[PL 211] Quibbling about 'everything' and 'anything'

In this interesting Rationally Speaking podcast, Graham Priest says that one of the major motivations behind paraconsistent logic is to challenge the logical orthodoxy according to which everything follows from a contradiction.

Priest says that students in Introduction to Logic are taught this orthodoxy. Why is it the case that, in classical logic, contradictions explode, i.e., that {P & ~P} implies everything?

In Introduction to Logic, Copi and Cohen give the following proof:

1. P
2. ~P
3. P v Q [by addition from (1)]
4. Q  [by disjunctive syllogism from (2) and (3)]

For instance, let P be 'Today is Sunday' and let Q be 'The moon is made of cheese', then:

1. Today is Sunday.
2. Today is not Sunday.
3. Today is Sunday or the moon is made of cheese. [by addition from (1)]
4. The moon is made of cheese. [by disjunctive syllogism from (2) and (3)]

Accordingly, Copi and Cohen say the following:

Inconsistent statements are not "meaningless;" their trouble is just the opposite. They mean too much: They mean everything in the sense of implying everything.

Based on the aforementioned proof, wouldn't it be more accurate to say that contradictions imply anything (i.e., any single proposition), rather than everything (i.e., all propositions)? In other words, wouldn't it be more accurate to say that {P & ~P} implies Q, rather than Q & R & S....? Or is this just a quibble?