# If A is true then B must also be true

What if in the statement, "If A is true then B must also be true," A & B are related but, "B must also be true," is not necessarily true because other possibilities exist? In other words, A being true means B could be true but not necessarily.

• What you mean by "what if"? – Kristian Berry Mar 14 at 23:48
• The possibility that “A and not-B” is true means that “A therefore B” is false. – Mark Andrews Mar 15 at 0:01
• Maybe you are thinking of something like □A→□B, if A is necessarily true then B is necessarily true. Then if A is true only contingently (in some possible world) B may not be (in the same world), i.e. □(A→B) does not follow. In other words, if A is true by accident B could be true, but not necessarily. "If A is true then B must also be true" is ambiguously phrased, it is unclear where the necessity symbol □ is intended to be, if anywhere at all. – Conifold Mar 15 at 9:51

It sounds like you might be interested in the following topics:

• Defeasible logic allows arguments to defeat other arguments, so that at first we may conclude C, and then later more evidence arises, "defeating" the first argument, and we conclude not C.
• Non-monotonic logic is the category of logic that defeasible logic falls into.
• Argument mapping diagrams informal arguments in which claims lend or detract support from other claims, rather than having a strict deductive implication.

By the end of your question, what you are getting at with "A being true means B could be true but not necessarily" can be formalized as "if A then possibly B":

``````A → ◊B
``````

The logic of "possibly" and "necessarily" is called modal logic. There's a lot of "what-ifs" that follow from that sentence. For example, if follows that, if B is impossible, then A is false:

``````□(¬B) → ¬A
``````