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alexandræ.fr · all of term logic in a cat-dog-animal-aquatic model
all of term logic in a cat-dog-animal-aquatic modeljuly 26th, 2026
in term logic, we don't have an inductive formal language construction unlike in most logics nowadays:
- for all categories \(A\) and \(B,\) we have that \(\textit{all }A\textit{ is }B\) is a logical formula, notated \(AaB.\)
- for all categories \(A\) and \(B,\) we have that \(\textit{no }A\textit{ is }B\) is a logical formula, notated \(AeB.\)
- for all categories \(A\) and \(B,\) we have that \(\textit{some }A\textit{ is }B\) is a logical formula, notated \(AiB.\)
- for all categories \(A\) and \(B,\) we have that \(\textit{some }A\textit{ is not }B\) is a logical formula, notated \(AoB.\)
- for all formulas \(p,q,r,\) an aristotelian argument is an ordered triple \(\langle p,q,r\rangle.\)
there are really two things we need to consider in finding a model equivalent to term logic as a whole: the 24 valid arguments, called syllogisms, and the square of oppositions. we notably need to make sure that we find a model that doesn't add more valid arguments nor does weird stuff like \(AaB\) being essentially equivalent to \(AiB.\) using an sat solver (z3 on python), we can find one such model to be the following :
$$
\begin{array}{c|cccc}
a &\rm cat &\rm dog &\rm animal &\rm aquatic\\\hline
\rm cat &\blacksquare& &\blacksquare& \\
\rm dog & &\blacksquare&\blacksquare& \\
\rm animal & & &\blacksquare& \\
\rm aquatic& & & &\blacksquare
\end{array}
\quad
\begin{array}{c|cccc}
e &\rm cat &\rm dog &\rm animal &\rm aquatic\\\hline
\rm cat & &\blacksquare& &\blacksquare \\
\rm dog &\blacksquare& & &\blacksquare\\
\rm animal & & & & \\
\rm aquatic&\blacksquare&\blacksquare& &
\end{array}
$$
$$
\begin{array}{c|cccc}
i &\rm cat &\rm dog &\rm animal &\rm aquatic\\\hline
\rm cat &\blacksquare&&\blacksquare&\\
\rm dog &&\blacksquare&\blacksquare&\\
\rm animal &\blacksquare&\blacksquare&\blacksquare&\blacksquare\\
\rm aquatic&&&\blacksquare&\blacksquare
\end{array}
\quad
\begin{array}{c|cccc}
o &\rm cat &\rm dog &\rm animal &\rm aquatic\\\hline
\rm cat &&\blacksquare&&\blacksquare\\
\rm dog &\blacksquare&&&\blacksquare\\
\rm animal &\blacksquare&\blacksquare&&\blacksquare\\
\rm aquatic&\blacksquare&\blacksquare&\blacksquare
\end{array}
$$
though, i had to restrict a little further to make a model that makes sense. indeed, there are smaller models still, but they can lead to things like all \(A\) not being \(A\) while all \(C\) is \(C.\) because i like weird stuff like this, here's one such model which is just as equivalent to term logic as the previous one:
$$
\begin{array}{c|ccc}
a &0 &1 &2\\\hline
0 & &\blacksquare&\\
1 & & &\\
2 & &\blacksquare&\blacksquare
\end{array}
\quad
\begin{array}{c|ccc}
e &0 &1 &2\\\hline
0 &\blacksquare& &\blacksquare\\
1 & & &\\
2 &\blacksquare& &
\end{array}
$$
$$
\begin{array}{c|ccc}
i &0 &1 &2\\\hline
0 &&\blacksquare&\\
1 &\blacksquare&\blacksquare&\blacksquare\\
2 &&\blacksquare&\blacksquare
\end{array}
\quad
\begin{array}{c|ccc}
o &0 &1 &2\\\hline
0 &\blacksquare&&\blacksquare\\
1 &\blacksquare&\blacksquare&\blacksquare\\
2 &\blacksquare&&
\end{array}
$$
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