Described as perhaps “the greatest empiricist of the 20thcentury” (Salmon, 1977a), the work of Hans Reichenbach(1891–1953) provides one of the main statements of empiricistphilosophy in the 20th century. Provoked by the conflict between(neo-) Kantian a priorism and Einstein’s relativity of space andtime, Reichenbach developed a scientifically inspired philosophy andan uncompromisingly empiricist epistemology. He was literate in thephysical science of his time, and acquainted with many of its mosteminent practitioners. Criticism and justification of scientificmethodology formed the core of almost all his philosophical efforts,which he promoted in a crescendo of books, in the journalErkenntnis, which he founded and edited with Rudolf Carnap,and within a group of philosophers, mathematicians and scientists heled in Berlin. His commitment to objectivity and realism in sciencetogether with his probabilistic justification of belief in scientificresults carried philosophical and technical difficulties that shapedmuch of the subsequent debate in philosophy of science.Reichenbach’s contributions cover large swathes of formalphilosophy, especially in philosophy of physics, logic, induction andthe foundations of probability, and his later work encompassedlinguistics, philosophical logic, and ethics. The fruits of some ofhis insights are only belatedly having their full impact. For example,several of the recent accounts of causality employ ideas that can betraced to Reichenbach’sThe Direction of Time.

Partitioning the work of a philosopher with views about almosteverything is indispensable in a retrospective, but inReichenbach’s case separation is especially artificial. From1915 until 1953, most of his philosophical essays entangle issues anddoctrines about probability, causality, physics, epistemology andmetaphysics. Reichenbach’s ideas about causality and probabilityare so intertwined that it makes little sense to discuss themseparately. Unavoidably, therefore, different aspects of the sameworks are discussed in several sections of the following essay.Reichenbach’s views on all of these topics changed so radicallyover time that there is no one “Reichenbach system,” andwe therefore offer a synoptic history of his views within each topic.All of our citations are from the English editions of hispublications, where available.

• 1. Life
• 2. Causality and Probability
  • 2.1 Doctoral Thesis
  • 2.2 Views Prior toThe Theory of Probability
  • 2.3The Theory of Probability (1935, 1949)
• 3. Epistemology and Metaphysics
  • 3.1 Early Views
  • 3.2 Mature Views: Experience and Prediction (1938)
• 4. Philosophy of Physics
  • 4.1The Theory of Relativity and A Priori Knowledge (1920)
  • 4.2 Early Writing on Relativity
  • 4.3Axiomatization of the Theory of Relativity (1924)
  • 4.4The Philosophy of Space and Time (1928)
  • 4.5 Quantum Mechanics and 3-Valued Logic
  • 4.6The Direction of Time (1956)
• 5. Logic and Modality
• 6. Ethics and Free Will
• 7. Reichenbach’s Influence
• 8. Retrospective
• Bibliography
  • Primary Literature
  • Secondary Literature
• Academic Tools
• Other Internet Resources
• Related Entries

1. Life

Born in Hamburg, Germany, in 1891, Hans Reichenbach was the second offour children of a half-Jewish but baptized father and a non-Jewishmother. In secondary school and at university he was active in thesocialist student movement. From 1910 to 1911 he studied civilengineering at the Technische Hochschule in Stuttgart, and then movedamong Berlin, Munich and Göttingen, studying physics, philosophyand mathematics with some of the eminences of the time, includingErnst Cassirer, Max Planck, Arnold Sommerfeld and David Hilbert.^[1] He wrote his doctoral dissertation largely on his own after theneo-Kantian Paul Natorp would not accept him as his student. Aftersearching for alternative advisors, his dissertation was finallyaccepted by Paul Hensel, a philosopher, and Max Noether, amathematician, in 1915 in Erlangen. Reichenbach was conscripted intothe army while completing his thesis. He served in the German armysignal corps on the Russian front until a serious illness sent himback to Berlin in 1916. He was relieved from active military duty in1917 to work as an engineer for a firm specializing in radiotechnology. In Berlin, Reichenbach attended Albert Einstein’slectures on relativity and statistical mechanics, which influenced himprofoundly, and inaugurated a life-long friendship between the twomen. He wrote several popular articles defending Einstein, especiallyin the context of the observations of the solar eclipse of 1919confirming the predictions of the general theory of relativity.

In 1920 Reichenbach became an instructor in physics, and eventuallyassociate professor, at the Technische Hochschule in Stuttgart.The Theory of Relativity and A Priori Knowledge (1920f) wasaccepted as his habilitation in physics. In this period Reichenbachmarried Elizabeth Lingener and they had two children, Hans Galama in1922 and Elizabeth (Jutta) in 1924. While in Stuttgart he developedcontacts with Moritz Schlick, Rudolf Carnap and Erwin Schroedinger. In1926, after much back and forth, he assumed a teaching position in“natural philosophy” at the University of Berlin, where heremained until Hitler came to power in 1933. During this timeReichenbach organized discussion groups on scientific philosophy,similar to those of theVienna Circle. The group around Reichenbach that developed out of the Society forEmpirical Philosophy and became known as the Berlin Group includedWalter Dubislav, Kurt Grelling, Wolfgang Köhler, Kurt Lewin andlater Carl Hempel, Reichenbach’s student. Together with membersof the Vienna Circle, Reichenbach initiated the publication of thejournalErkenntnis in 1930 as a forum for scientificphilosophy. Reichenbach and Carnap were the only editors after Schlickresigned in reaction to Reichenbach’s opening article (seeSection 4.4 below). In addition (and also for financial reasons), Reichenbach wasa frequent contributor of popular essays and a regular radio lectureron scientific topics.

With Hitler’s elevation, the views and methods of the BerlinGroup and Vienna Circle were branded Jewish philosophy, andReichenbach—who counted as Jewish to the National Socialists andwas in any case considered undesirable given his socialist writings asa student—was dismissed from his university position and fromradio work. He moved to Istanbul in 1933, where, as part of hisefforts to westernize Turkey, Mustafa Kemal (Atatürk) hadestablished a new university to attract intellectuals fleeing Europe.Reichenbach was joined there by 32 other German professors, notablyRichard von Mises, the mathematician whose views on probability musthave influenced Reichenbach, and Erwin Freundlich. Freundlich had beenEinstein’s assistant in Berlin on matters astronomical, led anunsuccessful German expedition to Russia to measure the gravitationaldeflection of starlight at the 1914 solar eclipse (the members of theexpedition became prisoners of war!), and later joined the“Einstein Observatory” in Potsdam.

Under a five-year contract which prevented him from accepting aposition at New York University arranged by Einstein and Sidney Hook,Reichenbach remained in Turkey until 1938, when, through the effortsof many people, most notably Charles Morris, Reichenbach moved to theUniversity of California, Los Angeles (UCLA) with his family. Hikingin the Swiss Alps shortly before his move to Los Angeles, Reichenbachsuffered a heart attack, which prevented him from teaching during thefirst months of his American appointment. (Reichenbach was an avidhiker and skier, but not physically impressive. His student, CynthiaSchuster, describes his appearance around 1950 this way: “short,almost rotund, stubby hands and feet, round face, snub nose, thickglasses, false teeth, a hearing aid, and a thin high-pitched speakingvoice.” (Reichenbach 1978, vol. I) His influence on students,she says, was a case of mind over matter.) With the US entry intoWorld War II in 1941, Reichenbach—as a resident Germanalien—was kept under effective house arrest, allowed to leaveonly to work and for medical purposes until he obtained Americancitizenship in 1943. During the war, Reichenbach was engaged inhelping to move members of his family out of Germany and to securepositions for colleagues at UCLA, in particular members of theFrankfurt School (Theodor Adorno, Max Horkheimer), who with Berthold Brecht and Thomas Mann became partof Reichenbach’s German intellectual circle in Los Angeles. In1939 he was reunited with Maria Moll, who had been his colleague inIstanbul. They married in 1946, the day after her divorce. Despitebitter discussions at UCLA about the McCarthy era California LoyaltyOath, in which each faculty member had to attest that they were notmembers of the Communist party, we are not aware of any writtenobjections from Reichenbach. He signed the oath in 1949/50, apparentlyunwillingly, as an understanding letter from Rudolf Carnap suggests(McCumber 2016, p.52). Reichenbach’s untimely death from anotherheart attack on April 9, 1953, prevented him from presenting theWilliam James Lectures at Harvard in the fall of the same year, andalso prevented his inclusion in a planned volume ofLivingPhilosophers in the series edited by Arthur Schilpp.

Further Reading. Much more detail about Reichenbach’slife can be found in the excellent biography by Gerner (1997). VolumeI of Reichenbach (1978) contains memories and comments from many ofReichenbach’s friends, colleagues, students and relatives.Similarly, see Maynes & Gimbel (2022) for interviews with MariaReichenbach and Elizabeth Austin (Reichenbach’s daughter).Further biographical detail can be found in Salmon’sintroduction (translated to German by Maria Reichenbach) to volume 1of Reichenbach (1977), in Salmon (1979), in Maria Reichenbach (1994),in Padovani (2008) and Kamlah (2013), whileThe Hans ReichenbachCollection (linked into the Other Internet Resources sectionbelow) at the University of Pittsburgh contains a wealth ofautobiographical notes, in particular references HR 014-33-08 and HR044-06-21 to HR 044-06-26. Damböck et al. (2022) discussesReichenbach’s involvement in the socialist student movement andHoffman (2007) and Milkov (2013) describe the historical developmentsof the Berlin Group. Irzik (2011) provides the same forReichenbach’s move to Istanbul. Verhaegh (2020) traces in detailReichenbach’s move to Istanbul and the decisions and hurdlesthat eventually took him to the U.S.. Avkiran (2021), who alsoincludes a wealth of references in Turkish, and Roure (2022) describeReichenbach’s time in Istanbul and explain his limited influenceon philosophy in Turkey.

2. Causality and Probability

2.1 Doctoral Thesis

Reichenbach’s doctoral thesis,The Concept of Probability inthe Mathematical Representation of Reality (1915b), contains manyof the themes that concerned him throughout his life, and anticipatedin some detail 21^st century philosophical discussions ofprobability relations between microscopic and macroscopic systems.Indeed, the development of his ideas about causality and probabilityfrom 1915 until the end of his life can be seen as a series ofreexaminations and reformulations of issues the thesis implicitlyposed and the solutions it explicitly offered.

The thesis develops an account of probability appropriate forscientific inference. It presents an argument to supplement whatReichenbach understands to be Kant’s transcendental principle ofcausality with a transcendental principle of probability. InReichenbach’s reading of Kant, the “principle ofcausality” asserts that every event is preceded by a cause thatdetermines it according to some universal law (see the discussion ofKant’s principle in Section 2 of the entry onKant and Hume on Causality). The principle is “transcendental” because it cannot beempirically established, but is instead a precondition for the verypossibility of empirical knowledge. Reichenbach’s claim is thatthere is a principle of probability that has an equal status: itcannot be empirically established, but it is a precondition ofempirical knowledge. It states that events are governed by aprobability distribution.

Reichenbach considers, and rejects, the subjective interpretation ofprobability advocated by the then prominent philosopher andpsychologist Carl Stumpf (1892a, b), and, less emphatically, theattempt at an objective interpretation advocated by Johannes von Kries(1886), a physiologist who had studied with Hermann von Helmholtz.According to Reichenbach, von Kries’s account of probabilityneeds to be freed from the principle of insufficient reason—thatmutually exclusive events of which we have no knowledge that woulddetermine differential probability are equally probable. As asubjective principle, Reichenbach claimed, it had no place in science.Reichenbach insists on an “objective” interpretation ofprobability—in the Kantian sense as about the world ofexperience—for which probability statements are synthetic, butnot verifiable, claims about the empirical world. His task, as he seesit, is to demonstrate that probabilistic statements are supported by aclaim—the “existence of a probabilityfunction”—which is a transcendental principle that isnecessary, and in combination with causal principles sufficient, forempirical knowledge.

Reichenbach’s technical argument is an adaptation of HenriPoincaré’s results on probability functions and what havesince come to be called “strike ratios” (Poincaré,1912, pp. 148–150). By considering events collected in ahistogram, in which alternate equally narrow columns are black andwhite, Reichenbach argues that as the number of (independent) eventsincreases (and the width of the columns decreases) the ratio of blackto white events within any interval of the abscissa will approximate1. Reichenbach’s general idea is that if a variableXis divided into two or more classes of very small intervals of equalwidth (inX units) juxtaposed in a definite order andproportion, then there will be a probability for the occurrence of avalue ofX within any particular class that is invariant overall Riemann integrable probability distributions forX. (TheRiemann integral of a function describing a curve is defined as thelimit of the sum of rectangles touching the curve as their widthapproaches 0.)

Reichenbach extends the analysis to error probabilities for physicalmeasurements. He then argues that since all physical measurements aresubject to error, knowledge of natural laws is possible only if errorsoccur subject to a probability distribution, a proposition that issynthetic but cannot be established empirically. Empirical knowledgethus requires both an a priori principle of causality for individualevents and an a priori principle of probability to ensure thatindividual events can be aggregated into general laws. No explicitinterpretation of probability is forwarded, although Reichenbachimplies that probability claims are about frequencies of causallyindependent events—a notion for which probability theoristswould later substitute the idea of independent, identicallydistributed events (see the entry onprobabilistic causation)— in observed and unobserved collections of cases.

