Popperian falsifiability  is a cornerstone of modern science and its philosophy. I will not question the importance of Popper’s work; it was a major source of inspiration for our work. I will not argue that there are weaknesses in the concept of falsifiability of scientific hypotheses. Popper made it clear in his work that the concept of falsifiability is theoretical and cannot be achieved in practice. His achievement was to demarcate empirical science, as the investigation of logically falsifiable statements, from metaphysics. Popper’s key thesis was proposed as follows:
“But I shall certainly admit a system as empirical or scientific only if it is capable of being tested by experience. These considerations suggest that not the verifiability but the falsifiability of a system is to be taken as a criterion of demarcation.* In other words: I shall not require of a scientific system that it shall be capable of being singled out, once and for all, in a positive sense; but I shall require that its logical form shall be such that it can be singled out, by means of empirical tests, in a negative sense: it must be possible for an empirical scientific system to be refuted by experience.”
As examples of logically falsifiable statements, take the following sentence from a review article on the Pathobiological Determinants of Atherosclerosis in Youth (PDAY) Study, which investigated the prevalence of atherosclerosis in 2876 subjects aged 15 to 34-years-old .
(1) “All subjects in this study had lesions in the abdominal aorta, and all except 2 white men had lesions in the thoracic aorta.”
From this sentence we can construct an example of a scientific statement that logically falsifiable:
(2) All humans between the ages of 15 and 34-years-old have atherosclerotic lesions in the abdominal aorta.
Or in a more logical form:
x = humans between the ages of 15 and 34-years-old,
α = x has atherosclerotic lesions in the abdominal aorta
Logically this statement (2) could be falsified by the example of one human aged 15 to 34-years-old who did not have any lesions in the abdominal aorta. On the other hand, the universal claim made by “all” makes this statement unverifiable as there is a potentially infinite number of humans.
If we go back to the sentence (1), we can see that the authors note that “all except 2” subjects had lesions in the thoracic aorta. This was not incorporated into the statement (2). But given that 2874 of 2876 subjects (99.9%) did have lesions in the thoracic aorta, is it reasonable to reject the claim (3) below?
(3) All humans between the ages of 15 and 34-years-old have atherosclerotic lesions in the thoracic aorta.
Certainly these two subjects appear to have falsified (3). If we accept the falsification of the claim in (3), then it appears that the most that can be claimed is (4)
(4) Almost all humans between the ages of 15 and 34-years-old have atherosclerotic lesions in the thoracic aorta.
Now (4) is no longer logically falsifiable and thus no longer scientific or empirical, as we can never be sure whether a subject without lesions in the thoracic aorta falsifies the claim or is one of the those excluded by the hedging word “almost”. According to Popper, science involves the empirical investigation of statements like (2) and (3), but not those like (4).
Popper was clear on the real world limitations and possible objections to his approach and discusses them explicitly:
“A third objection may seem more serious. It might be said that even if the asymmetry is admitted, it is still impossible, for various reasons, that any theoretical system should ever be conclusively falsified.”
… “I must admit the justice of this criticism; but I need not therefore withdraw my proposal to adopt falsifiability as a criterion of demarcation.” [1, p19-20]
“If falsifiability is to be at all applicable as a criterion of demarcation, then singular statements must be available which can serve as premisses in falsifying inferences. Our criterion therefore appears only to shift the problem — to lead us back from the question of the empirical character of theories to the question of the empirical character of singular statements.” [1, p21]
Popper is acknowledging that any evidential claim put forward to falsify a hypothesis is in itself a scientific hypothesis that by Poppers philosophy cannot be verified, only theoretically falsified at best, leading us into an infinite regress. Poppers philosophy offers demarcation of empirical science statements from other meta-physical statements, but does not offer definitive rejection of hypotheses by falsification. Falsification is only theoretical. Experiments never definitively falsify a hypothesis.
A simple example
An example hypothesis that meets the requirement of Popperian falsifiability is as following:
(1) John Smith has familial hypercholesterolemia
Familial hypercholesterolemia is a disease that causes LDL cholesterol (bad cholesterol) to be very high. As specified by the name, the disease is genetic and is passed down through families. There are many genetic defects that can cause the disease, but for this example let us assume that there is only one genetic defect that causes the disease and that (1) is denoting only the disease caused by that defect. Without this assumption the statement becomes unfalsifiable as we could always argue that while John Smith has none of the known genetic defects, he may have an as yet unknown defect that causes the disease.
If we run a genetic test on John Smith, we could falsify (1) with the following finding:
(2) John Smith does not have the familial hypercholesterolemia genetic defect
However, this falsification can never be definitive as (2) is in itself an empirical statement. All diagnostic tests are exposed to uncertainty and this is represented by their false positive or false negative rate. In this case, there is always a chance that John Smith’s diagnostic result was a false negative. That is, he may really have the genetic defect, but the test failed to identify it. No diagnostic test can definitely falsify a hypothesis. There is always some chance that the test itself is false.
Popper notes that empirical hypotheses “can never become ‘probable’: they can only be corroborated, in the sense that they can ‘prove their mettle’ under fire—the fire of our tests.” [1, p259]. This same concept applies symmetrically to apparently falsifying empirical evidence. Popper’s philosophy can’t even give us a probability of falsification. Popper ultimately only gives us a theoretical means for demarcating scientific from meta-physical statements.
Note that I am not a scholar of Popper and that this discussion is not complete. I believe that Popper addressed these points well in his work and you should refer to it for more detail.
Version 2.0, 19th October, 2016