Keynes and Carnap both demonstrated that logical probability requires a determinate measure function over a structured language, making imprecision a formal deficiency rather than a permissible feature.

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Carnap: Rudolf Carnap (1891-1970) was a highly influential German-American philosopher who believed that many traditional philosophical problems could be solved by carefully analyzing the language we use to talk about them. He pioneered the idea that philosophy should work closely with science and mathematics, using precise logical methods to clarify confused thinking. His work fundamentally changed how philosophers approach their discipline, making language analysis and logical rigor central to philosophical practice.
Formal deficiency(as used in logic): A flaw or weakness that breaks the rules or standards of a logical system, making it unable to work properly.
Imprecision(as used in logic and mathematics): Lack of exactness or clarity; when something is fuzzy, ambiguous, or doesn't have a single definite meaning.
Keynes(as a philosopher of probability and evidence): John Maynard Keynes was a 20th-century British philosopher and economist who developed ideas about how we should measure and think about probability based on the evidence we have available.

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