Further Reading. The thesis with an introduction to thearguments can be found in Reichenbach (2008). Re-statements and slightvariations of the arguments in the thesis can also be found inReichenbach (1920c), Reichenbach (1920e) and to a lesser extent inReichenbach (1930g) and chapter 9 of Reichenbach (1949f). A synopsisis also given in Padovani (2011), while Eberhardt (2011) provides acritical assessment of the thesis.

2.2 Views Prior toThe Theory of Probability

Reichenbach’s allegiance to Kantian formulations waxed and wanedand transformed over time, but while the terminology changed,Reichenbach long retained the essential claim of his doctoral thesisthat the establishment of any empirical law requires a superempiricalpresupposition about probability. In 1920, the argument andconclusions of the thesis were repeated with little qualification orvariation in two essays, “The Physical Presuppositions of theCalculus of Probability”, and “A Philosophical Critique ofthe Probability Calculus”. Again in “Causality andProbability”, in 1930, Reichenbach reprises the claims of hisdoctoral thesis, but substitutes the “principle ofinduction”—that observed frequencies will continue to holdin new cases—for the “principle of probability.” Theprinciple of induction is not given an explicitly a priori basis, thepossibility of which Reichenbach had come to doubt when studying withEinstein (see Reichenbach, 1920f). Instead, it is justified by anuneasy mixture of loose convergence arguments (foreshadowing thestraight rule, see below) and psychological habit. Withoutattribution, he dismisses a “conventionalist” account ofthe principle of induction, expressed in nearly Kantian terms, as“the principle of induction is not a statement about thephysical world, but merely constitutes an ordering principle ofscience.” He argues that such an account does not“justify” scientific preferences for simpler hypotheses(1930g, see Reichenbach 1978, vol. II, p. 340–341). Arbitraryconvention could choose any scientific hypothesis that captures thephenomena, but a descriptively accurate account of theory selection inscience must explain why simpler theories should be preferred. Thejustification of the preference for simpler hypotheses remainsunexplained in “Causality and Probability”, andReichenbach’s own conclusion seems to be essentially the Kantianprinciple he rejects, supplemented with an attempt to prove that theprinciple of induction is unchallengeable because, in his view,probability claims are generalizations over as yet unobserved casesand therefore presuppose the principle of induction:

We found that a probability statement is meaningful only if theprinciple of induction holds; therefore the statement that theprobability laws do not hold is itself meaningless unless theprinciple of induction holds. To say that probability laws do not holdis equivalent to predicting that the observed relative frequency ofsequences of events will not be preserved in the future, that theregularity implied by the principle of induction does nothold—and this statement has empirical meaning only if it can bedecided inductively, i.e., if the principle of induction holds. Thestatement that probability laws do not hold is self-contradictory andmakes no sense. (1978, vol. II, p. 343)

In a further break with the Kantian tradition, the role of causationas conceptually primitive is reconsidered. It is difficult to pinpointReichenbach’s position between 1915 and 1935 since his writingsmix epistemic and metaphysical issues. In his thesis (1915b) and in“Stetige Wahrscheinlichkeitsfolgen” (1929L) Reichenbach iscommitted to individual deterministic causal events. In “TheCausal Structure of the World” (1925d) probability is regardedas the more fundamental concept. In later years Reichenbach oftencited this paper as an anticipation of the indeterminism of quantumtheory, because his account allows the possibility that finer andfiner measures will not converge to deterministic laws (see Gerner,1997, p. 153). In “Causality and Probability” (1930g)causality explicitly refers to regularities in populations rather thanto particular events, a view whose source Reichenbach attributes toLudwig Boltzmann’s development of the theory of gases. Theseparation of causality and probability, and the recasting ofcausality as a higher level concept, required that Reichenbach find anew foundation for probability. Presumably due to the influence of hiscolleague Richard von Mises, Reichenbach moved towards a view ofprobability as a property of sequences.

Reichenbach’s 1925 essay “The Causal Structure of theWorld” (1925d) is also an early attempt to account for thedirection of time in terms of causal and probabilistic asymmetries.Reichenbach there introduces the notion of a “probabilityimplication” with 10 axiom schemes on propositional variablesinvolving both material implication and a new 2-place probabilityimplication connective. The axioms are evidently meant to supplementthose of propositional logic. No rules of inference are specified, butsubstitution and modus ponens are used. The axioms do not guaranteethat a probability implication,a ⊇b^[2], is the conditional probability ofb givena, oreven that the consequents of a collection of probability implicationswitha as antecedent satisfy the axioms of finiteprobability. Interpretation is difficult, since Reichenbach bothasserts that the probability of the consequent in a probabilityimplication can be between 0 and 1 inclusive (1925d, 1978, vol. II, p.89), but then disallows a probability implication because theconsequent has probability 0 (p. 92). Reichenbach’s thoughtseems to be thata ⊇b asserts that incircumstancea,b has a well defined probability,that is,a specifies something like what Ian Hacking (1965)later called a “chance setup,” or as Reichenbach mighthave put it, implies the existence of a probability function for{b, ~b}. Reichenbach uses the operation very much inthat way in his 1925 discussion of the direction of time, which hethinks can be founded on cases in whicha ⊇bis true butb ⊇a is false. Even under thisreading some of his axiom schemes have false instances, e.g.,(a ⊇b) ⊃ (a.c ⊇b), where the dot is ordinary conjunction. (The claim thatb has a probability distribution in contexta doesnot necessarily imply thatb has a probability distributionin contexta conjoined with contextc, sincec might make a distribution impossible.) Revised, probabilityimplication later became a foundational notion of Reichenbach’stheory of probability and the central concept of his approach toinductive logic.

Further Reading. For Reichenbach’s views on causalitysee also the discussion of “The Direction of Time” below and the separate entry on Reichenbach’sCommon Cause Principle.

2.3The Theory of Probability (1935, 1949)

Reichenbach continued to revise and elaborate his ideas aboutprobability in a series of papers in the early 1930s^[3] until, in 1935, hisThe Theory of Probability provided afuller statement of his developed view. Surprisingly, Reichenbach doesnot acknowledge help from Richard von Mises, his colleague in Berlinand Istanbul in the period and the mathematician whose views on thefoundations of probability were closest to his own and are oftendiscussed in the book. He does attribute part of the mathematical workin the book to Valentine Bargmann, who after finishing his doctoratein physics in Berlin fled to Switzerland in 1933 and later became anassistant to von Neumann and to Einstein at the Institute for AdvancedStudy, from where in the 1940s he also assisted Reichenbach’swork on quantum theory.

It is fair to say thatThe Theory of Probability was not wellreceived, drawing intense criticism from Karl Popper (1934), who hadread Reichenbach’s papers presenting a frequentistinterpretation of probability, C.I. Lewis (1952), Bertrand Russell(1948), and Ernest Nagel (1936, 1938). Kolmogorov’s measuretheoretic axioms for probability, which appeared in 1933, soonovershadowed Reichenbach’s formulation of the theory ofprobability.

TheTheory of Probability uses class terms—A,B,C—and individualvariables—x,y,z—as well asreal variables—p,q,u,r,w. Distinct individual variables are (unnecessarily)sometimes and sometimes not associated with distinct class names, butReichenbach’s formulae without iterated probability conditionalscan be read as universally quantified with a single individualvariable. The earlier 10 axiom schemes for probability implication(1925d) are replaced by 4 axiom schemes expressed inReichenbach’s abbreviated form as (1949f, p. 53–65):

1. Univocality: (p ≠q) ⊃ [(A⊇_pB) . (A⊇_qB) ≡ (~A)]
2. Normalization:
   1. (A ⊃B) ⊃ ∃p[(A⊇_pB) . (p =1)]
   2. ~~A . (A⊇_pB) ⊃ (p ≥ 0)
3. Addition: [(A ⊇_pB) .(A ⊇_qC) .(A.B ⊃ ~C)] ⊃∃r[(A ⊇_r (B∨C)) . (r =p+q)]
4. Multiplication: [(A ⊇_pB) . (A.B ⊇_uC)] ⊃ ∃w[(A⊇_wB.C) . (w =p∗u)]

Reichenbach’s intent with formula I is to say that where itexists, the probability has a unique value. Axiom II is meant toensure that the probabilities conditional on a nonempty set havevalues between 0 and 1 inclusive. Axiom III is Reichenbach’sversion of the requirement that the probability of the union ofmutually exclusive events is the sum of their probabilities. Axiom IVis essentially the chain rule of probability:P(CB |A) =P(C |BA)P(B |A). Axiom I is implicit in theKolmogorov axioms (see Section 1 (“Kolmogorov’sProbability Calculus”) in the entry oninterpretations of probability) since probability is taken to be a real valued function. Axiom IIcorresponds to Kolmogorov’s first and second axioms, thatprobability values are bounded between 0 and 1 inclusive. Axiom IIIamounts tofinite additivity because Reichenbach’slogic does not have infinite disjunctions: it is a finite restrictionof Kolmogorov’s third axiom, which postulates additivity ofprobabilities for countable, even infinite, disjoint sets. Axiom IV(at least its interpretation in terms of the chain rule) follows fromKolmorogov’s first three axioms. Reichenbach requires it as anadditional axiom, because of his mixture of logical and mathematicalnotation. Without the additional fourth axiom, Reichenbach could notswitch between logical conjunction and mathematicalmultiplication.

Reichenbach proceeds to show that finite frequencies satisfy his fouraxioms. He construes all probabilities as frequencies of sub-series ina larger series—or what is the same thing for finite sets,cardinalities of subsets in a universal set. Hence his probabilitiesare always with respect to a non-empty reference class, so theprobability that an event is in a classB is inReichenbach’s notation,P(A,B) whereA is the reference class—similar to the modern notationof the conditional probabilityP(B |A).But the probability logic axioms are so weak that many formalstructures satisfy them, and Reichenbach’s claim is a long wayfrom a representation theorem. He fails to provide the additionalconstraints on the space that probabilities are applied to, which inKolmogorov’s case are given by the assumption that the space isa sigma-field, i.e. a field closed under complementation and countableunion.

Reichenbach’s official definition of probability for infinitesequence pairs ⟨x_i,y_i⟩ withx_i inA, andy_i inB, for which the limitp of the relative frequency ofB inAexists, is as follows: “the limitp is called theprobability fromA toB within the sequencepair.” (1949f, p. 69). Constraints on the nature of thesequences are added later (1949f, section 30) that are aimed toformalize aspects of randomness. Probabilities of single cases are“fictive” or elliptical, to be understood as claims aboutthe frequency of a kind of case in an implicit reference class.Elsewhere, Reichenbach puts more emphasis on finite frequencies andeven suggests that limiting frequencies are simply a mathematicaldevice for justifying inductive procedures (‘A letter to B.Russell’, 1978, vol. II, p. 405–406). Once introduced, thelogical framework is dropped in the mathematical development ofprobability theory in the book.

In combination with the representation of probability relations byclaims in a quasilogical language, the limiting frequencyinterpretation creates fundamental formal problems that Reichenbachdid not foresee. Sets of limiting relative frequencies are not closedunder finite intersection; they are not closed under countable union;they do not satisfy countable additivity. They do not, in other words,form a sigma field, or a Borel field, or even a field. (See theentries oninterpretations of probability,the early development of set theory, andset theory.) These and several other mathematical difficulties ofReichenbach’s setup are described in van Fraassen (1979).^[4]

Reichenbach imposes two further axioms—the axioms of order(1949f, p. 137)—that are supposed to hold necessarily oflimiting frequencies for infinite time series. One is trivial,essentially asserting that a conditional frequency on lags(Reichenbach’s term for lags is “phases”) of aconstant “variable” can always be replaced by aconditional frequency on no lag of that variable. The second, however,appears to be a very strong stationarity principle that is notgenerally true: the probability of one variable conditional on aspecified common lag of other variables is invariant under all uniformtranslations of the lags. This axiom seems to derive directly fromReichenbach’s interpretation of the foundations of probability,in particular from Reichenbach’s assumption about normalsequences, described below.

In addition to an axiomatization, Reichenbach attempts to provide afoundation for probability claims in terms of properties of sequences,similar to von Mises. Reichenbach regarded von Mises attempt (vonMises 1919) at formally characterizing a “random sequence”as a failure and instead attempted to characterize a weaker sequenceproperty—a “normal” sequence. In his account ofnormal sequences Reichenbach retains (although the second only in aweaker form) two features that are deemed essential for randomsequences: the lack of “after-effect” and the invarianceof the limiting relative frequency under subsequence selection.Informally, the “invariance under subsequence selection”is supposed to capture the idea that the probabilities of events inany infinite subsequence selected from the original sequence by aprocedure based on the indices of events in the original sequencealone will be the same as the probabilities of events in the originalsequence. “The lack of aftereffect” is supposed to capturethe idea that given any initial segment of a sequence, one cannotpredict the probability of the next event any better than predictingit based on the limiting relative frequency of events in the infinitesequence. Reichenbach’s definition of lack of aftereffect is notbased on initial segments of sequences, but rather on subsequencesselected by a particular set of rules (1949f, p. 142). Reichenbachdefines a “selection”S as any rule thatdetermines for each member of a sequence whether it is a member ofS (p. 143). He intends by a “rule” literally anysubsequence.

