Hughes and Cresswell's definition of consistency

On page 46 of Hughes and Cresswell's A New Introduction To Modal Logic, we have the following definition of consistency.

Consistency : We shall say that an axiomatic system is consistent iff not every wff is a theorem of that system. In other words, a system is inconsistent iff every wff is a theorem.

The definition of consistency strikes me as having clear counterexamples. Consider a system S where wffs A, ¬A and ¬P are theorems but P is not. Here we have a contradiction even though not every wff is a theorem of S. What am I missing ?

Solution 1:

This was pretty much answered in the comments, but the resolution is that there is no such system $S.$ Classical logic has the principle of explosion, which says that for any sentences $A$ and $B,$ we can infer $B$ from $A$ and $\lnot A.$ How this manifests in deductive systems varies by presentation, but semantically it follows from the fact that $A$ and $\lnot A$ are not simultaneously satisfied in any interpretation, so (vacuously) $B$ holds in any interpretation that satisfies $A$ and $\lnot A.$

Thus H&C's definition of inconsistency is equivalent to the definition that a system is inconsistent if there is a sentence $A$ such that both $A$ and $\lnot A$ are theorems.

Still, it would probably be better to use the other definition, since it is closer to the "idea" of inconsistency, and there are nonclassical logics that aren't explosive, and so the two definitions are not equivalent in these contexts.