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I am reading Popper's "Conjectures and Refutations" (2002 edition) where he writes on page 54 in the footnote:

Thus the negation of any nonsense will be nonsense, and that of a meaningful statement will be meaningful. But the negation of a testable (or falsifiable) statement need not to be testable, as was pointed out, first in my "Logic of scientific dicovery" (e.g. pp. 38 f.) and later by my critics.

I understand the first part, but I have no clue about the seocnd part (the "testable"). I cannot imagine any example and unfortunately, I have no access to the book. I found something in the www, but I could not find any reference therin which explains the second statement.


Appendix thanks to many useful answers & comments:

Short version: I stop bothering about this statement as I think it is too vague and thus meaningless (I think). But I learnt a lot from the answers. :-)

Long version:

  • Although I already highlighted "Words are significant only as instruments for the formulation of theories, and verbal problems are tiresome: they should be avoided at all cost." (again from Popper, p. 37/38) you again showed how important this statement is. This said:
  • The statement appears in the context of his "criterion of demarcation" and here with respect to Wittgenstein’s "verifiability criterion of meaning". Thus, I cannot exclude that the statement was meant in the sense of @Agent Smith proposed.
  • Concerning verbal problems: it should be made clear whether "testable in theory / principle" is meant or "testable in practice". All statements which "hold for all time" are impossible to prove. Depending on the statement S or its negation NS, we cumulate evidence for S or for NS. But this was not my point for the question – I thought it was some "deep fact" which I did not see.
  • As discussed here: "... the scientific method for testing the statement 'Not all swans are white' is to examine the genetic structure of swans, determine what parts render them white and see if mutation can be evoked that would render them non-white. If such a mutation would not be survivable, all swans are white. A statement is unfalsifiable if you cannot identify a procedure that would disprove it. And this could. So this statement is falsifiable just like its opposite...". Again, this was not the example I looked for. I know that testing can be tricky.
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  • "Absence of evidence is not evidence of absence"
    – CriglCragl
    Jul 25, 2023 at 16:18
  • @CriglCragl but how does that answer my question? The example of "all swans are white" and its negation are both in principle falsifiable.
    – Christoph
    Jul 25, 2023 at 16:48
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    @CriglCragl The negation of that statement is not scientifically falsifiable. "All swans are white" can be disproven by observing a single non-white swan, but cannot be (practically) proven because doing so would require observing every swan to be white. Inversely, "Not all swans are white" can be proven by observing a single non-white swan, but can only be disproven by observing every swan to be white. Jul 26, 2023 at 1:23
  • There is a typo in the word "second" (written as seocnd) that I cannot edit because it is too short, someone with that ability please fix that typo. Jul 26, 2023 at 19:51

7 Answers 7

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A popular phrasing of the second law of thermodynamics is

Heat always moves from hotter objects to colder objects.

This statement is falsifiable. All you need to do is demonstrate a situation where this is not the case.

Its negation

It is not the case that heat always moves from hotter objects to colder objects.

is not falsifiable since no matter how many experiments you perform, you can never exhausted all possible setups.

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I've actually never read Popper but I ran into this recently while studying Marshall McLuhan's Laws of Media. McLuhan uses this definition to defend the scientific nature of his own claims. From what I understand, Popper is essentially trying to demarcate a difference between scientific claims and claims that are not scientific. In order for a claim to be scientific, Popper thinks, it must be "testable"/"falsifiable". So take a claim like "All bears are brown". This would be easy to test. You would just look at the color of bears fur. This would also be easy to falsify because as soon as you find a black bear or Polar bear the statement would be false (though still scientific). The negative of the statement "Not all bears are brown", would be difficult (I guess impossible?) to falsify, since you would need to experience every bear to determine that it is false. So it's just to say that the opposite does not need to be testable.

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  • The negation is "there is at least one bear which is not brown". Positive version: you believe in your claim as long as you do not find a non brown bear. If the latter happens your hypothesis is wrong. Negative version: you do not believe in the thesis until a non brown bear is found. To me there is no difference except perhaps different likelihoods which do not add anything useful. You agree?
    – Christoph
    Jul 25, 2023 at 16:11
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    A single case against the claim "all bears are brown" would falsify it ( thus giving it ‘conclusive’ falsifiability). But give a single case that would falsify the opposite (not all bears are brown)? If no single case can do that then it is not testable for Popper. Btw, I did find the Popper text that was referenced in your quote: philotextes.info/spip/IMG/pdf/…. I think it would be best to read through chapter two. I started it but got bogged down in his language. He is NOT easy to read.
    – ferris
    Jul 25, 2023 at 16:57
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Here’s a tongue in cheek example: The negation of a semidecidable set need not be semidecidable.

In particular, we can test whether a turing machine halts on a particular input (just run it) but we cannot test for its negation : the TMs failure to halt at time x is not indicative of its failure to halt overall.

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Reading the cited work might help, but I did and I think Popper's explanation could be clearer, so here's mine instead.

Lloyd: "What are my chances [of dating you?]"

Mary: "Not good."

Lloyd: "You mean “not good” like… one out of a hundred?"

Mary: "I'd say.. more like one out of a million."

Lloyd: "So you're telling me there's a chance?"

Statements in the form always or never which do not refer to logical or, to the best of our knowledge, physical impossibility, merely to probability indistinguishable from 1 or 0, are falsifiable. Mary means she will never date Lloyd. Lloyd knows that this could be falsified if Mary dates him, so he believes he has a chance. The audience laughs because we know this is not a reasonable interpretation.

