Problems with the full-specification approach
While Lakoff’s theory of lexical semantics has been hugely influential, there nevertheless remain a number of outstanding problems that have attracted a fair degree of attention in the literature. As we mentioned earlier, Lakoff’s full specification view had been criticised as it entails a potentially vast proliferation of distinct senses for each lexical item. For example, Lakoff’s approach entails that over has, at the very least, several dozen distinct senses. A proliferation of senses is not problematic per se because cognitive linguists are not concerned with the issue of economy of representation. However, the absence of clear methodological principles for establishing the distinct senses is problematic. In this section, we focus on two main problems that have been pointed out in relation to this issue.

Polysemy and vagueness: the role of context
The first problem concerns a failure to distinguish between polysemy and vagueness. A linguistic expression is vague rather than polysemous if context rather than information stored in semantic memory provides the meaningful detail about the entity in question. Consider the word thing. This expression could be used to refer to almost any entity or event, yet it seems unlikely that semantic memory links this expression to all the possible entities that it could refer to. Instead, the meaning of this expression is fully specified by context. Aless extreme example is the expression aunt, which can refer either to a maternal or a paternal aunt. While our knowledge associated with this expression contains this information, the distinction between these senses is fully dependent upon non-linguistic context. Therefore, while a polysemous expression relates to a range of conventional senses, a vague expression is characterised by a lack of conventional sense distinctions.
Based on proposals by Tuggy (1993), the distinction between polysemy and vagueness is illustrated in Figure 10.12. Polysemy is illustrated in Figure 10.12(a) and vagueness is illustrated in Figure 10.12(b). In the case of polysemy, A represents the central sense and other senses are represented by the boxes marked B and C. All the boxes are marked with bold lines which represent the idea that all three representations have equal degrees of entrenchment in memory (Chapter 4). The lines between the boxes indicate that the senses are related. In the case of vagueness, on the other hand, A licenses the interpretations designated by B and C: the arrows represent the idea that inter pretations B and C are ‘computed’ from the basic meaning A plus context. The dashed lines represent the idea that meanings B and C are not stored in semantic memory as distinct senses, but emerge from ‘on-line’ processing.

Given this distinction, it becomes clear that one of the reasons Lakoff’s full specification model results in such a large number of distinct senses is that the model fails to distinguish between polysemy (distinct senses stored in memory) and vagueness (meaning ‘filled in’ by context). Recall that Lakoff argued for at least six distinct senses associated with the ABOVE-ACROSS schema alone. This number rises to eight if we include the two image schema transformations resulting in endpoint focus. A number of cognitive semanticists have argued that this proliferation of senses results from a failure to take into account the role of context in determining meaning (‘filling in’ information). From this perspective, Lakoff’s full-specification model represents the opposite extreme of the monosemy approach by denying the role of context in meaning altogether. Some cognitive linguists have argued for a position somewhere between these two extremes. For example, Tyler and Evans (2003) argue that the examples in (13) do not represent distinct senses of over (one specifying contact and one specifying lack of contact):

Instead, Tyler and Evans argue that the interpretation of over with respect to contact or lack of contact derives from the integration of over with the other elements in the sentence. Our knowledge about birds (they can fly) and people (they cannot), provides us with the inference that birds do not come into contact with walls when crossing over them while people do. In other words, the linguistic context together with encyclopaedic knowledge provides the details relating to the presence or absence of contact. According to Tyler and Evans, over in (13) is vague with respect to contact. Tyler and Evans argue that while Lakoff’s position on polysemy as a conceptual phenomenon is correct, it is also important to take into account the crucial role of context in word meaning (recall the discussion in Chapter 7).
The polysemy fallacy: unconstrained methodology
The full-specification approach has also been criticised for a lack of methodological constraints. In other words, Lakoff provides no principled criteria for determining what counts as a distinct sense. This means that the polysemy account presented for over (or whatever lexical item we might apply the approach to) results purely from the intuitions (and perhaps also the imagination) of the analyst rather than actually representing the way a particular category is represented in the mind of the language user. This problem has been discussed in some detail by Sandra and Rice (1995) and by Sandra (1998). Sandra argues that to view all context bound usages of a particular lexical item as instances of polysemy is to commit what he calls the polysemy fallacy: just because lexical items can exhibit polysemy, it does not follow that all or even many distinct senses associated with a lexical item are instances of polysemy. Indeed, Sandra has even suggested that the lack of clear methodological principles underpinning Lakoff’s semantic network analysis undermines its status as a true linguistic theory. As he puts it, ‘what is lacking from the exercise is a set of scientifically valid [decision] principles’ (Sandra 1998: 371; original emphasis).