In topos theory, the notion of “being geometric” often comes up. Some examples are: geometric morphisms, geometric logic, and geometric theories. For instance, here's a quote from Steve Vickers' Locales and Toposes as Spaces:
... the logic is not at all ordinary classical logic. It is an infinitary positive logic known as geometric logic.
I have a vague intuition on this notion of geometricity: in locale theory one sees open sets as observable properties, and my understanding of geometricity is that when we categorify from opens to sheaves, the notion of observability generalises to geometricity. In this context, “geometric morphisms” make sense as they respect the geometric structure as a continuous function respects the observability structure. However, I don't quite understand the use of the adjective “geometric” for this.
How do I connect the intuition of observability with the notion of geometricity? If this approach doesn't make much sense, what is a good way to think about geometricity?
PS: I am aware that the paper of Vickers I linked to essentially aims to explain the spatial intuition in detail. However, it seems to take for granted the notion of being “geometric” without a familiarity with which I find hard to follow the paper.
To summarize the comments: geometric logic constitutes the logic, models of whose theories are preserved by geometric morphisms between topoi. Geometric morphisms are those appropriate to toposes viewed as generalized spaces, for instance, identifying the topos of sheaves on a topological space, or on a locale, with the space itself. Historically, toposes were first introduced by Grothendieck's school to model generalized algebro-geometric spaces, which explains why these morphisms are called geometric, rather than topological.