Popper on Strict and Numerical Universality
by Flemming Steen Nielsen
In his article, ”Evolutionistiske forklaringer og kritikken af historicismen” (Evolutionary explanations and the Critique of Historicism) (1), professor Mogens Blegvad raised a series of searching objections to Karl Popper’s famous critique of Historicism (2) – objections which any future treatment of historicism would do well to take into consideration. Not least, Blegvad described and criticised those of Popper’s arguments that are based upon his distinction between on the one hand stricly universal law-statements and singular statements of individual fact on the other. In the following (3) I shall attempt to clarify this distinction of Popper’s – a distinction absolutely central to his thought, but never really treated as such by his commentators and critics. Hopefully this could provide a basis for a detailed reply to some of Blegvad’s objections.
The main sources of the subject are Popper’s works Logik der Forschung (1934) and Die beiden Grundprobleme der Erkenntnistheorie (written before Logik der Forschung but published in incomplete form as late as 1979). Die beiden … must be considered the best of these and gives us by far the the most complete description we now have of Popper’s views on the subject. Very naturally I shall base the following primarily on this work.
The first of the two problems of epistemology that Popper discusses in his book is the problem of induction, which he treats by first presenting his well-known falsificationist and fallibilist solution and then giving very detailed logical and transcendental critiques of various alternative solutions under the following five headings: (1) naïve inductivism, (2) strict positivism, (3) apriorism, (4) probability positions and (5) pseudo-sentence positions (”Scheinsats-Positionen”). He argues that the first four can be explicated and criticized in terms of a formulation of the problem of induction which in a traditional manner distinguishes between ”general” (”allgemeine”) and ”particular” (”besondere”) propositions. In connection with the fifth position, however, he finds it necessary to subject this very distinction to a closer examination. The expression ”pseudo-sentence positions” refer to epistemological positions which in order to circumvent the problem of induction assert that law-statements are not genuine sentences with truth-values, but instead a kind of ”formulae for the construction of singular sentences” (in analogy with propositional functions), or ”tools or instruments for the construction of prognoses, which cannot be true, false, or probable, but at most more or less useful.” (4)
To be able to examine the scope and validity of this view, Popper asserts, it is necessary to introduce a distinction between two kinds of synthetic, universal statements. The following example could illustrate this distinction:
(a) The trajectory of all stone’s throws (Steinwürfe) are parabolae, …
(b) the trajectory of all stone’s throws that have hitherto been measured are parabolae. (5)
There has been a tendency – not least in Classical Empiricism and Logical Positivism – to overlook the difference between these two kinds of statement, i.e. the difference between universal law-statements and empirical generalisations; but Popper in this respect like so many other respects follows Immanuel Kant. Like Kant, Popper insists that this distinction is indispensable for any attempt to characterize the theoretical, nomothetic sciences in an adequate way. Kant speaks about ”strenge Allgemeinheit” in the first case, and ”komparative oder angenommene Allgemeinheit” in the second. (6) Popper’s terminology is as follows: Statement (a) in our example is a (synthetic) strictly universal statement; statement (b) is a (synthetic), numerically universal statement. As we shall see, numerically universal statements strictly speaking are singular statements. (7)
Why do for instance Logical Empiricists overlook or reject the distinction? Popper’s explanation is that they tend to accept only such distinctions as can comfortably find expression in the their favoured logic, i.e. the logic of Principia Mathematica: ”Die Logistik”. But the distinction beween strict and numerical universality cannot be expressed in this Logistik, Popper insists. In our example both statements can be formalized as the so-called ”general” or ”formal implication”:
(Ax) (Fx Gx). (9)
From a purely logical point of view the lack of the distinction is of no importance according to Popper; but for the purposes of epistemology and scientific methodology we most certainly need both kinds of statement.
As the statements both can be formalized as general implications, i.e. statements about all members of a class, of course the difference between tham cannot be a question of logical form. So it must be a question of (logical) ”content”, i.e. a difference between the concepts involved? In the statements. This means that we must similarly distinguish between universal and individual concepts. This distinction is, according to Popper, ”unambiguous and absolute.” (10)
It may surprise that Popper considers the distinction ”unambiguous and absolute”. An obvious objection would be the following: There is at least one sense of ”unambiguous” in which it would seem quite absurd to label it unambiguous. For is it not a fact that what in one context functions as an element of a class, i.e. as an individuum, in another might itself function as a class? If that is the case, would it not be more correct to characterize the distinction as ”relative”? Popper’s reply to this objection is: ”True, but irrelevant!”. A more thorough analysis of such terms as ”class”, ”element”, ”universal” etc. will make clear why:
The most important source of confusion in connection with these terms is that we confuse three different distinctions, namely those between (i) class and element, (ii) class and subclass, and (iii) universals and individuals.
