Popper Explains The Asymmetry Between Refutation and Verification

In my last post, I discussed various problems (or rather pseudo-problems) with Popper’s epistemology around his concept of refutation/falsification. (For our purposes, we will treat refutation and falsification as synonymous. [1]) In this post, I will quote Popper explaining in his own words what the actual asymmetry between refutation and verification is.

As it turns out, it has nothing to do with ‘verification’ always being about justifying a theory as true while refutation is always tentative. (i.e. The Absolute Verification Fallacy.) On the contrary, the asymmetry that exists between refutation and verification stems entirely from Popper’s realization that you can turn an explanatory theory into logical propositions.

Popper’s Asymmetry Between Refutation and Verification is Due to an Asymmetry that Exists in Logical Propositions

All quotes, unless specified otherwise, are taken from Realism and the Aim of Science p. 181-189 in the section entitled “The Asymmetry Between Falsification and Verification”:

There is, as pointed out in the L.Sc.D., a fundamental logical asymmetry between empirical falsification and verification. Although some of my critics have denied the existence of this asymmetry, their arguments were anticipated and fully answered in my L.Sc.D.

This fundamental asymmetry cannot, I think, be seriously denied: a set of singular observation statements (‘basic statements’, as I called them) may at times falsify or refute a universal law; but it cannot possibly verify a law, in the sense of establishing it. Precisely the same fact may be expressed by saying that it can verify an existential statement (which means falsifying a universal law) but that it cannot falsify it. This is the fundamental logical situation; and
it shows a striking asymmetry.

(p. 181. Emphasis mine.)

So right off the bat, Popper explains that the asymmetry between refutation and verification is not about certainty or justification but about the logical nature of universal statements vs existential statements.

Verification and Falsification are Logically Equivalent

Of the various objections which have been raised against my claim that this asymmetry exists—and thereby against my criterion of demarcation—the one which at first sight looks the most striking is the following. Whenever we falsify a statement we thereby automatically verify its negation, for the falsification of a statement a can always be interpreted as the verification of its negation, non-a. Accordingly, we can always speak, if we like, of verification instead of falsification, and vice versa: the difference between these two ways of putting things is merely verbal, and they are therefore, for logical reasons, completely symmetrical.

For example, if we can describe an empirical test as an attempted falsification or as a search for a negative instance (of the statement a), then we can also describe it as an attempted verification or as a search for a positive instance (of the statement non-a). Similarly, any obstacle to the verification of a statement a must, for logical reasons, also be an obstacle to the falsification of non-a, and vice versa. Thus it is said that it is difficult to verify an existential statement such as ‘there exists a perpetual motion machine’, that is to say, ‘a machine which continues to emit energy without ever absorbing energy from its environment’, since we might have to search the whole world for it (and incidentally to examine each candidate for an indefinite period of time); but obviously, it must be just as difficult to falsify the negation of this existential statement—that is, the universal statement ‘all machines which continue to emit energy must, after a finite time, absorb energy from their environment’. For the verification of the one statement is nothing but the falsification of the other.

Now it follows from these obvious premises, my critic concludes, that it is, for purely logical reasons, pointless to distinguish between falsification and verification, or between falsifiability and verifiability, or to demarcate a class of ‘falsifiable’ or ‘testable’ statements as ‘scientific’ and to distinguish them from another class of non-falsifiable (though perhaps unilaterally verifiable) statements, which are called ‘metaphysical’. Thus the falsificationist who asserts that the statement ‘all swans are white’ can be falsified will have to admit that every falsification or refutation of this universal statement will be equivalent to the verification, and acceptance, of the existential statement ‘There exists a non-white swan’. It must be wrong, therefore, to call the universal statement ‘scientific’ and the existential statement ‘metaphysical’. (The lesson usually drawn from this is that the distinction between scientific and metaphysical statements does not depend on such things as testability—that is, on a relation between statements—but rather on the concepts—observable or otherwise—which occur in the statements.)

p. 182. Emphasis mine.

