Significance tests, p-values, and falsificationism

You called?
entrance

lengthily show off how clever we are without ever trying to find out what the other person actually means
-That comment is insulting and applies in reverse: It displays your profound ignorance of both logic and the viewpoint of some experienced applied statisticians who understand the need for multiple and precise perspectives on real, complex problems (which is to say all problems in health, social and medical sciences). Logical and mathematical skeletons play a central role in describing systems and forming judgments and decisions about them, even though those skeletons are far from sufficient for that purpose. That you fail to see the geometric pictures and symmetries that underpin the fundamental meaning of mathematical statistics like P-values, and instead denigrate an argument you fail to comprehend instead of asking for clarification, only underscores your lack of qualification to be lecturing or even debating on this forum.

‘As discussed above, that’s false’ would also have to be called dogmatic. Which would be silly and no way to conduct a good-faith discussion.
You dodged the point: I backed up my statement with a detailed explanation of why I wrote it. You offer instead a series of ipse dixit pronouncements as if I should take you seriously when you clearly can’t understand and grapple with a detailed logical explanation of what I meant - something you did not supply for your own statements.

(Not to mention somewhat juvenile sneers about “Pope Popper”, which we should all be above, even though those are at least interestingly ignorant.)
That you would call out that indirect comment when you commit the above juvenile evasions and direct insults is an act of remarkable hypocrisy, even for yet another self-proclaimed authority on philosophy of science with no more than a shallow grasp of statistical theory and applications.

I will say your reaction is consistent with much of what I see from those who promote philosophy over practicality, and go so far as to confuse the two. Most of all, it confirms what I suspected from your Twitter posts: You are not only unfamiliar with most of the literature relevant to foundational debates in statistics; you are also unable to acknowledge your massive knowledge gaps, or see how in those gaps there is a coherent counterpoint to the naive Popperianism you promote. Your evasion is understandable, as to realize what you are missing would reveal how far short you fall in your comprehension of real-world statistics and the debates that have raged in it for centuries.

As for Popper: He was far from naive, and his own writings are worth reading because he would follow his outrageous openers with detailed explanations of why he made them, displaying how they were provocations for pursuit rather than slogans for lesser minds to adopt as catechisms. The same for Feyerabend, DeFinetti, and others with a theatrical bent. Popper also attempted to grapple with logical technicalities head on, rather than evade them with insults and excuses. My comment about Pope Popper was instead directed at those like you, who simply repeat Popper’s words without his substance or depth.

You need to read much more widely and deeply with a mind unshackled to Popper or anyone, and find your own voice if it is to amount to anything other than evangelical diatribe masquerading as philosophy, or as mere pretension posing as depth.

Why not start by defining what “falsificationism” even is? I might be wrong here, but I don’t think that’s a term Popper has ever used (definitely not in The Logic of Scientific Discovery). People just use “falsificationism” to refer to different subsets of Popper’s ideas. Is it a normative claim about how science should be done? A descriptive claim about how science progresses? An epistemological thesis about what can be known? A mix of everything? Gelman thinks he’s a “falsificationist” and I’ve seen people describing de Finetti as a “falsificationist”, so I’d say the term is so vague that it’s not very useful.

Relatedly, it’s almost taken for granted in these debates that frequentists are “falsificationists”, but I’m not so sure. People forget that Fisher distinguished two main problems in statistics: estimation (when there are well-defined alternative hypotheses and calculable likelihoods) and testing (when there are no clear alternative hypotheses and no clear likelihood). Testing problems do sound “falsificationist” in nature, but Fisher was more than happy to make positive claims (“claiming a discovery”) in estimation problems. (Also, as the Higgs Boson example by Sander shows, even testing can be used to make discoveries). The goal of Fisherian estimation is not to falsify a whole model or anything else, but to find out what’s evidentially supported by the data. There doesn’t seem to be anything falsificationist about it, whatever that is. So it seems that Fisher thought that both falsification and confirmation were important, albeit in different contexts. If he is a “falsificationist” just because he thought falsification is important in some situations, then the term is not very descriptive. It would be just as misleading to call him a “verificationist”.

3 Likes

A plea to everyone: keep it civil. Criticize ideas not people. Watch the adjectives when applied to persons. Lastly I hope someone can summarize key points from all sides in simpler language that what I read above.

