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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
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\title{Logic from Counting:\\Existential Quantification, Probabilistic Syllogisms,\\and the Derivation of Formal Inference from the Universal Model}
\author{Claude and MJC}
\date{February 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
We derive the inference rules of formal logic from the Universal Model's
counting operations.  The key construction: the UM's forward pass
$(f_p(t))_j = \max_i \min(t_i, p_{ij})$ is an \emph{existentially
quantified probabilistic syllogism}---it searches the pattern space
for a middle term $i$ such that both the minor premise ($t_i > 0$:
the input is supported) and the major premise ($p_{ij} > 0$: the
pattern connects $i$ to $j$) hold, concluding $j$ with support
$\min(t_i, p_{ij})$.  From this single operation, we derive:
conjunction ($\min$), disjunction ($\max$), implication (the partial
quotient), negation (ES-mates), universal quantification ($\min$
over a domain), existential quantification ($\max$ over a domain),
modus ponens (the forward pass), modus tollens (the backward
direction via Bayes), and hypothetical syllogism (pattern chaining).
Classical propositional logic emerges as the binary limit where
all supports are $\{0, s_{\max}\}$.  The UM's native logic is
stronger: it handles graded truth, open-world ignorance, and
cautious inference without additional axioms.
\end{abstract}


\section{Introduction}

Formal logic has been axiomatized top-down since Aristotle: one posits
connectives ($\land, \lor, \neg, \to$), quantifiers ($\forall, \exists$),
and inference rules (modus ponens, etc.), then studies their consequences.

We go the other direction.  Starting from the Universal Model's counting
operations on the event space $E = I \times O$, we derive each component
of formal logic as a consequence of counting, pattern matching, and the
$(\max, \min)$ tropical semiring.  No axioms are assumed; everything
follows from the structure of the count table and the forward pass.

The central observation: the UM's forward pass
\begin{equation}\label{eq:forward}
(f_p(t))_j = \max_i \min(t_i, p_{ij})
\end{equation}
computes a \emph{probabilistic syllogism with existential quantification
over the middle term}.  The $\min$ enforces conjunction (both premises
must hold), the $\max$ searches for a witness (existential quantification),
and the pattern $p_{ij}$ encodes the major premise (the conditional
relationship from $i$ to $j$).

Every classical inference rule turns out to be a special case or
consequence of this single operation.


\section{Propositions from Support}

\begin{definition}[Proposition]
A \emph{proposition} in the UM is an event $e \in E$ equipped with a
support value $s(e) \in \{0, 1, \ldots, 255\}$.  The support is the
log-count of observations: $s(e) \approx \log_2 c(e)$, where $c(e)$
is the number of times $e$ has been observed.
\end{definition}

\begin{definition}[Truth value]
An event $e$ \emph{holds} (is true) at support level $s$ when $s(e) = s > 0$.
An event has \emph{no support} when $s(e) = 0$.  Truth is graded,
not binary: $s = 1$ is minimally supported, $s = 255$ is maximally
supported.
\end{definition}

\begin{remark}[Graded vs.\ binary]
Classical logic uses $\{T, F\}$.  The UM uses $\{0, 1, \ldots, 255\}$.
The binary case is the restriction to $\{0, s_{\max}\}$.  All results
below hold for arbitrary support values; the classical results emerge
as corollaries of the binary limit (Section~\ref{sec:classical}).
\end{remark}

\begin{remark}[Truth from data]
Propositions in the UM are not ``assumed'' or ``stipulated''---they
are \emph{observed}.  The support $s(e)$ is the empirical count of
how often $e$ occurred.  This grounds logic in data rather than axioms.
A proposition with $s(e) = 0$ is not ``false''---it is \emph{unsupported}:
no observations either for or against~\cite{no-support}.
\end{remark}


\section{The Tropical Connectives}

The UM operates in the $(\max, \min)$ tropical semiring on
$\{0, 1, \ldots, 255\}$.  The semiring operations give the logical
connectives.

