Numbers in parentheses refer to the footnotes at the bottom of the writeup.
Why does Entropy Increase?
There is no rigorous theory that provides a complete explanation for why the state of certain thermodynamic systems will spontaneously change. The second law of thermodynamics(1) is the closest that science has come to providing an explanation for this observation. What is the second law of thermodynamics (from here on the SLT) and what can it tell us about this kind of change?
Many different statements of the second law have been formulated. After the discovery of statistical mechanics, even more statements of the second law appeared. The statements of the second law that appeared before (or during) the discovery of statistical mechanics (c. 1900) are sometimes referred to as classical formulations. Classical formulations of the SLT might then include the statements of Carnot, Clausius, Kelvin, Gibbs, and Planck. More recently, other scientists have attempted to restate the SLT without reference to statistical mechanics. I will refer to these statements of the second SLT, which include the work of Caratheodory, Lieb, and Yngavson (from here on LV) as neoclassical formulations. Here, Caratheodory and LV are included in the same category for logical rather than temporal reasons(2).
The purpose of this paper is to provide a critical comparison of the LV (a neoclassical formulation) and the classical formulations of the SLT. But before I make any comparisons, I want to make sure that I specify exactly what I am comparing.
To simplify the job of comparison, I will restrict the comparison of these formulations in general to a comparison between the classical statement of Clausius and the neoclassical statements of LV and Caratheodory. The logical structure for the development of this comparison will be roughly chronological: In section I, I will describe first the Clausius statement and discuss its classical counterparts before attempting in section II a description of the LV statement and how it leads naturally to a discussion of Caratheodory's statement, and finally, in section III, a comparison between the two. For the uninitiated second law historian, this development may require some patience.
I. The Classical Formulation
I will now state my working version of Clausius' statement of the SLT. It is a working version for several reasons. First, Clausius was German and wrote in German, so my version will be a translation. Second, Clausius published several different versions of the SLT during his career. Working with the assumption that he was making progress in his understanding of the SLT throughout the course of his career I have selected one of the later versions that he published. I will not go into a historical discussion of his development of this version. Third, Clausius relied on a rather vague notion of forces (Kraft) in his explanation of reversible and irreversible that I have replaced with more tangible explanations that are due more to Planck than to Clausius.(3)
For a system and its surroundings that cannot exchange heat with the rest of the universe:
∆S = Integral from Psii to Psif( dq / T)
∆Srev = 0 and ∆Sirr > 0
where: S represents the entropy function
Psii and Psif represent the system in its initial and final states
dq represents the heat the system exchanges with its surroundings
T represents the temperature of the surroundings.
∆Srev represents the path integral ∆S along a reversible path (a path along which the system remains within some neighborhood of thermodynamic equilibrium even in the limit where the size of that neighborhood becomes arbitrarily small).
∆Sirr represents the path integral ∆S along an irreversible path (a path along which the system is not within some neighborhood of thermodynamic equilibrium).
This statement of the SLT is probably the best classical formulation. As mentioned before, there are other statements (or parts of statements) provided by Carnot, Kelvin, Gibbs, and Planck. I will try to address each of these in turn, and in the process elaborate on this statement, which is due largely to Clausius.
Carnot made the earliest recognizable attempt at stating the SLT. I call it an attempt, because it is not completely clear that Carnot's results do constitute a full statement of the SLT. The Carnot statement(4) refers to the maximum efficiency of some engine that, working in a cycle, draws heat from a high temperature reservoir and expels heat to a low temperature reservoir. The condition of maximum efficiency is obtained in the limit that the engine works reversibly (here reversible can be taken to mean the same thing as it does in the Clausius statement).
Carnot's statement is correct, but it is much weaker than the Clausius statement. Carnot's statement is less abstract, and therefore less general, limiting itself specifically to engines that work in cycles. In fact, the Carnot statement can be reproduced by calculating ∆Srev over the cycle of Carnot's engine. Thus while Clausius' statement may not have been possible without Carnot's, the Clausius' statement is a stronger representative of the classical formulation of the SLT.
