What is the ``problem'' of Other Minds?

There is no doubt that we believe others to exist, and that there'd be other minds with similar feelings, and thoughts as our own. In fact, as Bertrand Russel stated in his essay The Analogy Argument for Other Minds, ``we find ourselves believing in them [other minds] when we first begin to reflect...what I am discussing is the possibility of a postulate which shall establish a rational connection between this belief [other minds] and data...'' So we can see here that the question is not one of skepticism. And that although many philosophers may argue the non-existance of other minds, it's only at a professional level. The ``problem'' or question is how exactly is it to be proved that other minds do indeed exist!

Our belief in the existance of other minds, that can have feeling and pain, is with such certitude, but seems to exist on no rational basis. Here is where the philosopher steps in, in an attempt to ground this belief onto a rational base. There have been various explanations to this ``problem'' of other minds, but much like arguments in other philosophical arenas, these solutions have their holes.