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On the Truth of Scientific Theories

from Asad Zaman

Popper’s view on falsification appear plausible, and are widely accepted. Let me first summarize them, and then pose a puzzle.
Summary:
1:  A theory S is a scientific theory only if there exist some potential observations which can disconfirm it.  If all observations are compatible with S, then S is a tautology, true by virtue of logic, and not an empirical scientific theory. In this case, theory S has nothing to say about empirical phenomena, since all phenomena are compatible with it.
2: A single observation in violation of the laws posited in S lead to falsification of S. All scientific theories are potentially falsifiable.
3: It is impossible to PROVE scientific theories — no matter how many confirming instances we observe, there always remains the possibility of a contrary observation tomorrow.
Prior to Popper, verificationists were searching for ways to validate scientific theories. However all efforts at proving the validity of induction failed. This led Popper to formulate the falsificationist view: Scientific theories can never be verified. Scientists attempt to falsify theories. If a theory resists strenuous efforts at falsification, this proves its robustness and suggests that the theory is good. The longer a theory survives, the better the theory.

Poppers views are also held in Statistical hypothesis testing, where we maintain that the Null Hypothesis can be rejected, but it cannot be accepted — in the sense of confirmation of truth. A sequence of trials may fail to reject the null hypothesis the a coin is fair (p=50%), but this failure to reject does not establish that the null hypothesis is true. For example, it may be that the null hypothesis is false, and p=50.01%, but this small difference was not detected in the trials.

The reason for this lengthy replication of what is well known is to ensure that we are all on the same page. Now to pose the logical puzzle:

A rejection of a theory S is logically equivalent to a confirmation — a proof of the negation of this theory. So rejection of S means affirmation of NOT S. SO certain types of theories can be proven. Similarly, certain types of null hypotheses can be proven true.

So a simple falsificationist view is logically impossible. If some theories can be proven false, then their negations can be proven true. To save Popper, we must consider different classes of theories, some of which cannot be proven true, while there are others which can. It is not at all clear how one would separate scientific theories into these two (possibly more) categories.
  1. Silwyson
    March 29, 2016 at 11:53 am

    “Popper’s view on falsification appear plausible, and are widely accepted”. This statement is false, even among philosophers. Philosophers are not scientists. It is important to practise science to test propositions about science and what scientists do. Popper fails to understand science, even granting the importance of falsification. Einstein’s letter to Popper in the last appendix of “The Logic of Scientific Discovery” refuted many fundamental assumption of Popper, who did not understand what Einstein was saying.

    • March 29, 2016 at 8:51 pm

      “Philosophers are not scientists.” are you saying that being a philosopher (of sorts) precludes one from being a scientist (of sorts)? An absolute claim, and the burden of proof is on you.

      “Popper fails to understand science” is again an absolute claim with no substantive argument to support it, so perhaps it is only an opinion, or worse, a prejudice. It is nonetheless conceivable that one could demonstrate that Popper made some implicitly contradictory claims about science (whatever ‘science’ means to you), in which case I am curious to read your proof.

      “Einstein’s letter to Popper in the last appendix of “The Logic of Scientific Discovery” refuted many fundamental assumption of Popper”. I do not see any fundamental challenge in the said letter. So if you do understand the argument of Einstein better that Popper (as you implicitly claim) and probably better than most economists, then please enlighten us.

      • Silwyson
        March 30, 2016 at 1:05 am

        “Philosophers are not scientists”. What I mean is that scientists do not bother to philosophize about science, because they would know that science is complex and cannot be reduced to a philosophy.

        A philosophy of science is merely a personal view of science. That view (e.g. Popper’s) differs from those of other philosophers (e.g. Wittgenstein, Lakatos, Feyerabend, etc.) For example, I disagree with Popper’s assertion that:

        “The empirical sciences are systems of theories. The logic of scientific knowledge can therefore be described as a theory of theories. Scientific theories are universal statements.”