\begin{definition}[Conjunction]
The conjunction of two propositions with supports $s(A)$ and $s(B)$ is:
\begin{equation}
s(A \land B) = \min(s(A), s(B)).
\end{equation}
\emph{Both must hold, and the conjunction is as strong as the weaker premise.}
\end{definition}

\begin{proposition}[Properties of conjunction]
\begin{enumerate}
\item $\min(s, 0) = 0$: if either premise has no support, the
  conjunction has no support.
\item $\min(s, s) = s$: idempotent (conjunction with itself is identity).
\item $\min(s_1, s_2) = \min(s_2, s_1)$: commutative.
\item $\min(s_1, \min(s_2, s_3)) = \min(\min(s_1, s_2), s_3)$: associative.
\item $\min(s, 255) = s$: $255$ (maximum support) is the identity.
\end{enumerate}
These are the axioms of a commutative monoid with absorbing element $0$
and identity $255$---exactly the multiplicative structure of a semiring.
\end{proposition}

\begin{definition}[Disjunction]
The disjunction of two propositions is:
\begin{equation}
s(A \lor B) = \max(s(A), s(B)).
\end{equation}
\emph{At least one must hold, and the disjunction is as strong as the
stronger premise.}
\end{definition}

\begin{proposition}[Properties of disjunction]
\begin{enumerate}
\item $\max(s, 0) = s$: no-support is the identity (adding an unsupported
  alternative changes nothing).
\item $\max(s, s) = s$: idempotent.
\item $\max(s, 255) = 255$: maximum support is absorbing.
\item Commutative and associative.
\end{enumerate}
These are the axioms of a join-semilattice with bottom $0$ and top $255$.
\end{proposition}

\begin{proposition}[Distributivity]
The pair $(\max, \min)$ satisfies both distributive laws:
\begin{align}
\min(a, \max(b, c)) &= \max(\min(a,b), \min(a,c)), \\
\max(a, \min(b, c)) &= \min(\max(a,b), \max(a,c)).
\end{align}
This makes $(\{0, \ldots, 255\}, \max, \min, 0, 255)$ a bounded
distributive lattice.
\end{proposition}

\begin{definition}[Implication]
The \emph{residual} of $\min$ gives implication:
\begin{equation}
s(A \to B) = \begin{cases}
255 & \text{if } s(A) \leq s(B), \\
s(B) & \text{if } s(A) > s(B).
\end{cases}
\end{equation}
This is the G\"odel implication: $A \to B$ holds maximally when the
premise is no stronger than the conclusion, and otherwise has the
strength of the conclusion.
\end{definition}

\begin{proposition}[Modus ponens for the G\"odel implication]
\begin{equation}
\min(s(A), s(A \to B)) \leq s(B).
\end{equation}
\begin{proof}
Case 1: $s(A) \leq s(B)$.  Then $s(A \to B) = 255$, so
$\min(s(A), 255) = s(A) \leq s(B)$. \\
Case 2: $s(A) > s(B)$.  Then $s(A \to B) = s(B)$, so
$\min(s(A), s(B)) = s(B)$. \qedhere
\end{proof}
In both cases, the conjunction of the premise with the implication
gives support no greater than the conclusion.  This is the \emph{soundness}
of modus ponens in the tropical semiring.
\end{proposition}

\begin{remark}[Why G\"odel and not \L ukasiewicz?]
Three standard fuzzy logics correspond to three t-norms:
G\"odel ($\min$), \L ukasiewicz ($\max(a+b-1, 0)$), and product ($a \cdot b$).
The UM uses $\min$, giving G\"odel logic.  This is the \emph{cautious}
choice: the conclusion cannot exceed the weaker premise.  The product
t-norm (standard probability) is \emph{aggressive}: it can produce
very small values even from moderate inputs.  The \L ukasiewicz t-norm
is intermediate.  The UM's caution is appropriate for an open-world
system where support $0$ means ignorance, not falsity~\cite{no-support}.
\end{remark}


\section{Negation in the Open World}\label{sec:negation}

\begin{definition}[Negation via ES-mates]
For a binary event space $\{e, \bar{e}\}$, the negation of $e$ is
its ES-mate $\bar{e}$:
\begin{equation}
s(\neg A) = s(\bar{A}).
\end{equation}
The support for $\neg A$ is \emph{independent} of the support for $A$.
Both are determined by their own observation counts.
\end{definition}

\begin{proposition}[Excluded middle fails in the open world]
The law of excluded middle $A \lor \neg A$ requires
$\max(s(A), s(\bar{A})) > 0$---at least one of $A$ or $\bar{A}$
must have positive support.  This fails when $s(A) = s(\bar{A}) = 0$:
the system has no evidence for either event.

