Inductive Logic and the Carnap Programme

Deductive logic settles whether a conclusion follows from premises by their form alone, with no appeal to what the sentences are about. The programme this article describes was an attempt to do the same for support: to define a function taking any evidence sentence and any hypothesis to a number measuring how far the first confirms the second, determined by syntax and nothing else.

It was pursued with more rigour than any comparable project in twentieth-century philosophy, and it failed in a way that taught more than a success would have.

Logical probability

The setting is a first-order language with finitely many predicates and countably many individual constants. A state description says, for every predicate and every individual, whether it applies. A structure description is an equivalence class of state descriptions under permutation of individuals, collecting those that agree on how many individuals have each combination of properties while differing on which.

Assigning probabilities to state descriptions fixes everything else, and the choice is where the content lies. Carnap's first candidate divided the probability equally among state descriptions, and he rejected it because the resulting function learns nothing from experience: observing a thousand ravens leaves the probability that the next is black exactly where it started. Equal weighting of possibilities is the principle of indifference applied at the wrong level, and it makes induction impossible rather than justified.

His second candidate, the function he called $c^{*}$, divides the probability equally among structure descriptions and then equally within each. Since a structure description with many individuals sharing a property contains fewer state descriptions than one with the properties spread out, this weighting favours uniformity, and the resulting function learns. The move is the essential one, and it is an admission: some inductive assumption must be built in, since no purely formal constraint delivers learning from experience.

The continuum

At $\lambda = 0$ the function is the straight rule, predicting the observed frequency exactly and learning as fast as possible. As $\lambda$ tends to infinity the function ignores the evidence entirely. Carnap's $c^{*}$ is the case $\lambda = k$. Every member of the family is a legitimate confirmation function by the formal criteria, and nothing in logic selects one.

That is the programme's central difficulty, stated in its own terms. The parameter is a free choice representing how cautious an inductive policy one adopts, and a logic whose central notion depends on a free parameter has not made confirmation a matter of form.

Exchangeability is the assumption underlying the whole continuum, and de Finetti's theorem is the reason the programme connects to Bayesian statistics rather than standing apart from it. Johnson had proved in the 1920s that exchangeability together with a sufficiency condition forces the continuum's formula, a result Carnap rediscovered. So the family is not arbitrary; it is the unique family satisfying two intelligible conditions. What remains arbitrary is the parameter.

Universal laws

The decisive difficulty is not the parameter.

The consequence is fatal to the programme as a theory of scientific confirmation. Newton's laws, conservation principles and every universal claim in science receive confirmation zero from any finite evidence and continue to do so forever. A theory of confirmation on which no amount of evidence raises the probability of a law above zero is not a theory of what scientists do.

Carnap's response was to redirect the programme toward instance confirmation, the probability that the next observed case will conform, which is well behaved and which he argued is what matters in practice. The move is defensible for prediction and it abandons the original target, since a scientific theory is asserted as a universal claim and not merely used to predict the next case.

Hintikka's systems are the constructive answer, and they cost something. They require a prior distributed over constituents, which are descriptions of the world's qualitative structure, and the distribution is a further free choice on top of the continuum's parameter. The problem of selecting a function on formal grounds is not solved but made larger, which is the honest summary of where the programme stands.

Next

The remaining articles return to the core. The first takes up a presupposition present in every quantified system so far, namely that the terms of the language denote, and examines what happens when they do not.