        For an explanation of this, see http://www.asepp.com/facts-and-economic-science/

        “Popper fails to understand science” when he tried to reduce uncertainty in quantum mechanics to a universal law, which Einstein refuted in his letter to Popper concluding:

        “I wish to say again that I do ot believe that you are right in your thesis that it is impossible to derive statistical conclusions from a deterministic theory.”

        The whole Popperian approach to understanding science is flawed. Falsification is a complex process, because a single falsification can itself be falsified. Dozens of experiments have been carried out over decades to try to falsify the fact that the velocity of light is constant (which a fundamental assumption of the theory of relativity).

        Popper’s view of science is not science. Science is not a step-by-step process like a computer program. For example, a scientific breakthrough which disrupts at least a sub-discipline of a subject often is random, not based on a systematic method. (Otherwise, it would not be a breakthrough or a surprise.)

        So it is important not to invalidate the relevance of science for economics by invalidating a particular flawed view of science. In the end, it is facts, not theories, which is the foundation of science. Many sciences, for example biology, may not have any universal laws at all.

    • Titanic
      March 30, 2016 at 1:59 am

      That’s a very interesting point. Other than buying “The Logic of Scientific Discovery”, is there any online copy of Einstein’s letter?

    • Harry Mann
      March 30, 2016 at 2:12 am

      @Silwyson

      Yours is a very interesting point and — it seems — it did not pass unnoticed.

      I wonder, other than buying “The Logic of Scientific Discovery”, is there any way one could read Einstein’s letter online. That could settle any doubts,

    • March 31, 2016 at 12:06 pm

      “What I mean is that scientists do not bother to philosophize about science, because they would know that science is complex and cannot be reduced to a philosophy.”

      You clearly cannot prove what every scientist “would know”. And is your statement not already an attempt to formulate a principle, to find a truth, and therefore already philosophical?

      “A philosophy of science is merely a personal view of science.”

      Just like there are discoveries in science there are discoveries in philosophy, which pursues ever greater consistency and efficiency of thought, and therefore of our system of meaning. When a philosopher discovers a logical dependency previously unrecognised, this may be verified and accepted or challenged and refuted by others, based on the same, fundamental laws of logic.

      “I disagree with Popper’s assertion that:
      “The empirical sciences are systems of theories. The logic of scientific knowledge can therefore be described as a theory of theories. Scientific theories are universal statements.”

      Experience does not become an object of science until a theory is formulated to define its relation to other experiences. A theory may in some way begin with experience, but experience (even outside of empirical science) is alway an element of a theory: as soon as you get up in the morning you already apply a system of theories about how your body works, what it needs, etc. these theories are internalise as everyday meaning, but they are inductive, deductive and they evolve subject to new circumstances. More complex, scientific theories are built on more rudimentary theories prevalent in our culture. So I think the statement of Popper can be plausibly extended to all existence.

      “Einstein refuted in his letter to Popper concluding:
      “I wish to say again that I do ot believe that you are right in your thesis that it is impossible to derive statistical conclusions from a deterministic theory.””

      This is not a refutation but a statement of opinion. And it is not a statement about a scientific fact but about logic, ergo, philosophy.

      “The whole Popperian approach to understanding science is flawed. Falsification is a complex process, because a single falsification can itself be falsified. Dozens of experiments have been carried out over decades to try to falsify the fact that the velocity of light is constant…”

      I do not recall Popper claiming that any attempt at falsification always constitutes a valid falsification. And I don’t even like Popper. While I recognise his contribution to philosophy of science his work is patchy in quality. I don’t like his style of argumentation, his over-examination of banal situations as if some great mystery was about to be revealed, and then nothing. And underneath all this he comes across as a naive realist.

      “Popper’s view of science is not science.”

      He never claimed it to be science, to my knowledge.

      “Science is not a step-by-step process like a computer program. For example, a scientific breakthrough which disrupts at least a sub-discipline of a subject often is random, not based on a systematic method.”