Ad (i) It must be admitted that the distinction class/element is relative in exactly the above sense. For instance, the concept ”iron” could be viewed as a class of physical bodies with certain properties in common. On the other hand, any of these bodies can be viewed as elements of the class ”iron”. But ”iron” can of course in another context be viewed as an element, namely as an element of a higher class ”metal”, where ”metal” is the class of classes of certain physical bodies [the following manner of exposition is mine, not Popper’s]:
metal – iron – a piece of iron
Such a string of class/element-related concepts we call a type hierarchy. Popper gives us the following example:
Type hierarchy (the example taken from Carnap, though somewhat changed): ”My dog Lux” is an element of the class ”dogs living in Vienna”, that class itself an element of the class of ”dog-classes in Vienna”; ”my dog Lux” is, however, also a class, namely, whose elements are ”the states of the dog Lux”; a single ”state of Lux” is (according to Carnap) ”a class whose elements are points in the world of experiences” etc. (11)
Ad (ii) The distinction between class and element must not be confused with another distinction, namely that between class and subclass (Überbegriff/Unterbegriff), a distinction which is also relative:
mammals dogs the dogs of Vienna
A string of class/subclass related concepts is called a hierarchy of concepts (”Begriffshierarchie”):
Hierarchy of concepts: ”In Vienna living Alsations”; ”in Austria living Alsations” etc. … ”in Austria living dogs”; ”dogs” … ”mammals” … ”animals”.- All these classes are of the same type, which can be seen from the fact that my dog Lux is an element of any of these classes. (Or from the fact that you can construct the general implication: ”x is a Viennese dog” generally implies ”x is an animal”.) (12)
Ad (iii) The third distinction, that between universal and individual concepts cannot be illustrated in a similar manner as the two others. Examples are:
naval battle the battle of Trafalgar
needle this needle
It is Popper’s thesis, then, that this distinction is unambiguous and absolute in exactly the sense in which the two others are not so. It ”cuts through” the type- and concept-hierarchies in an unambiguous manner:
Right through the types and the extensions runs a boundary in such a way that it runs through every type so that every type is divided by it into two parts. This boundary divides the whole system of concept extensions into two, namely the domain of universals (examples: ”the race of dogs”, ”A large, brown dog”) and the domain of individuals (examples: ”the races of dogs in Vienna”; ”my dog Lux”).
Each of the two domains contains type hierarchies, contains classes and elements; and each of the domains contains concepts of greater and smaller extent.
This boundary between universals and individuals is according to the present point of view unequivocal: Whereas one and the same concept in different contexts can function as a class or an element, as well as as a broader or a narrower class, we must be able to answer the question whether it is a universal or an individual unambiguously. (13)
What this last assertion means I shall discuss in quite a detailed manner in section VII. But first I shall present something the Popper never gives us, namely a systematic illustration of the way in which the distinction universal/individual cuts through the two other distinctions:
A. Type hierarchy of universals:
¬¬¬¬_ _ _ _ _
_ _ _ _ _
”metal” as a class of classes of bodies
”iron” as a class of physical bodies
a physical body as the class of its states
the states of a class of molecules
_ _ _ _ _
_ _ _ _ _
B. Concept hierarchy of universals:
_ _ _ _ _
_ _ _ _ _
body of metal
body of heavy metal
body of iron
body of cast iron
_ _ _ _ _
_ _ _ _ _
C. Typehiearchy of individuals
_ _ _ _ _
_ _ _ _ _
the animal classes of Vienna
dogs living in Vienna
my dog Lux
the states of Lux
_ _ _ _ _
_ _ _ _ _
D. Concept hierarchy of individuals
_ _ _ _ _
_ _ _ _ _
in Europe living mammals
in Austria living Mammals
in Austria living dogs
in Vienna living dogs
_ _ _ _ _
_ _ _ _ _
The distinction universal/individual cannot be defined in a non-circular way, Popper admits. Accordingly, he considers these concepts ”indefinable, logical primitives”. Nevertheless, it is possible to set up a ”simple and unambiguous criterion” for the application of them. There is an old logical rule to the effect that a given individuum cannot be characterized without the use of proper names or expressions functioning as proper names. This means that the individuum cannot be adequately characterized by general terms, but that proper names will have to be used. Universal concepts (Universalbegriffe) can therefore be defined as concepts which can be defined without the use of proper names, and individual concepts (Individualbegriffe) as concepts that need at least one proper name for their definition.
