Can "If A then B may be" be proven or disproven?

Is there a way to prove or disprove logically a statement of the form:

If A is true then B may be true.

Here is my attempt: Suppose A is true. Then, suppose I prove that B is also true. Then the statement is true. Suppose instead that I prove that B is impossible. Then A is false.

EDIT: Thanks a lot for all your responses, I learned something from every response. This is a much more complex question than I thought it was.

• Possibility (may, ◇) and necessity (must, □) are operators of modal logic interpreted by using multiple "possible worlds". Whether something is true will depend on a world you are in, so generally you may not be able to prove either that B is true or that it is false. To prove "If A then possibly B" (A→ ◇B) you have to find a world where A and B are both true OR prove that A is true in no world, to prove "If A then necessarily B" (A→ □B) you have to prove that in every world where A is true so is B. May 27, 2018 at 0:00
• Thank you for your comment. I didn't know about modal logic, so I appreciate your links and explanation. I am somewhat hazy about the difference between "If A then possibly B" (your phrasing), "If A then sometimes B" (Mark Andrews' phrasing), "If A then B may be true" (my phrasing), and "If A then B may not be true" (another phrasing). Is it just hair splitting, or are there logical differences between these statements? May 27, 2018 at 3:10
• "Sometimes" and "may" are formalized as "possibly" in modal logic, more or less. May 29, 2018 at 15:12

If A is true then B may be true.

The form is: "If A is true, then B is sometimes true". I take that to be the premise.

If it is shown that A is true, then it is indeed shown that B is sometimes true.

If it is shown that B is never true, then it is shown that A is false.

To clarify: Would you also interpret "If A is true, then B may not be true" as "If A is true, then B is sometimes true"?

Not necessarily. The statements "Some B are true" and "Some B are not true" are subcontraries. Both may be true at the same time but both may not be false at the same time. They are not equivalents.

• Thank you. This gets very close to the answer I was looking for. To clarify: Would you also interpret "If A is true, then B may not be true" as "If A is true, then B is sometimes true"? May 27, 2018 at 2:59

The logic of conditionals is considerably more complex than most elementary accounts provide for, and the literature on them is vast: it runs to scores of books and thousands of papers. There are several issues with your sentence that would require clarification.

1. Does the "may be" indicate an epistemic possibility? I.e. is it saying B may actually be true for all we know, but we're not certain? For example, "If Alice is not in her office, she may have gone home." If so, then the nearest formalism for the conditional is likely a statement of conditional probability such as "P(B | A) > 0" - where probability is interpreted in the Bayesian fashion as a degree of uncertain belief. I suspect that usually when we say "if A is true then B may be true" we mean that B has some significant degree of probability, and is not a mere possibility. So, the zero could be replaced by a larger, possibly non-specific number. This formalism has the major advantage over other formalisms of conditionals that it does a much better of job of capturing what is means for a conditional to be uncertain. "B | A" is not a truth function of A and B, and contrary to what some elementary logic texts suggest, most conditionals in ordinary English are not truth functions. Disproving a conditional of this kind would involve showing that it is in fact highly unlikely that B on the supposition A, or in simple cases, showing that A is true and B false.

2. Does the conditional express a relationship? "If Alice is a smoker, she may get lung cancer" would normally be understood to mean that we think there is some causal connection between smoking and lung cancer. Proving or disproving causal connections is notoriously difficult and typically involves being able to demonstrate the mechanism. A relation may also be evidential rather than causal.

3. Does the conditional express a simple logical possibility? "If Manchester United win their match on Saturday, they may become league champions", seems to express nothing more than that given the rules of the league and the current standing of the teams, a win will put the team in reach of the championship. This might be disproved if you could demonstrate that even with a win, the arithmetic shows they cannot catch the current leaders.

4. Does the conditional express a habit or rule? Sometimes, conditionals are implicitly quantified. For example, "If it snows, the flight may be cancelled" suggests that whenever it snows sometimes the flight is cancelled and sometimes not. Disproving this would be difficult unless you could show that in fact flights are never cancelled during snow.

5. Does the conditional express a physically indeterminate possibility? "If a photon hits this half-silvered mirror it may be reflected." Here we might want to say that even if the photon were not actually reflected, it might have been, i.e. there is (or was) a counterfactual possibility of it happening. Disproving this would require disproving the scientific theory behind the claim.

I'm not sure what your example is trying to show. The contrapositive of "if A is true then B may be true" is "if B is impossible then A is false". This is not in general always true or always false.

• Thanks for making clear how complex interpreting a (somewhat pathological) statement really is, and for correcting my attempt at the contrapositive! May 27, 2018 at 13:47

The problem is that "If A is true then B may be true." is not a logical statement, it is a statement in everyday language, and we must first figure out exactly what it means.

Joe says he saw James robbing the Louisville bank. If Joe never was in Louisville then Joe is lying, but if Joe was indeed in Louisville around the time of the robbery, then his statement may be true. It also may be false. So in this situation, the everyday language statement "If A is true then B may be true." actually means "If A is false then B is false".

• Right, I think this is what I was trying to get at, but you expressed it much better than I could: How to translate my statement into a logical statement. In my understanding you suggest one way to interpret my statement. Mark Andrew's interpretation is a bit closer to what I was looking for though. May 27, 2018 at 3:13
• @gnasher729 So we could say that "If A then B may" means "If A then there is a probability that B" which according to law of contraposition is equivalent to "If there is no probability that B (not B) then not A". In other words "If not B then not A". Jul 23, 2020 at 21:57