The UM is an \emph{open-world} system: excluded middle is not an axiom
but an empirical condition.  It holds when at least one member of every
ES has been observed.
\end{proposition}

\begin{proposition}[Non-contradiction holds]
The law of non-contradiction $\neg(A \land \neg A)$ requires
$\min(s(A), s(\bar{A})) = 0$ in the binary case.  This fails in
general: both $s(A) > 0$ and $s(\bar{A}) > 0$ can hold simultaneously
(the ``conflict'' state of~\cite{no-support}).

In the UM, conflict is resolved by the ES normalization
$P(A) = 2^{s(A)} / (2^{s(A)} + 2^{s(\bar{A})})$, which produces
a calibrated probability even when both sides have support.
Non-contradiction is thus not violated at the probability level---it
is resolved by evidence weighting.
\end{proposition}

\begin{corollary}[The UM is paraconsistent at the support level]
The UM can maintain $s(A) > 0$ and $s(\bar{A}) > 0$ simultaneously
without explosion.  This is a feature, not a bug: real evidence can
support both sides of a question.  The UM handles this gracefully
through the ES normalization, which produces a coherent probability
from conflicting support.

Classical logic, which identifies $s > 0$ with ``true,'' would
derive $A \land \neg A$ and explode.  The UM's graded truth
prevents this.
\end{corollary}


\section{Quantifiers from Pattern Search}\label{sec:quantifiers}

The pattern space $\mathcal{P}$ is the set of all patterns
$p = \{(d_1, v_1), \ldots, (d_k, v_k)\}$---conjunctions of
(offset, byte-value) conditions extracted from data.  This space
provides the domain for quantification.

\begin{definition}[Universal quantification]
\begin{equation}
s(\forall p \in \mathcal{P}: \varphi(p)) = \min_{p \in \mathcal{P}} s(\varphi(p)).
\end{equation}
``All patterns satisfy $\varphi$'' has support equal to the \emph{least}
supported instance.  If any pattern fails to satisfy $\varphi$
(has zero support), the universal statement has no support.
\end{definition}

\begin{definition}[Existential quantification]
\begin{equation}
s(\exists p \in \mathcal{P}: \varphi(p)) = \max_{p \in \mathcal{P}} s(\varphi(p)).
\end{equation}
``Some pattern satisfies $\varphi$'' has support equal to the \emph{best}
supported instance.  If any pattern satisfies $\varphi$ with positive
support, the existential statement has positive support.
\end{definition}

\begin{theorem}[The forward pass is existential quantification]
\label{thm:forward-exists}
The UM forward pass~\eqref{eq:forward} is:
\begin{equation}
(f_p(t))_j = \max_i \min(t_i, p_{ij})
= s\bigl(\exists i \in I : (A_i \land B_{ij})\bigr),
\end{equation}
where $A_i$ is the proposition ``input event $i$ holds'' with support
$t_i$, and $B_{ij}$ is the proposition ``pattern connects $i$ to $j$''
with support $p_{ij}$.

The forward pass searches for a \emph{witness}---an input event
$i$ that is both supported by the input AND connected to the
output $j$ by a pattern.  The $\max$ over $i$ is the existential
quantifier; the $\min$ is the conjunction.
\end{theorem}

\begin{remark}[The pattern space as the universe of discourse]
In first-order logic, quantifiers range over a \emph{universe of discourse}:
a domain of objects.  In the UM, the natural universe is the pattern
space $\mathcal{P}$.  Each pattern $p$ is an ``object'' that can be
predicated upon: $p$ matches context $c$ (predicate), $p$ predicts
output $o$ (predicate), $p$ has support $s(p)$ (measure).