Of the classical formulations, the statement due to Lord Kelvin (also known as William Thomson) is probably the closest to that of Clausius, both historically and logically. Their early work progressed along similar lines, and could be characterized as a more rigorous version of Carnot's statement(5). In each case, the system to which their statements applied is an engine working in a cycle. The minor differences that existed between the two at this stage are only that Clausius did not rely on a definition of absolute temperature in his development and that Kelvin's definition of reversible relied more on a temporal scheme(6). Both statements suffer from the same narrowness as Carnot's statement, and both possess that ambiguity inherent in the use of language. The next step for both was to attempt a mathematical restatement of this early work.
Kelvin had the first success in 1854 when he published the equation:
for reversible cyclic processes where heat Qi is exchanged with reservoirs at Ti
He regarded this equation as the first mathematical expression of the SLT, and it probably is precisely that. Ten years later (in 1864), Clausius himself addressed the problem, but extended the theory to include the possibility that the reservoirs varied in temperature during a cyclic process. He arrived at the result:
for a cyclic process with heat dq exchanged with reservoir at T equality holds for the reversible case
Clausius also regarded this as a mathematical expression for the SLT. A year later, Clausius took another important step. He extended his statement to include processes that do not complete a cycle, and gave the form of Clausius statement (including the first definition of an entropy function) that I have taken to be the best representative of classical formulations of the SLT. Lord Kelvin, then, preceded Clausius in developing a mathematical statement of the SLT; but it was Claudius that gave it in its most general form.
These mathematical statements were a great advance for the theory of thermodynamics. Ambiguities often result from attempts to use verbal statements of physical theory to produce quantitative predictions. These classical formulations, exchanging words for equations, were extremely useful in that sense. But ambiguities remain in the classical formulation, particularly in the slippery concepts of reversible and irreversible. To anticipate a little, one can imagine how a desire to make even these ambiguities mathematically precise might lead to the neoclassical formulations of the SLT. Before continuing with that point, I want to show why other statements of the SLT (those made after Clausius' statement of 1865) are not to be considered representatives of the classical formulation.
In addition to the work he did in thermodynamics, Max Planck was an important figure in the development of quantum mechanics and statistical mechanics. He was working at a time when the picture of the universe that physical theory painted was changing drastically, and he himself was partly responsible for these changes. We can forgive him then when we observe that his work on the SLT -- including especially his several definitions of reversible and irreversible -- seems a step backwards in terms of clarity and rigor. It is clear that Planck desired also to make the SLT more mathematically rigorous. However, his arguments seem to rely too narrowly on ideal gases. They may thus serve pedagogical purposes well, but it seems unlikely that he has strengthened the logical foundations of the SLT as expressed by Clausius with such arguments.
Worse yet: he seems to get himself into a serious problem with his definition of the system. One of Planck's statements of the SLT (note that we have returned to a verbal statement of the SLT!) starts off:(7) "Every physical or chemical process occurring in nature proceeds in such a way that the sum of the entropies of all bodies who sic participate in any way in the process is increased." The question that begs to be asked is: what bodies participate? This question is not easy to answer in general. Planck, who probably developed this statement of the SLT in order to avoid some of the problems associated with older notions of reversible and irreversible, seems to have exchanged one ambiguity for another. As I will show, this is a danger to all who attempt a rigorous statement of the SLT. Planck's statement is not the best classical formulation. Nevertheless, the problem of how the system and its surroundings will be defined remains. I will later discuss how it is both a strength and a weakness of the theory of thermodynamics.
The final statement of the SLT(8) that might be considered a classical formulation, is that statement generally called Gibbs' principle:
the variation in entropy with the energy of system held constant = 0
for an isolated system at equilibrium
Several observations can be made about this statement immediately. First, this is a logically narrower statement than that of Clausius, restricting itself to energetically isolated rather than adiabatically isolated systems, which need not be energetically isolated since work can be done (and hence energy transferred) by deformations of the system's volume.(9) However, it is not clear how much this would affect the balance between the overall logical strength of these statements since presumably, one could side step the issue by redefining the boundaries of the system.
Second, Gibbs has successfully avoided making explicit reference to the sticky concepts of reversible and irreversible. Instead, he asserts that the system is in equilibrium. This point is of great importance. If it holds true (and it does), then Gibbs' principle is the only classical formulation to avoid these nasty ambiguities, which have cropped up in all other classical formulations in one shape or another.
