Inference to the Best Explanation

GILBERT HARMAN

Gilbert Harman is professor of philosophy at Princeton University and is the author of Thought (1973), from which this selection is taken. Harman seeks to correct Goldman's causal account of empirical knowledge by replacing "cause" with "because" and by viewing the causal account as a special case of an explanatory account of knowledge. Inductive knowledge is inference to the best explanation. In the second part of his essay, Harman goes on to discuss undermining evidence to knowledge claims, modifying his formula for inferential conclusions to read "Y because X and there is no undermining evidence to this whole conclusion."

Gettier Examples and Probabilistic Rules of Acceptance

In any Gettier example we are presented with similar cases in which someone infers h from things he knows, h is true, and he is equally justified in making the inference in either case. In the one case he comes to know that h and in the other case he does not. I have observed that a natural explanation of many Gettier examples is that the relevant inference involves not only the final conclusion h but also at least one intermediate conclusion true in the one case but not in the other. And I have suggested that any account of inductive inference should show why such intermediate conclusions are essentially involved in the relevant inferences.


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Gettier cases are thus to be explained by appeal to the principle 

It is easy to see that purely probabilistic rules of acceptance do not permit an explanation of Gettier examples by means of principle P.  Reasoning in accordance with a purely probabilistic rule involves essentially only its final conclusion. Since that conclusion is higly probable, it can be inferred without reference to any other conclusions; in particular, there will be no inter- mediate conclusion essential to the inference that is true in one case and false in the other. . . .

The trouble is that purely probabilistic rules are incompatible with the natural account of Gettier examples by means of principle P. The solution is not to attempt to modify P but rather to modify our account of inference.

1. Knowledge and Explanation: A Causal Theory

Goldman suggests that we know only if there is the proper sort of causal connection between our belief and what we know. For example, we perceive that there has been an automobile accident only if the accident is relevantly causally responsible, by way of our sense organs, for our belief that there has been an accident. Similarly, we remember doing something only if having done it is relevantly causally responsible for our current memory of having done it. Although in some cases the fact that we know thus simply begins a causal chain that leads to our belief, in other cases the causal connection is more complicated. If Mary learns that Mr. Havit owns a Ford, Havit's past owner- ship is causally responsible for the evidence she has and also responsible (at least in part) for Havit's present ownership. Here the relevant causal connection consists in there being a common cause of the belief and of the state of affairs believed in.

Mary fails to know in the original Nogot- Havit case I because the causal connection is lacking. Nogot's past ownership is responsible for her evidence but is not responsible for the fact that one of her friends owns a Ford. Havit's past ownership at least partly accounts for why one of her friends now owns a Ford, but it is not responsible for her evidence. Similarly, the man who is told something true by a speaker who does not believe what he says fails to know because the truth of what is said is not causally responsible for the fact that it is said.

General knowledge does not fit into this simple framework. That all emeralds are green nei- ther causes nor is caused by the existence of the particular green emeralds examined when we come to know that all emeralds are green. Goldman handles such examples by counting logical connections among the causal connections. The belief that all emeralds are green is, in an extended sense, relevantly causally connected to the fact that all emeralds are green, since the evidence causes the belief and is logically entailed by what is believed.

It is obvious that not every causal connection, especially in this extended sense, is relevant to knowledge. Any two states of affairs are logically connected simply because both are entailed by their conjunction. If every such connection were relevant, the analysis Goldman suggests would have us identify knowledge with true belief, since there would always be a relevant "causal connection" between any state of true belief and the state of affairs believed in. Goldman avoids this reduction of his analysis to justified true belief by saying that when knowledge is based on inference relevant causal connections must be "reconstructed" in the inference. Mary knows that one of her friends owns a Ford only if her inference reconstructs the relevant causal connection between evidence and conclusion.

But what does it mean to say that her inference must "reconstruct" the relevant causal connection? Presumably it means that she must infer or be able to infer something about the causal connection between her conclusion and the evidence for it. And this suggests that Mary must make at least two inferences. First she must infer her original conclusion and second she must infer something about the causal connection between the conclusion and her evidence. Her second conclusion is her "reconstruction" of the causal connection. But how detailed must her recon- struction be? If she must reconstruct every detail of the causal connection between evidence and conclusion, she will never gain knowledge by way



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of inference. If she need only reconstruct some "causal connection," she will always know, since she will always be able to infer that evidence and conclusion are both entailed by their conjunction. I suggest that it is a mistake to approach the problem as a problem about what else Mary needs to infer before she has knowledge of her original conclusion. Goldman's remark about reconstruct- ing the causal connection makes more sense as a remark about the kind of inference Mary needs to reach her original conclusion in the first place. It has something to do with principle P and the natural account of the Gettier examples.

