Falsifiability is an essential concept in the philosophy of science. To convey the concept roughly, we can say that for an assertion to be falsifiable, it must be in principle possible to make an observation or do a physical experiment that would show the assertion to be false. For example, the assertion "All crows are black" could be falsified by observing a red crow.
The basic concept of falsifiability is simple, but, like all basic concepts in philosophy, its precise definition has been a matter of considerable disagreement. In particular, there has been great disagreement among the logical empiricists and philosophers of science who learned from them about when to consider a statement falsified. W. V. Quine is well-known for his observation in his influential essay, "Two Dogmas of Empiricism" (which is reprinted in From a Logical Point of View), that nearly any statement can be made to fit with the data, so long as one makes the requisite "compensatory adjustments." Hence, in the above example, one could say, for example, that it was in fact not a crow at all but some other kind of bird that was observed, or that the observer was mistaken that the color of the crow was in fact red (perhaps it was only painted red), etc.
In the philosophy of science, verificationism (also known as the verifiability theory of meaning) held that a statement must be in principle empirically verifiable in order to be meaningful. This was an essential feature of the logical empiricism of the so-called Vienna Circle that featured and essentially influenced such philosophers as Moritz Schlick, Rudolph Carnap, Otto Neutrath, Hans Reichenbach, Ludwig Wittgenstein, A. J. Ayer, and Karl Popper. Later, the leading theory of meaningfulness posited not verifiability but falsifiability as the criterion of meaningfulness (also known as cognitive significance). In other words, in order to be meaningful at least in a strict sense, it had to be in principle possible (but it has been a vexed question how the phrase "in principle possible" should be interpreted in practice) that we might produce some data that would show (or perhaps only tend to show) the proposition to be false. Critics of analytic positivism have pointed the inconvenient fact that this statement is not falsifiable.
Falsifiability has also been importantly connected not only with meaningfulness but also with scientific method: Karl Popper, for example, stressed that falsifiability is critical to the scientific method. If a hypothesis offered in explanation of some empirical phenomenon cannot be falsified, then the hypothesis is "unscientific" and pointless to test (all results will be, necessarily, positive, which proves nothing).
Claims about verifiability and falsifiability have been used to criticize various controversial views. On the view of some, for example A. J. Ayer, theism is not falsifiable; since God is typically alleged to be a transcendental being, beyond the realm of the observable, claims about the existence of God can neither be supported nor undermined by observation. This is, of course, a matter of controversy for anyone who places stock in natural theology--the argument from design and other a posteriori arguments for the existence of God. Logical empiricists like Ayer have held that claims about morality (such as "murder is evil" and "John was wrong to steal that money") are also not falsifiable and not part of scientific inquiry; their function in language is not even to state facts, on Ayer's view, but simply to express certain moral sentiments. Ayer has been far from unique in this view; see non-cognitivism.
There are other examples of theories, however, that are much less controversial as examples of unfalsifiable claims. Some so-called "conspiracy theories," at least as defended by some people, are essentially unfalsifiable because, whenever evidence is presented that tends to falsify the theory, it is explained away, for example, by positing that the evidence is fabricated and spread by the conspirators.
In philosophy, solipsism is often dismissed as unfalsifiable. Solipsism has it that the Universe exists entirely in one's own mind. This can straightforwardly be seen not to be falsifiable, because whatever evidence one might adduce that is contrary to solipsism can be, after all, dismissed as something that is "in one's mind." In other words, there is no evidence that one could possibly adduce that would be inconsistent with the proposition that everything that exists, exists in one's own mind. This view is somewhat similar to Cartesian skepticism, and indeed, Cartesian skepticism has been rejected as unfalsifiable as well by many philosophers.
Definitions and tautologies are typically regarded as unfalsifiable, but not merely on that account regarded as meaningless: in formulating theories of cognitive significance (meaningfulness), they are generally regarded as a special case. For example, "all bachelors are male" and "all green things are green" are necessarily true (or given) without any knowledge of the world. Proving mathematical theorems involves reducing them to tautologies, which can be mechanically proven as true given the axioms of the system or reducing the negative to a contradiction. These are unfalsifiable, because any evidence given is ignored in the proof process. How a mathematical formula might apply the the physical world, however, is testable.
The laws of physics are an interesting case. Occasionally it is suggested that the most fundamental laws of physics, such as "force equals mass times acceleration" (F=ma), are not falsifiable because they are definitions of basic physical concepts (in the example, of "force"). More usually, they are treated as falsifiable laws, but it is a matter of considerable controversy in the philosophy of science what ought to be regarded as evidence for or against the most fundamental laws of physics. Isaac Newton's original laws of motion in their original form were falsified by experiments in the twentieth century, and replaced by more exact theories that hold under more conditions (though Newton's theories are still close enough to be used practically without the modern updates). In the case of less fundamental laws, their falsifiability is much easier to understand. If, for example, a biologist hypothesizes that, as a matter of scientific law (though practicing scientists will rarely actually state it as such), only one certain gland produces a certain hormone, and then someone discovers an individual that has the hormone but lacks the gland, the hypothesis is falsified.
See also Occams razor