Please note that Halpern himself corrected his own first paper, which you cite. In a second paper (https://doi.org/10.1613/jair.644) he showed how to solve the difficulties he raised in his first paper. See also the work by Snow (https://doi.org/10.1111/0824-7935.00070), which again corrects Halpern's first paper and also some points by Paris (https://doi.org/10.1017/CBO9780511526596) and finds no serious difficulties with Cox's main points.
Please note that Halpern himself corrected his own first paper, which you cite. In a second paper (https://doi.org/10.1613/jair.644) he showed how to solve the difficulties he raised in his first paper. See also the work by Snow (https://doi.org/10.1111/0824-7935.00070), which again corrects Halpern's first paper and also some points by Paris (https://doi.org/10.1017/CBO9780511526596) and finds no serious difficulties with Cox's main points.