10 Likes

A good part of the 1982 Introduction to Realism and the Aim of Science is devoted to untangling “[a]n entire literature [that] rests on the failure to observe this distinction” between “the logical possibility of falsification in principle … [and] a conclusive practical experimental proof of falsity.” [xxii, emphasis in original] Part IV, starting on p. xxxi, begins like this [bold emphasis added]:

This may be the place to mention, and to refute, the legend that Thomas S. Kuhn, in his capacity as a historian of science, is the one who has shown that my views on science (sometimes, but not by me, called ‘falsificationism’) can be refuted by the facts; that is to say, by the history of science.

I do not think that Kuhn has even attempted to show this. In any case, he has done no such thing. Moreover, on the question of the significance of falsification in the history of science, Kuhn’s and my views coincide almost completely.

So I think we have there at least a clear statement on the status of ‘falsificationism’.

2 Likes

I think the questions being discussed are now more fruitfully addressed using the techniques of computational theory as Paul Thagard discussed in A Computational Philosophy of Science. It is also discussed in this link on computational epistomology.

If we translate “falsification” to “computable” the “scientific method” becomes a decision procedure (aka. algorithm) that can be examined rigorously with mathematical methods. I like to think of “computational theory” as simply applied logic.

This is closely related to Sander’s attempt to bring the power of modern mathematical logic to the questions of statistical methods in his posts above.

In his introductory text Mathematical Logic, Stephen Cole Kleene, addresses the paradox of using logic to study logic. The technique used is to compartmentalize the formal system being studied (the object language), from the logic used to study it (observer langauge or meta-language).

In statistics, there are at least 3 formal languages that can be used to discuss “scientific methods” (or “learning algorithms” if you prefer the computational philosophy):

  1. Bayesian Theory
  2. Frequentist Theory
  3. Information Theory.

Some of the most important results in statistics are mapping the formal languages of Bayesian Theory to those of Frequentist theory (and vice versa).

Waiting exploration is to use information theory as the meta-language to explore questions regarding the mappings between Bayesian and Frequentist theory.

4 Likes

I just have to mention that I recently encountered a lovely and very readable articulation of one view of the (proper) aims of science [1]. This in turn led me to read It made me wonder if Popper might be of little use to someone who has adopted (what Popper calls) an instrumentalist view of science, or of the role of their chosen discipline (statistics, say) within science. If we eschew an interest in theories (particularly bold ones) and in explanation, if there is no deeper reality than what is immediately presented in the data tables, then why why should Popper’s ideas matter?

  1. Popper KR. Ch 5: The Aim of Science. In: Objective Knowledge: An Evolutionary Approach. Rev. ed. Clarendon Press ; Oxford University Press; 1979:191-205. http://www.bretthall.org/uploads/3/1/2/9/31298571/karl_r.popper-_the_aim_of_science.pdf
4 Likes

[Response to Norris:] Excellent point to raise; if however I’m forced to label myself it would be as a perspectivalist or epistemic and methodologic pluralist. In doing statistics that plurality would indeed include foremost instrumentalism and its pragmatic relatives.

Here’s one reason why: Most of the so-called hypothesis and “significance” testing activity of a competent applied statistician in soft sciences concerns “theories” barely worth that label. Many hypotheses of enormous practical importance when laid out in a modern causal framework (like “do mRNA vaccines cause anaphylactic reactions?”) look embarrassingly trivial when framed in the “bold conjecture” discussion that it seems Popper is most often cited for. That’s because they do not challenge any legitimate existing (let alone “established”) theory; in fact they are often most plausible based on previous observations or theory.

The bold-conjecture framework suits truly startling breaks from received frameworks (e.g., relativity, continental drift, and jumping genes), where the primary challenge is in upending older and previously highly successful explanations that have ossified into dogmas or universal facts. In contrast, the challenge in testing philosophically trivial theories like medical side effects are a combination (often toxic) of highly technical challenges in study design and data analysis with human factors like statistical incompetence (including but not limited to my hobbyhorses of dichotomania, nullism and reification), conflicts of interest (fueled by liability concerns and amplified by our tort system), political biases, etc. Here it is the sociology of science as a scientific discipline that needs our utmost attention.

