Although there is an argument that something like the BCP can be defended on elementary grounds: It is an axiom of probability theory that tautologies should receive probability one; Hájek suggests that necessary truths also should receive probability one.
Another interesting application can be the design of optimum algorithm for MIMO antenna selection by joint minimization of receive probability of outage.
Given that the axioms are formulated such that the second axiom only requires that tautological sentences receive probability one, it is possible to satisfy the axioms, yet still be open to a sure loss.
Even with strengthening the second axiom to require that all necessary truths receive probability one, there is still a reading upon which the Converse Dutch Book theorem is false, since an agent will be vulnerable to a sure loss if she attaches a probability less than one to a known truth (or a probability greater than zero to a known falsehood).
Sometimes the second axiom is instead formulated as requiring that all logical truths receive probability one, but satisfying this constraint leaves open the possibility of a sure loss by betting against a necessary truth such as 'nothing is both red and green all over'.
If 'sure' means decidable, then neither the formulation that logical nor that necessary truths receive probability one will do, since there is no decision procedure for determining in general whether a given sentence is a logical truth, let alone a necessary one.
The choice for these probability functions is made such that the points belonging to the self class and all other classes receive probabilities close to 1 and 0, respectively, with a smooth transition as a function of generalized distance, as shown in Figure 3C.
Figure 5 Secondary channel capacity vs instantaneous received interference probability constraint, ξ.
The fact that the great majority of the calibration nodes received posterior probability values of 1.0 justified fixing their phylogenetic relationships in the divergence time analyses.
* = sites in bold received posterior probability support > 0.9.
