Conditionals, probability, and non-triviality

We show that the implicational fragment of intuitionism is the weakest logic with a non-trivial probabilistic semantics which satisfies the thesis that the probabilities of conditionals are conditional probabilities. We also show that several logics between intuitionism and classical logic also admit non-trivial probability functions which satisfy that thesis. On the other hand, we also prove that very weak assumptions concerning negation added to the core probability conditions with the restriction that probabilities of conditionals are conditional probabilities are sufficient to trivialize the semantics.