Philosophy of science can help if applied at appropriate levels of analysis, but can hinder when overemphasized at the narrow technical level needed to extract information from data (which I see as the focal purpose of statistical methods). I see that hindrance as exemplified in the logically absurd “philosophical” conflicts between avowed frequentists and Bayesians that plagued mid-20th century statistics and continued to plague “philosophy of statistics” to that century’s end. It took until the 1970s for many application-savvy statisticians (like Box, Cox, and Good) to realize that conflict was a distraction and endorse the value of having flexible perspectives and diverse tools (frequentist, Bayesian, likelihoodist, and more) which could even be used in tandem in the same problem.

If all that sounds a bit Kuhnian, that’s because it is (or at least neoKuhnian): As far as I have read, modern perspectivalism grew out Kuhn’s SSR, which I think crystallized the view (anticipated by Hume* and later dramatized by Feyerabend) that epistemology and especially philosophy of science is itself subject to the human factors including dogmatism, evangelism, intolerance, pretension to universal truth etc. that characterize the worst of science and religion; this happens especially when it drifts into fields beyond the expertise of its “experts”, such as statistical theory, applied statistics, metaphysics, and the largely hidden influences of morals and values on science and its methodologies (note the plural).

*For today’s example see Syll’s blog:
On sophistry and illusion | LARS P. SYLL (wordpress.com)

4 Likes

The bold-conjecture framework suits truly startling breaks from received frameworks (e.g., relativity, continental drift, and jumping genes), where the primary challenge is in upending older and previously highly successful explanations that have ossified into dogmas or universal facts. In contrast, the challenge in testing philosophically trivial theories like medical side effects are a combination (often toxic) of highly technical challenges in study design and data analysis with human factors like statistical incompetence (including but not limited to my hobbyhorses of dichotomania, nullism and reification), conflicts of interest (fueled by liability concerns and amplified by our tort system), political biases, etc. Here it is the sociology of science as a scientific discipline that needs our utmost attention.

Yes, Popper was so fixated about revolutionary science that he forgot that normal science even exists.

This is a good summary of what’s wrong with this view (by David Papineau):

In retrospect, Popper’s falsificationism can be seen as an over-reaction to the
demise of classical physics at the turn of this century. The replacement of Newton’s
physics by Einstein’s was a great surprise, and showed that the evidence underpinning
the classical edifice was far less firm that everybody had supposed. Popper’s mistake,
however, was to condemn all inductive reasoning for this failure. Maybe inductive
evidence will never suffice to lay bare the large-scale structure of space-time, or the
other fundamental secrets of the cosmos. But this does not mean that it can never
identify such more mundane facts as that cigarettes cause cancer.
(…)
Despite these manifest failings, Popper’s falsificationism is popular among
practising scientists. The reason is probably that Popper’s story best fits science at the
cutting edge of research. Most new ideas at the limits of knowledge do start life as
pure speculations, and it is true that they are distinguished from the musings of
madmen only by the precision which allows them to yield definite predictions. By
focusing exclusively on this aspect of science, Popper creates the impression that all
scientists, however workaday, are creative visionaries with minds of steel.

But speculative research is not the only kind of science, or even the most
important kind. There would be no point to science unless its conjectures sometimes
acquired enough inductive evidence to graduate to the status of established truths.
This is the real reason for testing hypotheses against predictions. The aim is not to
falsify them, but to identify those that can be turned into the kind of positive
knowledge that enables us to build bridges and treat diseases

2 Likes

1.An informational view of statistics goes back at least to Fisher’s landmark work on estimation. Many links between information theory and standard frequentist and Bayesian statistical theories were worked out in the mid-20th century soon after Shannon’s landmark 1948 paper, as covered in this 1959 book by one of the pioneers Amazon.com: Information Theory and Statistics (Dover Books on Mathematics): 9780486696843: Solomon Kullback: Books
-note that the Fisher information matrix is in the second-order expansion of the Kullback-Leibler information criterion (a divergence measure in information geometry which corresponds to using likelihood-ratio statistics in regular models).

Of course in the >60 years since there have been far more developments linking statistics and information concepts, including books such as Good Amazon.com: Good Thinking: The Foundations of Probability and Its Applications: 9780816611416: Good, Irving John: Books and Jaynes (which takes a very Bayesian view compared to Kullback) Amazon.com: Probability Theory: The Logic of Science: 9780521592710: Jaynes, E. T.: Books
I confess I am in no position to recommend a recent one and would gratefully receive such a recommendation.