Statements of not-always or not-never are not falsifiable, except when they can be expressed as logical or physical impoosibilities. Mary can falsify Lloyd's not-never hypothesis only by not-dating Lloyd until one of them dies, because Lloyd or Mary does not exist is logically incompatible with Lloyd dates Mary. If we rephrase this to be about Mary only ('Mary will never get that haircut'), then Mary can never have falsified the negation, since the only way to falsify it is to cease to be Mary.

For a more physical example: It is true, falsifiable, and not logically necessary, that no-one will ever spontaneously disintegrate into an explosion of subatomic particles because all their protons decided to quantum tunnel a few meters away all at once. The reverse is false, but unfalsifiable, since it is in principle logically possible. To falsify, you have to run the experiment until the last someone ceases to exist.

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Me two cents ...

A testable statement = A scientific statement (synonymous)

The negation of a testable statement = a nonscientific statement i.e. we're now in pseudoscience territory, notoriously untestable. I'm a cancer, but in some universe I'm a scorpio; me very November you see.

EDIT 1 (prompted by Christoph's comment)

There's an ambiguity then in Popper's statement. Did he mean what I think he meant (vide supra) or did he mean what the OP (Christoph) thinks he meant?

Let's take a simple claim and its negation, both testable (a scientific claim thus):

P = There's a pig in the hall. ~P = There's no pig in the hall

Notice that P can be tested - all we havta do is look around the hall.

~P can also be tested, in exactly the same way, but does it have to be testable. What's the default truth value of scientific claims? Is it unknown or is it false? Do we assume ~p or do we start off with p v ~p? If the former, the negation of a scientific claim needn't be testable (it's false).

This position is similar to atheism (they don't need to disprove god; all they have to do is refute theistic arguments).

Then there's you can't prove a negative, which is one of the standard responses against theists.

All in all, an issue of onus probandi (burden of proof). I've googled this topic hundreds of times with null results.

Moreover, the scientific method defined by testable hypotheses goes like this:
Hypothesis (H) --> Observation (O)

~O (observation is negative) --> ~H

We've just proved the negation of the scientific claim H.

In addition, scientific claims tend to be positive, loosely that something exists/is. We're interested in what is and not in what is not. No scientist goes around saying things like "peaches are not apples" or "gravity doesn't exist". A negation of a scientific claim would then, in most cases, fall into the category of claims of nonexistence. Put this together with my view on default truth value (false) and there's some sense to Popper's claim that negation of scientific claims don't have to be testable.

In a nutshell, if our zero (starting point) is ~p, why would we need a test to prove ~p.

Apologies if my post is haphazard. I'm responding intuitively.

EDIT 2

If the negation of a scientific claim could be tested, we could prove scientific claims like so:

~H --> O
~O
Ergo,
H

What happens then? The consensus, last I checked, was that scientific claims can't be proven.

EDIT 3

This is my final comment om the issue. Took a nap, cleared up me head.

The Scientific Method

  1. H (hypothesis) --> O (Observation)
    Two possibilities now:
    2a. O
    Ergo,
    3a. H
    The hypothesis H is not falsified or, using an older, discredited term, H is confirmed.

2b. ~O
Ergo,
3b. ~H
H is falsified, another way of saying ~H has been proven i.e. the negation of the scientific claim H has been proven.

Conclusion: H can only be "confirmed" or is not falsified but ~H can be proven.
That is to say a test can only falsify/"confirm", it can't prove and we need to prove ~H, in order to falsify H. To cut to the chase, we need to prove ~H and a test simply can't. So ~H (the negation of a scientific claim) needn't be testable, but it better be provable.

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  • Would you then not talk about the "opposite" statement instead of "negation"?
    – Christoph
    Jul 25, 2023 at 19:13
  • I edited my answer @Christoph Jul 26, 2023 at 4:50
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Kind of a weird answer to your question, but maybe it will interest you. In this post from Math Stack Exchange, the top-voted answer describes how one can think of the open sets of a topology as encoding so-called "semidecidable properties", as in @emesupap's answer. That answer is more focused on the interpretation in terms of computability theory, but I think the interpretation of empirical testability works just as well.

To be more explicit, if W is a mathematical set of "possible worlds" or "possible states of affairs" under consideration, we could define a topology on W in such a way that the open sets correspond to sets of possible worlds with the property that we can confirm that the true state of affairs belongs to that set through some observation/experiment. In a topology, the closed sets are the complements of the open sets, so the closed sets would correspond to falsifiable hypotheses - that is, collections of possible worlds with the property that some possible observation allows us to conclude that the real world is not among them.

Within this framework, the hypotheses that are both confirmable and falsifiable would be the clopen sets of the topology on the set of possible worlds.

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The answer following is a specific case of the question you have asked, but nonetheless, I believe it is relevant and interesting.

In Computational Complexity, the problems belonging to the NP-complete complexity class (wiki) are widely conjectured (see NP != coNP) to be easy to verify but their negations aren't. Consider one of the NP-complete problems, the traveling salesman problem (wiki). To check "There exists a path of length <= L that covers all cities," one can be presented with a path that satisfies this property. But take the negation, "There is no path of length <=L that covers all the cities," it becomes significantly harder to verify.

I took the complexity class of NP-complete rather than just NP because the class NP contains problems whose negation is easier to verify (i.e. P).

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