To disprove the idea that scientific theories have to be falsifiable, you need merely provide some examples of important scientific theories that could never be falsified. It is very easy to do this. Let's start with one of the most widely discussed and best-loved theories of modern times, the theory that life exists on some other planet. There is no way to falsify this theory, because the universe is too big.
The universe consists of billions of galaxies, and in each of these galaxies there are millions or billions of stars. Astronomers believe that planets are extremely common, and that a large fraction of all stars have planets. What would you need to do to falsify the belief that extraterrestrial life exists? You might think that you could do this in theory by launching some grand fleet of spaceships to search all planets. But such a task would be impossible. There would be too many planets to search, and it would take too long. The speed of light limits travel to other stars. So some grand fleet of spaceships would require billions of years to search all of the other planets in the universe.
Suppose that after, say, five billion years of exploration such a massive fleet of spaceships still reported no signs of extraterrestrial life. Would that falsify the theory that extraterrestrial life exists? It certainly would not. For in the billions of years that such an expedition had been operating, it would always be possible that extraterrestrial life had appeared in one of the places that had already been searched. So searching every single extraterrestrial planet in the universe for life (without finding any) would absolutely not falsify the claim that extraterrestrial life exists.
Galaxies (Credit: NASA)
Could we imagine, perhaps, that such an expedition might place cameras on every planet that it searched, to try to send back to Earth a signal allowing us to say that at this current moment none of them have life? That wouldn't work, because there is no signal that can travel faster than the speed of light. So, for example, if we got a signal from some camera that had been left on some planet 130,000,000 light-years away, and the camera showed no life, that would only prove that part of the planet did not have life 130,000,000 years ago. It would not prove that the planet does not now have life. Nor would it even prove that 130,000,000 years ago the planet had no life, for the planet might have life in some place that the camera was not observing. Besides the fact that you can't get live “showing it as it is right now” signals from planets many light-years away, there is the difficulty that you can't fill a planet with cameras, and there's all kinds of life that cannot be detected with cameras.
Very clearly, the important scientific theory that life exists on other planets is a theory that is not falsifiable. This simple fact destroys the credibility of the claim that a scientific theory has to be falsifiable. There is no logical need to write anything further on the topic. But just for the sake of overkill, I will give some further examples of scientific theories that are not falsifiable.
Another example of a very popular and widely discussed theory is the theory of natural abiogenesis, which is the theory that life naturally arose from non-life. It is theoretically quite possible to get a result supporting this theory. If a lab that was simulating early Earth conditions reported that life had spontaneously arisen from chemicals, that would be evidence supporting the theory of natural abiogenesis. But it is quite impossible to falsify the theory of natural abiogenesis. Even if you did a billion years of experiments trying to produce life from lab chemicals, and none of them were successful, that still would not prove that life had not arisen from chemicals because of some incredibly unlikely once-in-a-galaxy rare event.
A large class of scientific theories that cannot be falsified are those which describe realities on which our existence depends. Human biological existence has very many physical dependencies, so there ends up being quite a few theories describing realities on which our existence depends. To give one example, our existence depends on a strong nuclear force that binds together protons and neutrons in the nucleus of an atom. There is a theory corresponding to this reality, the theory that there exists a force that causes protons to be bound together in the nucleus, despite the very strong electromagnetic repulsion of their positive charges. Could we falsify this theory? No, we could not. We could only falsify it by observing that something on which our existence depends does not exist. But that could never happen. Similarly, the theory of electromagnetism is a theory describing the basic electrical repulsion and attraction on which biological chemistry depends. We cannot falsify such a theory, since that would involve observing the nonexistence of something that is a prerequisite for our existence.
Some other fundamental theories are the kinetic theory of matter (that gas consists of small particles in motion), the cellular theory of life (that cells are the fundamental building blocks of life), and the atomic theory of matter (that atoms are the basic building blocks of life). None of these theories can be falsified. Having made countless observations showing such realities, there is no way that future observations will refute them. We cannot imagine any observations that would cause us to believe that gases do not consist of moving tiny particles, or any observations that would cause us to believe that living things are not made up of cells, or any observations that would cause us to believe that rocks are not made of atoms.
The common expression “you can't prove a negative” is largely correct in suggesting that very many theories cannot be disproved or falsified. For example, you can prove that someone masturbates, but cannot prove that he does not masturbate.