So Popper upfront anticipated that his critics would claim that there was no asymmetry between verification and falsification because, logically speaking, every universal law is logically equivalent to its negation which is always an existential statement and vice versa. To falsify the theory “Bigfoot does not exist” is exactly equivalent to verifying the theory “Bigfoot exists.”

I would note here that Popper misses something important that Deutsch adds, namely that the negation of an explanation is not necessarily (but sometimes is) a good explanation itself. The negation of Newton’s theory is not a valid scientific explanation at all but General Relativity is. [2] But General Relativity is not the negation of Newtonian physics.

However, there are cases where the negation of an explanation is also an equally good explanation. This would never happen with a real scientific theory but doesn’t happen with really simple examples like “All Swans are White” where its rival theory is “There exists a non-white swan.”

For example, consider the theory “Bigfoot exists.” It could be legitimately thought of as an explanation for, say, why we find imprints of large feet and there are sometimes sightings of a big man-ape in the woods. But its negation “There is no Bigfoot” could just as much be thought of as an explanation for why we’ve never captured a Bigfoot so far. Again, this only ever happens when we’re dealing with really simple examples. Scientific theories are explanations that when negated, are never a good explanation. [2]

Now you may be wondering at this point if maybe Popper is just acknowledging the argument of his critics but not agreeing with them. But Popper quickly makes it clear that everything his critics are saying is true but merely irrelevant to his point.

In answer to this criticism, I wish to say that I accept all my critic’s premises as true—indeed, as trivially true; but I reject all his conclusions (which are stated in the last paragraph), none of which, incidentally, follows from his premises.

P. 183. Emphasis mine.

So if Popper is admitting that there is a logical symmetry between universal statements and existential statements — and by extension between refutation and verification — then what does he see as the fundamental asymmetry? Popper first makes it clear that it has nothing to do with the idea that verification is absolute and refutation isn’t. Refutation, if taken as absolute, is just as wrong as verification.

First I should like to get one point out of the way—a point with which I agree, since it belongs to my critic’s premises, but which nevertheless betrays a misunderstanding. I mean the reference, in the concluding part of my critic’s premises, to ‘obstacles’ or ‘difficulties’ which sometimes stand in the way of verifying a purely existential statement such as ‘there exists a perpetual motion machine’.

Of course, what my critic says is true—the empirical verification of this statement is for obvious reasons exactly as difficult—or exactly as easy—as the empirical falsification of the universal law which is its negation. But I have never worried about this ‘difficulty’, and I have never referred to it, or drawn any conclusion from it.

p. 183

So if refutation and verification don’t differ based on one being absolute and one not being absolute, then what do they differ on?

I do not call an isolated purely existential statement ‘metaphysical’ because it is ‘difficult’ to verify, but because it is logically impossible to falsify it empirically, or to test it. And I have of course always stressed that the logical impossibility of falsifying an existential statement of this kind is exactly the same thing as the logical impossibility of verifying its universal negation. My critic’s reference to obstacles or difficulties is therefore irrelevant. Moreover, it seems to betray a verificationist attitude: verificationists, it seems, cannot imagine any difficulty about purely existential statements, apart from the difficulty of verifying them.

p. 183. Emphasis mine.

So much for the Twit Rat idea that the asymmetry between refutation and verification is due to the impossibility of absolutely verifying something because nothing is certain. Refutations are no more certain than verifications and verifications can be tentative just like refutations can be. Further, you can easily switch between the two if desired.

Popper then goes on to admit that there is symmetry between verification and refutation just like his critics claim.

The point concerning the ‘obstacles’ or ‘difficulties’ may thus be dismissed as irrelevant; and I can proceed to other and perhaps more relevant points. No doubt one can say that problems of falsification and verification are in certain respects ‘symmetrical’.

p. 183

The issue isn’t that verification and refutation aren’t symmetrical in one being absolute and one being tentative but in the fact that they are symmetrical in this regard doesn’t mean that an asymmetry between them does not exist.

