What is the probablity that a 178cm man is tall (or that many items will cost between one pound and two pounds)?
Vagueness is a besetting problem in quantitative risk assessment and it’s often overlooked or ignored in the attempt to find one metric (probability) by which to measure uncertainty. Clearly it’s inappropriate to use probabilistic methods to assess whether or not someone is tall (unless you are using population data to assess, say, the probability someone is taller than a specified height). But many other situations are quantified with probability when they shouldn’t be. Philosopher Mark Colyvan argues that all situations with vague premises should be assessed non-probabilistically. Just as the tallness of a person or the number that constitutes ‘many’ is vague because there will always be borderline cases, so too we should not speak of the probability that, say, biodiversity is declining in a particular ecosystem. Both the measurement of biodiversity and indeed the limits of an ecosystem are vague in the same sense.
This is to highlight the logical assumptions of Cox’s theorem, that ‘Any measure of belief is isomorphic to a probability measure’. Not that Cox is wrong, just that he can be and is used inappropriately whare vague premises are involved.
Colyvan points to a strong claim:
no adequate defense of classical logic in domains employing vague predicates is possible. If I am right about this, then not only are non- probabilistic methods legitimate methods for quantifying at least some types of uncertainty, but are also required for the adequate treatment of uncertainty in any domain where vague predicates are used (2008: 651).
So shouldn’t we just speak more precisely? Wouldn’t this clear up the vagueness and allow us to have confidence in our probabilities? That question brings us to the punchline of this post. Colyvan produces a marvellous quotation from a book on Uncertainty (Morgan and Henrion):
They claim that [uncertainty due to linguistic imprecision] is “usually relatively easy to remove with a bit of clear thinking” (1990, p. 62). If it were so easy to remove, you would expect them to be able to state this thesis without appeal to at least four vague terms.
What is the probability that vagueness is here to stay?
In case you missed it: Certainty – I’m fairly sure we don’t need it.
Colyvan, M. (2008) Is probability the only approach to uncertainty? Risk Analysis 28.3: 645-652.
Cox, R. T. (1946). Probability, frequency and reasonable ex- pectation. American Journal of Physics, 14, 1–13.
Morgan, M. G., & Henrion, M. (1990). Uncertainty: A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis. Cambridge: Cambridge University Press.
Image credit: the vague shop by whyohwhyohwhyoh