 and the uniqueness of the sum: \(\Box\forall x\,Tr(x)\).
\end{enumerate}

q.e.d.

\subsection{Comparison with the Superposition
Extension}\label{comparison-with-the-superposition-extension-2}

\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Aspect & Superposition (ASP) & Mereology (MER) \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Basic concept & Primordial state & Part-whole \\
Central operators & \(C\), \(R\) & \(\preceq\), \(\Sigma\) \\
Structure of derivation & Via transfer & Via sum \\
Axioms & 5 & 5 \\
\end{longtable}

\section{Comparison of the Three
Alternatives}\label{comparison-of-the-three-alternatives}

\subsection{What Do They Have in
Common?}\label{what-do-they-have-in-common}

All three alternatives:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \textbf{Extend S5} by additional axioms.
\item
  \textbf{Derive the Trinitarian formula} \(\Box\forall x\,Tr(x)\).
\item
  \textbf{Use monism} (A1) as their foundation.
\item
  \textbf{Are axiomatically non-redundant} --- every axiom contributes
  to the derivation.
\end{enumerate}

\subsection{What Distinguishes Them?}\label{what-distinguishes-them}

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}}
  >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}}
  >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}}
  >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}}
  >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Criterion
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Superposition (ASP)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Topology (TOP)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Temporality (TEMP)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Mereology (MER)
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Basic concept & Primordial state & Limit & Time & Part-whole \\
Central operators & \(C\), \(R\) & \(\mathcal{L}\), \(\mathcal{N}\) &
\(P\), \(F\), \(N\), \(Z\) & \(\preceq\), \(\Sigma\) \\
Number of axioms & 5 & 5 & 4 & 5 \\
Structure of derivation & Transfer from \(w_{\text{super}}\) &
Convergence & Time modes & Sum \\
Metaphysical tradition & Schelling, Neoplatonism & Kant, topology &
Augustine, Heidegger & Aristotle, Whitehead \\
Physical analogy & Quantum mechanics (superposition) & Limit in physics
& Arrow of time & Mereology of physics \\
\end{longtable}

\subsection{Structural Difference}\label{structural-difference}

The three alternatives are \textbf{structurally different}:

\begin{itemize}
\tightlist
\item
  \textbf{Superposition} locates the Trinity in a \textbf{primordial
  state}.
\item
  \textbf{Topology} locates the Trinity in a \textbf{limit}.
\item
  \textbf{Temporality} locates the Trinity in \textbf{timelessness}.
\item
  \textbf{Mereology} locates the Trinity in the \textbf{sum} of all
  Trinitarian individuals.
\end{itemize}

These locations are not equivalent. They are different
\textbf{perspectives} on the same structure.

\section{Synthesis: The Trinity Is
Axiom-Invariant}\label{synthesis-the-trinity-is-axiom-invariant}

\subsection{The Main Result}\label{the-main-result}

\textbf{Theorem SYN:} The Trinitarian formula \(\Box\forall x\,Tr(x)\)
is \textbf{axiom-invariant} in the sense that it follows from at least
four systematically different extensions of S5.

\textbf{Proof:} Directly from Theorems TOP-A, TEMP-A, MER-A, and the
result of \texttt{modal5.md}. q.e.d.

\subsection{What Follows?}\label{what-follows}

\textbf{Consequence 1:} The Trinitarian structure is \textbf{not} bound
to a particular metaphysical language. It is formulable in different
languages.

\textbf{Consequence 2:} The choice of extension is \textbf{not}
determined by the target statement alone. It depends on metaphysical,
physical, or hermeneutic preferences.

\textbf{Consequence 3:} The question ``Which extension is the right
one?'' is \textbf{not} a metalogical question. It is a metaphysical,
physical, or hermeneutic question.

\section{What Follows from This?}\label{what-follows-from-this}

\subsection{The Trinity Is Robust}\label{the-trinity-is-robust}

The fact that the Trinitarian formula follows from \textbf{at least
four} systematically different extensions shows: It is \textbf{robust}
with respect to the choice of metaphysical language. It is not a
consequence of a particular axiomatics, but a \textbf{structure} that
recurs in different axiomatics.

\subsection{The Necessity of Supplementary
Structure}\label{the-necessity-of-supplementary-structure}

What \textbf{remains} is the necessity of a supplementary structure at
all. Pure S5 is not sufficient (according to \texttt{modal5.md}). The
supplementary structure is \textbf{unavoidable} --- but its
\textbf{form} is not unique.

\subsection{We Stand at the Beginning of
Infinity}\label{we-stand-at-the-beginning-of-infinity}

There could be \textbf{further} systematically different extensions that
we have not constructed. The question of \textbf{completeness} remains
open. What we have shown is: There are \textbf{at least four}. Whether
there are \textbf{exactly four} is an open question.

\section{Open Questions}\label{open-questions}

\subsection{Question 1: Is There a Canonical
Extension?}\label{question-1-is-there-a-canonical-extension}

We have constructed four extensions. Is there one that is
\textbf{canonical} --- i.e., one that is \textbf{distinguished} for
metaphysical or physical reasons?

