This volume gathers Rudolf Carnap’s final work on inductive logic, assembled with Richard C. Jeffrey and a small circle of collaborators and published as the first of a projected series. It is a technical book of philosophy—formal, axiomatic, and unfinished by design—written in the United States during the 1950s and 1960s, and it carries the marks of a research program still under construction when Carnap died in September 1970. The contents run from a substantial editorial introduction and postscript through Carnap’s two articles, “Inductive Logic and Rational Decisions” and “A Basic System of Inductive Logic, Part I,” alongside Jeffrey’s paper on probability measures and integrals and two contributions by Jürgen Humburg and Haim Gaifman on the principle of instantial relevance.
The book is not a treatise with a settled conclusion but a report from the workshop. Jeffrey’s introduction explains that Carnap’s Logical Foundations of Probability had been planned as the first half of a two-volume Probability and Induction, and that the second volume kept receding as its author’s ideas changed. The method once presented as the chosen system, the function c*, became one point in a continuum of inductive methods; then the continuum itself looked too narrow, because none of its members responded adequately to analogy by similarity. A new axiom system followed, and it too came to seem too narrow. What was meant to be a single definitive volume thus turned into a nonperiodical journal of work in progress, and the present book is its first issue.
From Credence to Confirmation
The Volume and Its Editor’s Introduction
The book opens as the first volume of Studies in Inductive Logic and Probability, edited by Rudolf Carnap and Richard C. Jeffrey. Carnap supplies two articles, “Inductive Logic and Rational Decisions” and “A Basic System of Inductive Logic, Part I”; Jeffrey contributes a piece on probability measures and integrals, and Jürgen Humburg and Haim Gaifman add papers on the principle of instantial relevance. Jeffrey’s introduction, written with Carnap’s approval, traces how the project outgrew its origins. Carnap’s Logical Foundations of Probability (1950) had been planned as the first of two volumes, with the second devoted to the c* system; The Continuum of Inductive Methods then showed c* to be merely one point in a continuum, and work at the Institute for Advanced Study with John Kemeny convinced Carnap that even the continuum was too narrow, since none of its functions responds adequately to analogy by similarity. A new axiom system in 1959 proved too narrow in turn. The projected Volume II receded by a decade, and the journal-like Studies was conceived as a surrogate reporting work in progress. It appears too late to serve that purpose, Jeffrey concedes, but the essential part survives in Carnap’s detailed presentation of his basic system—a foundation, not a finished structure.
Carnap’s Postscript and Working Notes
A postscript dated February 1971 records Carnap’s death the previous September, after he had revised all but the last two sections of the Basic System’s continuation; Lary Kuhns and Gordon Matthews took over the revision. Carnap’s own notes, appended to the introduction, recall the Princeton years. With Kemeny he extended his treatment of families of attributes to any finite number, studied de Finetti’s representation theorem, and heard Savage’s warning that a symmetric function in a finite language cannot always be extended to larger languages. Bar-Hillel’s April 1953 visit produced the semantical concept of information content, contrasted with Shannon’s statistical concept—only the latter, Carnap insists, connects with physical entropy. A conversation with von Neumann, who held quantum-theoretical concepts including entropy to be subjective, ended in friendly disagreement.
Coherence, Strict Coherence, and the Rule of Conditionalization
Rationality, the argument continues, demands that a credence function be coherent; de Finetti proved that coherence holds exactly when the function is a probability measure. Shimony’s stricter notion follows: a function is strictly coherent when no finite system of bets on molecular propositions can yield a net loss in some case and no gain in any. Requirement R2 adopts strict coherence, and a theorem—half Shimony’s, half Kemeny’s and Lehman’s—identifies strict coherence with regularity. Such axiom systems remain weak, delivering no numerical conditional probability except in trivial cases of 0 or 1. Requirement R3 governs how credence changes with experience: the new function depends only on the observational result E, and specifically equals the old conditional credence given E. The author notes that this idealizes away hope and fear, and that repeated conditionalization on successive data matches one-step conditionalization on their conjunction. Since the aim is a logic of induction rather than the psychology of actual reasoners, he assumes infallible memory, likening his subject to a robot rather than a fallible human. An initial credence function Cr₀, fixed before any data, determines every later function through conditioning on the accumulated knowledge Kₙ.
