Gerhard Tintner, La théorie probabiliste de Carnap et son application aux problèmes de l’économétrie (1957)

Published in Économie appliquée, Tintner’s short theoretical article examines what Carnap’s logical account of probability can contribute to econometrics. Its argument proceeds from a distinction between two meanings of probability, through their respective economic uses, to a sketch of Carnap’s formal machinery and classification of statistical inference. The central claim is prospective: although incomplete, inductive probability offers a framework for clarifying the evidential standing of economic theories and the disputed foundations of statistical methods.

L'idée fondamentale de la théorie de Carnap est la distinction entre deux conceptions de la probabilité ; la première est le degré de confirmation (ou probabilité inductive) : une idée purement logique qui exprime la probabilité des théories ou des hypothèses ; la deuxième idée est la probabilité comme fréquence empirique ou statistique, la conception usuelle des mathématiciens et des probabilistes.

English translation: The fundamental idea of Carnap’s theory is the distinction between two conceptions of probability; the first is the degree of confirmation (or inductive probability): a purely logical idea that expresses the probability of theories or hypotheses; the second idea is probability as empirical or statistical frequency, the usual conception among mathematicians and probability theorists.

This distinction matters because econometrics performs two different evidential tasks. It applies mathematical statistics to economic observations both to give numerical content to mathematical models and to test their adequacy. Determining the probability of an economic theory—Tintner’s example is Keynes’s theory—calls for inductive confirmation. Estimating the numerical frequency of economic phenomena from samples calls for statistical probability. The article thus separates the assessment of hypotheses from the measurement of frequencies without treating either as dispensable.

Tintner immediately qualifies the applicability of Carnap’s framework. The language available to it does not yet accommodate a central feature of mathematical economics:

Les modèles classiques de l'économie mathématique utilisent partout l'idée des variables de variation continue et la théorie de Carnap ne peut pas encore traiter ces problèmes.

English translation: The classical models of mathematical economics everywhere use the idea of continuously varying variables, and Carnap’s theory cannot yet address these problems.

The proposed bridge to economics is consequently selective. Carnap’s recent treatment of ordered magnitudes may illuminate the utility theories of Arrow and Stone, based on collections of goods and services; Tintner also recalls Arrow’s application to welfare economics. These are possibilities for analysis, not worked demonstrations. The limitation prevents the article’s enthusiasm from becoming a claim that Carnap already supplies a general foundation for continuous-variable econometric models.

A second motivation is disagreement within mathematical statistics. Tintner places Fisher’s foundational contributions alongside the criticisms of Neyman and Pearson, then identifies Wald’s decision-function theory as a purely pragmatic account of statistical method. Carnap’s research on estimation promises conceptual clarification across these debates. Tintner does not establish the superiority of one school; he suggests that a logical account of confirmation can help explain what their procedures mean.

The formal core represents empirical evidence by a sentence (e), a hypothesis by a sentence (h), and their conjunction by (e.h), within the language of the lower functional calculus. If (m(i)) measures a sentence in that language, the degree of confirmation is (c=m(e.h)/m(e)). Confirmation is therefore defined through the relation between explicitly formulated evidence and hypothesis. Yet the formula leaves a substantive foundational choice open:

Le problème de l'assignation des mesures n'est pas encore complètement résolu dans la théorie de Carnap.

English translation: The problem of assigning measures has not yet been completely resolved in Carnap’s theory.

Carnap provisionally gives equal probability to a class of sentences called “structure descriptions.” His subsequent work on estimation allows other measures, potentially preferable for particular statistical problems. Tintner therefore presents inductive probability as a developing framework with alternative specifications, rather than a procedure whose numerical conclusions follow uniquely from observations.

The classification of inference makes the consequences of those choices visible. Direct inference draws conclusions about samples from a known population and agrees with classical results. Predictive inference moves from a known sample to future samples, with results influenced by the theory’s logical conception. Inverse inference draws conclusions about an unknown population from a sample. Concerning this last case, Tintner emphasizes:

Encore peut-on constater que les résultats dépendent de la nature du langage et de la mesure adoptée.

English translation: Here again, one can observe that the results depend on the nature of the language and the measure adopted.

This dependence qualifies the earlier promise of clarification: the formulation of hypotheses and the chosen measure participate in determining the inference. Tintner then maps familiar statistical operations onto confirmation theory. Maximum confirmation corresponds to maximum likelihood; estimation by a confirmation-based mean is analogous to mathematical expectation; limits of confirmation approach Fisher’s fiducial limits and Neyman–Pearson confidence limits. His language marks correspondences and analogies, not demonstrated identities.

The final section distinguishes the scope of claims made in hypothesis testing. Universal inference concerns a law’s validity throughout a population. Modified universal inference requires validity for members outside the observed sample, but not necessarily within it. Instance confirmation instead concerns an individual outside the sample, avoiding the need to assert a law for a potentially infinite population; Tintner also names a qualified version without developing it. The article’s contribution lies in this conceptual differentiation: it connects econometric questions to distinct kinds of evidence, inference, and generalization while retaining the unresolved choices and technical limits of Carnap’s programme.