Nogot presents Mary with evidence that he owns a Ford. She infers that one of her friends owns a Ford. She is justified in reaching that conclusion and it is true. However, since it is true, not because Nogot owns a Ford, but because Havit does, Mary fails to come to know that one of her friends owns a Ford. The natural explanation is that she must infer that Nogot owns a Ford and does not know her final conclusion unless her intermediate conclusion is true. According to this natural explanation, Mary's inference essentially involves the conclusion that Nogot owns a Ford. According to Goldman, her inference essentially involves a conclusion concerning a causal connection. In order to put these ideas together, we must turn Goldman's theory of knowledge into a theory of inference.

As a first approximation, let us take his remarks about causal connections literally, forget- ting for the moment that they include logical connections. Then let us transmute his causal theory of knowing into the theory that inductive conclusions always take the form X causes Y, where further conclusions are reached by additional steps of inductive or deductive reasoning. In particular, we may deduce either X or Y from X causes Y.

This causal theory of inferring provides the following account of why knowledge requires that we be right about an appropriate causal connection. A person knows by inference only if all conclusions essential to that inference are true. That is, his inference must satisfy principle P. Since he can legitimately infer his conclusion only if he can first infer certain causal statements, he can know only if he is right about the causal connection expressed by those statements. First, Mary infers that her evidence is a causal result of Nogot's past ownership of the Ford. From that she deduces that Nogot has owned a Ford. Then she infers that his past ownership has been causally responsible for present ownership; and she deduces that Nogot owns a Ford. Finally, she deduces that one of her friends owns a Ford. She fails to know because she is wrong when she infers that Nogot's past ownership is responsible for Nogot's present ownership.

2. Inference to the Best Explanatory Statement

A better account of inference emerges if we replace "cause" with "because." On the revised account, we infer not just statements of the form X causes Y but, more generally, statements of the form Y because X or X explains Y. Inductive inference is conceived as inference to the best of competing explanatory statements. Inference to a causal explanation is a special case.

The revised account squares better with ordinary usage. Nogot's past ownership helps to explain Mary's evidence, but it would sound odd to say that it caused that evidence. Similarly, the detective infers that activities of the butler explain these footprints; does he infer that those activities caused the footprints? A scientist explains the properties of water by means of a hypothesis about unobservable particles that make up the water, but it does not seem right to say that facts about those particles cause the properties of water. An observer infers that certain mental states best explain some- one's behavior; but such explanation by reasons might not be causal explanation.

Furthermore, the switch from "cause" to "because" avoids Goldman's ad hoc treatment of knowledge of generalizations. Although there is no causal relation between a generalization and those observed instances which provide us with evidence for the generalization, there is an obvious explanatory relationship. That all emeralds are green does not cause a particular emerald to be green; but it can explain why that emerald is green. And, other things being equal, we can infer a generalization only if it provides the most plausible way to explain our evidence.

We often infer generalizations that explain but do not logically entail their instances, since they are of the form, In circumstances C, X's tend to be Ys.



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Such generalizations may be inferred if they provide a sufficiently plausible account of observed instances all things considered. For example, from the fact that doctors have generally been right in the past when they have said that someone is going to get measles, I infer that doctors can normally tell from certain symptoms that someone is going to get measles. More precisely, I infer that doctors have generally been right in the past because they can normally tell from certain symptoms that someone is going to get measles. This is a very weak explanation, but it is a genuine one. Compare it with the pseudo-explanation, "Doctors are generally right when they say some- one has measles because they can normally tell from certain symptoms that someone is going to get measles."

Similarly, I infer that a substance is soluble in water from the fact that it dissolved when I stirred it into some water. That is a real explanation, to be distinguished from the pseudo-explanation, "That substance dissolves in water because it is soluble in water." Here too a generalization explains an instance without entailing that instance, since water-soluble substances do not always dissolve in water.