The term ”proper name”, however, is itself indefinable, he admits, but it is possible to say quite a lot about how to use it in a fruitful and satisfactory way:
A proper name is a sign which if necessary can be directly attached to the object (for instance like a dog tag) and which if necessary is used once and for this object only. (If the object is such that an actual attachment is impossible – for example a name of a country and things like that – the proper name nevertheless can be ascribed to the nation’s borders; or it can be defined by veritable proper names like ”The Conference of February 8., 1893” [….]). Proper names are at the same footing as (demonstrative) references like ”this dog”, ”today” etc. 14)
Two ”guiding propositions” can be formulated in order to make quite precise the relation of irreduceability among universals and individuals:
(1) An individual object cannot unambiguously be characterized in its individuality by universal concepts alone, i.e. without proper names.
(2) A universal cannot be defined solely by proper names or by a class of individual concepts.
Guiding proposition (1) can be explained in the following manner: Let’s take the individual thing ”Lux”. If we attempt to characterize the dog Lux in general tems – i.e. universals – we soon discover that what we end up with will always be a class – not an individuum. We might try describing Lux as a poodle, a black poodle, a two years old poodle etc.; but unless we use proper names the result will always be a class. In fact, even if we narrow our description so much that only a single dog (or even no single dog) actually exists, we will only have arrived at a ”kind” of objects – a class!
By contrast, we can easily characterize an object in an unambiguous manner if we introduce proper names og terms functioning as proper names, for instance ostentative expressions. For instance, we can refer to it by applying expressions like ”Lux”, ”my dog Lux”, ”the dog which in 1930 carried dog tag no. 17948” etc. Even if we make use of space/time coordinates the use of proper names is implied:
Especially definite specifications of space and time make unambiguity possible. This is an important point. One must not overlook the fact that it must be specifications of a particular place or a particular moment of time; these again always involve proper names. The point of origin of a space/time coordinate system can only be determined by proper names (for example Greenwich or the Birth of Christ) or – what is actually the same – by direct (”demonstrative”) reference. (Only a reference to an ”individual coordinate system” specified in this way could work as ”principium individuationis”). Also, a particular human being, for instance Napoleon, can be characterized in an unambiguous way by giving his place and time of birth: but thereby individual concepts are being used. (15)
Two important aspects of Popper’s concept of an individual concept might seem to conflict with ordinary usage. First, he stresses that an individual concept according to his chosen way of speaking doesn’t have to be a well-defined physical body. Thus he considers ”The Battle of Waterloo” an individual concept , whereas ”iron cube with sides of 1 cm” is a universal concept. (16) Secondly, an individual concept need not refer to single ”objects”. ”All persons leaning out of a window in Copenhagen just now” or ”all persons who have ever been leaning out of this window” are individual concepts according to his definition. They need proper names or indexical terms to be formulated.
Analogously, according to the second guiding proposition universal concepts cannot be defined by proper names or by reference to a specific class of individuals. This principle is of the greatest epistemological importance – not least because it implies that even if universals may stand in a class/element-relation to individuals they cannot be ”reduced to” or ”constituted by” concrete classes of individuals.
