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If a model of ambiguity and vagueness (or ambiguity, polysemy, and vagueness) as well-defined categories, internally homogenous and with clear boundaries, does not work, what does? Categories based on prototypes have proved useful for analyzing many other linguistic distinctions (e.g., Lakoff 1987; Langacker 1987); perhaps they are appropriate for this one as well. I would suggest the following model within Langacker’s Cognitive Grammar framework.6

3.1. Categorization in Cognitive Grammar

Following Langacker (1987, etc.), we call what two cognitive structures have in common a “schema”, and represent its relationship to its elaborations (or subcases)

by arrows from the schema to each elaboration (Figure 1a); such relationships form the basis for categorization.7 _{Both schemas and their elaborations can coexist in a}

language; they exist to the degree that they are established (entrenched) in speak- ers’ minds through repeated usage. Well-entrenched structures, ceteris paribus, are more salient than less-entrenched structures, i.e., they occur more energetically. Entrenchment can be viewed as a kind of enduring salience, i.e., salience apart from relatively transitory effects such as directed attention or heightened activation due to contextual factors. Degree of salience is represented by the thickness and continuity of a box enclosing the structure in question (and secondarily by gray- scaling the representation of the structure itself): thus Figure 1b shows a salient structure A and a less salient structure B, both subsumed by marginally salient schema C; if transitory salience effects are not factored in, the degree of salience will be equivalent to the degree of entrenchment. Another parameter which we should distinguish is elaborative distance, which we represent by the length of the arrows from schema to elaboration:8 _{a schema is distant from its elaborations}

when relatively many specifications of the elaborations must be despecified to form the schema, and close when relatively few must be despecified. Thus in Figure 1c, C is at a greater elaborative distance from A and B than in 1b. (One might, for example, take A, B, and C in 1b to be CAT¬WEASEL¬and MAMMAL¬THAT¬EATS¬ MICE¬respectively, while in 1c they might be DOG¬STOOL¬and THING¬WITH¬LEGS¬ FOUND¬IN¬HUMAN¬DWELLINGS)

It will be readily seen that the Aristotelian definitional test amounts to looking to see if there is a schema subsuming two meanings and making that the definition. Many linguists seem to operate on the assumption that once such a schema is found, the subcases it subsumes may be safely ignored, regardless of their degree of entrenchment or salience. I am suggesting, however, that to the degree that they are salient they must not be ignored.

3.2. The characterization of ambiguity and vagueness

The prototypical case of ambiguity is where two semantic structures, associated with the same phonological structure (which is called their phonological pole), are both well-entrenched (and therefore salient), while there is no well-entrenched and elaboratively close schema, also linked to the phonological pole, subsuming them.9 _{This is represented in Figure 2a.}10 _{Prototypical vagueness, on the other}

hand, involves meanings which are not well-entrenched but which have a relatively well-entrenched, elaboratively close schema subsuming them, as represented in Figure 2e.11

Elaborateness (non-schematicity) of a semantic structure correlates with a smaller extensionality; the extension of a schema includes those of its elaborations. Thus Figure 2 corresponds to Figure 3, where Venn diagrams represent the extension of the structures involved.12 _{Again thickness and continuity of the lines (with gray-}

scaling) represents salience, and relative size represents schematic distance. 3a, like 2a, represents prototypical ambiguity, and 3e, like 2e, prototypical vagueness. It is perhaps more intuitively obvious from this sort of diagram that in the ambiguous case it is easy to separate the meanings and hard to unite them, whereas in the vague case it is relatively hard to separate them and easy to unite them.

It is important to realize that Figure 3 is a representation of the extension of the concepts (i.e., of the range of cases characterized by each concept’s cognitive specifications), not a representation of the extent of the cognitive specifications themselves (such as is implied by diagrams utilizing the “container” metaphor, e.g., Langacker 1987: 75, Figure 2.4a, and many other places). More extensive specifications (greater “elaboration”) corresponds to a lesser extension: thus a diagram like Figure 4, representing the extent of semantic specifications, would represent the same semantic structures as Figure 3 in quite opposite fashion (4a = 3a, 4c= 3c, 4e= 3e). The diagrams in Figure 4 make it clear that the schema is completely immanent to its elaborations (Langacker 1987: 180, 438–439): i.e., they contain all its specifications and more; the schema comprises those specifi-

cations which are common to the two subcases. Again relative size of a schema vs. its subcase represents cognitive distance: when the schema is much smaller than the subcase that means the subcase adds many specifications to those of the schema and is thus quite distant from it.