Instead of the principle “a scientific theory must be falsifiable,” a much better principle is “a scientific theory should be either verifiable or falsifiable.” In general, any theory that is either verifiable or falsifiable can be considered a scientific theory. The theory of extraterrestrial life is a scientific theory, because we can easily imagine some simple observational events that would verify it. The theory of natural abiogenesis is a scientific theory because we can easily imagine some simple observational events that could verify it.
The silly idea that a scientific theory must be falsifiable is one that was advanced by philosopher of science Karl Popper, because of ideological considerations. People such as Popper wanted to stigmatize theories that they disliked. So they advanced the strange, illogical idea that a scientific theory must be falsifiable, hoping that it would help brand certain types of theories and experimental inquiry as being unscientific.
And so, while scientists such as Joseph Rhine were piling up strong experimental evidence for the existence of ESP, as discussed here, thinkers such as Karl Popper were trying to sell the idea that a scientific theory must be falsifiable. This was ideologically convenient. It could now be claimed that the theory that ESP exists is not scientific, since there is no way that it could be falsified (even a million negative ESP tests would not prove that ESP does not sometimes occur). This kind of effort makes no sense, for the same sword invented to kill ESP kills just as effectively the theory of extraterrestrial life, the theory of natural abiogenesis, the kinetic theory of matter, and quite a few other things that ESP-loathing scientists may prefer to believe in.
I may note that a principle that Scientific American attributes to philosopher of science Karl Popper, the claim that “pseudo-science seeks confirmations and science seeks falsifications,” is bunk and fantasy. Mainstream scientists typically spend most of their time searching for confirmations, and spend almost no time looking to falsify their favorite theories. Typical mainstream scientists spend almost no time attempting to falsify the theories they most love, and they are also extremely bad about examining evidence that seems to contradict such theories. So, for example, neuroscientists spend endless hours trying to confirm their dogmas about brains storing memories and brains producing thoughts, but they pay almost no attention to the many facts that argue against such claims. Modern science academia is a conformity culture with cherished dogmas, and it is a culture that punishes and stigmatizes heretics and contrarian thinkers who attempt to discredit those dogmas. This is the exact opposite of an approach centered around falsification. The fact that scientists aren't very interested in falsifying things is shown by the fact that scientists find it is hard to even find a journal that will publish their negative experimental results (as discussed here), and also by the fact that when someone writes a scientific paper presenting evidence against a dogma cherished by scientists, he may find his paper retracted by a journal purely because the paper is daring to question a sacred cow (as happened in this case).
Karl Popper's essay “Science as Falsification” is a very revealing one, because it makes quite clear that Popper's claim about falsification was not at all something that came from a study of the way scientists actually behave, but was instead a claim that was contrived for the sake of a creating a weapon against a few theories Popper didn't like. Popper makes clear in the paper that he was bothered by the popularity of three theories: Freud's theory of psychoanalysis, Marx's theory of economics and history, and Adler's theory of psychology. Popper reveals in his essay that he invented his theory of falsification to combat such theories. It was actually a poor weapon against Marx's theory and Freud's theory, both of which actually are falsifiable. Freud's theory could be falsified if people became much crazier by going to Freudian psychotherapists, and Marx's theory could be falsified if capitalist countries all became blissfully happy and Marxist countries all became miserable failures.
The whole idea of creating a theory of how science works (“science as falsification”) based on a desire to create a weapon against theories you don't like is one that made no sense at all. Just as Marx and Freud were dogmatic thinkers who got far more influence than they deserved, Popper was a thinker who got far more attention than he deserved. Popper was bothered that the Marxists and Freudians claimed to see confirmations that weren't really confirmations. If people are claiming confirming evidence that isn't really there, a good way to handle that is to advocate tighter standards for confirming evidence, and to show how particular claims of confirming evidence are unfounded – not to be making untrue claims suggesting science is all about falsification rather than confirmation. Were we to follow Popper's bizarre claims such as “confirming evidence should not count except when..it can be presented as a serious but unsuccessful attempt to falsify the theory,” we would have to throw out a large fraction of the scientific results in textbooks.
Rather obviously untrue, the claim that scientific theories are all falsifiable is simply a rhetorical device, one that is occasionally trotted out by people as an argumentative weapon to use against theories they don't want to believe in (sometimes theories for which there is a great deal of evidence). It's a very ineffective weapon, because it's so easy to find examples of respected scientific theories that are not falsifiable.