Strawson has worked extensively on this problem of existence of others' minds. He first lays down his criteria of existence:
Strawson lays down the principle that one can ascribe experiences to oneself only if one is prepared to ascribe them to others. This is because a subject can meet this requirement only if they are able to pick out other subjects, and, Strawson holds, one cannot pick out non-spatial subjects.
Based on this criteria, he
concludes that when we self-refer we refer to an entity which has two sides or aspects, the physical and the mental, and not to a thing which possesses only the mental sort of feature, something else having the physical features
This clearly means that independent existence of the mental is not possible. But how do we even detect its dependent existence?
Since we can self-ascribe we must be able to other-ascribe, and that means that our methods for doing so must be adequate. As Strawson puts it, the criteria we employ for psychological ascription to others must be ‘logically adequate’.
There cannot, therefore, be a genuine problem of other minds.