Unfortunately, Gibbs does not manage to avoid these ambiguities entirely. While it is true that irreversibility is not explicitly mentioned, Gibbs is forced to introduce another concept that has its own Pandora's box of ambiguities: the virtual variation. I will not attempt a critical comparison of the logical tradeoff made here between irreversibility on one side and virtual variation on the other. Both have their advocates.(10)
Instead, I will attempt to trump Gibbs with his own words: "The laws of thermodynamics may easily be obtained from the principles of statistical mechanics, of which they are the incomplete expression." For my purposes, the truth or falsehood of this statement is not important. What is important is that Gibbs, having written this statement, has placed himself squarely in the camp of those thermodynamicists neither classical nor neoclassical -- the statistical mechanicalists. In short, I consider it wise to call the work of Gibbs statistical mechanics (or rather the foundation for it), even though it may have deep implications for both the classical and neoclassical formulations of the SLT. My only shred of evidence to support this view (aside from Gibbs' own words) is the fact that the entropy computed using Gibbs canonical ensemble is numerically equal to the entropy computed from Clausius' statement of the SLT for a system at equilibrium(12). So in this sense at least, a complete identification between the two statements is not possible until the principles of statistical mechanics are introduced by Gibbs.
Having addressed each of the important classical formulations in turn, I hope to have shown that Clausius' statement, as given above, is indeed the best representative of the classical formulation of the SLT. It is certainly suitable for any comparison that one might wish to make between it and another, more modern version of the SLT. To do justice to this truly beautiful statement of Clausius, I would like to point out a few of its particular strengths and weaknesses before continuing on to a description of more modern theories.
As is often the case in other matters, the strengths of the classical formulation of the SLT will carry with them certain weaknesses. It will be easier to discuss these peculiar strengths of the formulation if I choose a name for them. Following Gull, I will refer to them as the ontological and the epistemological strengths of the classical formulation. Later in section III, when I do compare the classical with the neoclassical formulation of the SLT, I will come back to these characteristics of the classical formulation.
Thinking about ontology means thinking about what exists(13). One special strength of the classical formulation of the SLT is that its ontology includes both a system and its surroundings. Almost all other physical theories in use by scientists today(14) limit their ontology to an isolated system. It is in this sense that some reductionists consider classical thermodynamics to be less rigorous than other theories. The circularity of Planck's various definitions of the thermodynamic system (and the mathematicians' discomfort with all of them) is probably a direct result of the ontological inclusiveness of the SLT. But rigorous or not, the fact is that the classical formulation of the SLT is in wider use, and is easier to apply, than practically any other physical theory known.
One way to explain the epistemological strength of the classical formulation of the SLT is to say that it does not know about anything except experiments. The classical formulation of the SLT is purely empirical, and makes no attempt to provide an answer to (for example) questions about what entropy is, how fast it increases, or why it increases. Rather it simply states that there is a quantity entropy that will never decrease in an experiment. In this sense, reductionists should give the classical formulation of the SLT another chance, as it ignores the pandemonium of the microscopic world for simple quantities like pressure, volume, and temperature: things almost everyone knows about.
The last remark that I will make before introducing the neoclassical formulations will be on one serious weakness. The problem is very specific, and it lies in the definition of ∆Sirr within Clausius' statement. What is the T in ∆Sirr ? The problem is that when the system is not in equilibrium, Tsys need not equal Tsurr. Therefore only ∆Srev is well defined in the Clausius' statement without making reference to some other fact about the system and its surroundings. This is a serious problem, and by itself it will provide us with the motivation to push on, and try to understand the more modern statements of the SLT.
II. The Neoclassical Formulation
The neoclassical formulation of the SLT that I will be most concerned with is the statement provided by LV(15). Published in 1999, the LV statement is the latest serious attempt to set the SLT on a formal, axiomatic foundation. The self-declared goals of LV are: "...to formulate the foundations of the theory in a clear and unambiguous way and to formulate... a theory in which there are well defined mathematical constructs and well defined rules for translating physical reality into these constructs."(16) Lieb and Yngvason are not the first to attempt such a formulation, and they point out themselves that they are part of a tradition that stretches back to Caratheodory(17), citing no fewer than twenty authors that have worked on an "order theoretical" formulation of the SLT. "Order theoretical" is for my purposes synonymous with what I am calling the neoclassical formulations of the SLT. Lieb and Yngvason explain what it means specifically: "...the emphasis is on the derivation of entropy from postulated properties of adiabatic processes."(18)
It is particularly interesting to note how LV make a distinction between "order theoretical" statements and the statement of Gibbs (Gibbs' Principle). According to LV, the Gibbs' statement is unsatisfactory for their purposes to the extent that it "...postulates the existence of an additive entropy function from which all equilibrium properties of a substance are then to be derived."(19) From the distinction they draw here it is clear that LV are not simply looking to give a rigorous statement of the SLT, they would like to derive it from first principles about adiabatic processes. I will not be able to give a definitive answer to the question of whether or not LV achieve their goal. Instead, I will attempt to give a description of the structure of their approach with particular attention to the way in which their arguments fit together into a form of the SLT. A few inferences about their statement of the SLT may then be drawn from the structural features of their logic.