Although we cannot simply deduce instances from this sort of generalization, we can often infer that the generalization will explain some new instance. The inference is warranted if the explanatory claim that X's tend to be Y's will explain why the next X will be Y is sufficiently more plausible than competitors such as interfering factor Q will prevent the next X from being a Y.  For example, the doctor says that you will get measles. Because doctors are normally right about that sort of thing, I infer that you will. More precisely, I infer that doctors' normally being able to tell when someone will get measles will explain the doctor's being right in this case. The competing explanatory statements here are not other explanations of the doctor's being right but rather explanations of his being wrong--e.g., because he has misperceived the symptoms, or because you have faked the symptoms of measles, or because these symptoms are the result of some other disease, etc..  Similarly, I infer that this sugar will dissolve in my tea. That is, I infer that the solubility of sugar in tea will explain this sugar's dissolving in the present case. Competing explanations would explain the sugar's not dissolving--e.g., because there is already a saturated sugar solution there, because the tea is ice-cold, etc. . . .

Another example is the mad-fiend case. Omar falls down drunk in the street. An hour later he suffers a fatal heart attack not connected with his recent drinking. After another hour a mad fiend comes down the street, spies Omar lying in the gutter, cuts off his head, and runs away. Some time later still, you walk down the street, see Omar lying there, and observe that his head has been cut off. You infer that Omar is dead; and in this way you come to know that he is dead. Now there is no causal connection between Omar's being dead and his head's having been cut off. The fact that Omar is dead is not causally responsible for his head's having been cut off, since if he had not suffered that fatal heart attack he still would have been lying there drunk when the mad fiend came along. And having his head cut off did not cause Omar's death, since he was already dead. Nor is there a straightforward logical connection between Omar's being dead and his having his head cut off. (Given the right sorts of tubes, one might survive decapitation.) So it is doubtful that Goldman's causal theory of knowing can account for your knowledge that Omar is dead.

If inductive inference is inference to the best explanatory statement, your inference might be parsed as follows: "Normally, if someone's head is cut off, that person is dead. This generalization accounts for the fact that Omar's having his head cut off is correlated here with Omar's being dead." Relevant competing explanatory statements in this case would not be competing explanations of Omar's being dead. Instead they would seek to explain Omar's not being dead despite his head's having been cut off. One possibility would be that doctors have carefully connected head and body with special tubes so that blood and air get from body to head and back again. You rule out that hypothesis on grounds of explanatory complica- tions: too many questions left unanswered (why can't you see the tubes? why wasn't it done in the hospital? etc.). If you cannot rule such possibilities out, then you cannot come to know that Omar is dead. And if you do rule them out but they turn out to be true, again you do not come to know. For example, if it is all an elaborate psychological philosophical experiment, which however fails, then you do not come to know that Omar is dead even though he is dead.

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Inference to the Best Explanation

4. Statistical Inference

Statistical inference, and knowledge obtained from it, is also better explicated by way of the notion of statistical explanation than by way of the notion of cause or logical entailment. . . .

Evidence One Does Not Possess: Three Examples

Example (1).

While I am watching him, Tom takes a library book from the shelf and conceals it beneath his coat. Since I am the library detective, I follow him as he walks brazenly past the guard at the front door. Outside I see him take out the book and smile. As I approach he notices me and suddenly runs away. But I am sure that it was Tom, for I know him well. I saw Tom steal a book from the library and that is the testimony I give before the University Judicial Council. After testifying, I leave the hearing room and return to my post in the library. Later that day, Tom's mother testifies that Tom has an identical twin, Buck. Tom, she says, was thousands of miles away at the time of the theft.  She hopes that Buck did not do it; but she admits that he has a bad character.

Do I know that Tom stole the book? Let us suppose that I am right. It was Tom that took the book. His mother was lying when she said that Tom was thousands of miles away. I do not know that she was lying, of course, since I do not know anything about her, even that she exists. Nor does anyone at the hearing know that she is lying, although some may suspect that she is. In these circumstances I do not know that Tom stole the book. My knowledge is undermined by evidence I do not possess.

Example (2).

Donald has gone off to Italy. He told you ahead of time that he was going; and you saw him off at the airport. He said he was to stay for the entire summer. That was in June. It is now July. Then you might know that he is in Italy. It is the sort of thing one often claims to know. However, for reasons of his own Donald wants you to believe that he is not in Italy but in California. He writes several letters saying that he has gone to San Francisco and has decided to stay there for the summer. He wants you to think that these letters were written by him in San Francisco, so he sends them to someone he knows there and has that per- son mail them to you with a San Francisco post- mark, one at a time. You have been out of town for a couple of days and have not read any of the letters. You are now standing before the pile of mail that arrived while you were away. Two of the phony letters are in the pile. You are about to open your mail. I ask you, "Do you know where Donald is?" "Yes," you reply, "I know that he is in Italy." You are right about where Donald is and it would seem that your justification for believing that Donald is in Italy makes no reference to letters from San Francisco. But you do not know that Donald is in Italy. Your knowledge is undermined by evidence you do not as yet possess.

