The problem of other minds is a philosophical problem traditionally stated as the following epistemological challenge raised by the skeptic: given that I can only observe the behavior of others, how can I know that others have minds? It is a central tenet of the philosophical idea known as solipsism; the notion that for any person only one's own mind is known to exist. Solipsism maintains that no matter how sophisticated someone's behavior is, behavior on its own does not guarantee the presence of mentality.
It remains possible, for example, that other people are actually nothing more than automata made out of flesh, referred to by philosophers as philosophical zombies. Perhaps the main argument offered against this possibility in the history of philosophy is the argument from analogy (other things have minds if they are sufficiently similar to us); it can be found in the works of John Stuart Mill, A. J. Ayer, and Bertrand Russell. The argument from analogy has faced scrutiny from the likes of Norman Malcolm who have issues with the "one case" nature of the argument.
More recently, it has come to be appreciated that the epistemological issue is intimately related to metaphysical and conceptual issues. This is best appreciated by considering the examples of type physicalism and philosophical behaviorism. According to the type physicalist, to be in a certain type of mental state is just to be in a certain type of physical brain state. So, if we can detect that another individual is in a certain type of physical state, then we can know that they are in a certain type of mental state. Thus, it seems that we can know, in a relatively unproblematic way, that other people are in certain mental states. In this case, the epistemological problem is dissolved by making a claim about the metaphysics of mind. Logical behaviorism, on the other hand, makes a claim about the nature of our mental concepts. This claim is that statements about mental phenomena can be analyzed into statements about behavior and behavioral dispositions. To be in a certain mental state, e.g. pain, is just to behave, or be disposed to behave, in certain ways. Since statements ascribing mental predicates to individuals make claims about nothing over and above their behavior, they can be verified to be true or false by observation of behavior. Thus, the behaviorist closes the conceptual gap between behavior and mentality which is responsible for the epistemological problem.
Metaphysical solipsists argue that there are indeed no minds but one's own and that attempting to prove the existence of another mind is futile. Proponents of this view argue that the world outside one's own mind cannot be known and indeed might be nonexistent. There are weaker versions of metaphysical solipsism, such as Caspar Hare's egocentric presentism (or perspectival realism), in which other persons are conscious but their experiences are simply not present. J.J. Valberg points out similar solipsistic aspects of his concept of the personal horizon.
The reductionist viewpoint, supported by John McDowell and others, has tried to tackle a part of this problem by putting forth certain modes of expression (such as being in pain) as privileged and allowing us direct access to the other's mind. Thus, although they would admit from the problem of pretense that at no one time can we claim to have access to another's mental state, they are not permanently unavailable to us.
Counter to the reductionist argument would be a more biological theory (and somewhat materialistic viewpoint). Take the eye and the perception of color. The light-sensing cone cells of the retina that respond to the portion of the electromagnetic spectrum designated as red are tuned similarly in every person tested, so we might expect all people to experience red in the same way. However, we also know that some people are missing certain (or all of) types of cone cells in the eye; thus giving rise to color blindness and other such ocular variances. Similarly, differences in the distribution of brain cells and dendritic connections (among many other potential variances) could give rise to different mental states for the same stimulus. Cross-culturally, when people have a word for red, they agree with other cultures on which wavelengths of light best fit the term "red" (the same wavelengths that primarily excite the cone cells which detect red, and the red/green channel to the brain). Yet even if human eyes and brains may be built in such a way that the same wavelengths stand out for everybody, still it is conceivable that for different individuals these wavelengths could evoke experiences that differ. In particular, one external stimulus may give different experiences to the same individual according to which eye is used.