We set out five basic requirements for a logical system to be adequate for the regimentation of deductive reasoning in mathematics and science. We raise the question whether there are any further requirements, not entailed by these five, that ought to be imposed. One possible reply is dismissed: that the logical system should allow one to infer any proposition at all from an inconsistent set—i.e., it should have as primitive, or allow one to derive, the rule Ex Falso Quodlibet. We then propose that the logic should be implosive: it should not allow an inconsistent set to have any consequences other than absurdity. This proposal may appear to be very radical; but we hope to show that it is robust against objections.