"uncertainty" has plural meanings, depending on context and discipline. It is really an umbrella term that covers several semi-related concepts associated with "lack of certainty" in reasoning. Varieties of uncertainty include vagueness, variability, imprecision, indeterminacy, undefined, unknowable, absence (i.e. missing information), error, and so on.

"Imprecise" has a more specific meaning than "uncertain". In numerical contexts, it basically means fewer significant digits. If the true value is "3.14159...." and the estimates are "3" vs. "3.14", then the first is imprecise compared to the second. Similar distinctions can be made in qualitative/structural/linguistic domains.

"Vague" combines imprecision with generality and corresponds to a range or region of values, and roughly corresponds to "fuzzy specification" or "fuzzy estimation". The adverb "about" is often used in vague expressions, i.e. "He is about 6 feet tall". (Notice we never hear "He is about 6.231193 feet tall".)

For more, see these books by Michael Smithson:

Uncertainty and Risk - http://www.amazon.com/dp/1844078515/

Ignorance and Uncertainty - http://www.amazon.com/dp/0387969454/

"uncertainty" has plural meanings, depending on context and discipline. It is really an umbrella term that covers several semi-related concepts associated with "lack of certainty" in reasoning. Varieties of uncertainty include vagueness, variability, imprecision, indeterminacy, undefined, unknowable, absence (i.e. missing information), error, and so on.

"Imprecise" has a more specific meaning than "uncertain". In numerical contexts, it basically means fewer significant digits. If the true value is "3.14159...." and the estimates are "3" vs. "3.14", then the first is imprecise compared to the second. Similar distinctions can be made in qualitative/structural/linguistic domains.

"Vague" combines imprecision with generality and corresponds to a range or region of values, and roughly corresponds to "fuzzy specification" or "fuzzy estimation". The adverb "about" is often used in vague expressions, i.e. "He is about 6 feet tall". (Notice we never hear "He is about 6.231193 feet tall".)

For more, see these books by Michael Smithson:

Uncertainty and Risk - http://www.amazon.com/dp/1844078515/

Ignorance and Uncertainty - http://www.amazon.com/dp/0387969454/

In ordinary language there is some variability and overlap in the usage of these terms. However, I believe that the core meanings of the terms are quite distinct. I now give definitions that will be largely agreed upon by those working in the field of semantics, from a logical and philosophical perspective.

Uncertainty is an epistemic concept. That is it concerns knowledge. A proposition is uncertain if it is not known whether it is true or false. More generally any property is uncertain if its value is not known. There may be degrees of certainty if one has partial knowledge relating to the likelihood that a proposition is true or that a property has a particular value.

Imprecision occurs where a quantitative property is measured or represented using a value system that is more course grained that the value scale inherent in the quantitative property itself. For instance we may give specify lengths in centimetres, although we are dealing with objects that in reality may have any of a continuous range of possible values. We may specify a degree of precision by specifying a unit (e.g. m, cm, mm) corresponding to the largest magnitude by which the actual value may differ from the measured or represented value.

A proposition or word is "vague" when there is no clear convention or specification of the conditions under which the proposition is true or the word is applicable. Vague concepts such as 'tall', 'short', 'near' or 'far' indicate that the value of a property is considered high or low but do not set any precise threshold. A vague proposition is one that includes one or more vague concepts and whose truth depends on how the applicability of these is interpreted. The word "vague" is sometimes used to indicate simple generality (as in "I will visit you within the next week") but its core meaning indicates not only generality, but also the lack of a clear borderline (as in 'I will visit you soon').

Although the terms "uncertain", "imprecise" and "vague" tend to be used in similar circumstances, their meanings are quite distinct. The outcome of a dice roll is uncertain, but no vagueness or imprecision is involved. One may be completely certain that the length of a metal rod is 1m to the nearest cm. We may be certain that Bill Gates is rich without specifying a precise minimum amount of cash required to be rich.