      Certainly. Science is full of vagueness, assumptions, bias… Like all systems of meaning it is affected by the unpredictable nature, or ‘singularity’, of thought. Scientific discovery is a dawn of new meaning of experience, a breakthrough in understanding of experience, which is not necessarily or even plausibly supervenient on something radically independent of the thinking subject.

      “So it is important not to invalidate the relevance of science for economics by invalidating a particular flawed view of science.”

      Science certainly has its place as a powerful mode of justification and technological creation, but not all science is good science, and some domains are essentially non-empirical (like value) and therefore incompatible with empirical science. Since economics deals with value, it’s definitions, attribution and relations between various forms of value, then empirical science might not be the most suited method of its justification, no matter how successful science is in other domains (like military technology).

      “In the end, it is facts, not theories, which is the foundation of science.”

      As I argued above, scientific facts can be said to exist only where there is a theory which is capable of predicting these facts. Facts without a ruling theory (which is not necessarily universal but can be local or contextual) are not yet scientific but only phenomenal.

      I appreciate your diligent attempt to address my objections to your first post.

  2. March 29, 2016 at 2:02 pm

    Actually, no theory can ever be proven or disproven (proven as a negation) by empirical observation alone, because every case of interpretation of what is observed/measured involves assumptions: at the very least, the assumption that what is observed is objectively real (independent of perspective or the individual state of mind). All proofs are inferential, not observational, because, as Popper also said: “All languages are theory-impregnated.” But my claim might be better substantiated by focusing on a central element of scientific logic: causation. Empirical evidence is, at best, evidence of correlation; the property of being a ’cause’ is never observed but only inferred based on perceived consistency of correlation. Causality cannot be proven.

    So what can be proven? Only logical relations, because the notion of proof is already a concept of logic. For example, the Sun ‘must’ rise tomorrow, because that is what tomorrow ‘means’: the concept of tomorrow is defined by the Sun’s rising and then setting, and if the Sun were not to rise there would be no tomorrow. The Sun’s rising tomorrow is a definitional certainty, an implicit tautology, but a tautology is not a valid epistemological theorem.

    Induction, although refutable in most formulations on purely logical grounds, is nonetheless defensible (and even provable provable) where it is so existentially fundamental that it is in effect definitionally fixed: X is always true, because if X was ever untrue the world as we know it would end. Conversely, by denying certain logical facts we might be denying our own existence or some other ontological commitment implied by our actions. In these contexts the problem is not strictly epistemological but instead gains validity of an ontological constant. This kind of proof is sometimes called (at least on account of Kant, Apel and Habermas) the ‘transcendental-pragmatic’ method of justification.

    • Silwyson
      March 29, 2016 at 11:59 pm

      You said: “Actually, no theory can ever be proven or disproven (proven as a negation) by empirical observation alone”. What is the following?

      Theory: All swans are white. Observation: A black swan exists.

      • March 31, 2016 at 10:04 am

        Good choice of example!

        But I also said: ‘because every case of interpretation of what is observed/measured involves assumptions: at the very least, the assumption that what is observed is objectively real (independent of perspective or the individual state of mind).’

        The proposition/claim that “A black swan exists” depends on whether what was observed was in fact a black swan. Since all observation depends on individual perspective and judgement, there is an inherent uncertainty in all observation. Also, no swan I know of is even black if blackness is precisely defined, as total absence of colour and grey shades. The said proposition relates to what we call ’empirical evidence’ but also to the meaning we equate with this evidence. The meaning is a state or property of observer, and some meaning only ‘signifies’ the external. This entanglement of conceptual, subjective content (marred by vagueness and essential incompleteness) in the objective/external is enough to render the real (if it is regarded as something independent of perspective or individual state of mind) at best unknowable but probably just an internal inconsistency of the system of beliefs of the observer.

  3. March 29, 2016 at 6:13 pm

    Asad, according to your (1), ‘Theory S is false’ is not a scientific theory, so surely Popper survives your argument?

