%
\label{def:truth4}
The \textbf{Truth4 lattice} is the set
\[
    \boxed{%
    \mathrm{Truth4}
    := \{\mathsf{T},\; \mathsf{B},\; \mathsf{N},\; \mathsf{F}\}}
\]
with the following interpretation
via bipolar evaluation $(\chi_+, \chi_-)$:
\begin{enumerate}
    \item $\mathsf{T}$ (\textbf{true}):
          $\chi_+(P) = 1$ and $\chi_-(P) = 1$.
          Both sectors confirm $P$.
          The predicate is \emph{fully determined and affirmative}.
    \item $\mathsf{F}$ (\textbf{false}):
          $\chi_+(P) = 0$ and $\chi_-(P) = 0$.
          Both sectors deny $P$.
          The predicate is \emph{fully determined and negative}.
    \item $\mathsf{B}$ (\textbf{both}):
          $\chi_+(P) = 1$ and $\chi_-(P) = 0$,
          or $\chi_+(P) = 0$ and $\chi_-(P) = 1$.
          One sector confirms and the other denies.
          The predicate is \emph{overdetermined} ---
          it is ``true and false simultaneously''
          from different spectral perspectives.
    \item $\mathsf{N}$ (\textbf{neither}):
          the predicate has no decisive witness
          in either sector.
          Neither $\chi_+(P)$ nor $\chi_-(P)$
          produces a conclusive value.
          The predicate is \emph{underdetermined} ---
          it is ``neither true nor false''
          due to insufficient spectral evidence.
\end{enumerate}
The encoding as pairs is:
\[
    \mathsf{T} = (1, 1),
    \quad
    \mathsf{F} = (0, 0),
    \quad
    \mathsf{B} = (1, 0) \text{ or } (0, 1),
    \quad
    \mathsf{N} = (\bot, \bot),
\]
where $\bot$ denotes the absence of a decisive witness.