Example (3).

A political leader is assassinated. His associates, fearing a coup, decide to pretend that the bullet hit someone else. On nationwide television they announce that an assassination attempt has failed to kill the leader but has killed a secret service man by mistake. However, before the announcement is made, an enterprising reporter on the scene tele- phones the real story to his newspaper, which has included the story in its final edition. Jill buys a copy of that paper and reads the story of the assas- sination. What she reads is true and so are her assumptions about how the story came to be in the paper. The reporter, whose by-line appears, saw the assassination and dictated his report, which is now printed just as he dictated it. Jill has justified true belief and, it would seem, all her intermediate conclusions are true. But she does not know that the political leader has been assassinated. For everyone else has heard about the televised announcement. They may also have seen the story in the paper and, perhaps, do not know what to believe; and it is higWy implausible that Jill should know simply because she lacks evidence everyone else has. Jill does not know. Her knowledge is undermined by evidence she does not possess.



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These examples pose a problem for my strategy. They are Gettier examples and my strategy is to make assumptions about inference that will account for Gettier examples by means of principle P. But these particular examples appear to bring in considerations that have nothing to do with conclusions essential to the inference on which belief is based.

Some readers may have trouble evaluating these examples. Like other Gettier examples, these require attention to subtle facts about ordinary usage; it is easy to miss subtle differences if, as in the present instance, it is very difficult to formulate a theory that would account for these differences. We must compare what it would be natural to say about these cases if there were no additional evidence one does not possess (no testimony from Tom's mother, no letters from San Francisco, and no televised announcement) with what it would be natural to say about the cases in which there is the additional evidence one does not possess. We must take care not to adopt a very skeptical attitude nor become too lenient about what is to count as knowledge. If we become skeptically inclined, we will deny there is knowledge in either case. If we become too lenient, we will allow that there is knowledge in both cases. It is tempting to go in one or the other of these directions, toward skepticism or leniency, because it proves so difficult to see what general principles are involved that would mark the difference. But at least some difference between the cases is revealed by the fact that we are more inclined to say that there is knowledge in the examples where there is no undermining evidence a person does not possess than in the examples where there is such evidence. The problem, then, is to account for this difference in our inclination to ascribe knowledge to someone.

Evidence Against What One Knows

If I had known about Tom's mother's testimony, I would not have been justified in thinking that it was Tom I saw steal the book. Once you read the letters from Donald in which he says he is in San Francisco, you are no longer justified in thinking that he is in Italy. If Jill knew about the television announcement, she would not be justified in believing that the political leader has been assassinated. This suggests that we can account for the preceding examples by means of the following principle.

However, by modifying the three examples it can be shown that this principle is too strong.

Suppose that Tom's mother was known to the Judicial Council as a pathological liar. Everyone at the hearing realizes that Buck, Tom's supposed twin, is a figment of her imagination. When she testifies no one believes her. Back at my post in the library, I still know nothing of Tom's mother or her testimony. In such a case, my knowledge would not be undermined by her testimony; but if I were told only that she had just testified that Tom has a twin brother and was himself thousands of miles away from the scene of the crime at the time the book was stolen, I would no longer be justified in believing as I now do that Tom stole the book. Here I know even though there is evidence which, if I knew about it, would cause me not to be justified in believing my conclusion.

Suppose that Donald had changed his mind and never mailed the letters to San Francisco. Then those letters no longer undermine your knowledge. But it is very difficult to see what principle accounts for this fact. How can letters in the pile on the table in front of you undermine your knowledge while the same letters in a pile in front of Donald do not? If you knew that Donald had written letters to you saying that he was in San Francisco, you would not be justified in believing that he was still in Italy. But that fact by itself does not undermine your present knowledge that he is in Italy.

Suppose that as the political leader's associates are about to make their announcement, a saboteur cuts the wire leading to the television transmitter. The announcement is therefore heard only by those in the studio, all of whom are parties to the deception. Jill reads the real story in the newspaper as before. Now, she does come to know that the political leader has been assassinated. But if she had known that it had been announced that he was not assassinated, she would not have been justified in believing that he has, simply on the basis of the newspaper story. Here, a cut wire makes the



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difference between evidence that undermines knowledge and evidence that does not undermine knowledge.

We can know that h even though there is evidence e that we do not know about such that, if we did know about e, we would not be justified in believing h. If we know that h, it does not follow that we know that there is not any evidence like e. This can seem paradoxical, for it can seem obvious that, if we know that h, we know that any evidence against h can only be misleading. So, later if we get that evidence we ought to be able to know enough to disregard it.