Although we find this point expressed both in Die beiden … and in Logik der Forschung, I prefer illustrating it by the far more elegant and clear treatment in the important article ”The Demarcation Between Science and Metaphysics” from 1955. (17) Here Popper has collected his arguments against Rudolf Carnap’s various theories of demarcation and meaning. In his critique of what he calls ”Carnap’s first theory of meaninglessness” Popper attempts to show that Carnap presupposes a particular, extremely simple kind of radical nominalism which can be shown to be untenable. According to this, all non-formative words (i.e. all words which are not logical constants) are names. This implies that not only proper names like ”Fido” are names, but that even a word like ”dog” strictly speaking is a kind of name, namely the name of, for instance, Fido, Candy and Tiffin. In a language constructed according to this assumption the meaning of general terms is given by an enumeration of the individual things denoted, i.e. through an enumerative definition. However, such a language can be shown to be totally inadequate as the language of science, Popper objects. This is because it has the absurd property that all its sentences are analytic – either analytic truths or contradictions. No synthetic sentence can be formulated in it, simply because the truth or falsity of all its sentences can be decided by a simple inspection of the enumerative definitions giving the meaning of the non-logical words used:
That this is so may be seen from our example. ”Fido is a dog” is true because Fido was one of the things enumerated by us in defining ”dog”. As opposed to this ”Chunky is a dog” must be false, simply because Chunky was not one of the things to which we pointed when drawing up the list defining ”dog”. Similarly, if I give the meaning of ”white” by listing (1) the paper on which I’m now writing, (2) my handkerchief, (3) the cloud over there and (4) our snowman, then the statement ”I have white hair” will be false, whatever the colour of my hair may be.
It is clear that in such a language hypotheses cannot be formulated. It cannot be a language of science. And conversely, every language adequate for science must contain words whose meaning is not given in an enumerative way. Or, as we may say, every scientific language must make use of genuine universals, i.e. of words, whether defined or undefined, with an indeterminate extension, though perhaps with a reasonably definite intensional ’meaning’. (18)
This also shows that any attempt to define universals enumeratively from individuals is doomed to failure. Let me add here that it would be no use to try a definition like such as ””dog” =def. ”Fido, Candy, Tiffin and all things similar to these”. For even a cat or a turtle are similar to Fido, Candy and Tiffin in some respects, of course. And a suggestion like ””dog =def. ”Fido, Candi, Tiffin and all things similar to these in respect to dogness” would of course lead us right back into the use of a universal concept.
These considerations are also destructive of any idea of a ”logical process of abstraction” which is supposed to make it possible for us to move from individual to genuinely universal concepts, although of course there is a method by which we can construct classes through ”abstraction” (- but these classes will remain individual concepts). (19) In fact we would do well to stop talking about a ”method of abstraction” or a ”process of abstraction” and to speak instead of a problem of abstraction in analogy with the classical problem of induction. The problem of induction arises from the relationship between singular end strictly universal statements. Strictly universal statements, according to Popper, are such that only involve universal concepts. Singular statements are such that involve at least one individual concept. The problem of abstraction, accordingly, is a problem about the relationship between universal and individual concepts. Both problems underline the hypothetical, tentative, almost groping nature of human knowledge. In science we work with genuine, strictly universal law-statements. These cannot be verified from our singular, experiental statements. Analogously, we have to make use of genuine, universal concepts; but we have no method – by way of ”reduction”, ”constitution”, or ”explication” – for once and for all securing an entire arsenal of absolutely unambiguous and, at the same time, concretely applicable universal concepts.
Now let us return to the question about what Popper might mean by characterizing the distinctions between universal and individual concepts – as well as that between universal and singular statements – as unambiguous. Some of his formulations might give the impression that what is meant is that a simple inspection of what we could call ”the grammatical character” of a given formulation makes it possible for us to decide once and for all whether a statement is strictly universal or not. The frequent talk of ”proper names” points towards an interpretation like that. Despite the somewhat heavy-handed formulations used by Popper when introducing the distinction in a general way, many of his later formulations make it clear that he has a more subtle view in mind. Far from maintaining that the grammatical form unambiguously reveals the universal or individual character of a statemement, he is quite aware that interpretation involving context almost always is necessary.
For instance, he points out, it is not possible simply to say whether a word like ”pasteurized” is an individual or a universal concept; for of course it can function as both:
”Pasteurized” can be defined either as ”treated according to Louis Pasteur’s instructions” (or something like that), or as ”heated to 80 degrees Celsius and kept at that temperature for ten minutes”. By the first definiton ”pasteurized” is an individual concept; by the second it is a universal concept. (19)
So context is decisive: In a historical discussion of the historical development of medical science, the expression might well be used as an individual concept, while in a theoretical discussion of the resistance of various micro-organisms it would be natural to use it as a universal concept. Another of Popper’s examples is the following, which introduces what we might call ”intended meaning”:
The use of the word ’mammals’ as an example of a universal name might possibly cause misunderstanding. For words like ’mammal’, ’dog’, etc., are in their ordinary use not free from ambiguity. Whether these words are to be regarded as individual class names or universal class names depends upon our intentions: it depends upon whether we wish to speak of a race of animals living on our planet (an individual concept), or of a kind of physical bodies with properties which can be described in universal terms. (20)
Both examples show that when Popper characterizes the distinction between individual and universal concepts as unambiguous he certainly does not think of ”unambiguity of a formulation” – a kind of unambiguity which would make it possible for us unambiguously to decide the question by sheer inspection of sentences. On the contrary, both ”pasteurized” and ”mammal” are ambiguous in that sense. We might express this important difference by saying that while the distinctions admittedly are ”grammatically ambiguous”, they are asserted by Popper to be ”logically unambiguous”. Exactly for that reason it is important to distinguish between their strictly universal and their individual uses. Popper might be said to defend the logical distinction by simply challengeing any opponent to explicate these and similar examples without applying it. In fact, this is exactly what Popper does in the epistemologi, methodological and metaphysical parts of his philosophy. Let me conclude this paper by illustrating this by offering a few more examples.
