We have discussed before (and in detail here) how Fisher, inventor of the wee P of which scientists boast (“Look how small my P is!” shouted the excited scientist), was deeply influenced by the ideas of logical positivism. The best known proponent was Karl Popper. His big idea was that no proposition (presumably except his own) could ever be verified, though they could always be falsified.
This explains the curious language Fisher used. A “null” “hypothesis”, which is a proposition by another name, could be “rejected”, i.e. “falsified” in some odd sense, but never “accepted”. One always “failed to reject” the “null”. That is, failed to “falsify.” Because, to Fisher, falsification was the only move in logic allowed.
This is, of course, silly. Because falsifying any proposition is always to truthify (if you will) it’s logical contrary. If you say “The null has been falsified”, and that is true, then “The contrary of the null is true” is itself true. You have violated Popper’s rule. It turns out propositions can be verified after all. Popper’s position, and therefore also Fisher’s, is irrational.
This sort of silliness was pounced on by David Stove in his book Anything Goes: Origins of the Cult of Scientific Irrationalism (1998; in the States; playing off Feyerabend’s title) also titled Scientific Irrationalism: Origins of a Postmodern Cult (2001; elsewhere); both reprints of Popper and After: Four Modern Irrationalists (1982; and now online).
Here is Stove dissecting a point in Popper’s The Logic of Scientific Discovery (1959). Stove shows how Popper “sabotaged” certain logical propositions (starting on page 65 in Anything Goes). Comments of mine are between curly brackets {like this}. Original edits by Stove are in square brackets [like this]. I have added paragraphifications to aid screen readability.
We couldn’t have had p-values foisted upon us without this sabotage, which this excerpt will show.
The propositions {sabotaged by Popper} were unrestricted statements of factual probability: that is, contingent unrestricted propositions of the form “The probability of F being G is = r”, where 0 < r < 1 {where the bounds are strict; i.e. 0 and 1 are barred}. For example, H: “The probability of a human birth being male = 0.9”. {Stove considered “factual” probability to be statements about observables, contrasted with “logical” probability, which are statements about non-observable propositions. We, however, say there is no difference. And, even more importantly, that all probability is conditional, just as all logic is.}
Concerning such propositions Popper had fairly painted himself into a corner. For he had maintained (1) that some such propositions are scientific; (2) that none of them were falsifiable (i.e. inconsistent with some observation-statement); while he had also maintained (3) that only falsifiable propositions are scientific. (The reason why (2) is true is, of course, that H is consistent even with, for example, the observation statement E: “The observed relative frequency of males among births in human history so far is = 0.51” {and where failure to grasp this is the cause of great and long-lasting probabilistic grief}).
Popper draws attention with admirable explicitness [Popper p. 191] to this—to put it mildly—contretemps. He puts it almost equally mildly himself, however. For he insists on calling the conjunction of (1), (2) and (3) a “problem” (“the problem of the decidability” [p. 196] of propositions like H); when in fact of course it is a contradiction. The reader can hardly fail to be reminded of Hume’s complaint about the absurdity of the “custom of calling a difficulty what pretends to be a demonstration and endeavoring by that means to elude its force and evidence”. But Popper’s ‘solution’ to his problem was far more remarkable than even his description of it, and indeed was of breathtaking originality.
It consists…in making frequent references to what it is that scientists do when they find by experience that s, the observed relation frequency of G among F’s, is very different from r, the hypothesized value of the probability of an F being G. What scientists do in such circumstances, Popper says, is to act on a methodological convention to neglect extreme probabilities (such as the joint truth of E and H); on a “methodological rule or a decision to regard […] [a high] negative degree of corroboration as falsification” [p. 2020], that is, to regard E as falsifying H. {You see the relationship to p-values.}
Well, no doubt they do {and they do; oh, they do}. But obviously, as a solution to Popper’s problem, this is of that kind for which old-fashioned boys’ weeklies were once famous: “With one bound Jack was free!”. What will it profit a man, if he has caught himself in a flat contradiction, to tell us about something that scientists do, or about something non-scientists don’t do, or anything of that sort? To a logical problem such as the inconsistency of (1), (2) and (3) there is of course—can it really be necessary to say this?—no solution, except solutions which begin with an admission that at least one of the three is false. But least of all can there be any sociological solution.