Inductive Logic Against Hume
Carnap aligns himself with Keynes and Jeffreys in holding that inductive logic grounds inductive reasoning, but breaks with those who treat induction as inferring the acceptance of a hypothesis. Hume’s objection—that a hundred cases of rain after a certain weather never warrant expecting rain tomorrow—cannot be answered on that reading. The proper result of inductive reasoning is instead a probability: from the total evidence Kₙ, the value 𝔠(H | Kₙ) = 0.99, and the agent’s credence matches it. This conclusion is analytic, resting on the definition of 𝔠 and on axioms each backed by a rationality requirement, yet it guides action because the factual premise supplies the evidence; credence alone fixes the rational value of each act, so the agent bets on rain at ninety-nine to one but no further. Hume’s critique of customary induction stands, and common sense is vindicated by reformulation.
The Basic System: Propositions, Models, and Axioms
The second article recasts inductive logic in set-theoretic terms. Probabilities attach to events or propositions, represented extensionally as sets of models, with complement, intersection, and union replacing negation, conjunction, and disjunction. Carnap lays out the notation of fields and σ-fields, partitions, and generated fields, then defines necessary, impossible, contingent, and inclusion. The 𝔠-function matters more than 𝔪, since its values represent rational degrees of belief.
Models as Functions and the Possibility Space
Carnap continues building the formal machinery of inductive logic by treating each model component Z as a function analogous to the earlier *Z, though Z may take values differing from *Z. These functions are usually handled as purely mathematical objects, yet no language confined to finite expressions can define them all. Since the actual case is one among the possible cases, *Z itself belongs to the set Z. A parenthetical aside notes that in a language of logical modalities, distinguishing extensional identity “=” from intensional identity “≡”, one could say more precisely that some mathematical function Z1 in Z satisfies *Z = Z1 without *Z1 ≡ Z; because Z was defined descriptively, it is not a mathematical function in the strict logical sense. In the nonmodal extensional language used here—essentially set theory—there is only one identity sign, so the sentence “Z1 = Z” is simply asserted, and hence *Z1 ∈ Z; but this identity is factually true rather than mathematically valid, like “The number of planets = 9.”
The set Z of all models of the language L, representing total possibilities, is then defined. Rather than a sequence {Z, Z, …}, a model is preferably represented by a single two-place function Z(m, i): definition D3-2 characterizes Z as the set of all two-place functions Z such that there is a sequence {Z1, Z2, …} with each Z in Z and Z(m, i) = Z(i) for every family index m. This set of models is also called the possibility space, and its elements are sometimes called points of the space.
Why Inductive Logic Fixes the Domain
Abstract mathematical theories customarily use a different form of model, specifying a domain of individuals whose cardinal number may vary from model to model—finite, denumerable, noncountable—as suits uninterpreted axiom systems like group theory and topology. Inductive logic, by contrast, finds it more useful to specify once for all, for each investigation, the domain of individuals and the families of qualitative attributes, holding these fixed across all models. Mathematical statistics works the same way: each investigation refers to a specified population, such as the inhabitants of Los Angeles on January 1, 1966, or the light bulbs produced by a given factory beginning tomorrow, with the characteristics to be observed specified in advance, so every possibility involves the same characteristics.
A clear distinction must be drawn between models and interpretations, often disregarded even by good authors. For any investigation a language L is assumed specified with a fixed interpretation the same for all models: in example (2-1), ‘P1’ means the color Red throughout. An interpretation assigns a meaning, an intension, to a predicate; models assign merely various possible extensions—sets of individuals—without changing that meaning.
Propositions and Truth Sets
The functions φ and .φ will be applied not to sentences in L but to propositions on L. Every sentence corresponds to a proposition, but not conversely; most propositions on L cannot be expressed in L. Speaking about sentences is therefore unnecessary, though sometimes useful for those more at home with logical language systems than with the set-theoretic apparatus of models and propositions.
