Logical Foundations of Probability is Carnap's most sustained and technically elaborate work — a comprehensive attempt to construct an inductive logic that would give a rigorous account of the degree of confirmation that empirical evidence confers on a hypothesis. The project arises from a fundamental tension in the Vienna Circle programme: logical empiricism holds that all meaningful statements are either logically true (analytic) or empirically confirmable, but the concept of confirmation itself — how much evidence raises the probability of a hypothesis — had never received a satisfactory formal treatment. Carnap distinguishes sharply between frequency probability (the statistical concept used in physics and statistics) and logical probability (a purely logical relation between a hypothesis and an evidence statement that measures the degree of rational belief the evidence warrants). He constructs a family of confirmation functions (c-functions) characterised by different values of the parameter λ, which governs the balance between the evidence and prior information. The limiting cases are c* (which gives maximum weight to the evidence) and the straightforward inductive method (which ignores prior probabilities entirely). The work is important both technically — it is the most rigorous realisation of Carnap's lifelong programme of inductive logic — and philosophically, as a sustained argument that there is a genuine logical concept of probability that is neither subjective (a degree of personal belief) nor purely frequentist (a property of long-run sequences), but an objective logical relation grounded in the structure of possible worlds.
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