Deductive Cogency

Where does this leave us? One moral of our discussion so far is that exceeding some probability threshold is neither necessary (because of the Preface Paradox) nor sufficient (because of the Lottery Paradox) for acceptability. The more general moral from these paradoxes, however, is that any account of acceptability should make acceptability accord with the principles appealed to above, (D) (and thus (&)) and (C). Putting these together, we get the following requirement on acceptability:

Deductive Cogency: If{H1, . . . , Hn}is a set of hypotheses each of which is acceptable, then

and

(ii) {H1, . . . , Hn}is consistent.

What the Preface and Lottery paradoxes in effect show is that the Threshold View violates

bothclauses of Deductive Cogency: The Preface shows that it violates (i), while the the Lottery shows that it violates (ii). So, in sum, the problem with the Threshold View is that it fails doubly to make acceptability deductively cogent.

Let me pause here to consider an objection to Deductive Cogency, in particular to clause (ii). It might seem that (ii) conflicts with the fact that there are cases of accepted scientific theories that are mutually incompatible. For example, classical electrodynamics and Bohr’s model of the atom are mutually incompatible, yet they were simultaneously ac- cepted by the scientific community at one point. Similarly, general relativity and quantum mechanics are widely taken to be mutually incompatible, yet both are currently accepted. Indeed, there may even be examples of theories that were once accepted but that are in- ternally inconsistent, such as Dirac’s original formulation of quantum mechanics, which used a mathematically inconsistent definition of the “Dirac delta function”. Many other examples of this kind can be given, and so there can be no denying that scientists have sometimes accepted inconsistent theories.17

However, these cases, interesting as they may be, do not conflict with (ii). For recall (from section 2) that we defined explanatory acceptability in terms of whether it isepistem- icallyreasonable to appeal to a theory in one’s explanations. That is compatible, of course, with scientists having excellent non-epistemic reasons to accept or explain with a set of theories that are, strictly speaking, explanatorily unacceptable. For example, it would be foolish at best to reject either general relativity or quantum mechanics without viable the- ories to replace them with, but that does not mean that appealing to both theories in one’s explanations is strictly speaking epistemically reasonable. What it shows is that given our

17_{I am grateful to John Roberts for calling my attention to these sorts of cases, and for pressing the objection}

epistemic limitations, we ought sometimes be prepared to sacrifice epistemic perfection in order to satisfy other important goals, including the goal of having informative theories.

So my claim is not thatDeductive Cogencyis an overriding or all-things-considered normative requirement on the set of hypotheses with which we should be prepared to ex- plain. Deductive Cogencyis best viewed as an ideal to which we should aspire – one that may very well be unreasonable to expect an agent to live up to in many circumstances, especially when due to our cognitive limitations we fail to see how to modify our theories in light of it. Kaplan put the point well in discussing an analogous requirement for (full) beliefs:

[...] you should want to satisfy Deductive Cogency in the following sense: you should view a demonstration that a set of beliefs violates Deductive Cogency as a criticism of that set of beliefs (as, presumably, you would not view a demonstration that a set of beliefs implies that the earth is not flat) - a criticism that can only be met by revising that set of beliefs. Deductive Cogency is best understood as simply a condition that your set of beliefs must satisfy on pain of being open to criticism. (Kaplan 1995, 118)

This,mutatis mutandi, is how I understand the nature ofDeductive Cogencyas a normative requirement upon the set of theories we should be prepared to appeal to in explanations.

To be sure, there are still those who conclude from the fact that the Threshold View conflicts withDeductive Cogency that the latter, as opposed to the former, should be re- jected.18 _{One might be tempted to plump for such a conclusion as a last resort given the}

plausibility of explicating acceptability in terms of probability and the enormously fruit- ful probabilistic approach in epistemology. But that would be too hasty, for the Threshold View is not the only way of explicating acceptability in terms of probability. It is true that developing an probability-based account of acceptability that isn’t just “rigged” to satisfy

Deductive Cogencybyad hocstipulation will not be easy. Fortunately, I will argue in what

18_{In the case of the threshold view of full belief, Kyburg (1961, 1970) and Foley (1979, 1992) both take}

follows that by replacing the “satisficing” approach of the Threshold View with an “opti- mizing” or “maximizing” approach, we open the door to a natural account of acceptability that turns out to accord withDeductive Cogencywithout anyad hocmaneuvers. The next section introduces the basic thought behind this idea.