Since every model represents a totally specific possible case, for any model Z in Z and any sentence S in L, either S is satisfied in Z or its negation is; when satisfied, S is “true in Z.” This truth relative to a model, unlike absolute truth, is purely logical: deciding whether S is true in Z requires only logical analysis of S and Z, no factual information. For any sentence S, the set of models in which S is true is called its truth set, Tr(S). A recursive definition, D3-3, follows: for an atomic sentence with the ith individual constant and the jth predicate of family F, Tr(S) = {Z : Z ∈ Z and Z(m, i) = j}; Tr(~S) = —Tr(S); Tr(S ∨ S′) = Tr(S) ∪ Tr(S′); Tr(S · S′) = Tr(S) ∩ Tr(S′); an identity sentence with two occurrences of the same constant has Tr(S) = Z, with two distinct constants Tr(S) = ∅; and a universally quantified sentence has as its truth set the intersection of the truth sets of its substitution instances, with the existential case following as a union. These rules determine Tr(S) for any sentence in L, and Tr(S) is analogous to the “range” of S in the earlier state-description framework, with theorems carrying over accordingly. Two sentences share a truth set exactly when they are logically equivalent.
Pure and Applied Inductive Logic
The relation between pure and applied inductive logic mirrors that between mathematical and physical geometry—a distinction credited with dissolving last century’s controversies about the nature of geometry and serving as a paradigm for pure versus empirical theories generally.
Extending Probability Functions and the Weaker Notion of Semiregularity
Carnap begins by asking how far a probability function can be pushed beyond the class on which it is first defined. If a function is specified only on a proper subclass of the molecular propositions and already satisfies the basic axioms, a theorem of R. Sherman Lehman (1955) guarantees it can be extended to the whole class while keeping those axioms intact; where the function is defined on the full class, a limit procedure grounded in σ-additivity yields a unique extension. Some writers, however, work with functions that abandon regularity in special cases, so Carnap introduces a weaker property he calls semiregularity, absent from his earlier publications and reserved for the treatment of representative functions and the λ-functions of the second volume. A function is semiregular when, for any finite family of attributes and any sample proposition, the probability of a new atomic proposition remains positive provided the attribute in question has actually been observed. Regular functions satisfy this automatically. The point of the weaker notion is that a semiregular function still behaves sensibly where a merely nonregular one would assign zero probability to an attribute already seen in the sample. The leading example is the straight rule, the function that sets the probability of the next case equal to the observed relative frequency, which corresponds to the parameter λ = 0.
Coherence, Decisions, and the Betting Argument
Pure inductive logic states axioms and derives theorems; applied inductive logic puts those theorems to work, judging the credibility of hypotheses and choosing rational decisions. Justifying an axiom belongs to neither, since it concerns the use of probability functions across all possible situations rather than the actual one. Carnap sets out a simplified decision schema: an agent with total observational knowledge at a given time faces a class of possible acts, a class of outcomes, and a class of states of nature he regards as possible and relevant, all drawn from the domain on which his probability function is defined. He knows which outcome each act produces under each state, knows his own quantitative utility function over outcomes, and can compute both the probability of each state given his knowledge and the contemplated act and the expected utility of each act. Bayes’ principle then directs him to choose an act whose expected utility is maximal. For a wager this yields the rule that it is rational to bet on a prediction only when the betting quotient does not exceed its credibility; the rule is approximate, holding only when the possible loss is small enough relative to the agent’s fortune that utility may be treated as linear in money. Arguing for a particular probability function directly is hard, so Carnap prefers to argue for the axioms: showing that a violated axiom leads, in some possible situation, to an irrational decision would give a strong reason for it. Arguments of this betting kind for the first five axioms go back to Ramsey and de Finetti.
Individuals, Permutations, and the Axiom of Symmetry
The ninth section turns to individuals. The language’s individuals are countable, each is named by a constant, and distinct constants name distinct individuals. Carnap then builds the machinery of permutations: displacement of an index, finite and bounded permutations, the induced mappings on models and on propositions, and isomorphism of models and of classes of propositions. A probability function is symmetric when it is invariant under every finite permutation of individuals, and a theorem carries symmetry back and forth between the credibility function and the measure function. Axiom A6 demands this symmetry, and Carnap defends it: if one sample proposition is isomorphic to another, it says exactly the same thing about possibly different individuals, so giving them different measure values would be arbitrary. Symmetry is required of credibility, not of credence, since a rational credence function may well rank a proposition about Mr. Smith above the analogous one about Mr. Brown; authors who read probability as credence are therefore right to restrict symmetry to special cases. The comparison with deductive logic is instructive, since invariance under permutation of individuals is normally taken for granted there. Coordinate languages, whose constants denote positions in a spatial or temporal series, are another matter, because distance may be logically determined and a permutation could alter the logical situation; symmetry will not be adopted for them. Carnap illustrates with die throws, where experience shows statistical independence, against meteorological observations at a fixed place, where rain following rain is far more frequent than rain overall. Accepting symmetry may rest on genuine independence, on neglecting small dependencies, or on the assumption that the real order lives in an unknown language related to the present one by a permutation of bounded displacement, which disturbs temporal order only locally and leaves limiting relative frequencies unchanged.