We are unable to reconstruct exactly what Reichenbach may haveintended, in particular since his definition of lack of aftereffect isdifficult to distinguish from the criterion of invariance undersubsequence selection. But we believe it is something close to thefollowing: A sequence ofB_i is“free from aftereffect” if (i) a subsequence is selectedbased on a rule of the form “For each indexi in thesequence, include the (i+k)^th element inthe sequence if thei^th element isB (or~B),” (ii) if the subsequence thus selected has thesame event probabilities as the original sequence, and (iii) if thisholds for all lagsk > 0. If our reconstruction iscorrect, this would distinguish Reichenbach’s account of lack ofaftereffect from that of invariance under subsequence selection,because the former includes subsequence selection rules that depend onthe values of certain items in the sequence, while the latter includesonly rules that are based on the indices. Two more definitions arerequired for the full picture. First, a selectionS of asubsequence of sequenceA belongs to the “domain ofinvariance” ofB, if the probability ofB (forall lags) inS is unchanged from the probability ofB inA, and if the same holds for any selectionS fromA with a lag. Second, a subsequence selectedby an algebraic rule that partitions the sequenceA—e.g. take every fourth element (see p. 144 fordetails)—is called a “regular division” ofA. Putting the pieces together, Reichenbach requires for the“normality” of a sequenceA that it be free ofaftereffect and that all “regular divisions” ofAare in the domain of invariance ofB. This condition ofregular divisions seems to underlie the stationarity expressed in thesecond axiom of order.

If random sequences are taken to satisfy (at least) the conditions ofthe lack of aftereffect and invariance of subsequence selection underany selection rule, then Reichenbach’s restriction ofsubsequence selection rules to regular divisions implies that the setof normal sequences is a proper superset of that of random sequences.Reichenbach accepts this weakening to avoid some of the difficultiesin characterizing a random sequence, and to broaden his earliernotions of probability to include sequences of trials, which might notbe perfectly independent. In later writings, he seems to suggest thatas long as the sequence converges, probability claims can be appliedto the component events.

A large section of the book is devoted to reconstructing classicalresults in the theory of probability as claims about relativefrequencies, including various continuous distributions andBernoulli’s theorem. Reichenbach claims that probabilities oncontinuous domains are “isomorphic” to limiting relativefrequencies, but, as van Fraassen (1979) notes, it is difficult to seeany sense in which that is true. The remainder of the book is notabout probability per se, but about its epistemological role.

Further Reading. A detailed attempt at the reconstruction ofReichenbach’s account of probability and its epistemologicalgrounding can be found in Eberhardt & Glymour (2011), whichincludes specific references to the original sources. The first halfof Rédei & Gyenis (2021) provides a very useful discussionof why Reichenbach’s axioms do not fit well withmeasure-theoretic accounts of probability. Hubert (2021) pinpointsseveral of the challenges with Reichenbach’s frequentist accountand develops his own solution.

3. Epistemology and Metaphysics

3.1 Early Views

Reichenbach began his philosophical career as a neo-Kantian, aperspective that is evident in his thesis and in his first book-lengtheffort (1920f) after his doctoral thesis, and that at least echoes inhis later work. The aim ofThe Theory of Relativity and A PrioriKnowledge is to reconcile Kantian theory in a limited way withthe theory of relativity by distinguishing two senses of synthetic apriori: Principles governing the content of experience can besynthetic a priori because they are necessary, transcendental truths,or because they are non-empirical principles that form part of how weconstruct our representation of reality, and are thus revisable.Reichenbach endorses the latter constitutive sense of the synthetic apriori, and sees his task as scrutinizing Kant’s principles inlight of the “new” reality described in the theory ofrelativity. Reichenbach’s—never clearlyspecified—process of a construction of a representation ishidden behind the idea of “coordination”. Particularprinciples, such as those of causality or probability, are supposed toestablish a correspondence betweensomething in experienceand our representation of it in the form of a mathematicallyaxiomatized scientific theory (à la Hilbert). What thatsomething is, Reichenbach never clearly describes. Heexplicitly rejects the proposal that it is just sensation, and headmits that what he is describing is a curious case of “thecoordination of two sets, of which one [...set’s] elements arefirst defined through the coordination” (p. 40, 1965a). While apriori in the constitutive sense, the coordination principles arecontingent, they could be changed if experience makes others moreconvenient. The picture is very much like that C.I. Lewis offers atabout the same time inMind and the World Order (1929).

Reichenbach’s account of the ascension from sense data toindividual things to scientific theories is via an account of testing,in spirit close to ideas that Hermann Weyl (1927) and Rudolf Carnap(1936) were later to advance, in which various hypotheses support oneanother, each functioning as an auxiliary in tests of others, an ideaGlymour (1980) later unsuccessfully tried to formalize as“bootstrapping.” The foundational character of sense dataand the view that objects and their properties and relations areconstructions endures for some years in Reichenbach’s thought,for example in his 1929 essay on “The Aims and Methods ofPhysical Knowledge” (1929g), where, however, the example andauthority is no longer Kant’s first Critique but ratherRussell’sOur Knowledge of the External World (1914)and Carnap’sThe Logical Structure of the World(1928).

By the late 1920s Reichenbach was moving away from neo-Kantianpositions and towards logical positivist views, allying himself withthe Vienna Circle philosophers, although he always maintained andemphasized in retrospect that the “empiricist philosophy”he pursued in the Berlin Group was much more focused and engaged withscience and did not fall prey to the positivist problems that came outof attempts to ground knowledge on sense data alone. He did not viewhis contemporaries uniformly. For Moritz Schlick, Reichenbach seems tohave had a somewhat condescending respect, for Ludwig Wittgenstein,who is one of the few philosophers whom he later criticizes by name inExperience and Prediction (1938c), and for Karl Popper, nonewhatsoever. Aside from Einstein, his deepest respect and closestintellectual alliances seem to have been with Kurt Lewin, KurtGrelling, Rudolf Carnap, Richard von Mises (although they do not seemto have got along personally) and Bertrand Russell, althoughReichenbach was not pleased with Russell’s criticism of hisviews in Russell’s last philosophical book,Human Knowledge,Its Scope and Limits (Russell, 1948; see Reichenbach 1978, vol.II, p. 405).

Further Reading. Chapter 2 of Ryckman (2005) provides a veryclear attempt at reconstructing Reichenbach’s struggle withKantian principles in the 1920s. Padovani (2008) provides a wealth oftextual detail and references for the same period, with a more focusednotion of Reichenbach’s notion of “coordination” inPadovani (2011, 2015, 2017). Chapter 6 of Milmed (1961) traces theKantian elements in Reichenbach’s epistemology and discusses theconflicts that arise, though the analysis concerns primarilyReichenbach’s mature views.

3.2 Mature Views: Experience and Prediction (1938)

By the 1930s Reichenbach abandoned foundationalism altogether andadopted an epistemological position closer to pragmatism than tological positivism. Reichenbach’s mature viewpoint, presented inExperience and Prediction (1938c) diminishes the status ofthe given; knowledge, belief and conjecture is built around hisconceptions of meaning, probability and convention. Coordination oflanguage and physical circumstances replaces his earlier coordinationof Kantian concepts and sensation. Reichenbach is a realist about theexternal world, but asserts that we can only have uncertain knowledgeabout it, inferred from sense data. Deliberation can reject the giftsof perception so involuntarily received. Claims about ordinaryobjects, and scientific claims about other kinds of objects, whethersense data or atoms, are probabilistic in nature and related byprobabilities, not by any kind of logical reduction. Reichenbach wasnot alone in this view at the time. In 1935, in the second issue ofAnalysis, Hempel, writing in English because Carnap could notat the time, described Carnap’s viewpoint in similar, if notquite identical, antifoundationalist terms.

The overall account seems to be something like the following: Languagerequires a coordination of words—or at leastsentences—with something signified. Scientific language,Reichenbach claims, requires “coordinative definitions”that specify physical procedures for measurement. A common example isthe “definition” of the Paris meter bar as a unit ofdistance. Definitions in the usual sense occur within a language, butReichenbach sometimes appears to intend something like an“ostensive” definition (he does not use the term). Hegives no account of how such an act can suffice to specify a rule, andhe recognizes instances in which it does not, for example that acoordination that measures time by the behavior of clocks does notsuffice to provide a rule for deciding relations of time measurementsbetween distant clocks. Various coordinative definitions may thusleave measures of other quantities or relations indeterminate, andthese must be specified by some stipulation or other. His chief, butnot only, example is the definition of simultaneity (see alsoRynasiewicz (2003), and Dieks (2009) for a clear discussion).

Once specifications of all relevant quantities are made, empiricalclaims are possible. There are, in Reichenbach’s view, tworelated ambiguities. First, different stipulations about measurementcan lead to apparently different empirical generalizations thatnevertheless are empirically indistinguishable—although justwhat “empirically indistinguishable” can consistently meanfor Reichenbach is problematic for reasons to be noted. Second, thesame total theory may be partitioned in more than one way into claimstrue by stipulation and empirical claims. Reichenbach’s solutionto these forms of underdetermination is that “equivalenttheories” say the same thing. The equivalence relation heintends is unclear. In earlier writings he suggests that empiricallyequivalent theories are those that have the same empirically testableconsequences; a later formulation is that empirically equivalenttheories are those that have the same (posterior) probability on anyobservations. The later characterization is as clear asReichenbach’s characterization of the probability of theoriesand their confirmation (see below). The earlier characterizationresults in a semantics for which no effective proof theory is possible(Glymour, 1970).

One puzzle about Reichenbach’s view of conventions is whycharacterizing them remained important to him, since they are, in hismature view, only a feature of the reconstruction of a theory, not anintrinsic logical or semantical feature of any proposition. Inprinciple, separating out different presentations of the same theoryand recognizing equivalence amid diverse conventions might be ofvalue, but except in cases such as simultaneity and gravitationaltheories with extra “absolute forces,” Reichenbach did notuse reconstruction to that effect, and the most important question ofequivalence in the physics of his era, the relation of wave and matrixmechanics, was solved in a quite different way by John von Neumann(1932). Reichenbach’s emphasis on locating conventions seemsinstead to be negatively motivated, a continuing prophylactic againstclaims that various principles are a priori.

The immediate description of the perceptual world is in terms ofenduring objects, their properties and relationships, and thatdescription is only probable. The world can be describedegocentrically in terms of “impressions” and “sensedata,” but, Reichenbach argues, ordinary descriptions of thingsare not equivalent to egocentric descriptions in terms of impressions,because, however elaborate, egocentric formulations do not entailobject claims, they only confer a probability on object descriptions,(and, of course, in parallel, given Reichenbach’santi-foundationalism, egocentric descriptions are only probable). Theargument is not in accord with Reichenbach’s own criterion forequivalent descriptions, but the conclusion is repeatedly emphasized.Reichenbach insists, sometimes in rather sharp language, that thelogical positivists and unnamed “pretentious” logiciansare simply wrong about locating the foundations of knowledge in sensedata or analytical truths.

Reichenbach’s mature views on the notion of analytic truth werecomplex. He no longer held with C.I. Lewis that there is at any timeany sort of Kantian a priori—but there is synonymy, there areequivalent descriptions, and the assertion of equivalence betweenequivalent descriptions is presumably a purely logical matter, henceanalytic. Although explicitly addressed to Carnap, C. I. Lewis wasquite possibly an equal target of Quine’s “Two Dogmas ofEmpiricism” (1951). (As Quine’s senior at Harvard, Lewiswas unlikely to be criticized by name.) In a brief essay on“Logical Positivism” in 1945, Russell gently but pointedlyridiculed Carnap’s disposition to resolve every apparentlyirresolvable dispute by appeal to linguistic relativism.Russell’s criticism, and Nelson Goodman’s criticism of thevery idea of synonymy (Goodman, 1949), but perhaps not Quine’s,could have been applied to Reichenbach.

All empirical claims are, according to Reichenbach, probabilisticjudgments based on relative frequencies in a reference class, orreached by induction. To the extent that reference classes rely onkinds, Reichenbach resorts to psychology: primarily, things are sortedinto kinds by the immediate perception of similarity, or by similarityin memory. Less primitively, theory and convention guide thedetermination of kinds. In practice he recommends the choice of thenarrowest reference class for which there are adequate statistics, arecommendation that is of no help (why, or why not, should we throwvarious astrological theories into the reference class for assessingthe prior probability of general relativity?).