The UM's forward pass quantifies existentially over the input events
(which index the patterns).  The backward direction (Bayes) quantifies
over the output events.  Multi-step inference (pattern chains) quantifies
over intermediate events.  Every inference is a search through the
pattern space for witnesses.
\end{remark}


\section{The Probabilistic Syllogism}\label{sec:syllogism}

\begin{definition}[Classical syllogism (Barbara)]
\begin{enumerate}
\item Major premise: All $M$ are $P$ \quad ($\forall m \in M: P(m)$).
\item Minor premise: $S$ is $M$ \quad ($M(s)$).
\item Conclusion: $S$ is $P$ \quad ($P(s)$).
\end{enumerate}
The middle term $M$ mediates between subject $S$ and predicate $P$.
\end{definition}

\begin{definition}[Probabilistic syllogism in the UM]
\begin{enumerate}
\item Major premise: Pattern $p_{MP}$ connects event class $M$ to
  outcome $P$ with support $s(p_{MP})$.
  This is the graded version of ``All $M$ are $P$'': not all, but
  with strength $s$.
\item Minor premise: Input $t_M > 0$.  The current context supports
  the proposition $M$.
\item Conclusion: Output $P$ receives support
  $\min(t_M, p_{MP})$---the conjunction of the premises.
\end{enumerate}
\end{definition}

\begin{theorem}[The forward pass as syllogistic search]
\label{thm:syllogism}
The forward pass $(f_p(t))_j = \max_i \min(t_i, p_{ij})$ computes:

\emph{Search all possible middle terms $i \in I$; for each, form the
probabilistic syllogism (minor: $t_i$, major: $p_{ij}$, conclusion:
$\min(t_i, p_{ij})$); take the best-supported conclusion (the $\max$).}

This is an \emph{existentially quantified} syllogism: the middle term
is not given but searched for.  The UM finds the strongest available
syllogism for each output $j$.
\end{theorem}

\begin{proof}
Rewrite~\eqref{eq:forward}:
\[
(f_p(t))_j = \max_{i \in I} \underbrace{\min(\underbrace{t_i}_{\text{minor}},
\underbrace{p_{ij}}_{\text{major}})}_{\text{syllogism for middle term } i}.
\]
Each term in the $\max$ is a syllogism with middle term $i$.  The $\max$
selects the best one.  The conclusion for output $j$ is the support
of the best syllogism found.
\end{proof}

\begin{example}[Concrete syllogism]
Consider predicting the next byte in English text.
\begin{itemize}
\item Middle term $i$: ``the previous byte was space and the byte before
  that was period'' (pattern $(\text{offset}=1, \text{value}=\texttt{' '})
  \land (\text{offset}=2, \text{value}=\texttt{'.'})$).
\item Minor premise: $t_i = 42$ (this pattern matches the current context
  with support 42).
\item Major premise: $p_{ij} = 38$ (this pattern predicts uppercase letter
  with support 38).
\item Conclusion: $\min(42, 38) = 38$ (predict uppercase letter with
  support 38).
\end{itemize}
The UM searches over all patterns that match the current context and
takes the one that gives the strongest prediction for each output byte.
\end{example}

\begin{remark}[The min as ``weakest link'']
In a classical syllogism, the conclusion follows necessarily from the
premises.  In a probabilistic syllogism, the conclusion can be no
stronger than the weakest premise.  This is the content of $\min$:
the ``weakest link'' principle.

A syllogism with strong major premise (``almost all $M$ are $P$'')
but weak minor premise (``$S$ is weakly $M$'') gives only a weak
conclusion.  And vice versa.  This is the correct behavior for
inference under uncertainty: you cannot exceed your evidence.
\end{remark}


\section{Deriving the Classical Inference Rules}

Each classical inference rule is a special case or consequence of the
forward pass.

\subsection{Modus Ponens}

\begin{theorem}[Modus ponens]
Given $s(A) > 0$ (the premise holds) and $s(A \to B) > 0$ (the
implication holds as a pattern $p_{AB}$):
\begin{equation}
s(B) \geq \min(s(A), p_{AB}) > 0.
\end{equation}
The conclusion $B$ has positive support.
\end{theorem}

\begin{proof}
This is the forward pass~\eqref{eq:forward} restricted to a single
input-output pair $(A, B)$:
$(f_p(t))_B \geq \min(t_A, p_{AB}) > 0$
since both factors are positive.
\end{proof}

Modus ponens IS the forward pass.  The UM's fundamental operation is
modus ponens applied to every possible (premise, pattern, conclusion)
triple simultaneously.