Representative Functions: MI, MS, and Their Sequences
The eleventh section develops representative functions for symmetric credibility functions. For a symmetric function, the measure of a sample proposition depends only on the tuple of counts, and this dependence is the representative function MI, which satisfies a normalization, a sum condition, and, for regular functions, positivity.
Instantial Relevance and Its Formulations
Carnap turns to a principle he takes to be central to ordinary inductive reasoning. Suppose evidence E_s is a state-description with k-tuple s, and consider two predictions about the next two individuals, H_j and H_j′. Customary reasoning holds that observing one instance of an attribute raises the probability of another instance of the same attribute. This is the principle of instantial relevance, or the positive relevance of one instance for another. Carnap splits it into a weak and a strong clause: the probability must not decrease, and it must actually change whenever the prior probability lies strictly between zero and one. With regularity the conditional clause can be dropped; with mere semiregularity it cannot, since a probability of one would force the second value to one as well. Carnap prefers to state semiregularity as an axiom while keeping the conditional formulation, so that the principle remains acceptable even to those who admit nonregular functions.
Two doubtful functions come in for scrutiny. The first, from the extended λ-system with λ = 0 and based on the straight rule, assigns probability one to the next instance regardless of evidence; regularity excludes it, but the conditional clause of the principle lets it slip through. The second, with λ = ∞, is regular yet makes all values for a given attribute identical across samples, so the principle excludes it. Carnap finds its inadmissibility obvious, yet Peirce, Keynes, and Wittgenstein all accepted it. He stresses that the prior evidence must not concern the individuals in the predictions; if it does, negative relevance can arise. Alternative formulations in terms of C and of the representative MI-function follow, with recursive notation for repeated instances.
Completing the Measure and Lebesgue Integration
The text then moves to a technical limitation: the class of measurable sets produced by the extension procedure need not be a σ-field, since countable unions may escape it. The fix is to declare countable unions of measurable sets measurable and assign them measure ∞ where necessary. The new class is the completion of the generated σ-field: a set belongs to it exactly when it differs from a set in that field by a set of measure zero. Lebesgue measure on Rⁿ is the leading example, built from products of intervals with volume as the measure; on the line it is ordinary length. When the whole space has finite measure the two senses of measurability coincide, and normalizing to one yields a probability space. Distribution functions on the line then determine Lebesgue–Stieltjes measures, with the familiar special cases: mass concentrated at a point, the discrete case (the St. Petersburg experiment among them), the uniform distribution, and mixed cases whose graphs must be continuous on the right.
Expectation, Product Measures, and Mixtures
Expectation is defined as the integral against a normalized measure, with the expected value of a bounded function lying between its extremes and linearity holding as usual. Integration in abstract spaces generalizes Lebesgue integration in Rⁿ; Riemann integrability coincides with Lebesgue integrability, continuity almost everywhere being the necessary and sufficient condition. Product measures multiply, and Fubini’s theorem reduces integration over a product to successive integrations under σ-finiteness. The extension theorem carries a countably additive measure from a field to its generated σ-field uniquely. Symmetric classes of events assign equal probabilities to intersections of distinct members; the Bernoullian case gives the nth power of the first. A measure built to learn from experience is contrasted with one that ignores evidence, and mixtures of symmetric measures remain symmetric though mixtures of Bernoullian measures generally are not. Stationarity is a special case of symmetry, preserved under mixing, and the strong law of large numbers is stated.
