Gerhard Tintner, Foundations of Probability and Statistical Inference (1949)

Gerhard Tintner’s journal article presents Rudolf Carnap’s inductive logic as a foundation for statistical inference independent of the practical consequences of decisions. Its motivating problem comes from econometrics: how might empirical evidence establish the probability of one economic theory relative to another, such as Keynesian and “classical” systems? Tintner offers a tentative route toward answering this question, beginning with competing foundations of probability, developing Carnap’s semantic apparatus, and applying it to prediction, estimation, hypothesis testing, and theoretical choice. His central distinction is between assessing evidential support and deciding what action that support warrants.

The opening survey separates frequency interpretations, set-theoretical probability, and logical accounts relating probability to propositions. Tintner then considers Jeffreys’s Bayesian approach, Fisher’s fiducial inference, Neyman–Pearson testing, and Wald’s decision theory. His objection to Wald is especially revealing: conflicting social ends make a single risk function difficult to construct, while the consequences of accepting a false scientific theory resist straightforward valuation.

What is desired is a “pure” theory of induction which would permit us to evaluate ultimately the degree of confirmation of given statistical hypotheses without any pragmatic considerations, as implied in Wald’s theory.

“Pure” here means independent of utilities or losses, not detached from evidence. Tintner accepts the usefulness of decision theory where gains and losses can be expressed monetarily, particularly in industrial applications. He nevertheless insists that the logical support evidence gives a hypothesis deserves separate treatment. Carnap’s probability₁ supplies that treatment as degree of confirmation; probability₂ concerns empirical frequency. The distinction allows a logical rule to evaluate factual evidence without making the rule itself an empirical assertion.

Section 2 constructs confirmation within a finite language of individuals and binary attributes. Conjunctions of attributes and their negations yield exhaustive Q-properties. State descriptions assign a Q-property to every individual; structure descriptions group states that differ only through exchanges of individuals. This grouping makes property counts, rather than individual identities, fundamental. Coin-toss examples and extensive tables render the construction explicit.

Carnap associates with each of the m structure descriptions the same measure 1/m.

This is the decisive weighting choice. Equal measure is assigned to structures and then divided equally among their constituent isomorphic states. It differs from treating every complete sequence of coin-toss outcomes as equally probable: homogeneous sequences receive more measure than individual mixed sequences. Tintner shows that, in the binary example, the resulting structure measures can also be obtained by integrating binomial probabilities over a uniform prior. The formal resemblance to Bayesian reasoning matters, although he grounds the measure in logical structure. Sentence measures sum the measures of compatible states, and confirmation is the conditional ratio (c(h,e)=m(e\cdot h)/m(e)).

A further conceptual move introduces “logical width”: the number of Q-properties included in a property. Section 3 distinguishes direct inference from population to sample, predictive inference from an observed to a future sample, and inverse inference from sample to population. Direct inference reproduces the familiar hypergeometric distribution. Prediction and inverse inference retain the language’s logical widths alongside observed counts.

Hence predictive inference, in distinction to direct inference, depends upon the semantical properties of the language.

For one future individual, the confirmation assigned to possessing a property is ((s_1+w_1)/(s+k)), combining the observed count with logical width. Without observations, relative logical width supplies the prior probability. Tintner argues that when the sample is sufficiently large relative to the language’s complexity, the influence of this semantic contribution diminishes and familiar frequency-based results emerge. Sections 4 and 5 extend the construction to attributes with several possible values and to exhaustive classifications into multiple properties.

Section 6 translates confirmation distributions into tentative procedures: maximum-confirmation, mean, and median estimates, together with confirmation intervals. Numerical examples show how the resulting estimates differ. Tintner draws analogies with maximum likelihood, fiducial limits, and confidence limits, but his organizing object remains the confirmation distribution conditional on evidence.

Sections 7–9 distinguish four objects of confirmation for a law asserting that every individual with one property possesses another. Universal inference concerns the whole population and loses all confirmation upon a counterexample. Modified universal inference concerns conformity throughout the unobserved remainder despite observed exceptions. Instance confirmation concerns one new individual, while qualified instance confirmation adds knowledge that this individual possesses the antecedent property. The distinction prevents the probability of avoiding a counterexample from being confused with the probability of the consequent given the antecedent.

Tintner applies these forms through an arbitrarily selected confirmation threshold and through comparative ranking:

We will choose the hypothesis with the highest degree of confirmation.

His rival coin-toss “laws” illustrate that the same evidence can favor one hypothesis under all four forms while producing different numerical degrees of support. Comparative preference therefore need not amount to acceptance at the selected threshold.

The concluding limitation governs the article’s relevance: the theory handles categorical attributes, not continuous variables, and cannot yet address the Fisher–Behrens problem or the quantitative languages central to economics and physics. Tintner’s achievement is consequently a worked demonstration of how logical confirmation could organize statistical inference within a restricted domain. Practical considerations remain important, but he seeks to distinguish them from the evidential relations on which scientific assessment rests.