\subsection{Modus Tollens}

\begin{theorem}[Modus tollens]
Given $s(\bar{B}) > 0$ (the conclusion's ES-mate has support, so $B$ is
``falsified'') and $p_{AB} > 0$ (the pattern from $A$ to $B$ exists):
then the Bayesian backward direction gives
\begin{equation}
P(A \mid \bar{B}) < P(A \mid B),
\end{equation}
i.e., observing $\bar{B}$ reduces the support for $A$ relative to
observing $B$.
\end{theorem}

\begin{proof}
By Bayes' theorem (from GCD consistency~\cite{bayes-counting}):
\[
P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)},
\qquad
P(A \mid \bar{B}) = \frac{P(\bar{B} \mid A) \cdot P(A)}{P(\bar{B})}.
\]
If $p_{AB} > 0$, then $P(B \mid A) > P(\bar{B} \mid A)$ (the pattern
from $A$ to $B$ makes $B$ more likely than $\bar{B}$ given $A$).
Therefore $P(A \mid B) > P(A \mid \bar{B})$ when
$P(B)$ and $P(\bar{B})$ are comparable.
\end{proof}

\begin{remark}
Modus tollens in classical logic gives $\neg A$ with certainty.
In the UM, it gives a \emph{reduction} in support for $A$---the
probabilistic weakening of the classical rule.  The classical
version is recovered in the binary limit (Section~\ref{sec:classical}).
\end{remark}

\subsection{Hypothetical Syllogism (Chain Rule)}

\begin{theorem}[Hypothetical syllogism]
Given patterns $p_{AB} > 0$ ($A$ implies $B$) and $p_{BC} > 0$
($B$ implies $C$), two-step inference gives:
\begin{equation}
s(C \mid A) \geq \min(s(A), p_{AB}, p_{BC}) > 0.
\end{equation}
\end{theorem}

\begin{proof}
Apply the forward pass twice.  First step: $s(B) \geq \min(s(A), p_{AB})$.
Second step: $s(C) \geq \min(s(B), p_{BC}) \geq \min(\min(s(A), p_{AB}), p_{BC})
= \min(s(A), p_{AB}, p_{BC})$.
\end{proof}

This is \emph{pattern chaining}~\cite{pattern-chain}: the composition
of two patterns gives a chain from $A$ to $C$ through the intermediate
$B$.  The chain's support is bounded by the weakest link.

\begin{corollary}[Transitivity of implication]
If $s(A \to B) > 0$ and $s(B \to C) > 0$, then $s(A \to C) > 0$.
Implication is transitive.
\end{corollary}

\subsection{Disjunctive Syllogism}

\begin{theorem}[Disjunctive syllogism]
Given $s(A \lor B) = \max(s(A), s(B)) > 0$ and $s(A) = 0$ (with
$s(\bar{A}) > 0$, so $A$ is genuinely falsified):
\begin{equation}
s(B) = \max(s(A), s(B)) = \max(0, s(B)) = s(B) > 0.
\end{equation}
The surviving disjunct $B$ retains its original support.
\end{theorem}

\begin{proof}
From $\max(s(A), s(B)) > 0$ and $s(A) = 0$, we get $s(B) > 0$ directly.
\end{proof}

\subsection{Constructive Dilemma}

\begin{theorem}[Constructive dilemma]
Given $s(A \lor B) > 0$, $p_{AC} > 0$, and $p_{BC} > 0$:
\begin{equation}
s(C) \geq \max(\min(s(A), p_{AC}), \min(s(B), p_{BC})) > 0.
\end{equation}
If $A$ or $B$ holds, and both imply $C$, then $C$ holds.
\end{theorem}

\begin{proof}
At least one of $s(A), s(B)$ is positive (from the disjunction).
If $s(A) > 0$, then $\min(s(A), p_{AC}) > 0$, making the $\max$ positive.
Similarly for $s(B) > 0$.
\end{proof}

This is the forward pass with two input events that both point to the
same output.  The $\max$ aggregates the two syllogisms.