Primary or fundamental inductive inference consists of taking observedrelative frequencies as probabilities, that is, as limiting relativefrequencies. This procedure, referred to as the “straightrule”, implies that one should take the current empiricaldistribution to resemble the limiting distribution, and thereforebehave accordingly. The justification of such taking, or“positing” in Reichenbach’s terminology, is that ifthere is a limiting relative frequency to a sequence, this procedurewill converge to it. Reichenbach notes inThe Theory ofProbability (1949f) that without further assumptions nothing canbe said about rates of convergence or about the warranted confidencethat an empirical distribution has converged. He further acknowledgesthat any procedure that estimates the probability to be the relativefrequency, plus any quantity that keeps the estimate between 0 and 1and that itself converges to 0, will also converge to the limitingfrequency if such exists. He there (misleadingly) treats suchalternative inductive rules as producing “equivalentdescriptions” to the straight rule and takes the choice as of noconsequence—but of course the equivalence is only in the limit.InExperience and Prediction (1938c) he instead dismissessuch alternative rules on the vague grounds that they are“riskier” than the straight rule.

The proposal of the straight rule goes back to the problem ofassertability of probability claims he discussed in his thesis.Reichenbach’s proposal is reminiscent of the law of largenumbers, that the empirical distribution of a sequence of independentand identically distributed trials converges in probability to thetrue distribution. But the law of large numbers depends onindependent, identically distributed trials. Reichenbach cannot resortto such assumptions if he wants to avoid circularity in his account ofinductive inference. Instead he introduces “posits” thatcan basically be understood as leaps of (tentative) faith that theempirical distribution is representative. Posits can be“blind” or “appraised.” Posits are blind whenthere is no data available to justify the posit. For example, if allone has is a sequence of measurements for which the empiricalfrequency distribution of events is given byF, then a blindposit might state that the probabilities specified byF arewithin ε, for some small ε, of the true distributionP. One has no reason to justify this claim, the posit isblind. However, if one had several sequences of measurements resultingin empirical frequency distributionsF₁,…,F_n, then the relative frequencies foundin each of these empirical distributions can, according toReichenbach, be used to get higher order distributional informationabout the original posit itself, and the posit therefore becomesappraised. So, if one has sequences of measurements of thegravitational constant from the Earth, the Moon, the 7 other planets,and, say, the Cavendish balance—10 sequences in total—andin all cases except for Mercury, the gravitational constant is withinsome small value ε ofg, then the higher orderprobability of the posit that the true value of the gravitationalconstant is within ε ofg is 9/10. How sure can onebe that the appraisal of the posit is accurate?—For thatReichenbach introduces again an even higher order blind posit. Onethus arrives at a hierarchy of posits, of which the lower levels areappraised posits, and the highest levels are blind posits. Just howthe integration of different levels is supposed to occur, remainsunclear and a point of criticism by Ernest Nagel (see below). Thegeneral idea appears to be that at the “data” level onecounts frequencies and integrates higher levels using Bayes rule.

The approach of estimating higher order distributional information onthe basis of subdividing the available sample is similar to the modernstatistical procedure of bootstrapping, although there re-samplingtechniques are used. Reichenbach argues that his procedure ensuresmore efficient convergence than does naïve application of thestraight rule alone. Faster convergence of the first levelestimate—the data-level—is somehow supposed to result fromsimultaneous convergence at all the different levels in this hierarchyof appraised posits—Reichenbach refers to this as the method ofcross-induction. But Creary (1969, Ch. 5) points out that there is noreading of such a conclusion about efficient convergence that followsfrom the premises Reichenbach assumes (though see Schurz (2021) for amore recent take).

Over time and place, the reasons Reichenbach gives for convergence offrequency estimates to a limiting value vary, including 1) his earlyview that convergence is synthetic a priori, or that 2) the principleof induction expresses a kind of psychological habit, 3) the semanticargument we quoted above in Section 2.2 that the negation of theprinciple of induction is meaningless, 4) a pragmatic vindication thatinduction using the straight rule works if any method of inductionworks, 5) a convergence argument based on higher order probabilities,or 6) that there is no guarantee, merely “posits.”

The probability associated with any foundational claim must beunderstood as a blind posit, i.e. as a good guess, which according toReichenbach can take into account pragmatic considerations. Thus,Reichenbach posits the existence of the external world as he thoughtit would make causal laws more “homogenous” (1938c). In arather mixed metaphor, Reichenbach compares our knowledge of theexternal world to seeing shadows of flying birds on the walls of acube in which one is confined. He argues that the patterns on thewalls, their regularity, would result in a high probability that thereare objects outside the cube producing the shadows.

Reichenbach’s eventual anti-foundationalism led him to a contestof opinions with C.I. Lewis, whose accounts of empirical learning andreasoning inMind and the World Order (1929), and inAnAnalysis of Knowledge and Valuation (1946) rested on aphenomenalist foundation of “qualia” (see the entry onqualia). Lewis, like Reichenbach a probabilist at heart, insisted that theassessment and revision of probabilities in the light of experiencerequires that some propositions obtained from experience be certain,or as he put it, “if anything is probable, something must becertain.” Reichenbach denied the claim, arguing that thedeliverances of experience directly provide, not certainty for claimsabout qualia, but only probabilities for claims about ordinary things.Those probabilities can be revised in the light of further experience.For the same reason, in his unremittingly critical review of KarlPopper’sThe Logic of Scientific Discovery (Popper1935), Reichenbach argues that probability assessments are essentialin the “falsification” of theories (1935e, 1978, vol. 2,p. 372). Lewis’ actual complaint appears to be based on amathematical error when computing conditional probabilities.Reichenbach pointed out Lewis’ error, but did not provide apositive example to refute Lewis’ concern (Peijnenburg &Atkinson, 2011). A more general problem that appears to have beenmissed in the discussion at the time is that since Reichenbach heldthat the probabilities of theories are to be assessed by Bayes rule,his position, unlike Lewis’, required a technical account of howclaims with non-extremal probabilities can produce changes in theprobabilities of others by some generalization of Bayes rule (see theentry onBayes Theorem). Since on Reichenbach’s view uncertainty is associated withobservation and perception and scientific theories are only confirmedwith a degree of probability, an account is required of how uncertaindata changes the probabilities of hypotheses logically remote from thedata. Much later, Richard Jeffrey (1983) provided an account ofBayesian updating that explicitly relates the probability of the datato the probability of the hypothesis confirmed—ordisconfirmed—by the data (see the entry onBayesian epistemology).

Bayes rule and the probabilities of theories posed another problempressed on Reichenbach by Ernest Nagel in a review inMind ofthe German edition of Reichenbach’sThe Theory ofProbability (Nagel, 1936; see also Nagel, 1938). Nagel’scourteous review found a series of difficulties withReichenbach’s theory, beginning with the logical status of the“probability implication” which Nagel thought was not,contrary to Reichenbach, of the same kind as material implication,because Reichenbach’s relation is not “extensional.”Nagel does not explain what he means, and Reichenbach’s extendedresponse in 1939 does not clarify the matter (1939b, 1978, vol. 2, p.388). In his response to Popper (1935e), Reichenbach had proposed twoways of evaluating theories, one of which is to count the relativefrequency of true statements among the consequences of a theory. Nagelpoints out that a theory 10% of whose tested predictions are falsewould scarcely have probability 0.9. Reichenbach does not attempt todefend this proposal in his reply to Nagel in 1938 (Reichenbach,1938a). That leaves the assignment of probabilities to theories byBayes Rule, which Reichenbach had also proposed, but Nagel observesthat if theories are to be evaluated by Bayes rule they must haveprior probabilities. How can there be a frequentist prior probabilityfor a theory, since we do not know to what reference class to assign aparticular theory and, whatever the reference class, according toReichenbach we do not and cannot know which theories in that class aretrue? As in several of his other responses to criticism,Reichenbach’s 1938 reply is unfortunately more defensive thanenlightening. About the reference class, Reichenbach says only“I do not think this is a serious difficulty as the samequestion occurs for the determination of the probability of singleevents” and recommends the choice of the narrowest referenceclass for which there are adequate statistics (though how the adequacyof a statistic for the truth or falsity of theories is assessed isnever explained). Reichenbach gave a sketch of such a procedure in his1935 reply to Popper, which Nagel pointedly does not think addressesthe problems. About the problem of the frequency of true hypotheses inthe reference class, Reichenbach says that we only need theprobabilities of theories in the reference class, not their truth orfalsity—a question-begging reply that Reichenbach must havesensed is unconvincing because he promises to address the issuefurther inExperience and Prediction (1938c). The discussionof the prior probabilities of theories in section 43 ofExperienceand Prediction is unsatisfactory for the same reasons.

Reichenbach’s most enduring distinction is between “thecontext of discovery and the context of justification.” ButReichenbach did not always allow the distinction, and the distinctionhe intended is not quite the one commonly attributed to him. In his1922 essay on “The Philosophical Significance of the Theory ofRelativity” (1922c) the (now conventional) distinction betweenthe context of discovery and the context of justification isformulated in other terms, and rejected:

It is important to follow the concepts by which the theory finds itsway step by step and to level criticisms at the theory from the sameintellectual path as was used in the creation of the theory. This workderives from such an attitude; indeed to renounce this method wouldachieve nothing but to promote traditional representations to a statusof absolute predominance. (Reichenbach, 2006, p. 97.)

In his review (Reichenbach, 1935e) of Popper’sThe Logic ofScientific Discovery, Reichenbach suggests that theories areordered by their prior probability, and the theory with the highestprior is further tested. InExperience and Prediction (1938c)and in his reply to Nagel (Reichenbach, 1938a), published in the sameyear, Reichenbach first formulates the distinction between the contextof discovery and the context of justification with regard tomathematics: the mathematical relations are what they are, and how wecome to recognize them is an entirely different, psychologicalmatter:

Theobjective relation from the given entities to thesolution, and thesubjective way of finding it, are clearlyseparated for problems of a deductive character; we must learn to makethe same distinction for the problem of the inductive relation fromfacts to theories. (1938a, p. 36– 37)

In other words, the distinction is supposed to be between objectiverelations among premises and conclusions, and subjective ways ofdiscovering those relations. The “context of discovery” isnot about search for hypotheses or about the order in which hypothesesare considered but about the search for the objective inductiverelation between a theory and a body of evidence. What littleReichenbach had to say publicly about how to search for promisinghypotheses was reported in his review of Popper’s book, and inhis dismissal in his reply to Nagel of Peirce’s“abduction” as a confusion of mathematical andpsychological relations. While Carnap later imagined an inductive,logically omniscient robot (Carnap, 1960), and Hempel much later (in“Thoughts on the Limitations of Discovery by Computer”,Hempel 1985) claimed that computerized generation of interestingscientific hypotheses is impossible, Reichenbach seems not to havegiven any thought to questions about how to come up with hypothesesand about better and worse ways to search through the space oflogically possible theories. He had one fundamental procedure forestimating true hypotheses: the straight rule, and one secondaryprocedure, whose application he never coherently explained, Bayes rulewith objective, frequentist, prior probabilities for hypotheses.

Further Reading. Putnam (1991) provides a helpful big pictureof Reichenbach’s mature views on metaphysics and epistemology.Chapter 7 of Milmed (1961) provides an overview of Reichenbach’sprobabilistic epistemology and views on induction. Salmon, in variouspapers (see the entry on theproblem of induction) tried in vain to save the straight rule, Reichenbach’spragmatic vindication of induction (see also Hacking, 1968). Eberhardt& Glymour (2011) try to reconstruct the details ofReichenbach’s probability logic and discuss some of thecriticism leveled at Reichenbach’s account in more detail.Fetzer (1977) and Háyek (2007) focus on the reference classproblem. Psillos (2011) disentangles Reichenbach’s argument forrealism. Galavotti (2011) provides a succinct attempt atreconstructing a coherent overall account of Reichenbach’sepistemology. Chapter 6 of Milmed (1961) provides one of the few gooddiscussions of Reichenbach’s views on logic. An entirecollection of articles in Schickore & Steinle (2006) explores thedifferent interpretations, historical connections and philosophicaloffspring of Reichenbach’s distinction of the context ofdiscovery and the context of justification.

4. Philosophy of Physics

Almost exclusively, Reichenbach’s scientific interest was inphysics. He regarded chemistry as an appendage, entirely reducible tophysics, and his only publication touching the subject is on MarcelinBertholot (Reichenbach, 1927a), the eminent French anti-vitalistchemist. Of biology, Reichenbach leaves open (at least in 1929) theserious possibility that life exhibits phenomena that cannot beexplained by physical laws, and of evolution he remarks that it hasdisappointed in failing to explain “the problem of life”(1929g, Section 2). He believed Freud’s psychoanalytic theorieswere scientifically warranted by Freud’s evidence, and in LosAngeles maintained friendships in the psychoanalytic community,including attending and sometimes lecturing at the PsychoanalyticInstitute. As a student Reichenbach showed some interest in psychology(Padovani, 2008, p. 13), however, except for brief passages inExperience and Prediction (1938c), Reichenbach publishednothing about psychology, and we have no evidence regarding howfamiliar he was with Freud’s publications. It seems unlikelythat Reichenbach knew Freud’s remarkably candid descriptionsbefore 1900 of his aggressive data collection procedures, and ofcourse Freud’s letters to Wilhelm Fleiss (Masson, 1986), whichreveal how little Freud thought of his own methodology, were unknownto Reichenbach.