\textbf{Possible answer:} The superposition extension is distinguished
by its proximity to quantum mechanics. But this distinction is
\textbf{physical}, not \textbf{metalogical}.

\subsection{Question 2: Are There Further
Extensions?}\label{question-2-are-there-further-extensions}

We have constructed four. Are there \textbf{further} systematically
different extensions?

\textbf{Possible answers:}

\begin{itemize}
\tightlist
\item
  An \textbf{information-theoretic} extension (information as basic
  concept).
\item
  A \textbf{process-philosophical} extension (process as basic concept).
\item
  A \textbf{phenomenological} extension (experience as basic concept).
\end{itemize}

\subsection{Question 3: What Is the Common
Core?}\label{question-3-what-is-the-common-core}

All four extensions derive the Trinitarian formula. What is the
\textbf{common core}?

\textbf{Possible answer:} The common core is the \textbf{necessity of a
supplementary structure} at all. Pure S5 is not sufficient. But the form
of the supplementary structure is variable.

\subsection{Question 4: Is Non-Uniqueness a
Defect?}\label{question-4-is-non-uniqueness-a-defect}

Is the fact that the Trinity follows from different extensions a
\textbf{defect} or a \textbf{strength}?

\textbf{Answer:} It is a \textbf{strength}. It shows that the
Trinitarian structure is not bound to a particular metaphysical
language. It is \textbf{axiom-invariant}.

\section{Conclusion: What Has Been
Shown?}\label{conclusion-what-has-been-shown}

The present investigation has shown:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \textbf{There are at least four systematically different extensions of
  S5} that derive the Trinitarian formula: superposition, topology,
  temporality, mereology.
\item
  \textbf{The Trinitarian formula is axiom-invariant.} It is not bound
  to a particular metaphysical language.
\item
  \textbf{The choice of extension is not determined by the target
  statement alone.} It depends on metaphysical, physical, or hermeneutic
  preferences.
\item
  \textbf{What remains is the necessity of a supplementary structure at
  all.} Pure S5 is not sufficient.
\end{enumerate}

The central claim of this treatise is:

\begin{quote}
\textbf{The Trinitarian structure is not bound to a particular
metaphysical language. It follows from at least four systematically
different extensions of S5. The choice of extension is a metaphysical,
physical, or hermeneutic question --- not a metalogical one.}
\end{quote}

The question of whether there is a \textbf{canonical} extension remains
open. It is the subject of \textbf{Modal 8} (Topological Modal Logic)
and beyond.

\section{Appendix A: The Three Alternatives in an
Overview}\label{appendix-a-the-three-alternatives-in-an-overview}

\subsection{Axioms}\label{axioms}

\[\begin{aligned}
\text{TOP} &:= \{TOP1, TOP2, TOP3, TOP4, TOP5\} \\
\text{TEMP} &:= \{T1, T2, T3, T4\} \\
\text{MER} &:= \{MER1, MER2, MER3, MER4, MER5\}
\end{aligned}\]

\subsection{Sufficiency}\label{sufficiency-1}

\[\begin{aligned}
& S5 + \text{TOP} \vdash \Box\forall x\,Tr(x) \\
& S5 + \text{TEMP} \vdash \Box\forall x\,Tr(x) \\
& S5 + \text{MER} \vdash \Box\forall x\,Tr(x) \\
& S5 + SP \vdash \Box\forall x\,Tr(x)
\end{aligned}\]

\section{Appendix B: Comparison of Extensions in a
Table}\label{appendix-b-comparison-of-extensions-in-a-table}

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}}
  >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}}
  >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}}
  >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}}
  >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Criterion
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Superposition
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Topology
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Temporality
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Mereology
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Basic concept & Primordial state & Limit & Time & Part-whole \\
Operators & \(C\), \(R\) & \(\mathcal{L}\), \(\mathcal{N}\) & \(P\),
\(F\), \(N\), \(Z\) & \(\preceq\), \(\Sigma\) \\
Number of axioms & 5 & 5 & 4 & 5 \\
Metaphysical tradition & Schelling & Kant & Augustine & Aristotle \\
Physical analogy & QM & Topology & Arrow of time & Physics \\
\end{longtable}

\section{Afterword}\label{afterword}

You have read this treatise. Perhaps the multiplicity of alternatives
surprised you. Perhaps the idea of axiom-invariance impressed you. Both
are fine.

The investigation has shown: The Trinitarian structure is
\textbf{robust}. It follows from at least four systematically different
extensions of S5. It is not bound to a particular metaphysical language.

This robustness is \textbf{not a defect}. It is an \textbf{insight}: The
Trinitarian structure is not a consequence of a particular axiomatics,
but a \textbf{structure} that recurs in different axiomatics.

What remains is the necessity of a \textbf{supplementary structure at
all}. Pure S5 is not sufficient. The supplementary structure is
unavoidable --- but its form is variable.

One final question:

\begin{quote}
\textbf{Do we dare the leap into openness to the limit --- even though
there are many paths that lead to it?}
\end{quote}

\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{document}