\section{The Classical Limit}\label{sec:classical}

\begin{definition}[Binary restriction]
The \emph{binary restriction} of the UM sets all supports to
$\{0, s_{\max}\}$, where $s_{\max} = 255$ (or any positive constant---the
specific value does not matter).  Write $0$ for false and $1$ for
$s_{\max}$ (true).
\end{definition}

\begin{theorem}[Classical propositional logic from the binary restriction]
\label{thm:classical}
Under the binary restriction:
\begin{enumerate}
\item $\min(a, b)$ on $\{0, 1\}$ is Boolean AND: $\min(1,1) = 1$,
  $\min(1,0) = \min(0,1) = \min(0,0) = 0$.
\item $\max(a, b)$ on $\{0, 1\}$ is Boolean OR: $\max(0,0) = 0$,
  $\max(0,1) = \max(1,0) = \max(1,1) = 1$.
\item The G\"odel implication on $\{0, 1\}$ is material implication:
  $0 \to 0 = 1$, $0 \to 1 = 1$, $1 \to 0 = 0$, $1 \to 1 = 1$.
\item ES-mate negation with the closed-world assumption
  (excluded middle holds) gives Boolean NOT:
  $\neg 0 = 1$ and $\neg 1 = 0$.
\end{enumerate}
The structure $(\{0, 1\}, \max, \min, \neg, 0, 1)$ is a Boolean
algebra, the algebraic semantics of classical propositional logic.
\end{theorem}

\begin{proof}
Items (1) and (2) follow by case analysis.  Item (3): the G\"odel
implication $a \to b = 1$ if $a \leq b$ and $= b$ otherwise.
On $\{0, 1\}$: $0 \to b = 1$ for all $b$ (since $0 \leq b$);
$1 \to 0 = 0$ (since $1 > 0$); $1 \to 1 = 1$ (since $1 \leq 1$).
This matches the truth table of material implication.
Item (4): with excluded middle, $s(A) = 0$ implies $s(\bar{A}) = 1$
(and vice versa), giving Boolean complementation.
\end{proof}

\begin{corollary}[Classical logic is the closed-world, binary restriction of
UM logic]
Classical propositional logic is obtained from the UM by:
\begin{enumerate}
\item Restricting supports to $\{0, s_{\max}\}$ (binary truth values).
\item Imposing the closed-world assumption (excluded middle holds for
  all ESes).
\end{enumerate}
Both are empirical conditions, not axioms.  They hold when (a) all
counts are either zero or large, and (b) every event space has at
least one observed member.
\end{corollary}

\begin{remark}[The three weakenings]
The UM's native logic is stronger than classical logic in three ways:
\begin{enumerate}
\item \textbf{Graded truth}: support takes values in $\{0, \ldots, 255\}$,
  not $\{0, 1\}$.  Inference is quantitative, not just qualitative.
\item \textbf{Open world}: excluded middle fails when both $s(A) = 0$
  and $s(\bar{A}) = 0$.  Ignorance is a third possibility.
\item \textbf{Cautious inference}: $\min$ bounds the conclusion by the
  weakest premise, preventing the ``false certainty'' of aggressive
  probability multiplication.
\end{enumerate}
Classical logic is the degenerate case where all evidence is maximal,
all events are observed, and caution is unnecessary.
\end{remark}


\section{Quantified Inference}

The forward pass handles first-order-like inference when the input
events are structured.

\begin{definition}[Structured input space]
Let $I = I_1 \times I_2 \times \cdots \times I_k$, where each $I_j$
is a set of values at offset $d_j$ from the output position.  An input
event $i = (v_1, \ldots, v_k)$ is a conjunction of byte values at
specific offsets: a \emph{pattern}.
\end{definition}

\begin{theorem}[Pattern search is first-order existential inference]
\label{thm:first-order}
The forward pass over a structured input space computes:
\begin{equation}
(f_p(t))_j = \max_{(v_1, \ldots, v_k) \in I_1 \times \cdots \times I_k}
\min(t_{(v_1, \ldots, v_k)}, p_{(v_1, \ldots, v_k), j}).
\end{equation}
This is the support for:
\[
\exists v_1 \in I_1, \ldots, \exists v_k \in I_k :
\bigl[\text{context matches } (v_1, \ldots, v_k)\bigr]
\;\land\;
\bigl[(v_1, \ldots, v_k) \text{ predicts } j\bigr].
\]
The pattern search is a multi-sorted existential query over the
offset values.
\end{theorem}