    • March 29, 2016 at 6:47 pm

      S is a theory that the coin is fair: p=50%. NOT S is the theory that the coin is NOT fair, p is NOT equal to 50%. If on a hundred trials we see 99 heads, we would reject S and we would accept NOT S.

      • March 29, 2016 at 6:50 pm

        On the other hand, if the coin displays a fair balance of heads and tails, one would be inclined to reject NOT S which says that the coin is not fair. In practical life, this often happens — that is, casinos routinely test their wheels for fairness and accept or reject the null.

      • March 29, 2016 at 6:56 pm

        NOT ‘the coin is fair’ covers all cases where the coin has a bias, however slight. What observations would disconfirm it?

  4. March 29, 2016 at 6:46 pm

    Consider the scientific theory that apples do not fall, but remain suspended in the air when dropped because the gravitational force is zero. Or, less radically, consider the scientific theory that the gravitational force is 5 m/sec-squared. The speed with which apples fall would refute both of these theories. I dont know how ontological/ epistemological/ philosophical considerations could play a role in preventing such refutations unless we take the a nihilistic postion that we can never know anything for sure because it is possible that we are brains floating in jars and being fed an illusory sensation stream.

    • March 29, 2016 at 7:01 pm

      Couldn’t brains in a jar have scientific theories about the sensation stream that they were being fed?

      • Dave Raithel
        April 1, 2016 at 1:21 pm

        Hillary Putnam might have thought so, at some time or other.

      • David Chester
        April 1, 2016 at 2:30 pm

        Only if they were your brains, Dave!

    • March 31, 2016 at 12:31 pm

      Your example theory is certainly refutable, but it is not refutable solely on empirical/experiential grounds: a refutation implicitly or explicitly calls upon an antecedent system of meanings, beliefs and theories that modulate our apprehension of experience and the phenomenal content of experience. Something metaphysical is always added to experience in order to make it a scientific fact as opposed to a merely subjective experience.

  5. March 29, 2016 at 6:49 pm

    Michael, I am not sure how transcendental-pragmatism would apply outside the social sciences. It seems to that mainstream economics rested on theories that were transcendentally-pragmatic until they were falsified by the crashes around 2008. Transcendental-pragmatism thus falls short of what many want from a concept of validity. For example, as Popper noted, one would at least hope that a serious attempt to falsify the assumptions had been made.

    Where transcendental-pragmatics seems to me useful notion would be to observe that some specific beliefs (such as those of some politicians) were only transcendentally pragmatic and had not been adequately challenged. Or have I misunderstood?

    • March 29, 2016 at 9:06 pm

      You are right in your scepticism, because many alleged transcendental-pragmatic ‘proofs’ have been shown to be false, or at least incomplete. This does not disqualify the method though but only its specific applications. No one can argue against the requirement to be logically consistent. I understand this method as follows, very briefly:

      Whenever we act we commit to certain beliefs, that is, we affirm those beliefs as true, and if those beliefs are negated by our statements, we are said to be committing a performative contradiction, from which follows that we are inconsistent and therefore in some way wrong. Now if we can show that all possible agents must commit to certain minimal beliefs in order to act (or exist) and if some of these beliefs are necessarily the same for all, we can also show that we all implicitly agree on certain truths. On this basis I would say that most of economic proofs do not satisfy the definition of transcendental-pragmatic justification. So yes, the scope of this method is very limited, but where and if it applies it is possibly the most powerful justification method of all.

    • March 29, 2016 at 9:13 pm

      One more thought… If transcendental-pragmatic justification is the only method capable of obtaining genuinely universal truths, then perhaps our failures of knowledge are not failures of theory but failures of our being. Perhaps we must evolve towards truth rather than ‘find it’ in the classical sense. Or perhaps we should give up on other truths. Just speculating….

      • March 30, 2016 at 1:05 pm

        Michael, At the risk of hijacking this thread, I agree that psychologically many people wish to act on beliefs, and that sociologically people are often held to be committed to the beliefs on which they are thought to have acted. This seems to me to be apart of the economic problem prior to the crashes. Once the crashes were recognized, it seemed to me that there was a general loss of confidence in and commitment to the previous beliefs, and a good thing too. In the UK Gordon Brown took decisive action, but this was not based on any theory of classical kind. More generally, it is characteristic of crises that you have to abandon old beliefs and then often act decisively before being able to form coherent theories/beliefs.

