In essence, fuzzy logic says only that we should account for the undeniable imprecision of the world by altering the system's operators, rather than by some awkward afterthought squeezed into the
probability calculus.
Carnap's basic point was simple: if P(h/e) > P(h), and h entails k, then there is nothing in the
probability calculus to show that P(k/e) > P(k).
If a person grants that probabilities are part of a continuum, it does not follow that he thinks about uncertainty the same way that a proponent of the standard
probability calculus does.
To take full cognizance of the crudity of such attitudes, we might need to eschew the use of real numbers and the
probability calculus altogether.
This question is addressed here, drawing lessons from an instructive series of attempts to reconstruct within the
probability calculus the classical problem of independent witnesses who corroborate each other's testimony.
But then P' is obtained by conditionalizing on e[double prime], and so P[prime](e[double prime]) = 1, and by the
probability calculus P[prime](e) = 1 also.
According to Edgington, probability may be interpreted as degree of belief - in the sense that degrees of (rational) belief are held to bear to one another relationships captured by the axioms of the standard
probability calculus (cf.
Qualified Reflection follows from the
probability calculus together with a few idealizing assumptions.
i) her beliefs are degrees of belief which conform to the laws of the
probability calculusHe infers, on this basis, that decision theory needs an epistemic measure of information quantity and quality (a measure of the "weight" of evidence) in addition to the
probability calculus. Anand offers the following axioms in his sketch of an alternative to SEU theory:
46) it is clear that Bennett installs the Equation as the cornerstone of his theory of indicatives, and that chapter carefully presents the necessary
probability calculus for its construction.
In the special case of observables that are part of the decoterence basis, this function will associate numbers with eigenvalues in a way that (to a very high degree of approximation) satisfies the
probability calculus.
The Lottery and Preface paradoxes show, he thinks, that EE rational belief is not closed under conjunction, and that it is not necessarily EE irrational either to have inconsistent beliefs or to have degrees of belief which violate the
probability calculus, or to make oneself vulnerable to a Dutch book.
