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Karl Popper maintained that empirical sciences should be based on the principle of falsifiability rather than verifiability for no amount of observations can guarantee veracity but a single contradicting observation is enough to falsify.

This emphasis promotes empiricism and fits well empirical sciences.

Assumptions are simply axioms with a different name. They carry a metaphysical burden. They are our building blocks, our groundwork, our operating cost. They are a necessary evil. So we should do well and at least pick acceptable assumptions. Falsifiability seems to be a good starting point (Necessary? Sufficient?)

The question now:

How falsifiable are assumptions based on probability or any other limit when x tends to infinity; how can one make infinite observations? How valuable, acceptable and warranted are non-falsifiable assumptions?

Example Statistical Models in Physics (Medical or Not).

How acceptable and warranted is the assumption of linearity in linear no-threshold model? How could such an assumption be empirically falsified? What would the observation be?

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    The issue is not simple... In principle, we do not "test" the "underlying assumptions" of the theory, but we test the axioms through the "falsifiable consequences". The rule is "if Axiom, then Consequence" and thus, by contraposition: "if not Consequences, then not Axiom". If we test the prediction derived from a theory and we falsify it, we are forced to reject the axiom (or some of them). Contra, see Quine's "holism". Nov 8, 2023 at 9:52
  • In other word, we formalize e.g. QM with probability: the test will affect the QM axioms and not the mathematical theory of probability. Nov 8, 2023 at 9:54
  • @MauroALLEGRANZA The prediction is a 5.5% probability of stochastic effects of radiation for every Sievert. How would a stochastic/probabilistic prediction be (empirically) falsified? Nov 8, 2023 at 10:42
  • Maybe with a very very different measured value wrt to the predicted one. Nov 8, 2023 at 10:47
  • Everything becomes false "when x tends to infinity". Nothing remains true forever. We're here now.
    – Scott Rowe
    Nov 8, 2023 at 11:30

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If I understand your question, by non-falsifiable assumptions you mean hypotheses that cannot be conclusively falsified, but which may be tested statistically. Assumptions such as these can indeed be acceptable and useful. Sometimes we cannot falsify a theory with a contrary observation; sometimes the best we can do is find data that disconfirms it to some degree.

There are different ways to do that. Within the frequentist framework of probability, significance testing is an attempt to determine how improbable it is that we would get a particular observation, or a more extreme one, given some hypothesis. Significance testing is often used with a null hypothesis in situations where there is reason to be believe that no effect is present. It is a common method, but it has been widely criticised and is overused and overrated. (See, e.g. The Cult of Statistical Significance by Ziliak and McCloskey, or the paper by Amrhein, Greenland and McShane, published in Nature 567, 305-307 (2019), in which the authors, together with more than 800 other signatories, call for a restriction on the use of significance testing and confidence intervals.)

Neyman-Pearson is another frequentist method that attempts to quantify the fit between hypothesis and data, based on possible false positive and false negative observations. This approach can in some circumstances be thought of as a kind of statistical analog of Popperian falsification. One advocate of this position is Deborah Mayo, in her book Error and the Growth of Experimental Knowledge.

Within the Bayesian framework, hypotheses may be confirmed or disconfirmed relative to rival hypotheses by calculating a likelihood ratio.

In each case, no finite number of observations will conclusively show a hypothesis to be false, but we may hope to be able to show that a hypothesis makes a poor fit with the data, or at least poorly by comparison with rivals.

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  • Non-falsifiable are not falsifiable (the negation if I may). Falsifiable is that which can be found false. Something is falsifiable if the logical (and metaphysical) possibility for an observation to falsify it exists. Non-falsifiable is everything else. Nov 8, 2023 at 20:20
  • Indeed, but there is an important difference between a proposition that is unfalsifiable because it is a tautology or because it has no empirical consequences, and a proposition that is unfalsifiable because it would theoretically require infinitely many observations to falsify it. The latter can only be partially disconfirmed statistically. I presumed from your mention of probability and limits that you had the latter in mind.
    – Bumble
    Nov 9, 2023 at 0:27
  • By non-falsifiable I meant both. But the question was how can the latter be falsified. Nov 9, 2023 at 15:16
  • +1 for a great answer. @GeorgeNtoulos - probabilistic hypotheses cannot be conclusively falsified, if that is what you are asking. They just become increasingly implausible. If you are willing to make a limiting argument from finite sample, then you can probably get there but it would depend on how you justify the limiting argument in the first place.
    – Annika
    Nov 12, 2023 at 20:58
  • Neyman-Pearson theory deals with statistical hypothesis tests and they used the term "significance". Although there are differences between Neyman-Pearson's and Fisher's (and other people's) views on statistical tests, I think it is misleading to present "Neyman-Pearson" as essentially different from significance testing. (Deborah Mayo has in fact written a lot about the connections and differences between Neyman-Pearson and Fisherian testing, acknowledging that these are strongly related.) Nov 12, 2023 at 22:20
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I read from your question that you accept for scientific theories the principle of falsification due to Popper. I also agree that assumptions or premisses of a theory are comparable to the axioms of an axiomatized theory.