A more explicit version of this interesting paradox goes like this.2 "If I know that h is true, I know that any evidence against h is evidence against something that is true; so I know that such evidence is misleading. But I should disregard evidence that I know is misleading. So, once I know that h is true, I am in a position to disregard any future evidence that seems to tell against h."  This is paradoxical, because I am never in a position simply to disregard any future evidence even though I do know a great many different things.

A skeptic might appeal to this paradox in order to argue that, since we are never in a position to disregard any further evidence, we never know anything. Some philosophers would turn the argument around to say that, since we often know things, we are often in a position to disregard further evidence. But both of these responses go wrong in accepting the paradoxical argument in the first place.

I can know that Tom stole a book from the library without being able automatically to disregard evidence to the contrary. You can know that Donald is in Italy without having the right to ignore whatever further evidence may turn up. Jill may know that the political leader has been assassinated even though she would cease to know this if told that there was an announcement that only a secret service agent had been shot.
The argument for paradox overlooks the way actually having evidence can make a difference. Since I now know that Tom stole the book, I now know that any evidence that appears to indicate something else is misleading. That does not war- rant me in simply disregarding any further evidence, since getting that further evidence can change what I know. In particular, after I get such further evidence I may no longer know that it is misleading. For having the new evidence can make it true that I no longer know that Tom stole the book; if I no longer know that, I no longer know that the new evidence is misleading.

Therefore, we cannot account for the prob lems posed by evidence one does not possess by appeal to the principle, which I now repeat:

For one can know even though such evidence exists.

A Result Concerning Inference

When does evidence one does not possess keep one from having knowledge? I have described three cases, each in two versions, in which there is misleading evidence one does not possess. In the first version of each case the misleading evidence undermines someone's knowledge. In the second version it does not. What makes the difference?

My strategy is to account for Gettier examples by means of principle P. This strategy has led us to conceive of induction as inference to the best explanation. But that conception of inference does not by itself seem able to explain these examples. So I want to use the examples in order to learn something more about inference, in particular about what other conclusions are essential to the inference that Tom stole the book, that Donald is in Italy, or that the political leader has been assassinated.

It is not plausible that the relevant inferences should contain essential intermediate conclusions that refer explicitly to Tom's mother, to letters from San Francisco, or to special television pro- grams. For it is very likely that there is an infinite number of ways a particular inference might be undermined by misleading evidence one does not possess. If there must be a separate essential conclusion ruling out each of these ways, inferences would have to be infinitely inclusive-and that is implausible.

Therefore it would seem that the relevant inferences must rule out undermining evidence one does not possess by means of a single conclusion,



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essential to the inference, that characterizes all such evidence. But how might this be done? It is not at all clear what distinguishes evidence that undermines knowledge from evidence that does not. How is my inference to involve an essential conclusion that rules out Tom's mother's testifying a certain way before a believing audience but does not rule out (simply) her testifying in that way? Or that rules out the existence of letters of a particular sort in the mail on your table but not simply the existence of those letters? Or that rules out a widely heard announcement of a certain sort without simply ruling out the announcement?

Since I am unable to formulate criteria that would distinguish among these cases, I will simply label cases of the first kind "undermining evidence one does not possess." Then we can say this: one knows only if there is no undermining evidence one does not possess. If there is such evidence, one does not know. However, these remarks are completely trivial.

It is somewhat less trivial to use the same label to formulate a principle concerned with inference.

There is of course an obscurity in principle Q; but the principle is not as trivial as the remarks of the last paragraph, since the label "undermining evi dence one does not possess" has been explained in terms of knowledge, whereas this is a principle concerning inference.

If we can explain "undermining" without appeal to knowledge, given Q, we can use principle P to account for the differences between the two versions of each of the three examples described above. In each case an inference involves essentially the claim that there is no undermining evidence one does not possess. Since this claim is false in the first version of each case and true in the second, principle P implies that there can be knowledge only in the second version of each case. So there is, according to my strategy, some reason to think that there is a principle concerning inference like principle Q That raises the question of whether there is any independent reason to accept such a principle; and reflection on good scientific practice suggests a positive answer. It is a commonplace that a scientist should base his conclusions on all the evidence. Furthermore, he should not rest content with the evidence he hap- pens to have but should try to make sure he is not overlooking any relevant evidence. A good scientist will not accept a conclusion unless he has some reason to think that there is no as yet undiscovered evidence which would undermine his conclusion. Otherwise he would not be warranted in making his inference. So good scientific practice reflects the acceptance of something like principle Q, which is the independent confirmation we wanted for the existence of this principle.