        I am not clear how one would ever come to view an empirical theory as ‘true’ or even assign a probability to it. I find it much more helpful to consider how well tested the theory is, and to compare the conditions under which it has been tested with those in which we may wish to apply it.

      • March 30, 2016 at 8:41 pm

        No amount of testing can result in certainty or even assigning a factual probability to future events to which a theory applies. This is the essence of the problem of induction. It is a problem which is in indeed insurmountable if one relies solely on empirical means (testing, observation); something more is needed and that is where the transcendental-pragmatic could be useful. It combines empirical testing with continuous logical evaluation of the terms of reference for consistency. It seems to me that science, as it stands today, is not concerned with the latter aspect beyond a narrow focus on experimental methodology, because it is not concerned with (or even believes) in objective truth. We live in times of extreme relativism in the intellectual domain, perhaps because technologists and various funding bodies realised they can produce a convincing reality by objectifying the underlying theories by means of technology. Technology now turns abstraction into reality without much resistance from the objective ‘real’ as it only needs internal consistency to function. And yet we know something is missing in technological approach, or rather, there is an inherent contradiction in this approach which undermines what we take to be real as well as its alleged usefulness. Perhaps because in the end “Scientists, technicians, and instruments are purchased not to find truth, but to augment power.” (Lyotard)

        A brief clarification re transcendental-pragmatic justification. Its strongest form does not draw conclusions from contingent beliefs/commitments but necessary beliefs/commitments, those beliefs that we profess by merely acting. For example, when I write this I affirm that I am not the only consciousness in the world, and since we all intend to communicate at times we all already agree that other beings like us exist in the world (a transcendental-pragmatic argument against solipsism). So we are dealings with very fundamental logical relations that cannot be abandoned by changing our contingent beliefs or attitudes. The method becomes weaker when we apply it contingent beliefs, like economic theories, and then must be augmented by some sort of verification, but in that case no certainty or real probability obtains in its predictions.

  6. JohnB
    March 29, 2016 at 8:53 pm

    Strictly logically, falsification is impossible, because you can’t prove a negative.

    On a practical level though – given that science is a practical process which has to make do with the limits placed on us in reality – it is practically possible to falsify things, and it’s all about determining ‘how much’ evidence (and at what quality), will do, before we consider something (for all practical purposes) as falsified.

    https://books.google.ie/books?id=8gXzHMtwRwYC&pg=PA37&lpg=PA38&ots=o9iJYkqPbR&focus=viewport&dq=falsification+%22prove+a+negative%22

  7. March 30, 2016 at 2:20 am

    Many disparate thought streams [red herrings] have arisen. The fundamental question under consideation is: Are scientific theories TRUE — This comes from the quote from Aumann posted by Lars Syll that TRUE/FALSE does not apply to scientific theories, which seems strange.

    Popper provides a new demarcation criteria to separate science from non-science — . He defines what it means to be a scientific theory. S is a scientific theory if it rules out certain empirical phenomena, and hence is falsifiable — some event can occur which would disprove the theory. If no possible events can occur which would contradict S, then S is compatible with all possible observations — it has nothing to say about observable events in the real world. Hence it is not a scientific theory.

    Note that this rules out certain AXIOMATIC constructions of science which have received some support on this forum earlier.

    One key claim of Popper is the scientific theories can never be proven. Thus the TRUE attribute can never apply to scientific theories. I dont see any way to avoid this problem.

    Now Popper does say that scientific theories can be falsified, so the FALSE attribute can be applied. This seems strange. If S is false then NOT S is true, at least in binary logic.

    Now we have the interesting example of Marsay — consider the sentence that this coin is NOT FAIR. This is the negation of S: The coin is fair. Assuming this allows for ANY p different from 50%, there is no way to prove this wrong with any finite sequence of observations.

