The p value is the probability of a false positive given that the null hypothesis is true. It is a hard mathematical fact. The p value is the only result that can be realistically calculated with mathematical certainty. The rest is up to the scientist to interpret judiciously. Alternatively, it is very speculative and debatable to try to expand that to a probability in general. I would prefer to start with a hard mathematical fact and interpret it judiciously than to muddy the picture immediately with unsubstantiated and debatable opinion regarding the overall probability of a particular result.