The Stanford article on inductive logic that Mauro referenced is a good, though lengthy, account; the short version is that the Bayesian approach to statistical problems requires specifying priors. These priors are not determined by the experimental evidence, but represent the state of play prior to the evidence being taken into account. At the time Fisher was working, there was no satisfactory answer to the question, where do these priors come from? Are they are just arbitrary assumptions? Fisher hoped that some kind of logic of induction could be derived that would avoid such assumptions. Such a logic would provide objective answers to questions of the form, how much evidential support does this proposition provide for that proposition?
The task was taken up by John Maynard Keynes and Rudolf Carnap, among others, who tried to derive a purely syntactic set of rules for a probabilistic theory of induction, incorporating the principle of indifference. Ultimately most theorists regard this project as a failure (although there are still defenders of the logical interpretation of probability). As Goodman showed with his "grue" example, there are no purely syntactic criteria that determine whether some hypothesis is projectible from known to unknown cases: it depends on what the terms mean and, on at least one popular account, whether they correspond to natural kinds. Also, as Duhem and Quine pointed out, hypotheses are underdetermined by data anyway, so there cannot be a purely objective way of saying how probable a hypothesis is given a set of data.
The result is that there is no general consensus on the best way to perform statistical inference. Classical frequentists use methods based on the work of Fisher, and Neyman and Pearson. Bayesians continue to use their methods, but have updated them to try to solve the problems with finding objective priors. Likelihoodists use a kind of Bayesian updating, but forego finding priors in favour of assessing the relative merits of competing hypotheses.