Logic is used by mathematicians and scientists at different levels. At the most elementary level, we have the "propositional calculus" which represents the standard logic applied by practically all mathematicians and scientists in the bulk of their research. At an intermediate level lie the induction principle and the diagonal argument of Cantor. A much more sophisticated example is the "non-standard analysis" used by some mathematicians, although shied away by most. These examples show that logic is applied in a significant way in mathematics and, therefore, in science.
I am not aware of any applications of the formal methods developed by the logical school of philosophy of probability in statistics or other sciences. It seems that a lot of intellectual effort was spent on a totally useless theory. If the logical theory of probability developed any applicable methods (unknown to me), it is a pity that no mathematicians or scientists ever noticed them.
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