There is a good summary of the Problem of Other Minds here at Standford's Encyclopedia of Philosophy. Although I don't believe they really pick up on the true application of the problem of other minds, which is determining if I can trust the objectivity of the world I perceive.
I'm coming to the conclusion that the problem of other minds needs to be understood as a trivial problem. And it really is better stated as "Do I perceive an objective world?" Some of the possibilities:
- I do perceive an objective world and other people who behave like they have a mind really do have other minds.
- I am really God and have constructed all of space and time, but am hiding that fact from myself (solipsism).
- I live in the Matrix. Some other people are really other minds (also plugged in), while some are simulated minds and/or computer programs (e.g. the Oracle, Agents, etc.).
- I am really a brain in a vat and this world is the simulation of someone controlling my reality in a variety of other ways.
If you want a deconstruction of anything other than #1 above it is simply Occam's Razor - what is the simplest explanation given the information we currently have, that doesn't create a higher order solution (with unseen/unknown/unknowable variables). Without some interaction with, or revelation from, other worlds/dimensions we must rule them out with this most basic rule of logic.
Note, however, that Occam's razor doesn't say that other solutions are not possible, it simply says that other solutions are not the best solution without any evidence supporting them.
The reason Occam's Razor is an effective deconstruction for any area that applies is that there is nothing practical that can be done with the alternatives, other than write a great movie script. Even if I am a brain in a vat - will that change the way I live? It shouldn't from a logical perspective. Not if you understand all the implications. I still have the same ability to think and interact with the universe (whether objectively real or simulated) and without any information on how to break out of the vat, my choices are no different.
Given that you are where you are, what are you going to do with your life? Even "ending your life" isn't a solution. Whoever is in control of you and the vat can just restart the simulation and plug you back in, or reincarnate you, or bring you to the next phase of the simulation - i.e. heaven/hell or whatever the afterlife is...
So, I take the position that any possible or real world, dimension, or force which does not interact with our world or reveal itself to us in some form or fashion, is irrelevant to my life. Granted, I believe this with a sort of blind faith.
Note that this is the basis for many worldview claims and clashes. The argument among adherents of various worldviews are generally arguing the validity of difference evidences for their faith. For example, is the the Bible real as a source of information/revelation that tells us about a greater reality than the objective world we directly experience. Is there such a thing as feeling the presence of God? Etc.
Skeptics attempt to deconstruct the evidence or even positively deny it (atheism, etc.). But in the case you describe, my examples #2-4 above - we have zero evidence for them, so they should be ignored. There is no reason even to debate among them. They can be trivially ignored, other than perhaps for use in thought experiments that attempt to discover some way of obtaining evidence about them.