Distinguishing between senses
Tyler and Evans provide two criteria for determining whether a particular sense of a preposition counts as a distinct sense and can therefore be established as a case of polysemy:
1. for a sense to count as distinct, it must involve a meaning that is not purely spatial in nature, and/or a spatial configuration holding between the TR and LM that is distinct from the other senses conventionally associated with that preposition; and
2. there must also be instances of the sense that are context-independent: instances in which the distinct sense could not be inferred from another sense and the context in which it occurs.
To see how these criteria are applied, consider the sentences in (14) and (15):

In (14), over designates a spatial relation in which the TR, coded by the hummingbird, is located higher than the LM, coded by the flower. In (15), over also designates a spatial relationship in which the TR, the helicopter, is located higher than the LM. In these examples, neither instance of over involves a non-spatial interpretation and both senses encode the same spatial relation. According to Tyler and Evans’s first criterion, then, the two instances do not encode distinct senses so the second criterion does not apply. The sense of over that is represented in both these examples is what Tyler and Evans call the ABOVE sense. According to Tyler and Evans, this is the central sense, a point to which we return below. Now compare the example in (16) with (14) and (15).

In (16), the spatial configuration between the TR and LM is not consistent with the ABOVE meaning in (14) and (15): in (16) the board is actually below the hole in the ceiling. In addition, there is a non-spatial aspect to this sense: part of the meaning associated with over in (16) relates to COVERING, because the LM (the hole) is obscured from view by the TR. This COVERING meaning is not apparent in examples (14) and (15). The presence of this non-spatial aspect in the sense of over in (16) meets the first assessment criterion stated by Tyler and Evans, which means we can now consider the second criterion. In doing so, we must establish whether the COVERING meaning is context-independent. Recall that if the meaning is ‘computed’ on-line, based on the central ABOVE meaning of over plus contextual and/or encyclopaedic knowledge, then this sense qualifies as vagueness rather than polysemy. Tyler and Evans argue that the meaning of over in (16) cannot be computed on-line, and is therefore context-independent. In other words, the knowledge that over in (15) has an ABOVE meaning does not allow us to infer a COVERING meaning from the context supplied by (16).
To elaborate this point, Tyler and Evans provide a different example in which the COVERING meaning is derivable from context. Consider example (17).

In (17), the TR (the tablecloth) is above (and in contact with) the LM (the table). The interpretation that the table is covered or obscured by the tablecloth can be inferred from the fact that the tablecloth is above the table, together with our encyclopaedic knowledge that tablecloths are larger than tables and the fact that we typically view tables from a vantage point higher than the top of the table. This means that the sense of over in (17) can be inferred from the central ABOVE sense together with encyclopaedic knowledge. This type of inference is not possible in (16) because the spatial relation holding between the TR and the LM is one that would normally be coded by the expression below (The board is below the hole in the ceiling), given our typical vantage point in relation to ceilings. The COVERING meaning of over in (16) must therefore be stored as a conventional sense associated with over, which means that we can conclude that this is an instance of polysemy.
It is worth observing that Tyler and Evans argue that examples like (17) which give rise to a ‘covering’ inference while conventionally encoding the ABOVE meaning of over– represent the means by which new senses are added to a lexical category. According to this view, when context-dependent inferences are reanalysed as distinct meanings (a process called pragmatic strengthening) a lexical item develops new senses. This perspective is somewhat at odds with Lakoff’s view that conceptual metaphor and image schema transformations hold a central place in meaning extension. By arguing that contextual factors can give rise to new senses, Tyler and Evans emphasise the usage-based nature of semantic change, adopting a position that owes much to the Invited Inferencing Theory of semantic change (Chapter 21).