The cornerstone of modern linguistic semantics is, of course, the concept of meaning. Alternative names of this concept are sense (Frege’s Sinn) and intension, and its content has been largely undisputed within linguistics proper, if we disregard so-called cognitivist approaches. Ever since (Frege, Zeitschrift Für Philosophie Und Philosophische Kritik 100:25–50, 1892) Über Sinn und Bedeutung, the aim of this concept was to capture the intuition that linguistic signs have a property that allows people to use them for making themselves understood, and that the property in question has to do something with the relationship of a sign to its intended referent (nominatum, denotatum, Frege’s Bedeutung). Unfortunately, no version of this approach to meaning is prepared to account for the cognitive aspect of acquiring and using linguistic signs. Frege himself, while recognising that signs, in addition to their meaning, are also associated with an image (Frege’s Vorstellung) in each speaker’s mind who is familiar with the sign, he seems to assume that images are irrelevant from the point of view of meanings, in the same way as the mental processes that take place in the audience’s mind. In general, it seems that modern formal semantics not only disregards all psychological phenomena related to semantics, such as associations, (non-conventional) figurative uses, phylo- and ontogenetic semantic changes etc., but it is also unable to offer a framework for treating them. This paper is not a manifesto in favour of cognitivist semantics. As a matter of fact, I will not be concerned with that alternative movement at all, because I believe they have also failed to establish a link between the cognitive aspects mentioned above, on the one hand, and the central concepts of mainstream semantic theories, on the other. (By “mainstream semantics” I mean those theories which, following Richard Montague’s work, think of the explanation of interpretation as a production line starting from a linguistic representation, going through disambiguation, then logical representation, then model theoretic interpretation.) Instead, I will focus on the origins and nature of the gap between the highly abstract concept of sense and the rather concrete mental aspects of language use. I will conclude that Frege’s original proposal about senses would be worth taking more seriously than his followers did, and that it has an inherently cognitive character, although Frege himself does not dwell upon that aspect, which is related to the interaction between speakers and addressees. Thus I claim that there is a way of filling the cognitive gap while being true to Frege’s ideas. Furthermore, I will argue that, under that view, models themselves must have a cognitive character, and the relations between senses and referents (speaker’s and hearer’s) are stochastic rather than deterministic.