Since concepts, in the field of cognition, perform a function similar to that of numbers in the field of mathematics, the function of a proposition is similar to that of an equation: it applies conceptual abstractions to a specific problem.
A proposition, however, can perform this function only if the concepts of which it is composed have precisely defined meanings. If, in the field of mathematics, numbers had no fixed, firm values, if they were mere approximations determined by the mood of their users ... there would be no such thing as the science of mathematics. [IOTE, 75]
I have already noted Rand's conflation of identity with understanding. In this passage Rand is guilty of conflating meaning with reference. This conflation is hardwired into the very warp and woof of the Objectivist epistemology. It is implicit in Rand's mania for establishing the "validity" of concepts. Remember, for Rand, concepts are knowledge; which means they must have a reference in reality (for if they did not "stand" for something in reality, they could not be regarded as knowledge.) If we follow the (implicit) logic in the Objectivist epistemology, unicorn is an "invalid" concept because it has no referent. For Rand a "valid" concept must have both a meaning and a reference.
This insistence on fusing meaning with reference confuses Rand about the nature of propositions. When Rand contends that "the function of a proposition is similar to that of an equation," she has stretched her mathematics analogy to the breaking point. There are important differences between equations and propositions that she is ignoring. Equations, for example, can be reversed: 2+2=4 expresses the exact same relation as 4=2+2. Propositions, on the other hand, cannot be so easily reversed. Sky is blue does not express the same thing as Blue is sky.. Of even greater significance is the fact that equations can only express relations of quantity. They cannnot express information contained in qualitative judgments -- judgments, moreover, which include all moral valuations and the discernment of human motivation. In other words, some of the most important knowledge is qualitative, rather than quantitative, in nature. Rand is generally not regarded as being part of the positivists axis in philosophy. On the contrary, she saw herself as a great enemy of positivism. Yet her insistence on bringing these crude mathematical analogies into her theory of concepts and propositions places her at the same level as the crudest quantity-worshipping positivists. There is more between heaven and hell than can be squeezed into a quantification-only philosophy.
It seems understandable why Rand got sucked into this sort of of quantification mania. Quantity-based judgments have one advantage over qualitiative judgments: numbers are more "precise." Mathematics therefore comes closer to Rand's ideal of "precisely defined meanings." Everyone knows exactly what two means. If I ask a waiter in a restaurant to bring me two glasses of water, he will know exactly what to get. If I ask the same waiter to give me a "warm" cup of coffee, he may not be entirely sure what I mean by "warm."
Rand's desire to look to mathematics for precision constitutes another one of her false ideals. The number two seems to have a more precise meaning than the term warm, but this greater precision is an epistemological illusion. In terms of sheer meaning, two and warm are "equally" precise. Precision is a built-in feature to meaning. Terms of thought mean what they mean, and only what they mean. You can't get any more precise than that. The lack of precision arises, not from the meaning, but from the reference. By confounding meaning and reference, Rand made it impossible for her to understand this.
If we regard concepts as meanings and meanings as precise, than the issue of vagueness involves to what degree our assertions, which are made up of meanings, aptly describe matters of fact. Rand seems to have been laboring under the (tacit) assumption that in order for knowledge to be "valid" (i.e., true of reality), it must never be vague or indistinct: it had to be "precisely" true. She also seems to have assumed (tacitly) that precise meanings are necessary in order to formulate precise, accurate statements about matters of fact. Both of Rand's (tacit) assumptions are wrong. Precision of meaning is no guarantee of accuracy of description. A fictional story could be described in exhaustive detail, with little if any ambiguity, while a true story could be described loosely, in the most generalized, indistinct way. Moreover, since all meanings are precise, imprecision of meaning cannot be the cause of vagueness or inaccuracy. The vagueness arises from whether the meanings convey precise information aptly describing the referent in question. If I say The water is hot, I'm speaking vaguely. But the vagueness comes, not from the meanings, but from the lack of specificity in the description. The term hot has an immaculately precise meaning. But the meaning itself doesn't contain much information. Therefore, when used in a description of matters of fact, the term hot is vague. However, this vagueness arises from the lack of information conveyed by the meaning of the term, not by the meaning itself.
Imprecision is largely a problem of communication. While Rand understood the role of concepts in communication, she tended to emphasize their importance in thought. Yet this emphasis gets it backwards. Vagueness is almost entirely a problem of communication. If I stick my hand in a tub of water and think, "This is hot," I know exactly what I mean, because I have actual experience of the water's temperture. But if I say to someone else, "The water in the tub is hot," that person will not know precisly what I mean. The only way to convey a more precise account of water's temperature is to measure it with a themometer. Then I could communicate more precise, accurate information: "The water is 109 degrees."
Imprecision in use of language does not necessarily indicate, as Rand implies, imprecision of thought. Vagueness can be an indication of several things. Often, it is merely a problem of articulation. Some people are not very good at expressing what they think or experience. Rand, as a writer, specialized in articulation, so it may have been difficult for her to relate to this issue. Imprecision can also arise when dealing with inexpressible subject matters. Not all knowledge is easily articulable. Some subjects, like quantum mechanics or abstract symphonic music, are difficult to describe and explain. Rand and her Objectivist philosophy seem entirely oblivious to the inarticulable. It's as if Rand and her disciples believe that what cannot be said or explained in words cannot be real.
The only time vagueness is an indication of "improper" or "bad" thinking is when vague terms are used either as cover for a lack of understanding or when people resort to equivocations when rationalizing their beliefs. In the first case, the trouble arises, not from imprecise meanings, but from an insufficient knowledge or understanding of some aspect of reality. If you don't understand a subject, knowing the "precise" or "true" meaning isn't going to help you describe it. The other issue, the one of rationalization, is potentially a bit more problematic; although even here, it's hardly the great evil that Rand would make of it. Most rationalizing involves making people feel good about themselves. People like to think better of themselves than is perhaps entirely deserved. It's a "human-all-too-human" failing which rarely has dire consequences. (If such "faking" of reality were as horrible as Objectivism makes it seem, then the human race would have disappeared long ago.)
So Rand's belief that "precision" of meanings is important for human cognition is vastly over-stated: precision is important largely for communication. And Rand's belief that precision arises from "proper" concept formation is flat-out wrong. Concepts, as meanings, are entirely precise in and of themselves. If, however, the concepts are very general (as some must be: remember unit-economy) and contain little precise information about their referents, then they will seem vague, particularly when used in communication.