For our purposes, however, what is important about the episode is the following. The pairs of propositions we are talking about are pairs such as E and H. As (2) implies, and as is in many cases obvious, E is consistent with H {again, if you can’t see this, you are confusing probability with decision, an exceedingly common and inveterate mistake}. But the logical word ‘falsifies’ or its cognates, applied to a pair of propositions, implies that their logical relation is that of inconsistency. So to say that E falsifies H would be to make a logical statement which is false, necessarily false, and obviously false.
So Popper will not say that. What he says instead are things which, however irrelevant to his problem, are at least true (even if only contingently true). Such as the following. That “a physicist is usually quite well able to decide” when to consider a hypothesis such as H “‘practically falsified'” [p. 191] (namely, when he finds by experience, for example, that E). That “the physicist knows well enough when to regard a probability assumption as falsified” [p. 204] (for example he will regard H as falsified by E). That propositions such as H “in empirical science […] are used as falsifiable statements” [p. 204]. That given such an observation-statement as E, “we shall no doubt abandon our estimate [of probability, that is, H] in practice and regard it as falsified” [p. 190].
Now I have pointed out countless times that, logically, “practically falsified” is equivalent to “practically a virgin”. Which is to say, not falsified and not a virgin. P-values when used to “practically falsify” a hypothesis, then, are not logical. They are decisions.
Scientists are allowed to make decisions, of course, but they should not be allowed to strip away uncertainty when conveying their results. Which is what p-value use does. (They also confuse uncertainty, implying, quite falsely, though in a vague sense which nobody can articulate, that the p-value is the probability the “null” is true.)
These are the very models of how to sabotage a logical expression by epistemic embedding, or of ghost-logical statements {made by embedding logical statements in empirical ones; such as “Statisticians regard small p as falsifying the null”; it sounds logical, but it is nothing more than a sociological statement}.
They use a logical expression, one implying inconsistency, but they do not imply the inconsistency of any propositions at all. They are simply contingent truths about scientists. Yet at the same time there is a suggestion that not only is a logical statement, implying inconsistency, being made, but that one is being made with which no rational person would disagree. This suggestion is in fact so strong as to be nearly irresistible, and it comes from several sources.
First, Popper’s references to a rule, decision, or convention, imply that when scientists regard E as falsifying H, they cannot be wrong: and they therefore serve to suggest that they are right. Second, there is the fact that scientists regard E as falsifying H, and that they are unanimous in doing so. How can a reader suppose that scientists, all scientists, are mistaken in regarding E as inconsistent with H? He might almost as easily suppose all philosophers mistaken in regarding a Barbara syllogism as valid. {Barbara: All X are Y; and all W are X; therefore all W are Y.}
Third, and most important of all: the reader’s own common sense—and it is his logical common-sense—emphatically seconds the statement of logic which here appears, by suggestio falsi, to be being made. He knows, as everyone (near enough), knows, that given E, it is rational to infer that H is false. And since scientists, as these statements report them, seem to be saying only very much the same thing, the reader is disposed to think that the scientists are right. And if they are right, it is clearly a point of logic on which they are right.
The suggestion, coming from all these sources, that a logical statement, and a true one, is being made, is so strong, in fact, that to many people it will appear perverse, or at least pedantic, to resist it. What is there, then, to object to, in the statement that scientists regard E as falsifying H?
Or: What is there, then, to object to, in statement that scientists regard “p < 0.05” as falsifying the “null”?
Simply that its suggestion, that a statement of logic is being made, is false; and that suggestio falsi is not better, but worse, the stronger the suggestion is. The statement is only a ghost-logical statement. It implies nothing whatever about the logical relation between E and H. A logical word, “falsifying”, is used indeed, but its implication of inconsistency is sabotaged by the epistemic context about scientists. This is cold-blooded murder of a perfectly good logical expression, in exchange for a handful of sociological silver about scientists.
What makes the case more unforgivable is that the logical expression here sabotaged is not only a strong or deductive-logical expression, but the one which is, of all deductive-logical words, Popper’s own particular favorite; and that he had just a few pages before undertaken that, however others might succumb to non-deductive logic, he never would, but that in his philosophy all relations between propositions of science would be “fully analyzed in terms of the classical logical relations of deducibility and contradiction” [p. 192].
Stove goes on, but we won’t. This is enough for us to see that what happens with p-values is nothing more than custom, and a custom having no logical weight.
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