Indefinite Integrals, Conditioning, and de Finetti
Indefinite integrals are countably additive and absolutely continuous, and the Radon–Nikodym theorem supplies the converse; Lebesgue decomposition splits a measure into singular and absolutely continuous parts, illustrated by a bent coin. Conditional probability becomes a random variable relative to a sub-σ-field, following Kolmogorov, Doob, and Billingsley. Billingsley’s betting justification yields the defining integral condition, and existence follows from Radon–Nikodym, with many versions differing on null sets. The simplest case is a finite partition; at one extreme the trivial field returns the unconditional probability, at the other the full field returns indicator values. A version constant on the level sets of a random variable recovers conditional probabilities in the old sense. The distinct conditional random variables behave almost everywhere like probability measures. De Finetti’s representation theorem then crowns the discussion: his representative function Φ, identical with MD in the simple language, is mathematically powerful for constructing inductive methods, though intuitive plausibility is best judged by inspecting the resulting C-values.
The Formal Core of Instantial Relevance
Carnap’s formulation of relevance principles rests on the representative functions M and C defined earlier. For attributes P₁ through Pₖ and natural numbers s₁ through sₖ summing to a positive s, an initial sample proposition is the conjunction in which each Pᵢ occurs exactly sᵢ times; symmetry lets any s distinct individuals stand in for the named ones. Writing E for such a proposition, M(s₁,…,sₖ) is the probability of E, and C(s₁,…,sₖ) the probability of P₁ holding for the next individual given E. The special cases yield the principles of nonnegative, positive, and bare relevance, and the vacuous sample with all sᵢ zero is identified with the necessary proposition Z.
De Finetti’s Theorem Does the Work
By the symmetry axiom, de Finetti’s theorem expresses M as an integral over the simplex A of k-tuples of nonnegative reals summing to one, against a Lebesgue–Stieltjes measure μ. A Cauchy–Schwartz argument then shows that the basic axioms plus symmetry imply weak nonnegative relevance — a special case of the earlier theorem. Equality in that inequality is exactly where bare relevance fails. From this the author proves Theorem 3: if bare relevance is violated for some tuple, then a corresponding C-value stays constant across every tuple in which nonzero entries remain nonzero. Theorem 4 shows that any violation of bare relevance is global: for fresh individuals b₁, b₂, … none mentioned in E, the probability of Mb_{n+1} given E conjoined with a run of matching and non-matching outcomes equals its probability given E alone. The proof turns on the fact that failure of the independence condition forces the mixing distribution F to be a step function, concentrated at a single point r.
The Principal Result
Theorem 5 closes the argument: the axioms A1 through A7 imply bare relevance. If bare relevance failed it would fail globally, making the conditional probability a constant r; Reichenbach’s axiom A7, applied to the pair {M, ¬M}, drives the probability of M on the next trial to 1 as the sample of M’s grows, forcing r = 1 and contradicting the regularity axiom A5. Combining Theorems 1 and 5, the axioms imply both nonnegative and bare relevance, whose conjunction is positive relevance. The principle of positive instantial relevance thus follows from A1–A7.
Gaifman’s Geometrical Setting
Haim Gaifman’s contribution recasts these questions in geometric terms. Meaningful expressions m(n₁,…,nₖ), with the nᵢ summing to at most n, are constrained by a normalization equation defining a hyperplane H_{k,n}, together with multinomial-type inequalities carving out a closed convex polyhedron M_{k,n} — the set of all probability functions on the relevant finite space. Its vertices correspond to the tuples summing to n. Weak instantial relevance becomes an inequality required wherever both sides are defined; strong relevance demands equality only when both sides vanish. Weak relevance implies a monotonicity property: if one tuple’s positive entries are matched by another’s, vanishing of the first forces vanishing of the second.
Faces, Vertices, and Extendibility
The set N_{k,n} of functions satisfying that monotonicity condition is convex, contains the interior of M_{k,n}, and also part of the boundary. Membership is characterized by a uniquely determined family of subsets J₁ through Jₖ, none containing another, with m positive on a tuple exactly when some Jᵢ covers the tuple’s positive coordinates. When the family is maximal the function is interior; otherwise it lies on a lower-dimensional face, and in the extreme case of a single set it coincides with one of k vertices. The sets W_{k,n} and S_{k,n} of weakly and strongly relevant functions are convex, with W_{k,n} closed and nonempty interior, and the interiors of W and S agreeing with S intersected with the interior of M. Since W is closed and properly contained in M, part of its boundary lies inside M’s interior whenever n exceeds 2. Finally, letting E_{k,n} be the functions extendible one level up and R_{k,n} those extendible to the infinite space while satisfying Reichenbach’s axiom, a compactness argument shows extendibility to every finite level yields extendibility to the infinite one. E_{k,n} is closed and convex within N_{k,n}; R_{k,n} is convex, meets the interior and every relevant face, and coincides with the interior of E_{k,n} relative to N_{k,n}.