The fact that there are certain symmetries here hardly precludes the existence of a fundamental asymmetry—any more than the existence of a far-reaching symmetry between positive and negative numbers precludes a fundamental asymmetry in the system of integers: for example, that a positive number has real square roots while a negative number has no real square root.

Thus one can certainly say that falsifiability and verifiability are ‘symmetrical’ in the sense that the negation of a falsifiable statement must be verifiable, and vice versa. But this fact, which I repeatedly stressed in my LSc.D. (where I even described universal statements as negative existential ones), and which is all that my critic establishes in his premises, is no argument against the fundamental asymmetry whose existence I have pointed out.

p. 183-184. Emphasis mine.

The actual asymmetry that exists between refutation and verification is a matter of logic, namely that universal statements are far more powerful than existential statements and thus are naturally of greater interest to scientists.

This asymmetry has a purely logical and also a methodological or heuristic aspect.

As to its logical aspect, there can be no doubt that a (unilaterally falsifiable) universal statement is logically much stronger than the corresponding (unilaterally verifiable) existential statement. For the following is a well-known logical rule. From a universal statement, pertaining to all things of a certain kind, or to all elements of a certain non-empty universe of discourse,

(1) All things have the property P, we can derive, for any individual thing a belonging to this kind or universe,

(2) The thing a has the property P; and from (2), in turn, we can derive

(3) There exists a thing that has the property P.

Thus (1) entails (2) and (3), and (2) entails (3). But (3) does not entail either (1) or (2), and (2) does not entail (1).

Or in other words, (1) is logically stronger than (2) and (3), and (2) is logically stronger than (3).

This is the source of the important asymmetry in the case of unilaterally falsifiable universal and unilaterally verifiable existential statements; and the situation is the same with more complex statements.

P. 184

Because universal statements are more powerful, they are of more interest in science. It’s not that you can’t have an explanation that is an existential statement (The ‘bigfoot exists’ example above is such an explanation!) it’s that it’s rare that a scientist would care much about such explanations because they are too weak to be useful in most cases.

Owing to their logical power, universal statements may be important as explanatory hypotheses: they may explain (especially in conjunction with singular initial conditions) singular events or statements. Purely existential statements, on the other hand, in isolation, or even in conjunction with singular statements, are usually too weak to explain anything.

This is why scientists are interested in universal hypotheses rather than in (isolated) existential hypotheses.

This leads us to the methodological or heuristic aspect of the asymmetry—to the difference between the critical or falsificationist attitude and the verificationist attitude.

P. 184

But can existential statements never play a role in science? Or do they sometimes play a role? Popper is clear that sometimes they do play a role in science, albeit a very limited role.

As to purely existential statements, [a falsificationist] is not interested in them because of their weakness, and because they cannot be falsified unless they form an integral part of a theoretical system. He is ready to admit them into science if they are entailed by an accepted basic statement; but even then their interest lies solely in the fact that their acceptance is equivalent to the rejection of their universal negations.

p. 185. Emphasis mine.

If we do ever discover bigfoot and put him in a zoo, we’ll then accept ‘Bigfoot exists at the San Diego zoo” as a basic statement. That will verify the theory that bigfoot exists.

That Popper has no real issue with a ‘verification’ like this might be a bit of a surprise to some. But elsewhere Popper is even more clear that verification is a necessary part of his epistemology when it comes to testing a prediction. He explains that predictions take the form of singular existential statements and thus must be verified.

…certain singular statements–which we may call ‘predictions’–are deduced form the theory [being tested]; …if the singular conclusions turn out to be acceptable, or verified, then the theory has, for the time being, passed its test: we have found no reason to discard it.

LOSD, P. 10 (Emphasis is Popper’s)

If you are still not convinced this is what Popper meant, here is what it says in the index of LoSD: “Verification of an existential statement, possible 10, 48-9&n…” (Note that he’s referring to the above quote on page 10.)