I find it a shame that information perspectives on statistics have been so neglected in the “soft sciences”, for I think theoretical and applied statistics have been at their best when devoted to extracting and summarizing information from data (whether for mere presentation or for input to decisions), rather than testing or placing bets (posterior probabilities) on statistical hypotheses (which are simplistic formalisms that seem inevitably confused with scientific hypotheses).

2.Thanks for bringing up computational epistemology! I have no expertise in the topic but have been following it for some years as a welcome attempt to move on from the 20th-century controversies and paradigms in which I was raised and educated. Kelly’s work cited in your link is mentioned in my 2017 article at https://academic.oup.com/aje/article/186/6/639/3886035 :
"nullism seems to reflect a basic human aversion to admitting ignorance and uncertainty: Rather than recognize and explain why available evidence is inconclusive, experts freely declare that ‘the scientific method’ treats the null as true until it is proven false, which is nothing more than a fallacy favoring those who benefit from belief in the null (29 = Greenland 2004 The need for critical appraisal of expert witnesses in epidemiology and statistics | Request PDF (researchgate.net) ). Worse, this bias is often justified with wishful biological arguments (e.g., that we miraculously evolved toxicological defenses that can handle all modern chemical exposures) and basic epistemic mistakes - notably, thinking that parsimony is a property of nature when it is instead only an effective learning heuristic (30 = Kelly 2011 Simplicity, Truth, and Probability - ScienceDirect ) or that refutationism involves believing hypotheses until they are falsified, when instead it involves never asserting a hypothesis is true (31 = Popper LSD 1959).

4 Likes

Will it, really? One way comes readily to mind: actually engage with anything I’m talking about. :wink:

Absolutely. But let’s not both-sides this when there is exactly one person here that your plea applies to. That person thinks it’s okay to insinuate another’s “lack of qualification”, that they “clearly can’t understand”, that they have “no more than a shallow grasp of” an issue (which that person hadn’t even talked about), that they are “unable to acknowledge…massive knowledge gaps”, and that they are “unfamiliar with most of the literature”. And that’s while he has also not engaged in a single thing I’m actually talking about. So he’s 2 for 2 with respect to the forum rules. :slight_smile:

So let me reiterate the invitation to discuss the actual ideas that I referenced in the OP. I’ll happily expand on what I said there and answer any questions regarding what they’re based on. And if any of the undoubtedly clever stuff (yes, I actually meant that) that’s already been said here should prove relevant to my points, then I’ll equally happily engage with that.

You cannot productively discuss statistical procedures without some comprehension of the mathematics behind them which often requires algebra and maybe a bit of calculus.

Richard Feynman in discussing the relationship of math to physics quoted Euclid, who said “There is no royal road” (ie. easy way) to learn geometry or physics. There is no royal road to learning statistics without mathematics, either.

1 Like

That’s certainly a relevant question. There are two ways of answering it, one negative and one positive. The negative way (‘what falsificationism isn’t’) can be seen in my reference [1]. Misconceptions about the idea abound, and nobody should be talking about it without at least making very clear how their use of the term and the idea corresponds to what Popper’s notion actually amounts to. Because very often, the people using the term don’t, in fact, know that.

If I had to give a positive sort of definition of the term, I’d say what I said in the OP: that it’s a methodology for learning from experience given that induction doesn’t work and that all our efforts are fallible. This includes a clear-eyed view of the aforementioned asymmetry between verification and falsification: the insight that singular statements may be verified or falsified, but that universal statements can only be falsified (always assuming valid logic). It extends into the realm of what Popper called “methodological rules”, which necessarily have to complement the purely logical analysis. (Cf. §11 of LoSD)

Of course, that doesn’t restrict anyone’s ability to call themselves (or somebody else) a “falsificationist”. That usually (in practice) just means that somebody endorses one or another of the elements of falsificationism as I defined it above. In that sense, Fisher would be a falsificationist for underlining the mentioned logical asymmetry: “Every ex­periment may be said to exist only in order to give the facts a chance of disproving the null hypothesis.”

