From d901c92b4cc17f36f3d23078173e8b2f17106fd7 Mon Sep 17 00:00:00 2001
From: Jacob Walchuk <jpw24@st-andrews.ac.uk>
Date: Sun, 6 Jul 2025 22:06:26 +0100
Subject: fixed all tables, paper 3 should be done

---
 papers/3.tex | 23 +++++++++++++----------
 1 file changed, 13 insertions(+), 10 deletions(-)

diff --git a/papers/3.tex b/papers/3.tex
index 52d364b..6b3f574 100644
--- a/papers/3.tex
+++ b/papers/3.tex
@@ -194,7 +194,7 @@ $\vdash \Box \forall x (Xx \ba x=y) \ra \neg \Diamond \exists z (Haec(X)(z) \wed
 $\vdash Haec(X)(y) \ra Tra(X)(y)$& \\
 \end{tabulary} % terrible hack but it fixed the rendering, I am very sorry Mohit
 \end{quote}
-Now I will show the first line. \\
+Now I will show the first line.
 \begin{quote}
 $\forall x (Xx \ba x=y), Haec(X)(z) \wedge z \neq y \vdash Xy \ba y=y \\
 \forall x (Xx \ba x=y), Haec(X)(z) \wedge z \neq y \vdash Xy \ba z=y \\
@@ -205,18 +205,21 @@ $\forall x (Xx \ba x=y), Haec(X)(z) \wedge z \neq y \vdash Xy \ba y=y \\
 \end{quote}
 Then, from $\vdash Haec(X)(y) \ra Tra(X)(y)$, we can observe that: \\
 \begin{quote}
-$\vdash \Box Haec(X)(y) \ra \Box Tra(X)(y) \ \ \ \ $(K) $ \\
-\vdash \Box \forall x (Xx \ba x=y) \ra \Box \Box \forall x (Xx \ba x=y) \ \ \ \ $(\textbf{4})$\\
-\vdash Haec(X)(y) \ra \Box Haec(X)(y) \ \ \ \ $(Chaining conditionals)$ \\
-\vdash Haec(X)(y) \ra \Box Tra(X)(y)$ 
+\begin{tabulary}{\textwidth}{Lr}
+$\vdash \Box Haec(X)(y) \ra \Box Tra(X)(y)$ & (K)  \\
+$\vdash \Box \forall x (Xx \ba x=y) \ra \Box \Box \forall x (Xx \ba x=y)$ & (\textbf{4})\\
+$\vdash Haec(X)(y) \ra \Box Haec(X)(y)$ & (Chaining conditionals) \\
+$\vdash Haec(X)(y) \ra \Box Tra(X)(y)$ &
+\end{tabulary}
 \end{quote}
 \subsection{The proof for (o-Tracking)} 
-
 \begin{quote}
-$\vdash Haec(X)(o) \ra Tra(X)(o)$ \ \ \ \ (Proved above) \\
-$\vdash \exists X Haec(X)(o) \ra \exists X Tra(X)(o)$ \ \ \ \ (Derivable from $\forall$ rule) \\
-$\vdash \Box \exists X Haec(X)(o) \ra \Box \exists X Tra(X)(o)$ \ \ \ \ (K)\\
-$\vdash \Box \exists X Tra(X)(o)$ \ \ \ \ (MP, o-Haecceity) 
+\begin{tabulary}{\textwidth}{Lr}
+$\vdash Haec(X)(o) \ra Tra(X)(o)$ & (Proved above) \\
+$\vdash \exists X Haec(X)(o) \ra \exists X Tra(X)(o)$ & (Derivable from $\forall$ rule) \\
+$\vdash \Box \exists X Haec(X)(o) \ra \Box \exists X Tra(X)(o)$ & (K)\\
+$\vdash \Box \exists X Tra(X)(o)$ & (MP, $o$-Haecceity) 
+\end{tabulary}
 \end{quote}
 
 
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