Reichenbach’s most original work on the foundations of physicsis in three books,Axiomatization of the Theory of Relativity(1924h),Philosophical Foundations of Quantum Mechanics(1944b), andThe Direction of Time (1956b). The firstattempts an empiricist aufbau of the special and (in less detail)general theories of relativity from experimentally accessible causalrelationships; the second essays a novel 3-valued logic for quantumtheory; and the third addresses a long standing problem in physics andmetaphysics.The Philosophy of Space and Time (1928h),Reichenbach’s most widely read, most elegant, and least originalwork on physics, combines an extension to spacetime theories ofHelmholtz’s and Poincaré’s conventionalism aboutgeometry with a description of Einstein’s special and generalrelativity theories from the point of view of theAxiomatization, much of which it repeats.

4.1The Theory of Relativity and A Priori Knowledge (1920)

Einstein’s lectures on relativity in Berlin in 1917–18,attended at the time by Reichenbach and a handful of others, provideda shock to Reichenbach’s viewpoint.The Theory of Relativityand A Priori Knowledge, published in 1920, is an attempt toidentify neo-Kantian doctrines that must be abandoned and toarticulate what can be salvaged. Twenty-five years before, in hisdissertation published in 1897, Russell had undertaken a similarproject in connection with non-Euclidean geometry, arguing thatconstant curvature is the geometrical synthetic a priori. In 1902,Poincaré’sScience and Hypothesis essentiallysupplemented Russell’s view with the opinion that the choice ofconstant curvature geometry is underdetermined; the choice of one suchgeometry rather than another can always be compensated by a change ofphysics to save the phenomena. Without mentioning Russell orHelmholtz, Reichenbach takes general relativity to have refuted bothPoincaré’s geometrical conventionalism and Kant’sgeometrical apriorism. Most of the book addresses generalepistemological issues discussed above rather than issues aboutphysicsper se.

4.2 Early Writing on Relativity

Between the publication ofThe Theory of Relativity and A PrioriKnowledge, in 1920, and the appearance ofThe Axiomatizationof the Theory of Relativity in 1924, Reichenbach published aseries of professional and popular essays expounding or defending thespecial and general theories and Reichenbach’s ownaxiomatization, which was published in outline in 1921 inPhysikalische Zeitschrift (1921d). The professional essayswere variously addressed to audiences of philosophers or physicists.They are interesting for our purposes chiefly because they statephilosophical positions that Reichenbach had previously rejected, orwould later come to reject, or because they offer interpretations ofthe theories that are problematic. Notable in the last respect isReichenbach’s view that general covariance is a substantiveclaim of general relativity: the laws of physics have the same“form” in every coordinate system and frame of reference.Reichenbach does not explain the idea of the “form” of aphysical law.

While Kantian perspectives—thea priori“constitution of the object”—still linger, the“Philosophical Significance of the Theory of Relativity”,published in French in 1922, signals a break with some ofReichenbach’s previous views (1922c). Perhaps influenced byEinstein’s “Geometry and Experience” (1921)Reichenbach no longer holds that, contrary to the conventionalistviews he attributes to Helmholtz and Poincaré, the theory ofrelativity has established a geometry as part of physics to whichthere can be no empirically adequate alternative. After introducingthe idea of a universal force affecting all objects no matter howinsulated, and noting that gravitation as usually conceived is such aforce, Reichenbach writes that: “The solution to the problem ofspace is…found only in this conception we callconventionalism…which goes back to Helmholtz andPoincaré.” (Reichenbach, 2006, p. 135). The idea thatempirically indistinguishable hypotheses differ onlyverbally—“equivalent descriptions” inReichenbach’s terminology—is already developed here.

In the 1920s Einstein and Herman Weyl each sought a field theory thatwould unify gravitation and electrodynamics. Reichenbach took part,chiefly in criticizing both Weyl’s and Einstein’s efforts.The onus of his criticism was that mathematical elegance is aninsufficient guide: the fundamental quantities of a theory should bemeasurable, indeed have “coordinating definitions.” In hisview field theories whose fundamental quantity is an affine connectiondid not. Giovanelli (2023) provides an insightful account of theexchanges on this topic among Einstein, Reichenbach and Weyl.

4.3Axiomatization of the Theory of Relativity (1924)

In 1921 Reichenbach published a very limited précis of theapproach to the theory of relativity he ultimately presented in 1924(Reichenbach, 1921d). TheAxiomatization is either a workvery much out of its time, or the times have not changed much.Reichenbach’s statement of purpose might have been written atany time in the last quarter of the 20th century in response toQuine’s holism and to Thomas Kuhn’s incommensurabilitythesis (see the entries onbelief and onThomas Kuhn).

It is not easy to arrive at…a judgment with respect to theaxioms of a theory. Usually the axioms, representing higher levels ofabstraction, are quite remote from direct sense perception…

In order to avoid this difficulty…It is possible to start withthe observable facts and to end with the abstractconceptualization…The empirical character of the axioms [aboutobservables] is immediately evident and it is easy to see whatconsequences follow from their respective confirmations ordisconfirmations.

Unfortunately…every factual statement, even the simplest one,contains more than an immediate perceptual experience: it is alreadyan interpretation and therefore itself a theory…The mostelementary factual statements, therefore, contain some measure oftheory….

Does there exist any confirmation other than that of the theory as awhole? …Let us assume that the theory by means of which weexplain a certain fact is false and is to be replaced by a differentone. It is nevertheless possible that the new theory, when used forthe interpretation of this one fact, makes hardly any differencequantitatively, whereas it leads to considerable changes with respectto other assertions…

The new theory has merely to satisfy the requirement that it will notresult in a practically noticeable quantitative difference whenapplied to these elementary facts…For this reason all axioms ofour presentation have been chosen in such a way that they can bederived from the experiments by means ofpre-relativisticoptics and mechanics.All are facts that can be tested without theuse of the theory of relativity…The particular factualstatements of the theory of relativity can all be grasped by means ofpre-relativistic conceptions; only their combination within theconceptual system is new. (p. 5–7)

The book is divided into two parts, one on special relativity and theother on general relativity, each in some respects dependent on theother. The first part presents a series of postulates about thebehavior of light, accompanied by a series of definitions. Theprimitives are a directed acyclic graph whose vertices are pointevents and whose edges represent the relation “a signal can besent from eventA and received at pointB,”and a partitioning of the event space by assignment of each event to apoint on some real line—intuitively, a world line—suchthat the set of events assigned to the same line are equicardinal withthe line and their ordering by signaling is the ordering of the realline, and such that no two such world lines intersect. Axioms aboutthe behavior of signals, especially “firstsignals”—meaning light signals—are intended tospecify sufficient constraints so that the coordination determines aninertial frame of reference in Minkowski space-time. Extensivediscussion is given over to the impossibility of experimentallydetermining a “metrical” simultaneity relation—thatis, of finding experimentally a unique simultaneity 3-space for aninertially moving observer. “Matter axioms” are then addedspecifying transport properties of rigid rods and the connection ofrigid rod distance measurements with round trip time measurements(although one of the “matter axioms,” Axiom VI.2, isentirely about clock transport). Reichenbach asserts that the fourmatter axioms that are restricted to inertial frames follow from hislight axioms. That is not true, and 60 pages later Reichenbachqualifies the claim to mean only that the relations follow from thegeneral theory of relativity.

Reichenbach very briefly discusses the classical tests of the generaltheory of relativity: the red shift of light emitted from the sun, thebending of starlight passing near the limb of the sun, and theanomalous advance of the perihelion of Mercury. He summarizes:“…only the deflection of light and the red shift canfurnish evidence for Einstein’s relation between gravitation andthe real metric…” (p. 170). A far more enlighteninganalysis of the bearing of the classical tests had been given byHarold Jeffreys in 1919 and taken over (without acknowledgement) byArthur Eddington inThe Mathematical Theory of Relativity(Eddington, 1924); Jeffreys’s approach eventually led to themodern parameterization of metrical theories of gravitation.

TheAxiomatization has three difficulties. It contains amathematical error discovered by John von Neumann in the proof thattwo frames of reference satisfying his five light postulates anddefinitions could not have a world line at rest in common. Reichenbachacknowledged the error in 1925, and again in 1928, without proposingany modification to obtain uniqueness. In an unflattering review,Herman Weyl gave a second reason (Weyl, 1924; see also Rynasiewicz,2005): the light cone structure of Minkowski space time does notsuffice to specify a unique set of inertial frames unaccelerated withrespect to one another, and hence neither do Reichenbach’saxioms. By introducing a singularity in the infinite past, a system ofaccelerated frames can be used. Weyl took this fact to defeat the verypurpose of Reichenbach’s attempted reconstruction of thetheories: to found them on observable relationships. But perhaps themost important problem is that much of Reichenbach’sconstruction, without the philosophical commentary, had beenanticipated by A.A. Robb, first in 1914 (A Theory of Time andSpace), and then in 1921 (The Absolute Relations of Time andSpace) for both special and general relativity. Reichenbach doesnot mention Robb in 1924 or, to our knowledge, later, even thoughRobb’s 1921 book was reissued with a different title in1936.

Reichenbach’s apparent unfamiliarity at the time with theEnglish language literature on relativity is notable and unfortunate.He cites only one essentially English language source, fromTheAmerican Journal of Mathematics, and seems not to know of thework of Jeffreys or Robb. He does appear to have readEddington’s 1921 paper on Weyl’s theory. Whatever thecause, theAxiomatization, which is one ofReichenbach’s most original technical efforts, would have been adifferent work, or none at all, had he taken account of thosedevelopments.

Further Reading. Section 4.4 of Ryckman (2005) provides amore detailed analysis of what Reichenbach was trying to do in thisbook. Dieks (2020) contrasts Reichenbach’s and Weyl’sfoundations of space and time.

4.4The Philosophy of Space and Time (1928)

Published only 4 years after theAxiomatization,ThePhilosophy of Space and Time (1928h) is an engaging unificationof a multitude of ideas Reichenbach had previously published,including the distinction between descriptive and inductivesimplicity, the synonymy of “equivalent descriptions,” thedistinction between differential and universal forces, the necessityof coordinating definitions and their “conventionality,”the conventionality of spatial metrics and of “metrical”simultaneity, and the account of the logical structure of relativitytheories in terms of properties of causal relations in theAxiomatization.

The introduction toThe Philosophy of Space and Time betraysthe Marxist inclinations Reichenbach retained from his student years.Science he says, has become mechanized, and develops so rapidly andautomatically that its practitioners cannot reflect on what they aredoing or why. (He might as well have said they are “alienatedfrom their labor,” but he did not.) A philosophicalcounterweight (he might as well have said “antithesis,”but he did not) is needed. The counterweight cannot be provided byindividuals announcing philosophical manifestos; it requires thatphilosophers organize into groups so that, collectively, they can keepup with, and exercise analytic control over, the products of thescientific machine. The Vienna Circle, and Reichenbach’s owngroup of philosophers and physicists in Berlin, are the models.Together, philosophers should produceresults, notmanifestos. This last demand was the conclusion of the editorialReichenbach placed in the first issue ofErkenntnis in 1930(Reichenbach, 1930a); Schlick and Carnap, his collaborators infounding the journal, refused to sign the editorial, Schlick resignedentirely as an editor.

The book begins with a clear and compelling discussion of theinterdependence of properties and laws, illustrated through therelations of alternative measures of length and alternativegeometries. The examples are then used to make two general claims,first that within the structure of complex theories there areidentifiable points that are definitions, and, second, thatalternative systems of definitions and laws that account for the same“facts” say the same thing, have the same content.Reichenbach’s “definitions” are not purely verbal,they are “coordinative definitions” that specify physicalprocedures for determining the values of quantities. The point isreinforced with a distinction between differential and universalforces. Universal forces influence all objects in the same way (wheretheory specifies what a “way” is) and cannot be shieldedagainst. Heat, for example, is a differential force because it affectsbodies of different compositions differently, whereas gravity is auniversal force. A space with curvature can be equally described as aspace without curvature but with a universal force. Without explicitstatement, Reichenbach takes for granted the view, later defended inExperience and Prediction (1938c), that talk in coordinativedefinitions about ordinary, middle-sized, nearby physical objects andprocesses is intelligible and legitimate.

Reichenbach has a complex discussion of visualization. He introducesHelmholtz’s idea that to visualize a geometry is to understandwhat experiences one would have in a world in which that geometryholds, and extends the idea to the visualization of spaces withcompact topologies, where, Reichenbach argues, Euclidean geometrycould be sustained but at the cost of causal anomalies. For example,in a toroidal space, travel on a geodesic would bring one back toone’s starting point, which, Reichenbach argues, would requireeither abandoning Euclid or else allowing “causalanomalies” in the form of duplicate worlds at regular distancesfrom one another. (By “Euclidean” geometry, Reichenbachmeans here a geometry on a manifold homeomorphic toR³, since there is a complete Euclidean metric fortoroidal 3-space.) Reichenbach gives other examples of causalanomalies but no general characterization.

