Find all models of given theory $\def\imp{\Rightarrow}$I have a theory $T=\\{p \imp \neg q, \neg q, r \imp q, r \imp \neg p\\}$ over $P=\\{p, q, r\\}$ I need to find all models of theory $T$. My question is whether I could use any systematic approach? Instead of just "inventing" how particular models look. My idea was to use the method of analytic tableaux. Where $$T((p \imp \neg q) \land \neg q \land (r \imp q) \land (r \imp \neg p))$$ would be a root. Then every branch without a conflict would give me one particular model.

While the following approach is hardly systematic, here's how I found all models very quickly and easily:

Given $T=\\{p \to \
eg q, \
eg q, r \to q, r \to \
eg p\\}$, observe that we already have one atom, $\
eg q$.

Because $r \to q \equiv \
eg q \to \
eg r$, it follows that $\
eg r$. So every model $M$ of $T$ must have $q^M=r^M=\bot$ ($false$).

The two formulas involving $p$ are $p\to\
eg q$ and $r\to\
eg p\equiv p\to\
eg r$. In any model $M$ of $T$, $(\
eg q)^M = (\
eg r)^M = \top$, so the value these formulas will be $(p^M \Rightarrow \top)$, which is $\top$ no matter what the value of $p$.

Thus,

* $(p=\top, q=\bot, r=\bot)$ is a model of $T$,
* $(p=\bot, q=\bot, r=\bot)$ is a model of $T$,



and there are no other possibilities.

Larger or more challenging theories can't be 'eyeballed' this easily, and for them, analytic tableaux provide the proper tool.