In this appendix I wish to outline and briefly comment upon John Pollock's official definition of objective epistemic justification, which goes as follows:

S is objectively justified in believing P if and only if S instantiates some argument A supporting P which is ultimately undefeated relative to the set of all truths.¹ (p. 189)

(We can recast the account of subjective justification to conform: S is subjectively justified in believing P if and only if S instantiates an ultimately undefeated argument supporting P [p. 188].)

This does not wear its meaning upon its sleeve; it is worth trying to understand it both for its own sake, and because in so trying we encounter Pollock's account of defeasibility—perhaps the fullest and most satisfactory account extant of that crucial but puzzling notion. To understand the definition, we must know five things: what an argument is, what it is for someone to instantiate an argument, what it is for an argument to be defeated, what it is for an argument to be ultimately undefeated, and what it is for an argument to be ultimately undefeated with respect to a set of truths. I shall try to convey the essential idea in each case, referring the reader to the text for details.

1. Arguments. The basic idea here is familiar from elementary logic, where an argument is a series of items (typically sentences), each member of which is either a premise or a consequence (by way of an argument form) of preceding items, which may be thought of as reasons for the item in question. Adapting this familiar idea, Pollock thinks of an argument as a finite series of epistemic states (not sentences). These states will include beliefs, being perceptually appeared to, seeming to remember something, and the like:

We must take arguments to proceed from internal states (both doxastic and nondoxastic states) to doxastic states [that is, beliefs], the links between steps being provided by reasons… reasons are internal states.…

Our epistemic norms permit us to begin reasoning from certain internal states without those states being supported by further reasoning. Such states can be called basic states. Paramount among these are perceptual and memory states. Arguments must always begin with basic states and proceed from them to nonbasic doxastic states.… [They] proceed from basic steps to their ultimate conclusions through a sequence of steps each consisting of a belief for which the earlier steps provide reasons. (p. 187)

So an argument is a series of epistemic states beginning from basic states and terminating in a belief; and for each belief in the sequence there will be a reason, which may be either another belief or a nondoxastic state (for example, being appeared to thus and so).

2. Instantiation. To instantiate an argument is to be in the basic states from which the argument begins and to believe the conclusion of the argument on the basis of that argument.

3. Defeat. What is noteworthy about Pollock's account of defeasibility is his recognition that it cannot be properly explained merely in terms of the relation between a single belief and a defeating belief or condition: “To handle defeasibility in a general way, we must recognize that arguments can defeat one another and that a defeated argument can be reinstated if the arguments defeating it are defeated in turn” (p. 48). Although he does not give an account of what it is for one argument to defeat another, he does say what it is for one proposition or belief to defeat another:

If P is a reason for S to believe Q, R is a defeater for this reason if and only if R is logically consistent with P and P&R is not a reason for S to believe Q. (p. 38)

(So let's say provisionally, if not wholly accurately, that a defeater argument for an argument A with conclusion C is an argument that supports a defeater of C.) Thus I meet someone at a party: he tells me his name is ‘Alexander Hamilton’, thus giving me a reason P (he says his name is ‘Alexander Hamilton’) for believing that that is his name; you take me aside and tell me that he is a notorious practical joker who is really named ‘Melvin Hamilton’ but invariably introduces himself as ‘Alexander’; you thus provide me with a defeater R (he is really named ‘Melvin’) for that reason P.

