Finding three formulas such that they are consistent as pairs, but inconsistent when all three are together. This problem and several others of a similar nature have shown up in my textbook in introductory logic:
"Give formulas $\phi, \psi, \sigma$ such that any pairing of them defines a consistent theory, but not all three together."
I know that, by definition, a theory $\Gamma$ is consistent if one cannot derive $\bot$ from $\Gamma$, using the rules of natural deduction. However, I'm not quite sure where to start. So far, we have only dealt with propositional logic. Is there a straight-forward way to handle these sorts of problems, using the syntax of propositional logic?
 A: A set of formulas is consistent if and only if there is a valuation for all of the formulas in the set to be true together.
Consider the formulas $\phi_{1}$, $\phi_{2}$ and $\phi_{3}$. We take $\phi_{3}$ as $$\phi_{1}\rightarrow\neg\phi_{2}$$
Then, the set $\{\phi_{1}, \phi_{1}\rightarrow\neg\phi_{2}\}$, hence, $\{\phi_{1}, \phi_{3}\}$ is consistent, for $\phi_{1}$ is true and $\phi_{2}$ is false. Notice that $\phi_{2}$ occurs in the compound formula $\phi_{3}$, but not in the set; compare this case to that of a set $A$ defined as $\{a, \{b\}\}$ for which $b\notin A$.
Likewise, $\{\phi_{1}, \phi_{2}\}$ and $\{\phi_{2}, \phi_{3}\}$ are consistent sets, but $\{\phi_{1}, \phi_{2}, \phi_{3}\}$ is not.
Consider now four formulas, $\phi_{1}$, $\phi_{2}$, $\phi_{3}$ and $\phi_{4}$. Take $\phi_{4}$ as $$\phi_{1}\rightarrow\neg\phi_{2}\vee\phi_{2}\rightarrow\neg\phi_{3}\vee\phi_{3}\rightarrow\neg\phi_{1}$$
If, as above, we evaluate the formula(s) left out to false, we get consistent sets of two and three formulas, but not of four formulas; $\{\phi_{1}, \phi_{2}, \phi_{3}, \phi_{4}\}$ is not consistent.
Following this pattern, we can inductively construct larger sets in the same fashion.
A: Consider, in a language with two unary predicates $P, Q$, the following sentences :

*

*$\phi$ states that there are precisely two elements that satisfy $P$.

*$\psi$ states that there are precisely two elements that satisfy $Q$.

*$\sigma$ states that there are precisely three elements in total, and that $P$ and $Q$ are disoint.