Let’s see, I wrote this: " ‘A single p-value doesn’t mean anything’ is false if taken literally, and embodies a confusion I find prevalent among P-values critics who fail to distinguish math objects from their various interpretations."
-that makes no mention of you personally; my sentence’s topic was instead the quoted statement and noted a general confusion the statement invites. I then explained what I meant at painstaking length. To which you replied that my explanation (which encapsulates published material) did “lengthily show off how clever we are without ever trying to find out what the other person actually means” - that’s just nasty in a direct personal way, especially when it would have sufficed to explain how I missed your intended meaning.

Regardless, you really ought to try and understand the math and logic of a criticism before you go off on the source challenging your claims. And you really ought to go back and read the literature to see how the issues you raise have been raised repeatedly and debated at length for at least a half century, without convincing any of us to become converts to falsificationism, refutationism, or whatever you want to call the philosophy you promote, even though we all agree it can be a useful perspective at times and has some wisdom worth noting. For just a small sample from epidemiology see
Buck C. Popper’s philosophy for epidemiologists. Int J Epidemiol 1975;4:159–168
Jacobsen M. Against Popperized epidemiology. Int J Epidemiol 1976;5:9–11.
Maclure M. Popperian refutation in epidemiology. Am J Epidemiol 1985;121:343–50.
Susser M. The logic of Sir Karl Popper and the practice of epidemiology. American Journal of Epidemiology 1986;124:711-718.
Eells E. On the alleged impossibility of inductive probability. Br J Phil Sci 1988;39:111–16.
Greenland S. Probability versus Popper: An elaboration of the insufficiency of current Popperian approaches for epidemiologic analysis. In: Rothman KJ (ed.). Causal Inference. Chestnut Hill, MA: ERI, 1988.
Susser M. Falsification, verification and causal inference in epidemiology: reconsideration in light of Sir Karl Popper’s philosophy. In: Rothman KJ (ed.). Causal Inference. Chestnut Hill, MA: ERI, 1988, 33-58.
Pearce N, Crawford-Brown D. Critical discussion in epidemiology: problems with the Popperian approach. J Clin Epidemiol 1989;42(3):177-84.
Karhausen LR. The poverty of Popperian epidemiology. Int J Epidemiol 1995;24:869–74.
as well as the Papineau article that Pedro cited,

and the two articles of mine I cited on Twitter:
Greenland S. Induction versus Popper: substance versus semantics. Int J Epidemiol 1998;27:543–548.
Greenland S. Probability logic and probabilistic induction. Epidemiology 1998;9:322–332.

The 1989 Pearce+Crawford-Brown article sums up the position of the unconvinced thusly: "The recent Popperian ‘trend’ has a positive aspect in that it has fostered deductive thinking, and exposed the shortcomings of induction. However, the restrictive Popperian framework actually inhibits discussion despite its veneer of ‘critical discussion’ ”. Indeed; a third of a century later it remains so, and it is remarkable how those who rally under the ideals of Popper have often done so by pairing the stated goal of critical rationalism with contemptuous dismissal of any criticism that is not framed and easily addressed in their terms.

This aggrieved sense of being unfairly ‘restricted’ by mostly unnamed oppressors (and indeed within the context of a ‘trend’) invites comparison with [1]. If calls for ecumenicism/pluralism are valid against criticism by the ‘big meanie’ Popper and his hordes of ‘little meanies’, why don’t they also apply against @ADAlthousePhD and comrades? What protects us from a post-hoc-power pluralism?

Recently I have resumed my old habit of worshipping reading Popper in the mornings. Chapter 15 in Conjectures & Refutations (CR) republishes [2], which has a wonderful passage I’ll quote at some length:

Dialecticians say that contradictions are fruitful, or fertile, or productive of progress, and we have admitted that this is, in a sense, true. It is true, however, only so long as we are determined not to put up with contradictions, and to change any theory which involves contradictions; in other words never to accept a contradiction: it is solely due to this determination of ours that criticism, i.e. the pointing out of contradictions, induces us to change our theories, and thereby to progress.

It cannot be emphasized too strongly that if we change this attitude, and decide to put up with contradictions, then contradictions must lose any kind of fertility. They would no longer be productive of intellectual progress. For if we were prepared to put up with contradictions, pointing out contradictions in our theories could no longer induce us to change them. In other words, all criticism (which consists in pointing out contradictions) would lose its force. Criticism would be answered by “And why not?” or perhaps even by an enthusiastic “There you are!”; that is, by welcoming the contradictions which have been pointed out to us.

