Bradley's Preservation Condition

Bradley (2000, p.220) offers the following as a "preservation condition" for conditionals:

If Pr(A) > 0, but Pr(B) = 0, then Pr(A → B) = 0.

He goes on to explain that all this means is "that one cannot be certain that B is not the case if one thinks that it is possible that if A then B, unless one rules out the possibility that A as well." (op cit)

Is this condition correct for all real conditionals (by "real" I simply mean to focus attention on conditionals that we find, as opposed to those we make up)? Moreover, is this condition similar to Jackson's robustness-condition for conditional assertion? In reading and rereading his explanation of the preservation condition I've come to wonder how different this is from Jackson's account as both can be explained in terms of modus ponens.

Recall that for Jackson (1987, pp. 28ff.), a conditional is assertible just in case it both has high enough probability on its own, and is robust with respect to its antecedent. This kind of robustness is explained by appealing to whether or not one's subjective probability regarding a conditional's acceptability is affected negatively were one to come to believe the antecedent. This rules out as unacceptable those conditionals whose truth depends solely upon the falsity of their antecedents. We would be wary of empolying such conditionals in modus ponens inferences.

Now look at the preservation condition and its explanation. The reason that you can't both doubt B and accept (even just a little that) A → B, unless you discount A, is that A and (A → B) would force B's truth (by modus ponens).

What is curious is that this preservation condition, unlike robustness wrt antecedent, seems to facilitate modus tollens as well. Accepting (even just a little that) (A → B) and denying that B, forces that you deny A (by modus tollens). If the preservation condition is true of all real conditionals, then Jackson's robustness condition is too narrow for assertibility. Instead, we would have to add two-sided robustness to Jackson's account (which I have advocated for other reasons).

References
Bradley, R. (2000), "A preservation condition for conditionals," Analysis 60.3, pp. 219-222.
Jackson, F. (1987) Conditionals Basil Blackwell.