    Therefore according to Poppers criteria. This is not a scientific theory, it cannot be proven wrong. But now the strange thing is that it CAN be proven right. So NOT S is a statement which has the potential to be labelled TRUE [empirically] but not FALSE. This shows that Popper’s theory of science requires some additional thought

    So apparently BINARY logic does not apply to scientific theories — only one of the two TRUE or FALSe can be proven for them. Or are there other ways to look at this?

    • March 30, 2016 at 1:13 pm

      Asad, I agree with Popper that one cannot prove a theory true, and with Michael that a theory can be ‘proved’ true to what some group believe. The challenge is to say something positive, useful and true that distinguishes good theories from bad. I interpret Keynes, Whitehead and Russell as saying that one can talk about the extent to which the theory has been tested (‘proven in use’) and the relevance of those tests to the intended application. Unfortunately, this is far from a binary, much less the simple criterion that Michael notes people and institutions want.

  8. David Chester
    March 30, 2016 at 10:57 am

    It seems to me that this discussion is too tied up with the semantics of word use. “Science” is a shortened meaning for “knowledge about particular facts within our universe”. These facts are provable or possibly hypothetical and waiting for proof. However to prove something in absolute terms is mathematically impossible, so we must be satisfied with imperfect proofs, that seem to be correct as far as our limited thinking powers can go.

    But having reached this level of understanding about a particular fact, we have then to judge it to be true. Proven facts are said to be true (even if there is no absolute certainty about them). In the case of facts which have been proved to incorrect, we call these false. Some facts remain unproven and are indeterminate at this time, but we hope to find a better result about them at a later stage of thinking.

    By taking this attitude, the gathering of scientific knowledge and its application to real-life situations makes sense. This is without being 100% certain or completely realistic, we can manage to understand a lot about our universe–and for many people this is sufficient. Practical matters arising is all we need from this knowledge.

  9. March 30, 2016 at 1:16 pm

    Life would be easier if economies didn’t have innovations, in which case naïve induction would be reasonable. My own view is that the difference between extrapolation and sound prediction does sometimes matter, and it can be important to know when. But this is a long way from coin tossing.

  10. March 30, 2016 at 3:32 pm

    so it seems that there is consensus that Aumann is right — TRUE/FALSE are not useful categories in terms of which scientific theories should be discussed — we have to be pragmatic. scientific theories may be useful, without being true.

    On the other hand, there is no such ambiguity about mathematical truth. We know that 2+2=4 is TRUE and 2=2=5 is FALSE and we dont have any need for pragamatic qualifications or an evolution towards truth.

    As opposed to this, the null hypothesis that a give coin is fair (has p=50% exactly) can never be proven true in any finite number of observations, if the coin is fair. An interesting Godelian situation — if the statement is false, we can find a convincing proof of its falsity in a finite number of trials. But if it true, then it cannot be empirically proven to be true.

    One important conclusion, with grave consequences, is that scientific methodology must be different from mathematical methodology.

    • March 30, 2016 at 11:25 pm

      Asad, If I may be pedantic. One could say that one should always be pragmatic : the question is, which pragmatism is appropriate? Sometimes true/false is pragmatic. Other times it is necessary to understand your last paragraph. Keynes addresses this in his Treatise, but not very accessibly.

    • March 31, 2016 at 11:41 am

      Thats a nice little summary of Godel’s theorem.

      It seems evident that math and science are different (though some people have suggested that actually math itself is a physical object not too different from atoms , genes, etc.).

      Math is purely axiomatic—based on defined symbolic relations.
      In science, one gas physical observation—signifiers and signified. These are assigned symbols and math relations. One can axiomatie sciences as suggested by Hilbert long ago, but it has essentially two sets of axioms—math ones, taken from math, and emprical ones taken from assigning symbols to observations .
      I wonder why this is seen to have ‘grave consequences’..