Notice that the scientist must accept some- thing like principle Q, with its reference to "undermining evidence one does not possess." For example, he cannot accept the following principle,

 

There will always be a true proposition such that if he learned that the proposition was true (and learned nothing else) he would not be warranted in accepting his conclusion. If h is his conclusion, and if k is a true proposition saying what ticket will win the grand prize in the next New Jersey State lottery, then either k or not h is such a proposition. If he were to learn that it is true that either k or not h (and learned nothing else), not h would become probable since (given what he knows) k is antecedently very improbable. So he could no longer reasonably infer that h is true.

There must be a certain kind of evidence such that the scientist infers there is no as yet undiscovered evidence of that kind against h. Principle Q says that the relevant kind is what I have been labeling "undermining evidence one does not possess." Principle Q's confirmed by the fact that good scientific practice involves some such principle and by the fact that principle Q together with principle P accounts for the three Gettier examples I have been discussing.

If this account in terms of principles P and Q is accepted, inductive conclusions must involve some self-reference. Otherwise there would be a regress. Before we could infer that h, we would have to infer that there is no undermining evidence to h. That prior inference could not be deductive, so it would have to be inference to the best explanatory statement. For example, we might infer that the fact that there is no sign of undermining.


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evidence we do not possess is explained by there not being any such evidence. But, then, before we could accept that conclusion we would first have to infer that there is no undermining evidence to it which one does not possess. And, since that inference would have to be inference to the best explanation, it would require a previous inference that there is no undermining evidence for its conclusion; and so on ad infinitum.
Clearly, we do not first have to infer that there is no undermining evidence to h and only then infer h. For that would automatically yield the regress. Instead, we must at the same time infer both h and that there is no undermining evidence. Furthermore, we infer that there is not only no
undermining evidence to h but also no undermining evidence to the whole conclusion. In other words, all legitimate inductive conclusions take the form of a self-referential conjunction whose first conjunct is h and whose second conjunct (usually left implicit) is the claim that there is no under- mining evidence to the whole conjunction.

Conclusions as Total Views: Problems

[We have seen] that we could use principle P to account for many Gettier examples if we were willing to suppose that induction always has an explanatory statement as its conclusion. On that supposition reasoning would have to take the form of a series of inductive and deductive steps to appropriate intermediate conclusions that there- fore become essential to our inference. However, certain difficulties indicate that this conception of inference is seriously oversimplified and that our account of Gettier examples must be modified. [I have] already mentioned a minor complication. There is a self-referential aspect to inductive conclusions. Instead of saying that such conclusions are of the form Y because X we must say that they are of the form Y because X and there is no undermining evidence to this whole conclusion.

Another difficulty, . . . the "lottery paradox," poses a more serious problem. . . . This paradox arises given a purely probabilistic rule of acceptance, since such a rule would have us infer concerning any ticket in the next New Jersey lottery that the ticket will not win the grand prize. We might suggest that the paradox cannot arise if induction is inference to the best explanatory statement, since the hypothesis that a particular ticket fails to win the grand prize in the next New Jersey lottery does nothing to explain anything about our current evidence. However, there are two things wrong with such a suggestion. First, the paradox will arise in any situation in which, for some large number N, there are N different explanations of different aspects of the evidence, each inferable when considered apart from the other explanations, if we also know that only N - 1 of these explanations are correct. So, the paradox can arise even when we attempt to infer explanations of various aspects of one's evidence.

Furthermore, inference to the best explanatory statement need not infer explanations of the evidence. It can infer that something we already accept will explain something else. That is how I am able to infer that the sugar will dissolve when stirred into my tea or that a friend who intends to meet me on a corner in an hour will in fact be there. Moreover, we can sometimes infer explanatory statements involving statistical explanations; and, if a particular ticket does fail to win the grand prize, we can explain its not winning by describing this as a highly probable outcome in the given chance set-up. So, if induction is inference to the best explanatory statement, we should be able to infer of any ticket in a fair lottery that the conditions of the lottery will explain that ticket's failing to win the grand prize; the lottery paradox therefore arises in its original form. But before attempting to modify the present conception of inference in order to escape the lottery paradox, let us consider a different sort of problem involving that conception.