So getting back to our theory “Bigfoot exists” if we found Bigfoot, yeah, we’re going to then admit that theory into empirical science by making it a basic statement (an observation accepted by the scientific community). But even then, the theory “Bigfoot does not exist” being refuted is also true and was the more powerful statement (due to being more constraining), so it’s hard to imagine a situation where verification becomes of special interest to scientists instead of refutation except when it happens to be the negation of a universal law. This is why Popper feels comfortable with the idea that he can nearly ignore verifications. [3]

So it makes perfect sense that Popper found (tentative) verification possible but uninteresting. There would never be a case where an existential statement being important to science wasn’t simultaneously also a refutation of a universal law. So Popper felt comfortable entirely downplaying verification other than in the case of basic statements (essentially verifying a prediction through a test/experiment.) This is the real basis for why Popper emphasizes refutation and falsification over verification. It has nothing at all to do with certainty being impossible. That is certainly true, but it doesn’t matter because any result of an experiment (what Popper calls a basic statement–which we saw above is a verifying instance) wouldn’t be certain either:

Another objection often raised against asymmetry is this: no falsification can be absolutely certain, owing to the fact that we can never be quite certain that the basic statements which we accept are true. …As to the claim that this fact refutes the asymmetry between falsification and verification, the situation is really very simple. Take a basic statement or a finite set of basic statements. It remains forever an open question whether or not the statements are true: if we accept them as true we may have made a mistake.

p. 185. Emphasis mine.

But again, Popper’s interest isn’t that it’s impossible to verify things. It’s that it’s specifically impossible to verify that which scientists are most interested in: general/universal laws.

Hence the asymmetry is that a finite set of basic statements, if true, may falsify a universal law; whereas, under no condition could it verify a universal law: there exists a condition wherein it could falsify a general law, but there exists no condition wherein it could verify a general law.

p. 185. Emphasis mine.

Summary and Conclusions

Popper’s famous asymmetry between refutation and verification has nothing to do with verification being about certainty. The Absolute Verification Fallacy is now refuted. The asymmetry is actually a logical consequence of scientists naturally being interested in universal statements/laws because they are far more powerful (i.e. more constraining) than existential statements.

Notes

[1] I am not here making the claim that Popper always used ‘refutation’ and ‘falsification’ as identical synonyms. See “The distinction between falsification and refutation in the demarcation problem of Karl Popper” by Nicolae Sfetcu. However, there should be no doubt that Popper did often use them as more or less the same thing. For example: “…a set of singular observation statements (‘basic statements, as I called them) may at times falsify or refute a universal law” (Realism and the Aim of Science, P. 181).

See also this quote from Popper where he clearly equates refutability and falsifiability.

Testability is therefore the same as refutability, or falsifiability.

Conjecture and Refutations, p. 266

[2] One might argue here that the negation of Newton’s theory is an explanation of sorts. Saying “why can’t light speed ever be exceeded due to the Michelson–Morley experiment?” can be “explained” by saying “Newton’s theory is wrong.” But note that this does not qualify as a “good explanation” as per Deutsch’s interpretation of Popper’s theory. So it would probably be more accurate to say that the negation of a good scientific explanation is not itself a good explanation rather than not an explanation at all. But this is all we needed to make sense of Deutsch’s argument.

[3] For now, let’s accept Popper’s logic here. But I would note that Penrose’s challenge (see also here) does seem to challenge Popper here, at least at first. The idea is this: suppose we did find a superpartner that verified supersymmetry and String theory. In that particular case, the negation of the theory (“there are no superpartners”) isn’t a theory we’re particularly interested in. The true competitor was actually regular old physics without symmetry. But that theory didn’t actually predict there were no superpartners. It just said nothing at all about the subject. Does this then refute Popper’s view that science is never interested in theories that are existential statements except in so far as their negation is refuted?

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