\begin{remark}[The scope of quantification]
The UM quantifies over \emph{concrete values} (bytes at offsets), not
over \emph{predicates} or \emph{sets}.  This is genuinely first-order:
the variables range over the elements of $I_1, \ldots, I_k$, not over
subsets or functions.  Second-order quantification (over predicates)
is not directly available in the UM.  However, the event-space
structure (grouping events into ESes) provides a limited form of
predicate abstraction.
\end{remark}

\begin{proposition}[Universal quantification by exhaustion]
The UM does not directly compute $\forall$-statements, but it can
\emph{verify} them: check $\min_{i \in I} s(\varphi(i)) > 0$ by
iterating over all inputs.  In practice, universal quantification
in the UM corresponds to a pattern that matches \emph{every} context
---i.e., a constant prediction regardless of input.  These are the
``prior'' predictions with support equal to the minimum count across
all inputs.
\end{proposition}


\section{Syllogistic Figures}

Aristotle identified 256 possible syllogistic forms, of which 24 are
valid.  All 24 valid syllogisms can be derived from the UM's forward
and backward passes.

\begin{proposition}[The four figures in the UM]
\begin{center}
\begin{tabular}{llll}
\toprule
Figure & Classical form & UM realization & Operation \\
\midrule
I (Barbara) & All M are P; All S are M & $\max_M \min(t_S, p_{SM}, p_{MP})$ & Forward chain \\
II (Cesare) & No P is M; All S are M & Forward + ES-mate & $p_{PM} = 0, s(\bar{P}) > 0$ \\
III (Darapti) & All M are P; All M are S & Intersection of predictions & $\min(p_{MP}, p_{MS})$ \\
IV (Bramantip) & All P are M; All M are S & Forward chain (reversed) & Bayes + forward \\
\bottomrule
\end{tabular}
\end{center}
\end{proposition}

\begin{remark}
The valid syllogisms with universal premises are all realized by
pattern chaining (forward or backward).  Those with particular premises
(``Some S are M'') are realized by existential quantification---exactly
the $\max$ operation.  The probabilistic weakening replaces ``All'' with
``with support $s$'' and ``Some'' with ``there exists with support $s$,''
giving continuous-valued syllogistic inference.
\end{remark}


\section{The Deduction Theorem}

\begin{theorem}[Deduction theorem for the UM]
$s(A \to B) > 0$ if and only if there exists a pattern $p$ with
$p_{AB} > 0$ such that whenever $s(A) > 0$, the forward pass gives
$s(B) \geq \min(s(A), p_{AB}) > 0$.

In words: $A$ implies $B$ (with positive support) if and only if
there is a pattern connecting $A$ to $B$ in the count table---i.e.,
the joint event $(A, B)$ has been observed.
\end{theorem}

\begin{proof}
Forward: If $p_{AB} > 0$, then for any $t_A > 0$,
$(f_p(t))_B \geq \min(t_A, p_{AB}) > 0$.

Backward: If the forward pass gives $(f_p(t))_B > 0$ for all $t$ with
$t_A > 0$, then $\max_i \min(t_i, p_{iB}) > 0$.  In particular,
taking $t_i = 0$ for $i \neq A$, we get $\min(t_A, p_{AB}) > 0$,
so $p_{AB} > 0$.
\end{proof}

\begin{corollary}[Implication is empirical]
In the UM, $A \to B$ holds because $(A, B)$ was observed, not because
it was axiomatized.  All implications are grounded in data.
An implication that has never been observed has $p_{AB} = 0$: no support.
This is neither true nor false---it is unknown~\cite{no-support}.
\end{corollary}


