Is the negation of a non-falsifiable statement falsifiable?

Question

In Popper's "Logic of Scientific Discovery", falsifiability is set up as a criterion for determining the purview of science. The canonical example is the statement "All swans are white", which is falsifiable, and therefore scientific, but coincidentally false.

Reading some criticisms of the SETI program, it occurs to me that falsifiability is frequently a property held by a statement, but not by it's negation. For example, "Not (All swans are white)" == "There exists a non-white swan" is not a falsifiable statement, since no quantity of white swans would rule out the possibility that there exists under some stone uncovered a non-white swan.

This is all well and good, but it seems to make the criterion moot, since this implies that not only is the negation of a falsifiable statement sometimes unfalsifiable, but the negation of an unfalsifiable statement is sometimes falsifiable:

An oft-cited example here surrounds the existence of some arbitrary deity, say, Flying Spaghetti Monster (FSM), who operates in mysterious ways and is beyond the reckoning of we mere mortals. The statement "FSM exists" is unfalsifiable, since observations that FSM does not exist here nor there may be simply rebutted by assertions that FSM exists everywhere, or some such. OTOH, the statement that "FSM does not exist" IS falsifiable, since FSM could show up knocking one day with her noodly appendage.

Now, the scientific method for testing the statement "All swans are white" is to literally travel about looking for non-white swans. By a naive application of this logic it seems as though the scientific method for testing the statement "FSM does not exist" is to search for FSM. This implies that those engaged in the study of and search for FSM are, in fact, scientists. Of course, these same individuals may take exception to the claim that they are attempting to prove FSM's NON-existence, but this is beside the point.

Surely such an obvious hole in the falsifiability demarcation criterion has not gone unaddressed. Is this a misapplication of the notion of falsification? What is the problem with this reasoning?

Answer 1

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.

"The FSM does not exist" is falsifiable by potential observation, in spite of its opposite not being so. Unlike swans, we do not know the structural requirements for producing an FSM, and we cannot rule out their occurrence. But we would know one if we met him.

But just as in the swan example, the route to finding him is not to just go out and look around. We can narrow the odds with known theories, and we can determine pretty quickly that the existence of this thing is not a good bet. We can try inventing live spaghetti... We don't expect certainty unless we are very lucky. But that is the norm for scientific evidence.

And we clearly have examples like the ones that originally motivated Popper, the cornerstones of theories like Marx and Freud. Reichian psychoanalysis claims that Orgone is blue. But Orgone is nonsense. Nonsense could be blue if it wanted to, but you would never know. So this is unfalsifiable either way.

We have found that all the permutations exist. So negation and falsifiability are not logically related.

The criterion is not about questions or topics, it is about theories. Anti-pastafarianism is a scientific theory, if an obviously silly one. (The more specific the counterexample necessary, the weaker the theory, and this is awfully specific. A criterion you could only meet with immense luck is still a criterion, but is not very good science.) That does not make Pastafarianism into a scientific theory. The logic is not symmetric just because we like symmetry -- it says what it says. The opposite theory is a separate theory to which the standard applies or does not apply.

This is how Dawkins can consider atheism scientific, and theism nonscientific. He can claim his high odds are based on observations, and atheism is scientifically likely based on the absence of many things one might expect to proceed from God. I don't think this is honest, because I do not believe Dawkins would admit it if he did see God personally. But if you assume God is like the FSM, and you could not help but identify Him given the evidence, then that still only makes atheism a potentially scientific position, not theism.

Theism becomes a scientific position only if you could identify what evidence would necessarily make you stop believing in God. Since his opponents can't do that and stay Christians, his is the only side with a scientific approach.

Answer 2

This is a question about predicate denial.

Certainly in Fregean logic, where predicate denial is taken as the unary "it is not the case that", for a predicate P, P¬x is logically equivalent to ¬Px.

So the question becomes, is the Fregean notion of predicate denial the correct notion to apply to the predicate "is falsifiable".

Here, one might argue that "is falsifiable" is a vague predicate. With vague predicates, statements such as "x is neither P nor not-P" are often judged true. This gives a different view of predicate denial since the law of the excluded middle may not hold.

Answer 3

TL;DR: If you remember about Popper's asymmetry between the contrast of your proposition, and the termination upon being proven false, for your given proposition [this procedure is called falsification], you will find by duality that the complementary proposition will be subject to some sort of verification instead.

Verification is not falsification but the opposite, so this question can be answered: exactly in the case that your non-falsable statement is verifiable. This is mostly the case of existence questions (e.g. Higgs Boson). Once you verify the Higgs boson exists, you are right.

Another example: For atheists, God's existence is a matter of verification rather than falsification: I will not accept that as true until you bring God to me, or bring me to God.

Let me explain this:

FSM does not exist Is not falsifiable (it is not even well defined) until you define what kind of existence are you talking about.

On the other side, "There exists a non-white swan" or "Not (all swans are white)" is falsable. This means that you can pick a method of experimentation (which may not seem practical, although possible).

Okay, I will change the sentence. I will say there is a tortoise of the species that can be found on Galapagos Islands (instead of swans) which is black. How could I verify that? Going to Galapagos (and the authorized zoos around the world that breed those species of tortoises) and checking the color.

This effort is huge, yet possible. Even if you don't know how to get there, currently available knowledge would let you develop or find the means of performing your observational experiment (I think that, right now, chinese government could ask this question in their country and actually experiment to track all swans' colors!!!).

In partcular, this statement you asked for is quite stationary: Take a snapshot right now, and you will answer yes or no for both of them.

However, although these two cases are not good examples (since both of them are falsifiable by the same mean, despite the negation taking a huge time), you could find examples when you may say the complementary propositions are not falsifiable.

Let's start by stating Popper's asymmetry in this matter. Popper talks about contrast (falsification) in this way:

• You need to have a method of experimentation / observation that could determine whether your proposition is true or false in such experiment.
• If false, your proposition is plain false and you should state another one (which could be similar, avoiding the false cases by some sort of comprehension or enumeration, but still being a different proposition).
• If true, and your proposition pretends to live across time and is not a proposition scoped to the current instant and circumstances, your proposition is good so far. However it cannot be proven definitely true, but just so far each time. The mostly known example is the study of gravity across all our history of physics, which involved a lot of different theories across the time, when the latter ones superseeded the former ones.
• If true, and your proposition does not pretend to live across time (i.e. is not that... general) but is just a one-time proposition who pretends to be valid just now or during the experiment's lapse, you can safely consider it true.

Most of the scientifically useful propositions satisfy the 1st (They have a method to test the truth) and 3rd point (they want to be generic across time, and want to be... true). So they can be expressed in a shorter way:

• If, by the appropriate experimental mean, you find that your proposition is false, then it is false.
• Otherwise, the proposition is good so far.

Now take the complementary proposition. You will have two analogous points:

• If you found your proposition was false during experimentation, it is false, definitely. This makes the complementary proposition... true.
• Otherwise, since your proposition is good(apparently true) so far, the complementary is bad(apparently false) so far.

Said this, you could somehow remind yourself about Adolph J. Ayer who talks about verification instead of falsification. The concept of verification is rarely used to generate a strong and useful concept on science (Popper himself argues a lot against verification), but you can quickly grasp here (after all this is the dual proposition!!) that falsification of a proposition is exactly as powerful as verification of its complementary, since it is the dual procedure.