A set of propositions is consistent if and only if **it is possible for all the members of the set to be true at the same time**. A set of propositions is inconsistent if and only if it is not consistent.
> A finite set of propositions $\Gamma$ is truth-functionally consistent if and only if there is at least one truth-assignment in which all propositions of $\Gamma$ are true.
$ \{P, Q\} $ form a consistent set because there is at least one assignment when both propositions are true. In fact there are two (corresponding to each disjunct) but one is sufficient.
> In terms of logical derivation, a finite $\Gamma$ of propositions is **inconsistent** in a system of derivation for propositional logic if and only if a proposition of the $P & \sim P$ is derivable from $\Gamma$. It is **consistent** just if this is not the case.
A [contradiction](Logical%20truth%20and%20falsity.md#logical-falsity) contradiction has very important consequences for reasoning because if a set of propositions is inconsistent, every and all other propositions are derivable from that set.
Here we want to derive some proposition $Q$. If we can derive a contradiction from its negation as an assumption then, by the [negation elimination](Negation%20Elimination.md) rule, we can assert $Q$. This is why contradictions should be avoided in arguments, they 'prove' everything which, by association, undermines any particular premise you are trying to assert.
