A theory is a set of propositions which are taken or assumed true. Presumably these propositions are not self-contradictory, though in complex theories, who can say? It is not necessary that any one person know each of the propositions, or even that the set is closed (as we’ll see). A complex theory contains more propositions than a simple one. There is no need to be more precise than this except to stress that a theory is its propositions. Change any one of them changes the theory into a new theory. With that we are done, really. But to expand.
Theories don’t make predictions. We do. This is because sets of propositions don’t do anything; they’re just propositions. Predictions are propositions themselves, chosen because they are of interest to us and constructed conditional (derived, deduced, inferred, etc.) on a theory. We hear things like, “Given my theory of the weather, tomorrow’s high will be 70oF.” This is usually shortened to “Tomorrow’s high will be 70oF” which the conditioning left implicit.
Now if the high temperature tomorrow is anything but 70oF, the theory is falsified. The theory said some thing would occur: it did not: the theory is false. End of story.
But nobody (except a pedant) understands the phrase “will be 70oF” to mean “will be precisely, exactly, to the nth decimal place 70oF.” Words mean something. Our task is to translate the prediction into the vernacular. That means adding some fuzz around 70oF; or, in other words, by taking the phrase to mean “There is a good (but not perfect) chance the high will be 70oF”. And, of course, “good (but not perfect) chance” puts us on probabilistic grounds.
In this case, the theory said something might happen. If it didn’t, the theory is not falsified. How can it be? One of the things the theory said could happen did happen. This cuts both ways. If 70oF obtains, the theory is not completely validated, either. That is because the theory also said it could have been 69oF or 71oF, albeit with a smaller chance, and these did not occur.
Whether the theory is useful depends on the decisions we make given the prediction; which is to say, on how we have (or would have) acted on the predictions as they stand (or stood). This is an entirely different topic, the gist of which is that a theory useful to one man may be useless to another. For example, climate models have been extraordinarily useful to Al Gore, but they haven’t treated not-yet-tenured critics awfully well.
You have it by now: if the predictions derived from a theory are probabilistic then the theory can never be falsified. This is so even if the predictions have very, very small probabilities. If the prediction (given the theory) is that X will only happen with probability ε (for those less mathematically inclined, ε is as small as you like but always > 0);, and X happens, then the theory is not falsified. Period. Practically false is (as I like to say) logically equivalent to practically a virgin.
Most theories, even though stated in deterministic terms, are actually meant, and are surely taken, in a probabilistic cast. This is because of the presence or suspicion of measurement error, imperfect specifications, and a host of other reasons which accompany any theory in practice. Entire fields are nowhere near falsifiable in the sense that the predictions associated with them are probabilistic or “fuzzy”: biology (the theory of natural selection and every single evolutionary psychology theory), sociology, economics, psychology, education, and such forth. But even a broad range of theories within more rigorous fields, like physics and chemistry, are also not falsifiable. (The reader can apply on his own the arguments given here to discern whether his favorite theory can be falsified.)
The sole way a theory can be falsified is if it states, in no uncertain terms, boldly and forthrightly, that X cannot happen, that the probability of X is 0, exactly 0. Then if X happens, à la mort, else not. Falsified is akin to mathematical proof: it is undeniable. Yet, as we shall see, it is still denied.
All theories are therefore trivially falsifiable (in logic). For example, we can derive from most any theory the prediction that the probability X = “The existence of a twelve-and-a-half footed half-duck half-snake that speaks (what else?) French on Uranus” is 0. But since we have to travel to that frigid locale in order to verify X, we will never learn whether the theory is falsified in fact because, of course, since X is contingent it might (conditioned merely on the premises which identify its contingency) be true. Therefore, we are interested in falsifiability in practice.
There are four main situations in which the call of “Not falsifiable!” is heard. The first is where the theory isn’t the theory, where a kind of three-card Monte trick has been played. The second is when there really is no theory; or, rather, there is a theory but it is infinitely malleable. The third is when the theory is protected from reality by erecting a barrier of contempt for outsiders in front of it. The fourth is when the theory is true but not loved or desired.
These four are not mutually exclusive, but in the examples to come, for simplicity, I’ll treat them that way.
Update See the discussion below. I do not claim that no theory can be falsified. Some can. All continuous statistical models of discrete reality are falsified by definition. But many theories cannot be falsified.