In The Poverty of Historicism Popper stresses the importance of distinguishing between theoretical and historical sciences. Theoretical physics is, of course interested in finding and testing universal laws; the historical sciences are interested in actual, singular, or specific events, rather than in laws and generalisations. (21) A historical explanation takes all kinds of universal laws (for instance those of economics) for granted when attempting to explain its singular statements. Both kinds of science use the hypothetical-deductive model when trying to offer, test, and predict causal explanations:
In the sense of this analysis, all causal explanations of a singular event can be said to be historical in so far as the ’cause’ is always described by singular initial conditions. And this agrees entirely with the popular idea that to explain a thing causally is to explain how and why it happened, that is to say, to tell its ’story’. But it is only in history that we are really interested in the causal explanation of a singular event. In the theoretical sciences, such causal explanations are mainly means to a different end – the testing of universal laws. (22)
The distinction is also, as might be expected, of great importance in connection with Popper’s discussion of the question whether the Theory of Evolution gives us reason to believe that there is such a thing as a law of evolution. Here Popper distinguished between (a) a theory about what might be called ”the Darwinian or Neo-Darwinist mechanism of evolution” and what he calls (b) ”the hypothesis of biological evolution” as a theory about an individual, though enormously complex occurrence. (23) (b) is not, as many seem to believe, a universal law, as many believe. Rather it must be viewed as a rather complex singular historical statement quite analogous to, for instance, ”Charles Darwin and Francis Galton had a common grandfather”. So even if an expression like ”all vertebrates” might look like a universal concept, in this context it is used as an individual concept, for ”all vertebrates” refers only to all vertebrates existing on Earth – ”… rather than to all organisms at any place and time which have the constitution which we consider as characteristic of vertebrates”. (23)
What we call the evolutionary hypothesis is an explanation of a host of biological and paleological observations – for instance, of certain similarities between various species and genera – by the assumption of the common ancestry of related forms. This hypothesis is not a universal law, even if certain universal laws of nature, such as laws of heredity, segregation, and mutation, enter with it into the explanation. It has, rather, the character of a particular (singular, specific) historical statement. (24)
This view of Popper’s might be seen as challenging anybody who disagrees to develop a version of the hypothesis of evolution in a form not using ”proper names or terms functioning as proper names”.
An analogous situation is found in a well-known cosmological debate between Popper, Adolf Grünbaum, and others. (25) At a particular stage of the discussion, Popper defends the use of egocentric particulars such as ’I ’, ’here’, and ’now’. Admittedly, they do not have a place in theoretical physics in a narrower sense; but this doesn’t mean that they don’t rightfully belong in cosmology:
”The present state of the surface of the moon suggests that …” is a phrase which is fully legitimate in science, though it is not likely to occur in theoretical physics. ”The present age of the universe” is a perfectly good term in cosmology, and one which it would be quite unnecessary and pedantic, if not downright misleaning, to replace by ”the age of the universe on October 14, 1970”. In other words, the past, present and future are perfectly good terms in cosmology and astronomy, two excellent examples of (to some extent historical) physical sciences. It is fully legitimate to remind the astronomer that what he observes, in certain cases, is the state of star 1,000 years ago, or of a gallaxy 100,000 years ago, where ”ago” is just a synonym of ”before the present”. The fact that these notions do not occur in theoretical physics, and that we replace them by names and dates in history, does not show that they are to be expunged.
