I believe I am understanding your question correctly. If I do not adequately address your problem, please just tell me.
You have given a specific hypothetical above with people and candidate preferences. You do realize that you have given us a hypothetical that can be true with what we are given. Your hypothetical would in fact make B a true statement. But that is simply a could be true situation. Is the hypothetical you posited enough for us to infer that in all situations? Something that is inferrable is something that must be true. Your situation does not have to hold true. It could be true that 3 people prefer A and 2 people prefer B, with all five people having the same concern: child obesity.
From the statements given on page 445, we know only a couple of things:
-More people prefer A to B.
-People prefer A some Budget
-If you prefer B ---> Budget
Do we know for a fact that there is at least one different concern among people that prefer B to A? No. They could all care about the budget.