For many purposes, speakers will “filter out” specifications below a certain level of salience. If the threshold of salience is set higher than the level of salience of any uniting schema in 2a/3a or of the elaborate structures (the subcases) in 2e/3e, 2a/3a will have two completely separate meanings attached to the same phonological pole, while 2e/3e has only one meaning. (A rough visual analog to imposing such a threshold can be achieved by squinting more or less tightly at Figures 2–6.13_{) This corresponds exactly to the traditional characterizations of}

the ambiguity/vagueness distinction.

3.3. The in-between cases: Polysemy

Since the differences between Figure 2a and 2e (3a and 3e) are gradual, we can expect to find cases in the middle, where a schema exists but is not salient and/or is distant (2b/3b), or where the subcases are somewhat salient but not so much so as the schema (2d/3d), or even where both the schema and the subcases are salient (2c/3c). These of course are the polysemous cases discussed in section 2.1, the border straddling cases, with meanings both clearly separable and clearly united. But the important thing is that the differences among the categories are gradual, not absolute.

Note that if a high enough threshold of salience is imposed (if a speaker “squints” tightly enough), the schema in 2b (3b) is filtered out, leaving a con-

Figure 4. The ambiguity–vagueness cline showing extent of semantic structures Figure 3. The ambiguity–vagueness cline showing extensionality of semantic structures

figuration which is effectively the same as 2a (3a). Similarly, if the threshold of salience is set above the level of salience of the sub-cases in 2d (3d), 2d (or 3d) becomes effectively equivalent to 2e (3e). This, together with the influence of contextual salience mentioned below (3.4.3), makes it even more impossible to draw absolute boundary lines between the categories of ambiguity, polysemy, and vagueness.

3.4. Fitting the data to the model 3.4.1. Straightforward cases

This model fits with the sorts of phenomena we find in languages. The case of bank “financial institution” and “edge of a river” fits very nicely at the ambigu- ous end of the spectrum: the two meanings are well established, and the nearest schema subsuming the two would be something like “Thing”, which is quite distant and is not linked to the phonological pole /baenk/ (5a, cf. 2a). At the vague end (5e, cf. 2e) we can place aunt, with the non-salient subcases “father’s sister” vs. “mother’s sister” subsumed at a minimal elaborative distance under the concept “parent’s sister”, which is clearly better established and thus more salient than they as a meaning of aunt. An example of 2c (5c) would be the notions of artis- tic painting (including painting in oils and watercolors as in (i–ii) above) and of typical utilitarian painting (subsuming (vi–viii) and others): both well-established concepts, but clearly united under an also well-established schema not greatly distant from the subcases.

3.4.2. Flexibility

Figure 2 of course oversimplifies things. Although elaborative distance between a schema and its elaborations tends to correlate inversely to entrenchment of the schema, the parameters are not absolutely parallel; Figure 2 does not represent this. Also one typically is dealing not with two or three meanings but with many more, arranged in multiply overlapping hierarchies – cf. Figure 6, of which 5c is an abbreviation, with 5c=6a–c. (Figure 6 itself is of course far from exhaustive of the meaning of paint.) Furthermore it is relatively rare for the subcases in ques- tion to have the same degree of salience; they are more likely to differ along this parameter, as in 1b–c (and, again, 6). These would all be valid criticisms if the presentation in Figure 2 were intended as absolute, but they are in fact predicted by the model, which has built into it the flexibility to accommodate them. Granted that Figure 2 is incomplete in this way, the main point nevertheless remains valid: To the degree that a group of meanings and any schema subsuming them approaches the configuration given in 2a, we can call it a case of ambiguity, and to the degree that it approaches 2e, it is vagueness.