Our present conception of reasoning takes it to consist in a series of inductive and deductive steps. We have therefore supposed that there are (at least) two kinds of inference, inductive inference and deductive inference; and we have also supposed that reasoning typically combines inferences of both sorts. But there is something fishy about this. Deduction does not seem to be a kind of inference in the same sense in which induction is. Induction is a matter of inductive acceptance. On our current conception of inference, we may infer or accept an explanatory statement if it is sufficiently more plausible than competing statements, given our antecedent beliefs. On the other



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hand, deduction does not seem in the same way to be a matter of "deductive acceptance." So called deductive rules of inference are not plausibly construed as rules of deductive acceptance that tell us what conclusions we may accept, given that we already have certain antecedent beliefs. For example, although the deductive rule of modus ponens is sometimes stated like this, "From P and If P, then Q, infer Q," there is no plausible rule of acceptance saying that if we believe both P and If P, then Q, we may always infer or accept Q Perhaps we should stop believing P or If P, then Q rather than believe Q.

A contradiction logically implies everything; anything follows (deductively) from a set of logically inconsistent beliefs. Although this point is sometimes expressed by saying that from a contra- diction we may deductively infer anything, that is a particular use of "infer." Logic does not tell us that if we discover that our beliefs are inconsistent we may go on to infer or accept anything and everything we happen to think of. Given the discovery of such inconsistency in our antecedent beliefs, inference should lead not to the acceptance of something more but to the rejection of some- thing previously accepted.

This indicates that something is wrong in a very basic way with our current conception of inference. We have been supposing that inference is simply a way to acquire new beliefs on the basis of our old beliefs. What is needed is a modification of that conception to allow for the fact that inference can lead as easily to rejection of old beliefs as to the acceptance of new beliefs. Furthermore, we want to avoid the supposition that deduction is a kind of inference in the same sense in which induction is inference, and we want to avoid the lottery paradox.

Inference to the Best Total Explanatory Account

Influenced by a misleading conception of deductive inference, we have implicitly supposed that inductive inference is a matter of going from a few premises we already accept to a conclusion one comes to accept, of the form X because Y (and there is no undermining evidence to this conclusion). But this conception of premises and conclusion in inductive inference is mistaken. The conception of the conclusion of induction is wrong since such inference can lead not only to the acceptance of new beliefs but also the rejection of old beliefs. Furthermore, the suggestion that only a few premises are relevant is wrong, since inductive inference must be assessed with respect to every- thing one believes.

A more accurate conception of inductive inference takes it to be a way of modifying what we believe by addition and subtraction of beliefs. Our "premises" are all our antecedent beliefs; our "conclusion" is our total resulting view. Our conclusion is not a simple explanatory statement but a more or less complete explanatory account. Induction is an attempt to increase the explanatory coherence of our view, making it more complete, less ad hoc, more plausible. At the same time we are conservative. We seek to minimize change. We attempt to make the least change in our antecedent view that will maximize explanatory coherence.

The conception of induction as inference to the best total explanatory account retains those aspects of our previous conception that permitted an account of Gettier examples, although that account must be modified to some extent. On the other hand, the new conception does not suppose that there is deductive inference in anything like the sense in which there is inductive inference, since deductive inference is not a process of changing beliefs. Furthermore, the new conception accounts for the fact that inference can lead us to reject something previously accepted, since such rejection can be part of the least change that maximizes coherence.

Finally, the new conception avoids the lottery paradox. Inference is no longer conceived of as a series of steps which together might add up to something implausible as a whole. Instead, inference is taken to be "a single step to one total conclusion. If there can be only one conclusion, there is no way to build up a lottery paradox.

Consider the case in which there are N explanations of various aspects of the evidence, each very plausible considered by itself, where however it is known that only N - I are correct. Competing possible conclusions must specify for each explanation whether or not that explanation is accepted. A particular explanation will be accepted not simply because of its plausibility when considered by itself



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but only if it is included in an inferable total explanatory account. We will not be able to infer that all N explanations are correct since, I am assuming, that would greatly' decrease coherence.

Similarly, we will be able to infer that a particular ticket will fail to win the grand prize in the next New Jersey lottery only if there is a total resulting view containing this result that is to be preferred to alternative total views on grounds of maximizing coherence and minimizing change. The claim that a particular ticket fails to win will be part of an inferable total view only if that claim adds sufficiently more coherence than do claims that other tickets fail to win. Otherwise that total view will not be any better than a different total view that does not contain the claim that the first ticket fails to win.