Even with Helmholtz available, Reichenbach is concerned to disputefurther the very idea that Euclidean geometry has a privileged statusin mathematical visualization that indicates some intrinsic limitationof thought. If humans had somehow been transported to a non-Euclideanworld, he says, they would have non-Euclideanvisualizations—surely a puzzling counterfactual. The very ideathat “pure visualization” has some normative content is amistake, he argues: “the normative function of visualization isnot of visual but of logical origin.” (p. 91). We do not presumeto know just what Reichenbach meant here, but for a plausibleelucidation one might turn to recent philosophical work on the logicalrelations implicit in geometrical diagrams.

Reichenbach’s discussion of space and time begins with a nearirony. Having argued against Kant’s effort to justify the basicsof Newtonian physics a priori, Reichenbach proceeds to argue that wecan knowalmost a priori that Newtonian theory is false. Forexample, Newton does not distinguish between the length of a movingsegment and its length at rest, but according to Reichenbach, this isana priori mistake: “the measurement of a segment witha measuring rod thatmoves relative to it requires theformulation of a new concept…The length of a movingline-segment is the distance between simultaneous positions of itsendpoints.” (p. 155) And further: “It follows from thenature of the extended concept of length that the length of a movingsegment is generally different from its rest-length.” To saywhich is the true length is “nonsense” (p. 157). Thus isNewton refuted.

Most of the rest of the book is a restatement of theAxiomatization with more extended examples and fewer proofs,and explicitly construed as a reduction of space time relations tocausal relations supplemented with definitions. The fundamental causalrelation to which Reichenbach appeals must be asymmetric and allowreidentification of objects or their features. Reichenbach thinks toobtain the asymmetry via a criterion of causation that invokes boththe preservation of something material and the asymmetry ofintervention. A “mark” placed at the beginning of a causalprocess can be received at the end of the process; Reichenbach’sexample is the color of a light signal. The asymmetry is that if thebeginning of the process is marked, the end of the process bears themark, but if the end of the process ismarked, the beginningdoes not bear the mark. The analysis thus depends on a distinctionbetween observing a mark and the asymmetric relation of causing amark. Reichenbach recognizes that for his empiricist, causalconstruction of the space metric, signals must be reidentifiable, arelation he notes that Kurt Lewin had discussed (in his 1922Habilitationsschrift) as “genidentity” (see also Padovani(2013)). Unsurprisingly, Reichenbach offers no guide toreidentification, which is obtained in all cases, including those oflight signals, from elaborate convictions—or at leasthabits—about all kinds of details and regularities of the world.Had Reichenbach thought it necessary to his constructions in theAxiomatization first to establish empiricist criteria ofgenidentity, the construction would not have begun.

Further Reading. Ryckman (2007) gives a more general overviewof Reichenbach’s philosophy of physics. Wilholt (2012, pp.44–48) provides a useful description of Reichenbach’sconventionalism and Giovanelli (2016) discusses in detailReichenbach’s early views on the geometrization of physics bygeneral relativity which were at some point supposed to form part ofthe (untranslated) appendix toThe Philosophy of Space andTime. Einstein’s brief review of the book can be found inEinstein (2021). Weinert (2023) discusses Reichenbach’s causaltheory of time and argues that it is an enthropic theory, since themarks should be understood as irreversible processes. The collectionby Lutz & Tuboly (2021) contains several articles that providecontext for Reichenbach’s philosophy of physics.

4.5 Quantum Mechanics and 3-Valued Logic

Reichenbach’s discussion of quantum mechanics contains a (then)standard presentation of the formalism of the theory, including theBorn rule and, without stating it, the Projection Postulate (see theentry onquantum mechanics). It is marked, however, by Reichenbach’s empiricism aboutmeaning, by his frequency interpretation of probability, by athree-valued “quantum logic” and by a curious neglect ofthe major problems and of some the most important previousfoundational literature. For later philosophical readers, the missingpiece in Reichenbach’s discussion of the theory is themeasurement problem (see the section on the measurement problem in theentry onphilosophical issues in quantum theory): he gives no account of how the dynamical equation of the theory canaccount for uncorrelated measurement and object systems becomingcorrelated in a measurement interaction.

There was a considerable history of “quantum logics”before Reichenbach’s work (see the entry onquantum logic). Martin Strauss had published a theory similar in spirit, but not indetail, to Kochen and Speckers’ partial Boolean algebras (whichappeared only thirty years later; see the entry on theKochen-Specker theorem), characterized by the fact that propositional connectives were allowedonly among simultaneously “verifiable” propositions.Reichenbach objects that some quantum mechanical propositions whoselogical form contains no propositional operators may be unverifiable.Paulette Fevrier had also introduced a three-valued logic, to whichReichenbach makes a similar objection. Strikingly absent is anyreference to Birkhoff and von Neumann’s 1936 paper on thelogical structure of quantum mechanics, which introduces a logicalstructure for orthocomplemented lattices and projective geometries. Wecan scarcely believe Reichenbach did not know of it and the neglect isdifficult to explain. Reichenbach seems to know of everything else,and he was in correspondence with Bargmann at the Institute forAdvanced Study. Reichenbach cites von Neumann’sMathematicalFoundations of Quantum Mechanics (1932), but only with regard tovon Neumann’sno hidden variable theorem (see the entryonquantum logic and the section on the measurement problem in the entry onphilosophical issues in quantum theory).

Reichenbach’s quantum logic has three values,True,Indeterminate, andFalse.Indeterminate isthe value of propositions that quantum theory implies cannot beassessed to be either true or false. Three unary propositionalfunctions are defined, one corresponding to classical negation, aswell as seven binary functions, including classical disjunction,conjunction and equivalence.

Reichenbach gives no axiomatization and no rules of inference(although passages suggest he intended an analog of modus ponens tohold for his two new conditional functions) and of course he thereforeprovides neither soundness nor completeness theorems. Nor does hecharacterize the class of functions that may be defined by compositionof his propositional functions. He does derive various properties ofthe new connectives from their definitions.

Reichenbach was quite explicit that logic is not empirical. “Therules of logic cannot be affected by physical experience.” (p.102). His logic withIndeterminate as a value of propositionsis the same as logic with “not empirically meaningful” asa value. In Reichenbach’s view that isthe logic, as itwere, with or without quantum mechanics. It is only that, beforequantum mechanics, the principle of equivalent descriptions precludedany need to pay attention to theIndeterminate value.

4.6The Direction of Time (1956)

As early as 1925, Reichenbach had essayed a probabilistic account ofthe direction of time (1925d) and in 1931 he still endorsed it,rejecting on familiar reversibility grounds proposals to rely onentropy increase. (1931i, 1978, vol. I, p. 336). His final work,almost completed at his death, was on the direction of time.

Reichenbach did not doubt that we have a definite psychologicalseparation of time into past and future, corresponding respectively toevents that can and cannot be remembered. His concern inTheDirection of Time is for a physical basis for the sameasymmetry:

The problem which the physicist faces can be formulated as follows.The elementary processes of statistical thermodynamics, the motionsand collisions of molecules, are supposed to be controlled by the lawsof classical mechanics and are therefore reversible, themacroprocesses are irreversible, as we know. How can thisirreversibility of macroprocesses be reconciled with the reversibilityof microprocesses? (p. 109)

The issues of a specific physical explanation of the psychologicalasymmetry, and of how there can be a changing psychologicalpresent—a moving “now”—were to have beenaddressed in an unwritten final chapter.

Reichenbach offers three accounts of the direction of time. The firstis that the direction of time in a region of space-time is the mostcommon direction of increasing entropy among multiple nearly isolatedsystems in that region. The idea, as Reichenbach notes, isBoltzmann’s. The second answer is that the direction of time isthat in which causal events turn high entropy systems into lowentropy, ordered systems. In the reverse direction of the sameprocess, he says, from low entropy to high entropy, the sequence ofevents is not causal but “purposive.” Thus stepping insand makes a shape (low entropy) from a flat surface (high entropy).The reverse process, removing the foot from the sand followed by thefoot impression filling with sand is not causal, but purposive. We donot pretend to understand this argument. The third argument intends toproduce a direction of time from causal directions inferred frommacrostatistics. This is the perhaps the aspect of the book most oftendiscussed, and it both breaks new ground and occasionally stumbles inthe furrows.

Reichenbach argues that spatially and temporally close coincidences oftwo causal processes suffice to determine that the coincidence iseither a common cause or a common effect of two other events. (Hepostpones the issue of how it can be known that there is any causalconnection between remote events.) He claims, correctly, that thisimplies that a causally connected network—an acyclic directedgraph—of events can have only two orientations; reversing anyedge requires reversing all other edges to yield an equivalentdirected graph. In an anticipation of 21st century discussions(Woodward, 2003), he considers whether interventions can disambiguatecause from effect in such a network, and argues that they cannot, oncounterfactual considerations that are not entirely clear to us.

Reichenbach’s discussion of the relations of probability andcausality began in his doctoral thesis, and continued in his 1925paper (Reichenbach 1925d) which, untilThe Direction of Time,he often cited himself with approval. InThe Direction ofTime Reichenbach acknowledges that the 1925 claims were incorrectand formulates a new “Principle of the Common Cause” thisway: “If an improbable coincidence has occurred, there mustexist a common cause” (p. 157). For any instance, he says, theconclusion of the principle is not certain but only probable, the moreprobable the more frequent the coincidences. He formulates fourconditions he says are satisfied when an association of simultaneousoccurrences ofA andB is due to a common cause,C, simultaneous with neitherA norB:C is not independent ofA or ofB, butA andB are independent conditional onCand also independent conditional on ~C. He says the principleis sometimes used in a weaker sense, requiring only that theprobability ofA and ofB be raised by theoccurrence ofC, but he says, that the independence canusually be established “by the use of a more detaileddescription [of] the causeC” (p. 161, n.2).Reichenbach does not say that his conditions are sufficient forC to be a common cause ofA andB, and asArtzenius has noted, they are not (Artzenius 1993 and the entryReichenbach’s common cause principle).

Using the principle, Reichenbach considers how to distinguish theconjunctive fork open to the future and the conjunctive fork open tothe past, structures I and II, respectively, below:

Considering only binary variables, Reichenbach says in structure I,the common cause case,A andB are not individuallyindependent ofC (the probability of each is greaterconditional onC than conditional on ~C), butA andB are independent of one another conditionalonC and on ~C. He recognizes what has come to becalled the “collider phenomenon”: in contrast to structureI, in a conjunctive fork open to the past (structure II),AandB are independent but dependent conditional onC, their joint effect (p. 163). In modern terminology, thisstructure is an “unshielded collider”. Reichenbach sensesthe tension between the collider phenomenon in conjunctive forks opento the past and his Principle of the Common Cause, which posits thatother than in exceptional cases a dependency between events orvariables is due to a common cause. In a conjunctive fork open to thepast there is an association, albeit conditional, that is notexplained by a common cause. Reichenbach does not regard two or morevariables with a common effect as unusual or an exceptional case.Instead, he claims that a conjunctive fork open to the past cannotoccur—it is logically possible but not scientifically possible.Whenever two or more variables,A andB, cause athird, he claims, there must also be a common cause ofA andB (structure III). Colliders (in current terminology) arepossible in nature, but conjunctive forks open to the past are not.His argument is that to allow otherwise would be to allow unscientificfatalism, which is inconsistent with thermodynamics (p. 162). Thedirection of time is given by conjunctive forks open to thefuture.

In this passage Reichenbach is close to a profound statisticalinsight, but ultimately muddles the explanation. Using an example withthe structure III he writes: “For instance, the spouting of thegeysers may have the effect that two clouds are formed which mergeinto one large cloud. Then the occurrence of this large cloud is aneffectE which satisfies [the probability properties ofstructure I]” (p. 162). This is an important mistake. Instructure III, representing the causal structure of the geyser cloudexample,A andB are independent conditional onD, but for almost all probability distributions,AandB arenot independent conditional onCandD. That is, III implies (for almost allprobability distributions) that none ofA,B,C,D are pairwise independent, thatA,B are independent conditional onD, and thatA,B are not independent conditional onCor onC andD. If the directions of edges in III arereversed, as in IV, the same relations hold, but withC andD interchanged. The direction of the causal relations betweenthe eruption and the formation of the large cloud is thus encoded inthe conditional probability relations. Yet Reichenbach mangles therelationships in a confused passage:

…if there exists a conjunctive fork with respect to a commoneffectE [as in structure II, withE in place ofC], the simultaneous occurrence ofA andBis more probable than a mere chance coincidence. Consequently, ifthere were no common causeC [D in structure III],the common effect would establish a statistical dependence betweenA andB and explanation would be given in terms of a“final cause”. …we regard final causes asincompatible with the second law of thermodynamics and consider suchforks impossible….this means: The principle of the common causedoes not exclude, throughout, a statistical dependence with respect toa common effect; but it does exclude such dependence if there existsno common cause. (p. 162)

Thus Reichenbach seems to think that in II above,A andB are independent conditional onC, but in III,A andB are dependent conditional onC andthe absence ofD. Reichenbach’s confusions on thesepoints is understandable—well into the 1990s much the same couldbe heard from several statisticians. In order to see the correctrelations, Reichenbach would have had to have explicitly formulatedthe factorization of the joint distribution for binary variables thatis implicit in his other discussions, and computed the jointdistribution of independent causes conditional on values of a commoneffect. He did not. Alternatively, with linear systems it is trivialto calculate the partial correlations of the causes controlling forthe effect, but Reichenbach undoubtedly did not know the formulas.Hartry Field (2003) argues that Reichenbach’s adherence to thePrinciple of the Common Cause blocked a better macroscopic causalaccount of the direction of time that would allow conjunctive forksopen to the past and use principles connecting causal direction withconditional probabilities such as those in the PC algorithm (Spirteset al., 1991).

