If then else
Bare indicative conditionals are bewildering, but they become surprisingly well-behaved if we add an 'else' clause.
Intuitively, 'if A then B' doesn't make an outright claim about the world. It says that B is the case if A is the case – but what if A isn't the case?
An 'else' clause resolves this question. 'If A then B else C' makes an outright claim. It says that either B or C is the case, depending on whether A is the case. That is: the world is either an A-world, in which case it is also a B-world, or it is a ¬A-world, in which case it is a C-world. For short: (A∧B)∨(¬A∧C).