Confirmation & the meaning of meaning

Inductive logic, run as arithmetic

Carnap's Logical Foundations of Probability (1950) fixes a language by its atomic sentences; a state-description is one complete assignment of truth-values to those atoms, and a proposition's range is the set of state-descriptions in which it holds. Every notion below is a fact about ranges: confirmation c(h, e) = m(h & e) / m(e), analyticity is range-equals-all, and the verifiability criterion sorts a sentence by whether it has confirmation conditions at all. Distilled from the Chapter 9 book pass and the c-function test suite.

Confirmation is a number you can compute

Propositions are written as func(sd) predicates over a declared vocabulary. A hypothesis confirms itself completely (c(h, h) = 1); given p, the confirmation of p & q is exactly 0.5 — half the p-worlds also have q. That is a Bayesian update, run to the digit.

The lambda-continuum: does the method learn?

The deepest move in The Continuum of Inductive Methods (1952): one parameter tunes how much evidence moves belief. The logical measure m-dagger weighs every state-description equally, so evidence is inert — two positives leave the third at the prior 0.5. Carnap's preferred c* (lambda = 2) instead learns, raising it to 0.75; as lambda climbs it slides back toward m-dagger.

Relevance: evidence that confirms, disconfirms, or does neither

Carnap's relevance is the shift c(h, e) − c(h): positive means e supports h, negative means it undercuts it, zero means it is idle. Because p and q are independent atoms, learning q tells you nothing about p — relevance is exactly 0. This is the graded successor to a yes/no verification.

Analyticity is a fact about range

In Meaning and Necessity analyticity is not a mystery — it is range-equals-all. A tautology holds in every state-description (l_true), a contradiction in none (l_false), and deductive entailment is range-inclusion (l_implies). Confirmation theory subsumes deduction as its limit case: when e entails h, c(h, e) = 1.

The Scheinsatz test: meaningless, not false

The verifiability criterion sorts every sentence into three bins: analytic (L_true, confirmation 1.0), synthetic (meaningful, confirmation 0.5), or — with no verification conditions in either bin — a Scheinsatz, Carnap's pseudo-statement. "The Absolute is incapable of evolution" is not false; it makes no claim that could be true or false. diagnose runs the classifier.