Now a (propositional) defeater R for a reason P to believe a proposition Q is a defeater for me only if I believe R. This reveals a certain incompleteness in the definition of defeater: for of course (as Pollock recognizes)² I can believe a proposition to different degrees. I believe R with great fervor; I also believe P, but much less firmly; then (if no other beliefs are relevant) I do not have a reason for believing Q. It is consistent with this, however, that if I believed P fervently and R much less firmly, then I would still have a reason for believing Q. I believe that the moon is sometimes visible from the floor of Deepwater Canyon on the basis of the rather dim and infirm belief that I once saw it from there; I learn (beyond the shadow of a doubt) that you (the world's premier expert on deep canyons) believe that the former is never visible from the latter; then I no longer have a reason for believing that it is. But suppose I believe very strongly that I saw the moon from the canyon (I was there just last night and distinctly remember seeing it); even if I come to believe that you think it is not visible from there, I may still have a reason for thinking that it is. As things stand, therefore, P&R will on some occasions be a reason for S to believe Q and on other occasions not; we must add a qualification to the definition of defeater, specifying that whether a potential defeater R of P actually defeats P will depend (among other things) upon the relative strengths of belief in P and R. Whether a given strength of belief in P&R constitutes a reason for believing Q will presumably depend upon the relevant norms; so some of them will have to take the form if you believe P to degree d₁ and R to d₂, then you may believe Q (to d₃); but if you believe P to d₃ and R to d₄, then you may not believe Q to d₁.³

Further, Pollock points out that it is not only propositions or beliefs for which there can be defeaters. It may seem to me that the sheet of paper on the table is red; taking ‘reason’ in a broad sense, says Pollock, being in this state gives me a reason for believing that the sheet of paper is red. But if you point out that the sheet of paper is being illuminated by red light, then you provide me with a defeater for that reason. He therefore broadens his definition of ‘defeater’:

If M is a reason for S to believe Q, a state M^∗ is a defeater for this reason if and only if it is logically possible for S to be in this combined state consisting of being in both the state M and the state M^∗ at the same time, and this combined state is not a reason for S to believe Q. (p. 176)⁴

4. Being Ultimately Undefeated. Here we must quote Pollock's account in full:

Every argument proceeding from basic states that S is actually in will be undefeated at level 0 for S.… Some arguments will support defeaters for other arguments, so we define an argument to be undefeated at level 1 if and only if it is not defeated by any other argument undefeated at level 0. Among the arguments defeated at level 1⁵ may be some that supported defeaters for others, so if we take arguments undefeated at level 2 to be arguments undefeated at level 0 that are not defeated by any arguments undefeated at level 1, there may be arguments undefeated at level 2 that were arguments defeated at level 1. In general, we define an argument to be undefeated at level n + 1 if and only if it is undefeated at level 0 and is not defeated by any arguments undefeated at level n. An argument is ultimately undefeated if and only if there is some point beyond which it remains permanently undefeated; that is, for some N, the argument remains undefeated for every n > N. (p. 189)

Restricting attention to arguments S instantiates, perhaps we can simplify this as follows. An argument A is defeated at level 1 if (I suppress the ‘only if ‘s’) some argument defeats it; A is undefeated at level 1 in case no argument defeats it; hence if A is undefeated at level 1, it is undefeated at every subsequent level and is ultimately undefeated. A is defeated at level 2 in case some argument undefeated at level 1 defeats it; if A is defeated at level 2, therefore, it is defeated at every subsequent level (since it is defeated by an argument undefeated at every subsequent level). A is undefeated at level 2 if it is not defeated by an argument undefeated at level 1—if, in other words, any argument that defeats it is defeated at level 1 (that is, defeated by some argument or other). In general, A is defeated at level n + 1 if it is defeated by some argument undefeated at level n; A is undefeated at level n + 1 if it is not defeated by any argument undefeated at level n. A little reflection shows that an argument is ultimately undefeated if it is undefeated at every even-numbered level and is also undefeated at some odd-numbered level. Some arguments not defeated at any even level are nonetheless not ultimately undefeated; such arguments are defeated at every odd level and undefeated at every even (for an example, see p. 220).

