“…Neither paper provide any argument for defining the Boolean extension to include B3, the reason for including it, it seems, is to get a so-called reduced semantics. 15 It should, however, be noted that B3 is rather different from the other two Boolean axioms: whereas B1 and B2 simply express that any instance of a Boolean excluded middle is entailed by every formula and a Boolean contradiction entails everything, B3 expresses that any relevant conditional either fails to be true, or the Boolean material conditional is true, or to put it equivalently; either the premises of any instance of modus ponens holds, or its conclusion does. Since B3 is equivalent to A ∧ (A → B) = B, and A → B A = B is a derivable rule of even CBB, it follows that the axiom is derivable in any logic with the pseudo modus ponens axiom A ∧ (A → B) → B which, again, is interderivable in BB with the rule of contraction, i.e.…”