(I ask you: Is there a more delightful expression than “There you are!” for capturing the spirit of Chang & colleagues’ posthocpowerism?) :joy:

How do we plead for pluralism without abandoning this basis for progress? In various places, Popper identifies as obscurantist certain metaphysical attitudes that hold back progress. For example, in “Three Views Concerning Human Knowledge” (CR, Ch.3), he attacks Cotes’s essentialist view of Newton’s gravitational theory as such:

That it was obscurantist is clear: it prevented fruitful questions from being raised, such as ‘What is the cause of gravity?’ or more fully, ‘Can we perhaps explain gravity by deducing Newton’s theory, or a good approximation of it, from a more general theory (which should be independently testable)?’

No other word so fully captures my own sense of Biostatistics’ failure to make progress in dose-finding methods, which is the area of my most intense engagement with the discipline. As much as I would like to analyze this obscurantism solely as a philosophical problem, such that it opens the door to rational persuasion, I have ultimately come around to a view somewhat like yours here, Sander, that a more comprehensive sociological (incl. economic) outlook is needed:

  1. Chang DC, Stapleton SM. Response: The Proliferation and Misinterpretation of “As Safe As” Statements in Surgical Science: A Call for Professional Discourse to Search for a Solution. Journal of Surgical Research. Published online August 2020:S002248042030500X. doi:10.1016/j.jss.2020.03.074 PubPeer

  2. Popper K. What is Dialectic? Mind. 1940;49(196):403-426. What is Dialectic? on JSTOR

From an interested observer not trained in philosophy, my current take on the cumulative discussion to date is that falsification is an interesting and useful concept in the abstract but not in the doing of science. I view this similarly as how I view causal inference when not applied to randomized experiments: interesting, and useful for infinite sample sizes, but not as useful as it appears once real finite datasets are involved.

2 Likes

What about criticism, though? It seems to me that the crucial contribution you make in this post is to propose a variable selection method that embeds a critical principle. You have approached the variable selection problem from the perspective of fallibilism (another name for Popper’s critical rationalism — cf. RAS p.xxxv), and advanced a method that objectively reminds the user “but you could be wrong!”

2 Likes

I used to have an interest in (academic) philosophy of science; I’d say that I share Sander’s taste for Feyerabend’s pluralism. The parts that are still interesting for the philosophically inclined are Foundations of Mathematics (or Meta-Mathematics) and mathematical logic itself, but some mathematicians who work in foundations are less interested in logic in favor of things like combinatorics (like Doron Zeilberger from Rutgers).

An interesting philosophical discussion involves the relative merits of Zermelo-Frankel set theory with Choice Axiom aka. ZFC and law of excluded middle (LEM) or varieties of constructive logic that do not assume LEM. Since Godel, an important distinction is made between what is “true” and what is “provable”, which gets to the heart of the Popper quote above.

The Curry-Howard-Lambeck correspondence (the mapping of math proofs to algorithms) addresses the notion Popper referred to above.

Much of science has had its start in questions philosophers asked. If we took much of the work of the logical positivists and substituted “verifiable” or “falsifiable”" for “decidable”, we get the essentials of computational theory.

The mathematical tools of game theory are used in logic as they have been in statistical decision theory. IIRC the logician Jakkko Hintikka was the editor of a scholarly journal where Jerzy Neyman published an important paper on the applications of hypothesis tests. Despite being written in 1977, it is still worth reading.

2 Likes

I would submit that you should never take anything seriously that Papineau says or has said about Popper. He simply doesn’t know what he’s talking about and will misrepresent Popper’s ideas as egregiously as anyone. You want examples? You quoted some:

“Popper creates the impression that all scientists…are creative visionaries”. No, he actually didn’t. Not only can’t Papineau distinguish a normative framework from a description of science in practice, he also doesn’t understand that Popper never said that scientists had to consciously want to overthrow theories. That’s a fabrication by people who never bothered to read the actual text (of LoSD in this case) and much rather ran with their prejudices.

“There would be no point to science unless its conjectures sometimes acquired enough inductive evidence to graduate to the status of established truths.” I mean, where do you even start? It’s ignorant of what Popper argued should be the aim of science, it’s ignorant of the way Popper explained knowledge can grow in a falsificationist methodology, it’s ignorant of how Popper’s view necessarily involves choice between competing theories…

1 Like