3.4.3. The influence of context

Given this model, the influence of context comes for free. For salience is not a static characteristic, but a dynamic one. Among the factors enhancing it is entrenchment, as previously noted, but also degree of activation produced by context.14 _When

a cognitive structure has itself been recently activated, whether in the linguistic or in some other cognitive context, or when structures closely linked to it have been activated, it retains a residual activation, and it will reactivate at a higher level of energy.15 _{Thus when the word painter or even more especially the phrase} be a painter is used as in sentence (2) above, the specifications attached to -er “one who does the verb professionally”, and to the phrase be a V-er, enhance the salience of those portions of the meaning(s) of paint which have to do with innate abilities and skills employed, purpose for which done, and so on. This moves

Figure 6. Paint

the configuration of 5c towards one more like 2b, increasing the salience of the elaborations at the expense of the schema. The fact that the same two readings of paint feel more ambiguous in (2) than in (1) is thus to be expected. In sentence (3), in contrast, the focus is on the even spreading of the color in painting, which characterizes both subcases and therefore the schema as well; thus the differences between the subcases are not especially activated and the schema (i.e., what is common to the subcases) is rendered more salient. This moves the configuration of 5c towards one more like 2d, which explains why all the senses of paint test as vague by sentence (3).16 _{When paint is used intransitively, as in (1), the nature of}

the object is less salient: When it is used transitively, as in (4) or (5), it is rendered more salient, and must match in the two cases construed for identity-of-sense anaphora to function normally.

3.4.4. Puns

Puns like (6) can also be explained: They are a special case of context enhanc- ing the salience of meanings. In general puns involve a double context, in which two meanings associated with a particular phonological pole are rendered salient. Thus saying that a pirate burying his gold by the river is putting his money in the bank involves the context of the river edge, which renders salient that meaning of bank, but also the phrase put (your) money in the bank, which strongly activates the financial institution meaning. The two meanings are overlaid on each other, producing the semantic discomfort we call a pun. The situation in (6) is similar: The construction paint X’s portrait strongly activates, and thus renders salient, a meaning like (i) (Figure 6d), while paint Y color¬activates something like (vi–ix) (6e). Both meanings are thus rendered salient by the context, yielding a configu- ration like 2b, and overlaid on each other. In exactly the same way, the phrase prepare your turkey for Thanksgiving dinner renders salient one meaning of pre- pare, while the continuation by breaking it to him gently renders another meaning salient, and the superimposition of the two salient meanings constitutes a pun. 3.4.5. Diachronic change

And, finally, the model clearly shows the distinction between ambiguity and vagueness to be a gradual one, with synchronic variation of meanings along the parameters of differentiation. This brings ease to the problem of diachronic change from one category to the other. There is no hard and fast boundary that a form needs to jump all at once: it can straddle the fence indefinitely, shifting its weight back and forth, before gradually moving more to one side than the other.

4. Summary

I commend this model to you on the following grounds:

(1) The theoretical constructs involved are well motivated on independent grounds (see extensive argumentation in Langacker 1987 and elsewhere).

(2) The traditional view of the ambiguity/vagueness distinction is seen as essentially correct; its intuitive characterizations fit the model nicely. It was only wrong if the categories were made absolute.

(3) The value of the traditional linguistic and definitional tests is affirmed: they clearly and correctly distinguish prototypically vague from prototypically ambiguous structures. The different behavior of ambiguous and vague structures in puns also fits into the model.

(4) The model correctly allows for a range of in-between (polysemous) cases, neither ambiguous nor vague in the prototypical sense (although in the cases we examined they are predictably vague by the definitional test).

(5) The influence of context (including the possibility of puns between related senses of a polysemous form) is accounted for automatically.

(6) The gradual nature of the ambiguity/vagueness distinction allows for grad- ual rather than abrupt diachronic shifting from one category to the other.

Notes

1. Quine’s logical test is difficult for me to apply in these cases (as in most others). Applying it, for instance, to examples mentioned below, I could say, of a case of painting stripes in a parking lot (x) or even more of swabbing iodine (xii), “She’s painting, but she’s not (really) painting”. By that I would mean that what she is doing may be properly called painting, but it is not painting proper, i.e. it is not the prototypical kind of painting. This might be taken as indicating ambiguity, but it is not very satisfactory; it does not precisely yield the “painting in one sense but not the other” reading that Quine was after. Also it does not work for more prototypical cases – e.g. (i), or (vi) or (vii). In general, however, to say of a person in any of these situations “She’s painting, but she’s not painting”, with no special intonation, seems infelicitous if not anomalous. This rather indicates vagueness. On the other hand, “This is the bank but it’s not the bank” is far from felicitous for me as well. Kempson (1977: 129), for rather different reasons, also concludes that “the characterization of ambiguity as the simultaneous assignment to a sentence of the values true and false has not provided a criterion for deciding unclear cases; it merely accentuates the point of disagreement.”