Reichenbach continues with an illustration of “screeningoff” and a definition of “causally between” forevents. The illustration is a causal chainA →B →C.A andC areindependent conditional on the occurrence ofB. He does notsay that “screening off” requires thatA andC are also independent conditional on ~B, but hisstatement of the Principle of the Common Cause strongly suggests asmuch. He defines “causally between” as follows:

Definition 1. An eventA₂ is causally between theeventsA₁ andA₃ if therelations hold:

1. 1 >P(A₂,A₃)>P(A₁,A₃) >P(A₃) > 0
2. 1 >P(A₂,A₁)>P(A₃,A₁) >P(A₁) > 0
3. P(A₁.A₂,A₃) =P(A₂,A₃)

(p. 190)

Recall that the dot is conjunction, andP(A₂,A₃) inReichenbach’s notation isP(A₃ |A₂) =P(A₃.A₂) /P(A₂) in thecurrently conventional notation. Reichenbach goes on to claim that acommon effect of two causes is causally between its two causes, thusA₁ →A₂ ←A₃, which is obviously false if absence of anycausal connection implies independence. Throughout his life,Reichenbach implicitly makes this assumption in his examples aboutcausal relations. With such an assumption, it follows thatP(A₁.A₃) =P(A₁) P(A₃)≥P(A₁,A₃) =P(A₁)P(A₃)/P(A₃)=P(A₁), contradicting (1) in thedefinition. One may take these and other examples as evidence thatReichenbach did not have in mind instances of what is now called thecausal Markov condition—that conditional on its relativelydirect causes a variable is independent of variables that are not itseffects—or, alternatively, that he was simply confused about theproperties of “colliders”—common effects—incausal graphs. We prefer the latter interpretation.

Causal direction in a network is now determined by supposing anasymmetrical intervention is available, which Reichenbach calls a“mark.” Marks are assumed to be passed down causal chains,and to imply an increase in the probability of each downstream event.Using the time order so obtained, “causal relevance” isdefined as:

Definition 2: An eventA₁ is causally relevant toa later eventA₃ ifP(A₁,A₃) >P(A₃) and there exists no set of eventsA₂⁽¹⁾,…,A₂⁽ⁿ⁾ which are earlier thanor simultaneous withA₁ such that this set screensoffA₁ fromA₃. (p. 204)

Definition 2 provides a clear anticipation of a proposal made 20 yearslater by Patrick Suppes (Suppes, 1970).The Direction of Timeends with a discussion of quantum statistical mechanics focused onissues of the identity of particles through time.

5. Logic and Modality

Reichenbach entertained non-standard logics as early as 1925 in theform of “probability logic.” In theTheory ofProbability, probability logic amounts to no more than theassignment of probability values to formulas in the propositionalcalculus. Later, a three-valued logic was introduced for quantumtheory, as discussed above. Reichenbach’s major effort in logic,however, is inElements of Symbolic Logic, published in 1947,but begun as lecture notes for courses during Reichenbach’sTurkish years. The book is notable chiefly for the extended anddetailed effort to formalize universal logical structures ofconversational languages within the limits of first order logic andtype theory. Reichenbach’s knowledge of German, English, French,and especially, Turkish, helped to make his proposals linguisticallyserious, and the result, a detailed, and in some respects quiteoriginal, logical grammar, including accounts of adverbialmodification, tense and modality, is substantially richer than relatedlogical efforts of his contemporaries. Other topics of interest, suchas vagueness, are not discussed—Reichenbach perhaps consideredvagueness a mistake rather than a topic for logical analysis.

In 1948 Reichenbach circulated an unpublished manuscript (1948e, 1978,vol. 1, p. 409–428), touching on Hilbert’s program andGödel’s theorems. Reichenbach was not persuaded thatGödel’s proof of the impossibility of proving consistencywithin a sufficiently strong formalized language was of anyphilosophical significance.

…a more profound analysis shows Hilbert’s program to beunshaken, and independent of Gödel’s results.

To prove the latter statement first, let us inquire into thesignificance of the theorem stating that the proof of consistency isonly to be given in the metalanguage. What then would happen, if themetalanguage should turn out to be inconsistent? This would lead tothe consequence, not that our deduction of the statement ofconsistency is incorrect, but that the contrary of this statementcould also be deduced; and this indeed would make our statementvalueless. Now let us assume for a moment that Gödel’ssecond theorem did not hold, or with other words, that Gödel hadproved the contrary of his theorem.

This would mean that the proof of consistency of the languageL could be given withinL. A simple analysis showsthat this would not improve the situation, since in this case ourproof of consistency ofL would be of value only if we weresure thatL is consistent. In caseL were notconsistent, we could also deduce the statement of the consistency ofL, with the qualification that then the negation of thestatement were deducible too. Thus if the consistency ofLwere deduced withinL, this fact would not prove theconsistency ofL. (p. 409–410)

Hilbert’s meta-proof of consistency, by contrast, Reichenbachthinks is genuinely important. The consistency of any“interpreted” formalism can be judged empirically if thereis empirical evidence for the claims made within the formalism, andReichenbach argues that the proposition within the formalism thatasserts that the interpreted formal system is consistent has anempirical probability (the only kind Reichenbach allows, of course) atleast as high as that of any other claim in the language. But formathematics, the language must be separated into the part concernedwith rational numbers, which can be interpreted so that measurementsgive rational numbers as values, and the part concerned with real orcomplex numbers, which correspond to no empirical measurement.“Since all theorems of applied mathematics are deducible fromthe subsystem [of physical interpretation of the field of rationalnumbers]…it is only this subsystem which is verified by theinterpretations.” (1978, vol. I, p. 423) So we cannotempirically confirm any claims that are properly about real or complexnumbers, so we cannot empirically confirm their consistency. Hence weneed Hilbert’s program of metamathematics, which, byarithmetizing the claims of mathematical languages, reduces claims oftheir consistency to claims in finite mathematics about themanipulations of symbols—a claim thatcan be confirmedempirically (see the entry onHilbert’s program). We leave to the reader to diagnose how this separatist account fitswith Reichenbach’s insistence that Bayes theorem suffices toestimate an empirical probability for theoretical claims that are not“directly” empirically confirmable.

InThe Theory of Probability, modal necessity is identifiedwith universal quantification and possibility is identified with jointexistential quantification of a propositional matrix and existentialquantification of its denial. At least the spirit of this account isretained in Reichenbach’s later discussions of modality. Hisaccount of modality inElements of Symbolic Logic (1947c) wasdeveloped as a separate work inNomological Statements andAdmissible Operations (1954e), and is the basis for hisdiscussion of possibility and necessity in his essay on freedom of thewill. Completed before his death and published soon after,Nomological Statements and Admissible Operations was reissuedin 1976 under the title,Laws, Modalities andCounterfactuals, with an invaluable expository foreword by WesleySalmon. Reichenbach’s own presentation is a nearly impenetrablemix of conditions on truth, entailment, logical form, andverifiability, which surely contributed to its lack of influence.Little if any discussion of it is to be found in the decades ofliterature on conditionals and counterfactuals since itspublication.

In contrast to Carnap, who was at work on modality at about the sametime, Reichenbach understands the task of his theory to be to explainthe logical form and content of subjunctives, especially ofcontrary-to-fact conditionals, and, at the same time, to explain howempirical evidence can warrant some counterfactuals, warrant thedenial of others, and leave still others undecided. He thereforeregards the ability of the theory to account for our common sensejudgments of the truth or falsity of sentences involving subjunctives,counterfactuals, laws and modals as critical to its evaluation.

Reichenbach’s theory is founded on an account of natural laws.He views the logical form of all declarative sentences—modal,counterfactual or otherwise—as specifiable in an extensionalfirst order or typed language. Modality is a property of sentences,not part of their content, and modal sentences therefore involve botha declarative sentence and a meta-language claim about that sentence.As in Carnap’s account, all modality isde dicto.Reichenbach is initially concerned with distinguishing conditionalsthat, as indicative sentences, are true because of the falsity oftheir antecedents, from true (or at least assertable) subjunctiveconditionals that have false antecedents, or antecedents not known tobe true. The former contain uses of propositional connectives (or“operations” in Reichenbach’s terminology) that areinadmissible. Reichenbach’s strategy is to characterize“admissible operations,” that is assertable, subjunctiveconditionals, and their truth values, by their deductive relationswith indicative “nomological statements.”

Fundamental or “original” nomological statements are thoselogically equivalent to a true sentence (with no terms that“essentially refer” to particulars) in prenex form with atleast one universal quantifier and such that no logically strongersentence in the same vocabulary is true. Derivative nomologicalstatements are logical consequences of original nomologicalstatements, but not all derivative nomological statements are“admissible.” To avoid conditionals true because of falseantecedents (or sentences true because of irrelevant disjuncts),Reichenbach essentially requires that derivative nomologicalstatements be the logically strongest sentences in their vocabularythat are consequences of original nomological statements. Statementsare graded: Original nomological statements have grade 3, derivativenomological statements grade 2, and other statements grade 1. Acounterfactual conditional can only be true—orassertible—if its antecedent has a grade at least as high as itsconsequent. Rather elaborate further conditions on logical form areimposed to avoid counterexamples. Counterfactuals about particularsare understood to be true if the corresponding indicative conditionalsare instances of true nomological generalizations.

Modal claims of necessity are construed as indicatives combined with ameta-claim that the indicative is nomological. Possibility claims havecorresponding meta-claims asserting that the denial of the indicativeis not nomological. Almost as an afterthought, Reichenbach notes thatquantification must be over possible as well as actual objects, but heprovides no logical mechanism for specifying suchde remodalities.

6. Ethics and Free Will

Reichenbach’s discussion of free action and free will is anattempt to reconcile our judgments that some actions are done freelyand others are not with a scientific and materialist conception of theworld. An action is free if there is a prior circumstance in which a“volition” of the actor causes the action, and in that,otherwise the same, circumstance a volition to act otherwise wouldwith high probability have brought about a different action.Reichenbach goes to some lengths to explain just how the volition mustcause the action in order to be free, but the conditions are open tofairly simple counter-examples. He also makes no attempt to relatefree action to moral responsibility, or its absence to innocence.

The Rise of Scientific Philosophy (1951a),Reichenbach’s last and most successful popular book, presentshis broad philosophical viewpoint in an accessible way. The Englishprose is more fluent than in earlier works, and occasionally almost aspithy as Russell’s. Four chapters of the book provideReichenbach’s review of the history of philosophical pretensionsto a priori knowledge in metaphysics, epistemology and ethics. Writingof Kant, Reichenbach instructs: “His cognitive a prioricoincides with the physics of his time; his moral a priori, with theethics of his social class. Let this coincidence be a warning to allthose who claim to have found the ultimate truth.” (p. 61) OfHamlet, Reichenbach writes “To be or not to be—that is nota question but a tautology.” (p. 250) Much of the remainder ofthe book consists of selective popular science summaries, amidsimplified restatements of the views inThe Philosophy of Spaceand Time (1928h), andExperience and Prediction (1938c).The news in the book is the extended discussion of ethics. In hiscontribution to the Schilpp volume (1939a) on John Dewey, Reichenbachhad written at some length, and with considerable disdain, aboutDewey’s ethical theory. (His personal relations with Dewey areunknown to us, but it is possible that Reichenbach knew anddisapproved of Dewey’s enthusiasm for World War I.) Aboutethics, Reichenbach was at least as pragmatic as Dewey, but aboutmetaethics, and in particular about the logical form of ethicalsentences, he was in close accord with Charles LeslieStevenson’s imperativism. To assert “X isgood” is just to assert “I approve ofX: Do so aswell!” InThe Rise of Scientific Philosophy he insiststhat ethical statements express “volitional decisions,”without truth values, that are not subject to empirical knowledge. Theempirical issues of ethics are only the causal questions of relationsof means to ends. Reichenbach allows a place for logic in reasoningfrom ethical premises to ethical conclusions, but he insists that thecharacteristic feature of ethical statements, and the properconclusion of ethical reasoning, is a call to action. Whatever elsethey are, ethical claims are imperatives. His recommendation forresolving fundamental ethical disagreements is not philosophy orscience, but “social friction.” In keeping with hispolitics, Reichenbach’s last practical advice was the same asJoe Hill’s: Organize! But Reichenbach’s deepest ethicalinjunction was implicit in his most popular book: to form beliefs, tojudge them, to change them, to weigh actions, to distinguish real frommerely verbal differences, by the canons of scientific philosophy.