    • April 1, 2016 at 5:58 pm

      Asad, if I too might be a little pedantic … we know that 2 + 2 = 4 in DECIMAL arithmetic, but in modulo base 3 [clock] arithmetic 2 + 2 = 1 and in base 4, 2 + 2 = 0! Such statements are only as unambiguous as the interpretation of their numerals. As Erich Fromm put it and the mathematicians developing Algol68 realised, the First Commandment of the Mosaic Law is also the first rule of logic: thou shalt not mistake the image for the reality.

      Gauss’s term ‘modulus’ dates back to 1801 (E T Bell, ‘Development of Mathematics’, 1945, p.191). I never came across it at school, nor indeed until working on Algol68 c.1970, but W W Sawyer, brilliant developer of the teaching of modern mathematics, explores the idea in ch.1 of his ‘Concrete Approach to Abstract Algebra’ (San Franscisco: W H Freeman & Co, 1959, p.23).

  11. Dave Raithel
    April 1, 2016 at 1:33 pm

    I have a theory about coins. Call it S. Theory S is not a single statement, but the conjunction of statements, s1, s2.. sn. I claim that if S is true (the conjunction of statements), flipped coins behave in such and such ways. I define a “fair coin” as one which, when flipped, is as likely to land on either side, and very unlikely to land on its edge. I can never prove there are fair coins, I can only show you coins that behave as S predicts. After that, we can only grow bored flipping coins over and over and over again. And when I find a coin that never lands heads-up, or lands on its edge, then I have to choose: Some “s” of S is not true, or the coin is “not fair.”

    Now what, exactly, is the problem?

  12. April 1, 2016 at 3:45 pm

    Good question — I was assuming without mention some statistical background. First we make a rather drastic, possibly untrue, assumption that a sequence of coin flips is like an IID sequence of Bernoulli trials with p(H)=p p(T)=1-p. Now Let S(p) be the scientific hypothesis that the probability is p. S(50%) is the hypothesis that the coin is fair. Suppose that S(p) is not true and that S(p*) holds, where p* is any value different from p. Then S(p) can be falsified with arbitrarily high probability in a finite number of observations. That is, if D=p-p* is the difference, and the number of trials N is such that SE=SQRT(0.5*0.5/N) is less than D/6, then the observed frequency of heads in N trial will be more than 3 Standard Errors away from 50% with probability 99.7%. Thus we will confidently reject the null hypothesis within N trials. — Of course, practically speaking the N could be so extremely large that it would be impossible to carry out so many trials, but let us ignore this possibility for the moment.

    This means that when S is false, then there exists a finite number of trials N such that we can reject S with 99% confidence of not making a type I error. In Poppers terminology, we could say that when S is false, then it can be falsified.

    However if S is a SHARP Null Hypothesis that p is exactly equal to 50% and S is TRUE; then it can never be proven true. No matter how many trials N we carry out, there will be small interval of values around 50% which will all be strongly compatible with the observations. Thus we can have stronger and stronger faith that S is likely to be true, but we can never actually prove it with any finite number of observations

    The situation that you are implicitly describing is one where we dont have a sharp Null. If our null is that P=50% plus or minus epsilon, where epsilon is a small number, then this hypothesis would be provable (if true) in the sense of being confirmable in a finite number of observations with high degree of confidence.

    This kind of resembles Popperian ideas regarding falsification as the hallmark of scientific theories, but is substantially more complex and differentiated. Taking into account the idea that large values of N might not actually be observable — we cannot carry out a trillion coin flips so this should not qualify as an observable — would lead to even more interesting classifications of theories with respect to their provability or dis-provability on the basis of observations.

    As far as I know, philosophers of science have not considered these complexities related to Poppers falsificationist account of science, so there is room here for an interesting note for publication in a Philosophy of Science journal.

    • April 2, 2016 at 9:21 am

      Popper dealt with determinism/indeterminism on multiple occasions, see in particular
      (1990).

      References
      Popper, K. R. (1990). A World of Propensities. Bristol: Thoemmes.

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