2. Traditional treatments of polysemy, however, tend to concentrate on cases in which the senses involved are not as closely related as those I am interested in: e.g., they would be more likely to discuss whether the nominal and verbal senses of paint can

be considered the same lexeme than to consider the distinctions among verbal senses I list in (i–xii).

3. Lyons (1977: 554) considers such in-between cases with the verb play: “Could we delete the second occurrence of the form plays in [She plays chess better than she

plays the flute]? And what about [?He played scrum-half in the afternoon and Ham-

let in the evening]?” Lyons concludes that “It may well be that the whole notion of discrete lexical senses is ill-founded.”

4. “1paint vb” vs. “2paint n”. The treatment is thus exactly parallel to that of “1painter

n... one that paints” (with subcases for artists and for housepainters and such) vs. “2painter n [... prob. fr. MF¬pendoir ...]: a line used for securing or towing a boat” vs. “3painter n ... [alter. of panther]... COUGARv¬three meanings with little or noth- ing in common beyond designating a thing of some sort. Cf. Lyons’ (1977: 21–22) comment, “The fact that there are two separate entries means that the compilers or editors of the dictionary have decided that two distinct lexemes are involved (and not one lexeme with two meanings”).

5. Langacker’s discussion of the notion “lexical item” (1987: 388) is apropos: “A cat- egory is coherent to the extent that its members are densely linked by well-entrenched categorizing relationships of minimal distance. ... The coherence of a category is naturally a matter of degree ... To the extent that a semantic network with common symbolization [i.e. linked to a common phonological pole] approximates a coherent category, we can reasonably speak of a lexical item. Despite its convenience, however, this construct is more a descriptive fiction than a natural unit of linguistic organization. Not only is coherence inherently a matter of degree, but also the definition allows a single network to be divided into lexical items in multiple and mutually inconsistent ways. I regard this as a realistic characterization of the phenomena in question.” 6. Most of this discussion is prefigured to some extent in Tuggy (1981: 56–59, 72), and

I have been using the essence of the diagram in Figure 2 at least since 1984. Never- theless, and although Langacker (1988: 137–139) credits my work, the ideas involved rest crucially on constructs of his theory, and in any case they came to me at least in part through discussions with him, Sue Lindner, Mary Ellen Ryder, Jeff Burnham, Barbara Levergood, and others. Thus I can claim no absolute credit for them. 7. For the discussion of this paper an essentially Aristotelian definition of schema is

adequate, in which the schema contains only (and all) material common to the sub- cases, and any concept which includes that material is a subcase. This, while not the only possible conceptualization, is doubtless the prototypical one, and is that described repeatedly by Langacker (e.g. 1987: 68, 132–138). The main point of this paper with respect to it is that (i) the existence of such a schema as part of the meaning of a lexical item does not preclude the existence of its subcases: in fact the subcases may well be more salient than the schema; and (ii) such variations in salience among a schema and its subcases account for the non-absoluteness of the distinctions between ambiguity, (polysemy), and vagueness.

A separate question is whether this conception of a schema is sufficient. (This is one of the important issues addressed by Geeraerts 1993.) Specifically, can a schema contain “either/or” specifications, or lists of alternative characteristics which do not have anything in common, or can it consist of specifications which are not charac- teristic (e.g. can one speak of a “schema” uniting checkmate and pawn, consisting of

the game of chess, which figures saliently in the meaning of both but characterizes neither)? Although I do not argue it here, I believe such “schemas” (if the term is appropriate to use for them) do indeed serve to unite concepts into categories and thus are relevant to the concerns of this paper. Langacker (e.g. 1987: 69, 92–93) speaks at length of a relationship of “partial schematicity” or “extension” in which there is conflict between the specifications of the “schema” and the “elaboration”, using it to describe the relationship between a prototype and non-prototypical members of the same category, or between the literal and figurative senses of a metaphor, or a gram- matical pattern and an ill-formed instance of that pattern; the prototypical kind of schematicity is the limiting case along a parameter of compatibility, in which there is no conflict of specifications. I would expect that there are other parameters involved in the other cases of non-prototypical “schemas” (those involving alternatives or lists, non-characteristic specifications, etc.), and that the prototypical schematicity will again prove to be the limiting and most salient case. In any case, I believe that the