  1. Having said that, the answer to your question

    How valuable, acceptable and warranted are non-falsifiable assumptions?

    is obvious: Non-falsifiable assumptions are neither valuable, nor acceptable, nor warranted for a scientific theory.

    For this point I do not see any relevance between assumptions on a probabilistic basis and those on a different basis.

  2. The specific question, whether a linear no-threshold model is a correct scientific explanation for a sample of observed events, must be decided by the specific methods of the domain under investigation.

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Karl Popper maintained that empirical sciences should be based on the principle of falsifiability rather than verifiability

The common-sense attitude, which is that of most people when dealing which everyday situations, but also that of scientists, is to start from factual assumptions, i.e., observations. For example, one would normally start from the assumption that the sun is the brightest object visible in the sky during the day. From this, humans typically conceive a theory, what we could call a theoretical assumption. For example, that the sun turns around the earth. Why not the other way around? Well, we observe the sky for several days and we invariably conclude that the sun certainly appears to be turning around the earth. Thus, adopting assumptions already involves verification. It is definitely true that the sun is the brightest thing visible in the day sky, and that it seems to turn around the earth.

From our assumptions, we may derive all sorts of logical consequences. If we do, we can use them to falsify our theory if we can verify that at least one consequence is false. Doing this again involves verification.

Why would we choose to falsify the theoretical assumption rather than our factual assumption? Well, facts are often just more certain than the theory. More fundamentally, all theories have to be based on some facts. We cannot think that the sun turns around the earth unless we already have a notion of what the sun and the earth are.

So, I'm afraid that verification comes first.

Yet, verification may be mistaken, which is why falsification comes in handy. However, it would be wrong to think that falsification itself cannot turn out to be wrong, too. We could perhaps, for example, be happy to have falsified the theory that Jim was the murderer, but later falsify our falsification by finding some more compelling evidence that he really was the murderer. Our falsifications are usually themselves complexes of facts and theory, and they can often for this reason be themselves falsifiable, if not necessarily falsified.

How falsifiable are assumptions based on probability

All our assumptions about the physical world are based on observations and are therefore fundamentally probabilistic and falsifiable. Which is also why we can doubt even that which we think is most certainly true.

when x tends to infinity; how can one make infinite observations?

We don't know how complex is reality, but if it was infinitely complex, no amount of cogitation of a human brain would possibly get to a true science. No amount of verification and falsification would ever be able to help us produce true science. For one thing, conception would take longer and longer as we proceed to complexify our theories to match the complexity of the world.

Still, we usually assume that reality is not infinitely complex, which means that successive models based on observation, and a logical analysis involving verification and possibly falsification, should get science closer to be true of the real world.

However, even if reality is not infinitely complex, which seems at least more plausible, we still don't know how complex it is, so that while science may get closer to be true, there might not be enough time left in the universe for us to produce true science.

If we further assume that reality is reasonably simple, which is a completely unwarranted assumption, we can still hope to produce true science, and we will do it using verification and falsification. The more complex the reality, the longer it will take to produce a true science of it, but part of current science is very plausibly already true even if we don't know that it is.

We also cannot tell in advance what kind of model will be true of reality. Maybe the basic concepts of General Relativity, though apparently verified, are in fact completely wrong. Logical reasoning at least seems to guarantee certainty that the consequences of a theory fit our basic assumptions, but we conceive our theory to fit our assumptions, so we cannot then use the theory to verify the assumptions, and so our assumptions will always be possibly false, even when in fact true.

Infinity in the size of the universe may or may not be a problem. Simply, if the universe has a size which is infinite, we won't be able to either verify or falsify that our theory applies everywhere in it. We wouldn't even be able to verify our basic assumption that the universe is consistent. And, there is no good reason to assume that the universe is finite in its dimension.

Still, what really matters is to be able to use science to help humanity survive, and any improvement in our scientific theories will help in this respect, even if we don't ever produce a completely true science because the universe, maybe, is either infinitely complex or infinitely big.

Metaphysics may be fun, but it won't help humanity survive... unless reality is really very, very different from what very nearly all humans believe that it is.

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@Bumble gave a great answer on in what sense we can and cannot falsify statistical hypotheses (i.e., things that only become certain in the infinite limit).

I'd like to add a meta-consideration that @Mauro ALLEGRANZA alluded to in the first comment. Quine, Duhem and others have argued for the underdetermination of scientific theories with respect to the available data. In particular, the Duhem-Quine Thesis states that you cannot test hypotheses in isolation, as there are auxillary hypothesis that are (often unconsciously) held to be true so that the results are interpreted as a test of the main hypothesis under study.

However, Quine, argued that one can always save a favorite hypothesis if one is willing to tack on ever more extreme auxiliary hypotheses. For example, accounting for the motion of the stars in the geocentric view required complex theories of how the celestial spheres move to account for observed epicycles. In contrast, if we view all of us as orbiting a central star, the picture simplifies considerably. Regardless, both theories will be well-supported by the data on stellar motion.

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The reason that science is based on falsifiability rather than verifiability is that a hypothesis or theory must be tested not confirmed. A null hypothesis is often contrary to the expectation from the theory. If it were not capable of being falsified it would be useless.

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