5. Conditionality. We need just one more idea to complete the account of objective epistemic justification. Say that an argument A is conditional on Y for S (where Y is a set of propositions) if A proceeds from basic states S is in together with doxastic states consisting of believing members of Y (p. 189). Then,

Roughly, a belief is objectively justified if and only if it is held on the basis of some ultimately undefeated argument A, and either A is not defeated by any argument conditional on true propositions not believed by S, or if it is, then there are further true propositions such that the initial defeating arguments will be defeated by arguments conditional on the enlarged set of true propositions. (p. 189)

More explicitly,

We then say that an argument instantiated by S… is undefeated at level n + 1 relative to Y if and only if it is undefeated by any argument undefeated at level n relative to Y. An argument is ultimately undefeated relative to Y if and only if there is an N such that it is undefeated at every level n relative to Y for every n > N. (p. 189)

A person S, therefore, is “objectively justified in believing P if and only if S instantiates some argument supporting P which is ultimately undefeated relative to the set of all truths” (p. 189).⁶ As we saw above, objective justification, Pollock believes, is very close to knowledge; we need add only a small qualification to accommodate what Pollock, following Harman, sees as “the social side of knowledge.”

“We are ‘socially expected’ to be aware of various things: what is in our mail, what is repeatedly announced on television, what any sixth grader has learned in school, and the like. A proposition is ‘socially sensitive for S’ if and only if it is of a sort S is expected to believe when true” (p. 192). The final definition of knowledge, therefore, goes as follows:

S knows P if and only if S instantiates some argument A supporting P which is (1) ultimately undefeated relative to the set of all truths and (2) ultimately undefeated relative to the set of all truths socially sensitive for S.⁷

I mention a small problem—one that probably requires no more than a little more Chisholming. Suppose P is a proposition that happens (unbeknownst to me) to be true: can I know (on this account) that I do not believe P? Suppose that

P Caesar had scrambled eggs for breakfast the morning he crossed the Rubicon

is in fact true, but suppose I don't believe it, having no views at all on the subject (as in fact I don't) and believing that I don't believe P. For me to be objectively justified in this belief, according to the official definition, I must instantiate some argument that (a) supports the proposition

Q I do not believe that Caesar had scrambled eggs for breakfast the morning he crossed the Rubicon

and (b) is ultimately undefeated relative to the set of all truths (p. 189). Recall that an argument is conditional on the set of all truths for me if it proceeds (in accordance with my norms) from basic states I am in together with doxastic states consisting of believing members of the set of truths. Now presumably the same norms are to be used both in an argument proceeding from the doxastic states I am in and an argument proceeding from members of the set of truths. But among my norms is the following: if you are considering a proposition A but do not believe it, then you may believe that you do not believe A. Among the truths, of course, is P itself; so relative to the set of truths there will be for me an argument (call it ‘A’) from the doxastic state of believing P to the denial of Q (call it ‘−Q’); this argument will proceed by the above norm.

Can this argument for −Q be ultimately defeated, as it would have to be for me to have an ultimately undefeated argument for Q? I don't see how. An argument (relative to the set of all truths) for Q (hence an argument that defeats the argument for −Q) would be, for example, Q, therefore Q (perhaps another such argument would be one from the basic state of not-believing P to Q). Call this argument ‘B’. Arguments A and B are so related, as a little reflection shows, that (if there are no other arguments defeating either) each is defeated (by the other) at all the odd-numbered levels; hence neither is undefeated at any odd-numbered level and hence neither is ultimately undefeated. The same will go, so far as I can see, for any other argument for Q. No such argument will be ultimately undefeated; any argument defeating A will itself be defeated at all the odd-numbered levels by A. But then Q won't be objectively justified for me, so that a necessary condition of my knowing it, according to Pollock's account, cannot be fulfilled. Yet surely I do know that I don't believe that Caesar had scrambled eggs for breakfast the morning he crossed the Rubicon.

No doubt this problem requires no more than a bit of tinkering; perhaps, for example, we must say that the norm just mentioned is to be applied only to doxastic states I am in, not to members of the set of truths. More exactly, what is needed is some generalization of that suggestion, since the problem will reappear if the norm in question may be applied to steps of an argument that are dependent upon basic steps that are not doxastic states I am in. Reflexive propositions about what one believes (or does not believe) clearly require special treatment when taken in the context of the account of an argument conditional on the set of all truths.