Nor are they expungeable. It is perfectly true that astronomers can use coordinates instead of speaking of the Great Nebula in Andromeda. But the coordinates go back to the axis and equator of the Earth, to here-and-now terms. (The Earth changes its axis in time; and although we may speak of the north pole as the sky on ”October 14, 1970” we must not forget that ”October 14, 1970”, though in many respects preferable to ”now”, refers to a zero date which is highly conventional and anthropomorphic. Nobody claims to know even the precise year of the birth og Jesus Christ.)
Thus my thesis is that notions like ”the present” are needed, if not in theoretical physics, at any rate in physical science. But I want to claim even more. Theoretical physics uses all the time spatiotemporal variables; and without applications in which these variables are specified (in the last instance with the help of ”here and now”), they would have no reasonable function whatsoever. (26)
The importance of Popper’s distinction is seen elsewhere in his philosophical writings. To mention only a few examples, the impossibility of reducing strictly universal law-statements to a finite number of singular statements is of course decisive for his epistemological deductivism and fallibilism. The distinction is also used by him in his arguments concerning free will. (27) Likewise, of course, in his critique of determinism and in his radical emergentism with its daring idea that our natural laws might not, after all, be strictly universal because they after all seem to be the result of an ’evolution’ of our individual, open universe. (28)
Translated with some changes from: ”Begrebet streng universalitet hos Karl Popper”, FILOSOFISKE STUDIER, vol. 9, Copenhagen 1987, pp. 125-142, Copenhagen 1987.
Dedicated to Mogens Blegvad
(1) FILOSOFISKE STUDIER, vol. 5, Copenhagen 1982, pp. 7-32.
(2) See especially Karl Raimund Popper: The Poverty of Historicism, 1944-45, 1957; K.R.Popper: The Open Society and Its Enemies, 1945.
(3) A shorter version of this paper was read at the Polish/Danish philosophical seminar, Copenhagen 1983.
(4) K.R.Popper: Die beiden Grundprobleme der Erkenntnistheorie. Aufgrund von Manu-skripten aus den Jahren 1930-33 herausgegeben von Troels Eggers Hansen, Tübingen 1979, p. 159 ff.
(5) Ibid. p. 228.
(6) Immanuel Kant: Kritik der reinen Vernunft, 2. Aufl. 1787, p. 3.
(7) In Logik der Forschung, § 13 Popper uses the expression ”spezifische Allgemeinheit” for Kant’s ”strenge Allgemeinheit”. In the English version (The Logic of Scientific Discovery, 1959), he prefers the expression ”strict universality”.
(8) See for instance Rudolf Carnap: ”Eigentliche und uneigentliche Begriffe”, Symposion Vol. I, 1927; Der logische Aufbau der Welt, 1928, p. 213.
(9) Logik der Forschung, § 14.
(10) Die beiden Grundprobleme, p. 234.
(11) Ibid. pp. 233-34.
(12) Ibid. p. 233.
(13) Ibid. p. 234.
(14) Ibid. pp. 234-35.
(15) Ibid. pp. 235-36.
(16) Die beiden Grundprobleme, pp. 238-41; Logik der Forschung, 4. Ausg. 1971, pp. 27-28.
(17) K.R.Popper: Conjectures and Refutations, 1963, Ch. II.
(18) Conjectures and Refutations, p. 226.
(19) Logik der Forschung, p. 37 note 1.
(20) The Logic of Scientific Knowledge, p. 59.
(21) The Poverty of Historicism, pp. 143-147.
(22) Ibid, p. 144.
(23) Ibid. p. 107 note.
(24) Ibid, § 30. For a closer treatment of this distinction as well as the problem of the respective falsifiability of these two kinds of ”theory of evolution”, see my ”Karl Poppers som evolutionistisk filosof”, (”Karl Popper as an Evolutionist Philosopher”) in: Niels Bonde, Jesper Hoffmeyer and Henrik Stangerup: Naturens historiefortællere, II: Udviklingsideens historie, Copenhagen 1987, Ch. 15.
(25) Adolf Grünbaum: ”Popper’s Views on the Arrow of Time” and Popper: ”Grünbaum on Time and Entropy” in Schilpp (ed.): The Philosophy of Karl Popper, p. 775 and p. 1143.
(26) Ibid. p. 1143.
(27) For instance, Die beiden Grundprobleme der Erkenntnistheorie, Dritte Auflage pp. 481 ff; The Open Universe, pp. 41 ff, pp. 128 ff.
(28) The Open Universe, p. 143.