On page 12 of their paper, LV provide a very useful hint to the reader looking for how their statement of the SLT fits together: "The existence of an entropy function is equivalent to axioms A1-A6 in conjunction with CH, neither more nor less is required." The "axioms A1-A6" are assumptions about the mathematical thing that they call "the order relation", and include "Reflexivity", "Transitivity", "Consistency", "Scaling Invariance", "Splitting and Recombination", and "Stability"(20). In a fashion characteristic of mathematicians, we are told: "All these axioms are completely intuitive."(21) While this may be true in general, it is far from true in the case of this particular author. In order to proceed, I will have to assume that these axioms are indeed good assumptions, and look elsewhere for insights into their statement of the SLT. Fortunately, I do not need to look far. The other component that they list as a requirement for the entropy function (namely ¡"CH" or the "Comparison Hypothesis") is the source of a serious problem.
Lieb and Yngvason are well aware of the importance that the CH has in their statement of the SLT. In the abstract for their paper they state: "One of the main concepts that makes everything work is the comparison principle(22) (which, in essence(23), states that given any two states of the same chemical composition at least one is adiabatically accessible from the other) and we show that it can be derived from some assumptions about the pressure and thermal equilibrium." The fact that they intend to derive the comparison principle is a relief at this point, since the distinction between the comparison principle and common verbal statements of the second law(24) is so fine -- the distinction being simply that the directionality of the connection between the states remains ambiguous in the comparison principle. One may surmise that distinctions of this sort are not particularly fine to mathematicians in general, but for me at least this looks suspiciously like question begging! But for the moment, LV are off the hook since they do intend to provide a derivation of their comparison principle.
How then is this comparison principle derived? On page 39, LV write (italics theirs): "...our chief goal in this section and the next is to derive CH from the other axioms." The first axiom that they give as necessary in their derivation of the comparison is "Irreversibility". Care must be taken here, because "irreversibility" does not mean the same thing to LV that it does to Clausius, and I will discuss this further in section III. Lieb and Yngvason have already explained that "irreversibility" is their replacement for Caratheodory's Principle(25) (italics theirs: "We shall replace it by a seemingly more natural idea, namely the existence of irreversible processes. The existence of many such processes lies at the heart of thermodynamics. If they did not exist, it would mean that nothing is forbidden, and hence there would be no second law."(26) Unfortunately their identification of irreversibility -- which is necessary in their derivation of the comparison principle -- with the SLT completes a tight logical circle.
The question of how this circularity will affect their result overall is very difficult, and as I have said before, I cannot claim that I fully understand their work. But the consequences that this circularity has for my inquiry into the similarities and differences between the neoclassical and classical formulations of the SLT is clear: the work of LV does not constitute a new statement of the SLT. Instead, it relies on the statement by Caratheodory. For this reason, it may have been better to consider Caratheodory's statement of the SLT as the best representative of neoclassical formulations.
I will not here attempt an in-depth analysis of Caratheodory's statement of the SLT, although it is certainly worthy of this treatment. Instead, keeping in mind that my overall goal is a comparison of the neoclassical and the classical formulations of the SLT, I will make a few observations about what I find to be the most striking features of this statement, which can be stated as follows: In every neighborhood of each state there are states that are inaccessible by means of adiabatic changes of state.(27)
My first observation is an observation about what Caratheodory needs to make his statement. He needs states (by which he means states in thermodynamic equilibrium) and he needs the idea of a neighborhood not unlike that which appeared in Clausius' statement of the SLT. Also note that this "neighborhood" is actually a concept borrowed from real analysis in mathematics. The second observation is that the logical character of this statement is such that it gives negative rather than positive results, i.e. given a state, it says what others states are inaccessible rather than what other states are accessible. With these observations I leave off my brief description of the neoclassical formulations of the SLT.
