Intentionality — the orientation of mental states to objects (things, properties, states of things, events) — has been considered a hallmark of the psyche since Brentano’s time. In this article, we consider the problem of intentionality from the second-person approach, or the standpoint of intersubjectivity. Our analysis shows that intentionality is intrinsically projective. The projective nature of intentionality is related to internal objects that play a crucial role in fixing the person’s subjective experience and serve as a fulcrum in the (...) development of the person. The internal object can be treated as a set of properties and tropes. The logic of intentionality proposed by Graham Priest and the theory of primary (that is, belonging to the Freudian system «unconscious») psychological attitudes developed by Linda Brakel created the preconditions for seman- tical analysis of projective intentionality. In the article, we rely on the logic of projective intentionality that reorients the resources of modal logics and semantics of possible worlds to the investigation and formalization of primary thinking. Considering the problem of mental existence within the framework of the second-person approach, we show that Wittgenstein’s reasoning about the «beetle in a box» does not refute the thesis of the privacy of mental meanings. Finally, involving the possible world semantics, we develop a neo-Aristotelian approach to the ontology of mental objects. (shrink)