Further Reading. In a somewhat unusual description ofReichenbach’s work, McCumber (2016) characterizesThe Riseof Scientific Philosophy as “Cold War philosophy”that prudently avoids the traps of its time.

7. Reichenbach’s Influence

No complete list of Reichenbach’s doctoral students appears tobe available. In Turkey there was no graduate program in philosophy,but his student Nusret Hizir continued to develop Reichenbach’sideas in Turkey. After coming to UCLA in 1938 Reichenbach had at leastsix Ph.D. students known to us. W. Bruce Taylor studied with himbetween 1949 and 1953, but we do not know about his subsequentcareer—he listed no academic affiliation in 1976. Melvin Maronand Norman Martin are listed by the Mathematics Genealogy Project ascompleting their Ph.D.s with Reichenbach in 1951 and 1952,respectively. Martin was an expert in computer design and architectureand became professor at the University of Texas where he taught until1990 in three departments: Philosophy, Computer Science, andElectrical and Computer Engineering. We have no further record ofMaron. Cynthia Shuster became a professor at Washington StateUniversity, from where she was fired during the McCarthy era for“corrupting the youth” by inviting Robert Oppenheimer tospeak on campus. She continued her career at the University of Montanauntil her death. Hilary Putnam, who was at Princeton and then Harvarduntil his retirement, besides other important contributions lessdirectly connected to Reichenbach, combined Reichenbach’semphasis on learning in the limit with the theory of computation tocreate the foundations of computational learning theory, which remainsa major theme in theoretical computer science. Ruth Anna Putnam,Professor of Philosophy at Wellesley College and Hilary Putnam’swife, was not Reichenbach’s doctoral student, but was profoundlyinfluenced by his undergraduate courses. She typed up his notesforThe Direction of Time and later completed her Ph.D. underthe supervision of Rudolf Carnap (Maynes & Gimbel 2022, Lerner2004). Until his death, Wesley Salmon, who taught at Washington State,UCLA, Northwestern, Brown, Arizona and Pittsburgh, was the philosopherwho most prominently and loyally developed and defendedReichenbach’s views, especially but not exclusively his views onprobability and on the justification of induction. Carl Hempel tookhis doctorate in Berlin with Reichenbach. After moving to the UnitedStates, he taught at Yale, Princeton and thenPittsburgh. Hempel’s work on confirmation was discussed byReichenbach but had no connection to his own. Early in his careerHempel’s thought was more closely connected with Carnap’slogical approaches, while his later views were more closely alliedwith those of Thomas Kuhn, and in general his intellectual andpersonal relations with Reichenbach do not appear to have beenclose.

Some of Reichenbach’s ideas have reemerged in recent philosophywithout notice of the connection. Michael Strevens’BiggerThan Chaos (Strevens, 2003) reprises the views and arguments ofReichenbach’s doctoral thesis without the Kantian gloss. GilHarman and Sanjeev Kulkarni’sReliable Reasoning (2007)adopts a view of induction very close to Reichenbach’s.

Reichenbach’s views on underdetermination in physics weredeveloped extensively by Adolf Grunbaum, but as a metaphysical ratherthan epistemological thesis. The issue of the conventionality ofsimultaneity relations has attracted a large philosophical literature.Reichenbach’s Principle of the Common Cause has attractedextensive philosophical comment, much of it devoted to purportedcounterexamples to a strict universal claim that Reichenbachexplicitly denied. Without reference to Reichenbach, the principle wasrestated in the 1950s by Herbert Simon (1954) as a claim about theexplanation of “spurious” correlations. Under the name“Markov condition,” in the early 1980s Reichenbach’sprinciple was generalized by several statisticians, most notably TerrySpeed (Kiiveri & Speed, 1982), and today plays an essential rolein representation and search for causal relations. We do not knowwhether Simon, who had been Carnap’s student at the Universityof Chicago, knew of Reichenbach’s ideas; the statisticians verylikely did not. Perhaps not coincidentally, the senior author of thisentry, who helped to develop the directed graphical representationinto search and prediction procedures for causal hypotheses, studiedwith two of Reichenbach’s doctoral students, Schuster andSalmon. As noted above, Reichenbach anticipated formulations ofprobabilistic causal relevance advanced by Patrick Suppes inAProbabilistic Theory of Causality (1970), and the use of“marks” can be found in a somewhat different guise inSalmon (1984) and Dowe’s (2000) work on causality.

With the exception of discussions of the “conventionality”of simultaneity, Reichenbach’sAxiomatization seems tohave had little influence on subsequent work, whereas Robb’s,rediscovered in the 1970s, has attracted some development from bothphilosophers and physicists. The interventionist account of causationin theAxiomatization, and more clearly inThe Philosophyof Space and Time, has been developed in various ways by several21st century writers, without explicit debt to Reichenbach. MichaelFriedman (2001) has attempted to revive the quasi-Kantian viewpoint ofReichenbach’sThe Theory of Relativity and A PrioriKnowledge emphasizing the “relativized a priori.”

A considerable literature in linguistics has pursuedReichenbach’s ideas about the logical form and semantics ofconversational language, especially about tense and the logical formof adverbial modification (see Binnick in the Other Internet Resourcessection below for a bibliography). Kamp (2013) lays out in some detailthe enormous influence in linguistics of Reichenbach’s shortcomments inElements of Symbolic Logic (pp. 289–298)that recognized the distinction betweenspeech time,reference time andevent time in the logicalanalysis of tense. Thus, in the past perfect (“had slept”)the event time precedes reference time, which precedes speech time,while in the simple past (“slept”) the event timecoincides with reference time, while both precede speech time (seealso Malatesta, 2016). Though Reichenbach never provided a precisedefinition of reference time, the subsequent analysis of referencetime and its relation to event time led to the development of formalsemantics that could handle a rich representation of context (e.g.Kamp’s own Discourse Representation Theory). In an ambitiousbook-length study of Reichenbach’s theory, McMahon (1976) takesReichenbach’s proposal to be a progenitor of Chomsky’stheory of syntax, and attempts to supplement Reichenbach’saccount with appropriate re-write rules. Reichenbach’s owndiscussion contains no explicit generative grammar or computationalmodels. More recently, Seiler & Weber (2022) suggested thatReichenbach’s reference time is not always as sharply defined ashis writings suggest, and propose a modified account that permits anunderspecified reference time.

8. Retrospective

Hilary Putnam (1991) has praised Reichenbach’s work as“one of the most magnificent attempts by any empiricistphilosopher of this or of any other century” and called for morehistorical study of his work. We do not disagree, but a candidphilosophical retrospective assessment of any major philosopher isbound to find flaws. As Putnam emphasizes, in the end Reichenbachtried to found epistemology and metaphysics on probability relations,but he evaded or dismissed coherent and pointed challenges from ErnestNagel and others as to how his conception of probability could servethe purposes he required of it. Reichenbach’s work repeatedlyignored or discounted the contemporaneous or prior efforts of othersthat address the issues that concerned him, efforts that are in one oranother way as good as, or importantly better, than his own. That istrue with respect to Robb with regard to the causal construction ofspace-time relations; it is true with respect to Kolmogorov withregard to the theory of probability; it is true with respect toBirkhoff and von Neumann with regard to quantum logic. The effect hasbeen to make much of Reichenbach’s best-known work something ofa scientific and philosophical eddy rather than a main current.Reichenbach nonetheless was a central figure in forming the mainstreamof 20th century philosophy of science as a crossdisciplinary studydevoted to reconstructing and “justifying” receivedscience rather than to proposing novel scientific frameworks or novelmethodologies. At least in that fundamental respect, he remained aKantian throughout his career.

Bibliography

Primary Literature

Chronological List of Reichenbach’s Publications

Secondary Literature

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• Creary, L., 1969,A Pragmatic Justification of Induction, ACritical examination, Ph.D. thesis, Philosophy Department,Princeton University.
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• Kiiveri, H., and T. Speed, 1982, ‘Structural analysis ofmultivariate data: A review’, in S. Leinhardt (ed.),Sociological Methodology, San Francisco: Jossey-Bass.
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• –––, 1946,An Analysis of Knowledge andValuation, La Salle: Open Court.
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• Lutz, L. and A.T. Tuboly (eds.), 2021,Logical Empiricism andthe Physical Sciences, New York: Routledge.
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• –––, 2011, ‘The Concept of Probability inthe Mathematical Representation of Reality’,HOPOS: TheJournal of the International Society for the History of Philosophy ofScience, 1(2): 344–347.
• –––, 2011, ‘Relativizing the Relativized APriori: Reichenbach’s axioms of coordination divided’,Synthese, 181(1): 41–62.
• –––, 2013, ‘Genidentity and Topology ofTime: Kurt Lewin and Hans Reichenbach’, in Milkov and Peckhaus(eds.),The Berlin Group and the Philosophy of LogicalEmpiricism, Dordrecht: Springer, 97–122. (See alsoPadovani, 2008, Ch. 5.)
• –––, 2015, ‘Reichenbach on Causality in1923: Scientific inference, coordination, and confirmation’,Studies in History and Philosophy of Science (Part A), 53:3–11.
• –––, 2017, ‘Coordination and Measurement:What we get wrong about what Reichenbach got right’, in M.Massimi, J.-W. Romeijn, and G. Schurz (eds.),EPSA15 SelectedPapers (The 5th conference of the European Philosophy of ScienceAssociation in Düsseldorf), Cham: Springer, pp. 49–60.
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• Poincaré, H., 1902,La Science etl’hypothèse, Paris: Flammarion, 2004. EnglishtranslationScience and Hypothesis, London: Walter ScottPublishing, 1905.
• –––, 1912,Calcul desprobabilités, Paris: Gauthier-Villars.
• Popper, K., 1934,The Logic of Scientific Discovery,London: Routledge, 2002.
• Poser, H., and U. Dirks (eds.), 1998,Hans Reichenbach,Philosophie im Umkreis der Physik, Berlin: Akademie Verlag.
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• Quine, W.V.O., 1951, ‘Two Dogmas of Empiricism’,The Philosophical Review, 60: 20–43.
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• Robb, A.A., 1914,A Theory of Time and Space, Cambridge:Cambridge University Press.
• –––, 1921,The Absolute Relations of Timeand Space, Cambridge: Cambridge University Press.
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• Russell, B., 1914,Our Knowledge of the External World,London: Allen & Unwin.
• –––, 1945, ‘Logical Positivism’,Polemic, 1: 6–13.
• –––, 1948,Human Knowledge, Its Scope andLimits, New York: Simon Schuster, London: Allen & Unwin.
• –––, 1897,An Essay on the Foundations ofGeometry, New York: Dover, 1956.
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• –––, 2007, ‘Logical Empiricism and thePhilosophy of Physics’, in A. Richardson and T. Uebel (eds.),The Cambridge Companon to Logical Empiricism, Cambridge:Cambridge University Press.
• Rynasiewicz, R., 2003, ‘Reichenbach’sε-Definition of Simultaneity in Historical and PhilosophicalPerspective’, in Stadler (ed.),The Vienna Circle andLogical Empiricism, Berlin: Springer, pp. 121–129.
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• Salmon, W., 1977a, Introduction inCollected Works, inReichenbach 1977a, Volume 1.
• –––, 1977b, ‘Laws, Modalities andCounterfactuals’,Synthese, 35: 191–229.
• –––, 1979,Hans Reichenbach, LogicalEmpiricist, Dordrecht: D. Reidel.
• –––, 1984,Scientific Explanation and theCausal Structure of the World, Princeton: Princeton UniversityPress.
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• Schilpp, A., 1939.The Philosophy of John Dewey,Evanston: Northwestern University Press.
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• –––, 1892b, ‘Über die Anwendung desmathematischen Wahrscheinlichkeitsbegriffes auf Teile einesContinuums’,Sitzungsberichte der philosophisch-historischenKlasse der Bayerische Akademie der Wissenschaft, 4 (1893):681–691.
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Other Internet Resources

We include the links below for further reference, but note that someof the online resources contain some factual errors in content. We didour best to avoid such errors here, but welcome any comments andcorrections. We also welcome suggestions for further links.

• The Hans Reichenbach Papers, Archives of Scientific Papers, University of Pittsburgh.
• Binnick, R.,Project on the Bibliography of Tense, Verbal Aspect, Aktionsart, and Related Areas, see papers related to Reichenbach’s work on language.
• Hans Reichenbach, entry in Wikipedia.
• Hans Reichenbach, entry in the Internet Encyclopedia of Philosophy.
• Hans Reichenbach, biography of Reichenbach in MacTutor (University of St.Andrews).
• Hans Reichenbach, Mathematical Genealogy Project.

Related Entries

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Acknowledgments

We would like to thank Flavia Padovani for her comments and furtherreferences on a draft and the update of this entry, as well as theanonymous reviewer(s). We welcome further corrections andcomments.