Section III. The Classical and Neoclassical compared
I will begin my comparison of the classical with the neoclassical formulations of the SLT by returning to the categories I introduced in discussing some of the peculiar characteristics of the classical formulations of the SLT, namely the ontological and the epistemological.
As I have already discussed in section I, the ontological inclusiveness of the classical formulation is appealing in the sense that few physical theories allow for interaction between a system and its surroundings. In particular, the neoclassical formulation(28) does not seem to share this characteristic, for while it admits the existence of other states, it is the purpose of the formulation to designate a relationship between them and in any case the collection of states together is assumed to be isolated. This might be seen as a strength in the classical formulation then, and a weakness in the neoclassical. I have also already mentioned how this strength may also be a weakness. In particular, the desire to include both the system and its surroundings makes the definition of the system mathematically difficult, if not impossible. In this respect, it is not surprising that the ontological "strength" of the classical formulation is dropped for the mathematical rigor of the neoclassical. This seems to be one of the true tradeoffs made between the two formulations.
The difference in the conception of "irreversible" in the classical and neoclassical formulations of the SLT is probably related to the ontological tradeoff made between the two. The classical concept of "irreversibility" relies on the existence of non-equilibrium states of the system. The neoclassical formulation of the SLT deals only with equilibrium states, and hence its concept of "irreversibility" is different, and has to do more with the "inaccessibility" of certain equilibrium states of the system. I pointed out at the end of section I that the classical concept of "irreversibility" leads to a serious problem for the classical formulation, namely that the T of the system is not well defined along irreversible paths. The neoclassical formulation seems to have avoided this problem, but at a certain cost. The neoclassical formulation, while probably more mathematically sound, is abstract to the point of obscurity(29). So again one is faced with a tradeoff.
There are some problems that neither the classical nor the neoclassical formulation escape. For instance, it is a known experimental fact that the entropy of certain materials is a discontinuous function (for example in a phase transition). Both the classical and the neoclassical formulation of the SLT rely on real analysis(30). There should be some part of the formulation that accounts for discontinuities of this kind, and there is none.
Furthermore, neither the classical nor the neoclassical formulations of the SLT can escape the logically negative character of their statement. In both cases, the important implication is for what may not happen. This is probably not a serious weakness, since from a very broad perspective, scientific theory in general may only provide negative results(31).
But what about the other category that I discussed in section I, that of epistemology? The great epistemological strength of the classical formulation of the SLT was that it relied entirely on experiment. Thus, the classical formulation of the SLT can only be overthrown when experiments stop displaying the behavior that it describes. This is not likely to happen. Nevertheless, the classical formulation is unsatisfying to the extent that it does not provide insight into questions about how or why systems behave according to the SLT. In fact, it may obscure the answers to these questions with the ambiguities inherent in its idea of irreversible paths and non-equilibrium states. Here again, there appears to be a tradeoff between the classical and the neoclassical formulations. While the classical formulation is extremely unlikely to be overturned by experiment as long as it is understood the way it is now, nobody has rigorously understood the classical formulation in the way that the neoclassical formulation is rigorously understood.
To this author then, it is not clear that one of the two formulations is superior. In some respects they share weaknesses, and in others their strengths and weaknesses seem to balance, canceling with each other. In particular, neither of these formulations gives much insight into questions about what exactly entropy is and why exactly it increases. And yet it does. It will be the job of future generations of thermodynamicists to sharpen the SLT and extend the range of its validity.
(1) Except for when I must resort to the use of acronyms, I will adopt the stylistic convention of leaving the names of theories and laws uncapitalized. The meaning of this use of capitals is ambiguous to me.
(2) The justification for the introduction of the neoclassical category of formulations for the SLT, as well as the justification for the choice of Clausius' statement of the SLT as representative of classical formulations will appear later, when I describe these statements and compare them to other neoclassical and classical formulations of the SLT.