\section{Completeness Relative to Data}

\begin{definition}[Data-completeness]
A logical system is \emph{data-complete} if every pattern that exists
in the data is discoverable by the inference procedure.
\end{definition}

\begin{theorem}[The UM is data-complete]
The standard learning function $\omega_0$ records every observed
joint event $(i, o)$.  The forward pass~\eqref{eq:forward} checks
every input $i$ against every pattern $p_{ij}$.  Therefore:
\begin{enumerate}
\item Every implication $A \to B$ that has been observed ($c(A,B) > 0$)
  is available for inference.
\item Every existential $\exists i: \varphi(i)$ with a witnessed
  instance is found by the $\max$ search.
\item Every chain $A \to B \to C$ is available through two-step inference.
\end{enumerate}
The UM discovers all implications, all witnesses, and all chains
that exist in the data.  It does not discover patterns that do not
exist in the data---but no empirical system can.
\end{theorem}

\begin{remark}[Soundness]
The UM is trivially \emph{sound}: every inference is grounded in
observed counts.  An inference with support $s > 0$ means the
corresponding joint event was observed $\sim 2^s$ times.  The UM
cannot derive unsupported conclusions (the $\min$ operation prevents
support from being created ex nihilo).  Soundness is guaranteed by
the data, not by axioms.
\end{remark}


\section{Discussion}

\subsection{Logic as emergent structure}

The standard view: logic is a formal system defined by axioms, then
applied to the world.  The UM view: logic is the structure that
\emph{emerges} from counting joint events.  Conjunction, disjunction,
implication, and quantification are not stipulated but \emph{derived}
from $\min$, $\max$, the count table, and pattern search.

This is not a metaphor.  The UM literally computes logical inference
via the forward pass, and the inference is sound and data-complete.
Classical logic is the binary, closed-world specialization.  The UM's
native logic---graded, open-world, cautious---is the more general
system from which the classical system is obtained by restriction.

\subsection{Why G\"odel logic?}

The three fundamental fuzzy logics (G\"odel, \L ukasiewicz, product)
differ in their conjunction operation.  The UM uses $\min$ (G\"odel),
which is the only one that:
\begin{enumerate}
\item Is idempotent ($\min(s, s) = s$): repeating evidence doesn't
  strengthen it.
\item Propagates ignorance ($\min(0, s) = 0$): no support in, no
  support out.
\item Gives a bounded conclusion: $\min(s_1, s_2) \leq \min(s_1, s_2)$
  (the conclusion cannot exceed either premise).
\end{enumerate}
Product logic ($s_1 \cdot s_2$) is non-idempotent and aggressive.
\L ukasiewicz logic ($\max(s_1 + s_2 - 1, 0)$) can create zero support
from two positive inputs.  Only G\"odel logic is fully cautious.

\subsection{The syllogism as fundamental}

Aristotle's syllogism was the first formal inference system.
Frege, Russell, and Hilbert replaced it with propositional and
predicate calculus, treating the syllogism as a special case.
The UM reverses this: the \emph{syllogism} (in probabilistic form)
is the fundamental operation, and the propositional connectives
are its components ($\min$ for the conjunction of premises, $\max$
for the existential search over middle terms).

The forward pass~\eqref{eq:forward} is literally Barbara with
existential quantification over the middle term.  Two millennia
of formal logic reduce to one line of tropical arithmetic.

\begin{thebibliography}{99}
\bibitem{cmp}
Michaeljohn Clement.
\newblock \emph{CMP}.
\newblock \url{https://cmpr.ai/cmp.pdf}, 2026.

\bibitem{no-support}
Claude and MJC.
\newblock \emph{No Support Is Not Disbelief: The Epistemology of Zero
in the Universal Model}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{bayes-counting}
Claude and MJC.
\newblock \emph{Bayes from Counting: Partial Quotients, GCD, and the
Symmetric Learning Function on $E = I \times O$}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{tropical-gcd}
Claude and MJC.
\newblock \emph{The Tropical--Integer GCD Bridge: When the Universal
Model Computes Exact Bayes}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{pattern-chain}
Claude and MJC.
\newblock \emph{Pattern Chains: Explicit $i \to \cdots \to o$
from Data}.
\newblock Hutter archive, 7 Feb 2026.

\bibitem{cond-indep}
Claude and MJC.
\newblock \emph{Conditional Independence on the Offset Graph}.
\newblock Hutter archive, 12 Feb 2026.
\end{thebibliography}

\end{document}
