Modern Bayesian approaches succeed by explicitly acknowledging prior choice as a modeling decision, not pretending logic alone determines them—suggesting the framework survives Carnap's setback.

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Bayesian approaches(as used in probability and statistics): A way of making decisions or drawing conclusions by starting with what you already believe (a 'prior'), then updating those beliefs as you get new evidence—named after mathematician Thomas Bayes.
Carnap's setback(as used in philosophy of science and logic): Carnap's failure to solve the problem of how logic alone could determine which starting assumptions are correct—a famous unsolved problem he ran into.
Modeling decision(as used in philosophy of science and statistics): A deliberate choice you make about how to set up a problem or system—acknowledging it's a choice you're making, not an objective fact handed down by logic.
Prior choice(as used in probability and decision-making): The starting assumptions or beliefs you decide to use before looking at new data; in Bayesian thinking, you have to actively choose what to assume at the beginning.

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