(3)Judging from the information provided by Uffink (see note 5 below) in his discussion of Clausius' statement, my version is closest to the statement Clausius set forth in: Clausius, R. Ueber verschiedene fur die Anwendung bequeme Formen der Hauptleighungen der mechanische Warmetheorie, (1865). English translation by R.B. Lindsay, in Kestin J.The Second Law of Thermodynamics. Dowden, Hutchinson and Ross, Inc. Stroudsburg, PA. (1976) pp. 162-193
(4) As it is set out for example in Callen, H.B. Thermodynamics and an Introduction to Thermostatics.2nd ed. John Wiley and Sons. 1985. pp. 118-120.
(5) The early statements of both Clausius and Kelvin appear in Uffink, Jos. Bluff your way in the Second Law of Thermodynamics. Available online: http://xxx.arXiv.org/abs/cond-mat/005327. p. 19
(6) However, there is an argument about whether these two early formulations give consistent results for systems at negative thermodynamic temperatures. See Uffink, ibid. p. 20 and especially footnote 26.
(7) The original German (and this translation) are provided by Uffink, ibid. p. 41 n.
(8) Uffink points out (ibid. p.43) that Gibbs did not himself call this a statement of the SLT. This is consistent with the argument I give for why Gibbs' Principle should not be considered the representative of the classical formulation of the SLT.
(9) Uffink also points this out: ibid. p.43 n.
(10) Callen makes a kind of attempt to explain the connection between Gibbs' principle with its virtual variations and Clausius' statement of the SLT (as I have given it here). See postulates II and III in Callen, ibid. pp.27,28
(11)Kittel,C. Kroemer, H. Thermal Physics. Second edition. W.H. Freeman and Co. New York. 1980. p. 57
(12) On this exciting point, see E.T. Jaynes. "Gibbs vs. Boltzmann entropies" American Journal of Physics. Vol 33. p. 391-8 and Gull, S.F. "Some Misconceptions about Entropy" in Buck, B., Macaulay, V. ed. Maximum Entropy in Action. OUP. 1991. Gull includes a discussion of the other pieces one needs (aside from Gibbs' Principle) in order to find a numerical value for the entropy from first principles. Gull's discussion was also helpful to me in formulating my ideas about ontological and epistemological strengths of the thermodynamics.
(13) Although not at all related to the topic of this paper, the philosophically minded reader unfamiliar with the work of W.V.O. Quine is encouraged to see "On What There Is" in From A Logical Point of View. Harvard University Press. 1961.
(14) This applies to the Gibbs' Principle, too.
(15) Available online: http://xxx.arXiv.org/abs/cond-mat/9708200. The references to this paper that follow will be designated simply by: LV p. #
(16) LV p. 3
(17) LV hold that Caratheodory's statement of the SLT can be shown from a subset of the axioms that they use to give their own statement (see LV p. 86). Working with the assumption, I will for now consider the LV statement as representative of the neoclassical formulation of the SLT.
(18) LV p. 3.
(19) LV p.2
(20) The discussion of these axioms begins on p. 20.
(21) LV. p.12
(22) For LV, the comparison hypothesis (CH) becomes the comparison principle after it is derived.
(23) On the use of this word in factual literature, see Fischer, D.H. Historians' Fallacies. Harper & Row, NY. 1970.
(24) From Uffink p.2: "A common and preliminary description of the second law is that it guarantees that all physical systems in thermal equilibrium can be characterized by a quantity called entropy, and that this entropy cannot decrease in any process in which the system remains adiabatically isolated."
(25) Which itself is considered a statement of the SLT for the purposes of this paper. If the reader will grant that it is, then it is not even necessary to accept their statement for the logical circle to be complete.
(26) LV p.32
(27) This is Born's version of Caratheodory's principle, as given in Uffink, ibid. p.53
(28) I will consider Caratheodory's Principle as the representative of neoclassical formulations of the SLT from here on, for reasons stated in section II.
(29) This point is made beautifully by Walter, as quoted in Uffink, ibid. p.49: "A student bursts into the study of his professor and calls out: 'Dear professor, dear professor! I have discovered a perpetual motion of the second kind!' The professor scarcely takes his eyes off his book and curtly replies: 'Come back when you have found a neighborhood U of a state xo of such a kind that every x U is connected with xo by an adiabat.'"
(30) Although LV make an attempt to avoid this problem in their paper, they still prove to be in need of Caratheodory's Principle and hence are not completely above it.
(31) Karl Popper gives an excellent argument for this point with his discussion of "falsifiability".