What the System Is For
The technical machinery serves a philosophical argument, and the argument is worth separating from the formalism. Carnap calls the theory of his 𝔪- and 𝔠-functions “inductive logic,” aligning himself with Keynes and Jeffreys against the tradition that treats inductive inference as the acceptance of a new proposition. That tradition, he thinks, cannot answer Hume: however many times a weather situation has been followed by rain, the prediction that it will rain tomorrow is not rationally warranted as an assertion. Carnap’s move is to change the conclusion rather than defend it. Inductive reasoning does not output the hypothesis; it outputs a number, the probability of the hypothesis given the evidence, and the reasoner asserts that this number is 0.99 and that his own degree of belief matches it. The step from the analytic value of the 𝔠-function to the thinker’s credence is licensed by his having chosen that function, a choice guided but not dictated by axioms, each backed by a requirement of rationality. Hume’s objection loses its grip because nothing contingent has been asserted beyond the observational record.
The requirements themselves are where the book’s ethical flavor lives. A credence function must be coherent, in de Finetti’s sense, which is to say it must be a probability measure; it must be strictly coherent, so that no system of bets exposes the agent to certain loss; and it must update by conditionalization, so that changes in credence depend only on observational results and not on hopes or fears about the outcome. Carnap concedes that these demands fit a natural human being poorly. He asks us instead to imagine a robot with perception, memory, and decision-making organs, and suggests that rules found for the robot can then advise humans “as rational as their limited abilities permit.” The infallible memory and the initial credence function assigned before any data arrives are idealizations in the same spirit—not psychology, but logic.
Symmetry, Analogy, and the Shape of the System
Two ideas do most of the constructive work. The first is symmetry: a credence function should be invariant under permutations of individuals, because if one evidence proposition is isomorphic to another it says exactly the same thing about possibly different individuals, and to weight them differently would be whimsical. Carnap is careful to restrict this to credibility rather than credence—a man may rationally think it likelier that Mr. Smith is over thirty than that Mr. Brown is—and to exempt coordinate languages, where positions in space or time carry logical structure that permutation would disturb. The second is analogy by similarity, the concern that pushed Carnap beyond the continuum of inductive methods in the first place. Attribute spaces are given distance functions, sometimes phenomenological ones drawn from psychological experiments on color and pitch, and the width of a region and the distance between regions become the features that a satisfactory inductive method must respect.
Around these ideas cluster the book’s characteristic constructions: families of attributes and their partitions, models as points in a possibility space, truth sets defined recursively, and the representative functions—𝔐I, 𝔐S, and the C-functions—that stand in for a probability function when the language is symmetric. The distinction between pure and applied inductive logic, modeled on the difference between mathematical and physical geometry, lets Carnap develop the calculus abstractly and then ask separately how a language should be interpreted for a given investigation. The requirement that primitive attributes be qualitative rather than positional, and the care taken to separate locational from descriptional features, show how much of the philosophy sits inside the formalism.
An Unfinished Foundation
The book’s own metaphor for itself is architectural. Carnap calls his system “basic” in two senses: many questions are touched only lightly, with notes in his shorthand left for later volumes, and the structure is a foundation rather than a completed building, “not immune from revision.” The postscript makes the situation plain. Carnap had finished revising all but the last two sections of the continuation when he died; Lary Kuhns and Gordon Matthews took up the revision, and Jeffrey’s introduction appears as a draft Carnap approved but did not fully revise. What remains is a record of a program at a particular moment—the invariance requirement of the 1950 system abolished, the completeness of language no longer demanded, quantitative magnitudes now admitted into the observational language through finite intervals, and de Finetti’s representation theorem woven into the apparatus. For readers interested in how logical probability was actually built, rather than in the polished summary of a finished doctrine, the incompleteness is the point.
