Social cognition has received much attention in fields such as neuroscience, psychology, cognitive science, and philosophy. Theory-theory (TT) and simulation theory (ST) provide the dominant theoretical frameworks for research on social cognition. However, neither theory addresses the matter of how the concepts of "self" and "other" are acquired through the development of human and nonhuman agents. Here, we show that the internal representations of "self" and "other" can be developed in an artificial agent only through the simple predictive learning achieved by deep neural networks with the superposition mechanism we herein propose. That is, social cognition can be achieved without a pre-given (or innate) framework of self and other; this is not assumed (or is at least unclear) in TT and ST. We demonstrate that the agent with the proposed model can acquire basic abilities of social cognition such as shared spatial representations of self and other, perspective-taking, and mirror-neuron-like activities of the agent's neural network. The result indicates that the superposition mechanism we propose is a necessary condition for the development of the concepts of "self" and "other" and, hence, for the development of social cognition in general.