โ Scribed by Chris Mortensen
No coin nor oath required. For personal study only.
As many authors from Popper onwards have noted, it would be highly desirable to have a theory of nearness to the truth. Therefore, the results of Miller and Tichy ([6], [11]), apparently demonstrating the unsatisfactoriness of Popper's qualitative theory of verisimilitude, led to considerable research attempting to find a better account of nearness to the truth. Almost without exception, this research concentrated on modifying the definition of verisimilitude with an eye to escaping MiUer-Tichy-style limitative results, while retaining the base of classical logic on which the results also depend. In Mortensen [7], however, it was shown that the Miller-Tichy result could be escaped while retaining Popper's original theory, b y modifying the logical base on which scientific theories are constructed. Needless to say, the interest of such a result varies with the plausibility of the logical base. 1 It is therefore gratifying that the new logical base could be any one of the usual relevant or relevance logics developed originally by Anderson and Belnap, 2 since it is independently arguable that these logics represent an improvement over classical logic as candidates for the logic of ordinary inferential situations, and especially those of science. It should also be emphasised that while in [7] it was shown that the letter of the Miller-Tichy theorem could be escaped in relevant logic, it was pointed out there that it was still an open question whether some analogue of the theorem which did not rely on classical assumptions was provable. As we see later, this turns out to be the case.
Arguably, choice of logic is governed partly by pragmatic considerations. 3 Hence the 'relevance program' is devoted to investigating what kinds of changes the move to a relevant logical base might make to mathematics and science, especially their foundations. In this connection it is worth mentioning Meyer's result that Peano arithmetic on a relevant logic base can escape at least some versions of G6del's Second Incompleteness Theorem. 4 Accordingly, if the relevant logics have the pragmatic advantage over classical logic -that the mathematics and
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