here are my thoughts ... in 'a if, and only if, b' makes sense because it's both 'a if b ... if b then a' and 'a only if b ... if a then b' ... can someone break 'a if, but only if, b' down for me like that? Thanks!

Understand the difference between the following:A if B.This means that if B occurs then A will occur. But it does not mean that if C occurs then A will not occur, or if D occurs then A will not occur, etc. In other words, it's possible for A to occur even if B does not occur.A if, but only if, B.This means that if B occurs then A will occur. In this situation, though, if C were to occur then A would not occur, and if D were to occur then A would not occur, etc. In other words, it is not possible for A to occur if B does not occur.