This is a rather complicated matter and depends not only on the probabilities involved but also on how we conceive the situation and, in particular, on what claims we are interested in.3 We can see this by considering the conditions under which we can make inferences that give us knowledge. For example, if we are simply interested in the question of whether a particular ticket will win or fail to win, we cannot include in our total view the conclusion that the ticket will fail to win, since it would not be correct to say that in such a case we know the ticket will lose. On the other hand, if we are primarily interested in a quite different question whose answer depends in part on an answer to the question of whether this ticket fails to win, we may be able to include the conclusion that it does fail in our total view, since we can often come to have relevant knowledge in such cases. Thus, we might infer and come to know that the butler is the murderer because that hypothesis is part of the most plausible total account, even though the account includes the claim that the butler did not win the lottery (for if he had won he would have lacked a motive). Or we might infer and come to know that we will be seeing Jones for lunch tomorrow even though our total view includes the claim that Jones does not win the lottery (e.g., because if he won he would have to be in Trenton tomorrow to receive his prize and would not be able to meet us for lunch).

However I am unable to be very precise about how our interests and conception of the situation affect coherence or indeed about any of the factors that are relevant to coherence.

Inference and Knowledge

Having seen that induction is inference to the best total explanatory account, we must now modify our account of knowledge and of the Gettier examples.

One problem is that we want to be able to ascribe several inferences to a person at a particular time so as to be able to say that one of the inferences yields knowledge even though another does not. Suppose that Mary has evidence both that her friend Mr. Nogot owns a Ford and that her friend Mr. Havit owns a Ford. She concludes that both own Fords and therefore that at least one of her mends owns a Ford. Nogot does not own a Ford; but Havit does. We want to be able to say that Mary can know in this case that at least one of her friends owns a Ford, because one inference on which her belief is based satisfies principle P even if another inference she makes does not. But, if inference is a matter of modifying one's total view, how can we ascribe more than a single inference to Mary?⁴

Principle P seems to be in trouble in any event. It tells us that one gets knowledge from an inference only if all conclusions essential to that inference are true. Since one's conclusion is one's total resulting view, principle P would seem to imply that one gains no knowledge from inference unless the whole of one's resulting view is true. But that is absurd. A person comes to know things through inference even though some of what he believes is false.

A similar point holds of premises. It is plausible to suppose that one comes to know by inference only if one already knows one's premises to be true. However, one's total view is relevant to inference even though it always contains things one does not know to be true.

The key to the solution of these problems is to take an inference to be a change that can be described simply by mentioning what beliefs are given up and what new beliefs are added (in addition to the belief that there is no undermining evidence to the conclusion). Mary might be described as making all the following inferences. (1) She rejects her belief that no friend of hers owns a Ford. (2) In addition she comes to believe that one of her friends owns a Ford and accepts a story about Nogot. (3) As in (2), except that she

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accepts a story about Havit rather than one about Nogot. (4) She does all the preceding things. All the inferences (1)-(4) might be ascribed to Mary by an appropriate reasoning instantiator F. ⁵  Mary knows because inference (3) adds nothing false to her view. Given Mary's antecedent beliefs, there is a single maximal inference she makes, (4), which is the union of all the inferences she makes. An inference is warranted only if it is included in the maximal warranted inference. That is why the lottery paradox does not arise even though we allow for the possibility that Mary makes more than one inference.

In order to handle the problem about premises we must require, not that the actual premises of the inference (everything Mary believes ahead of time) be known to be true but only that the inference remain warranted when the set of antecedent beliefs is limited to those Mary antecedently knows to be true and continues to know after the inference. More precisely, let (A) B) be an inference that rejects the beliefs in the set A and adds as new beliefs those in the set B; let B U C be the union of the sets B and C, containing anything that belongs either to B or to C or to both; and let be the empty set that contains nothing. Then principle P is to be replaced by the following principle P*,  which gives necessary and sufficient conditions of inferential knowledge.

P* 5 comes to know that h by inference (A) B) if and only if (i) the appropriate reasoning instantiator F ascribes (A) B) to S, (ii) S is warranted in making the inference (A,B) given his antecedent beliefs, (iii) there is a possibly empty set C of unrejected antecedent beliefs not antecedently known by S to be true such that the inference B U G) is warranted when antecedent beliefs are taken to be the set of things S knows (and continues to know after the inference (A,B) is made), (iv) B U G contains the belief that h, (v) B U G contains only true beliefs. ⁶

Reference to the set C is necessary to cover cases in which S comes to know something he already believes. Part (v) of P* captures what was intended by our original principle P.

Summary

Our previous conception of induction as inference to the best explanatory statement fell short in three ways. It failed to account for the rejection of previously held beliefs, it failed to avoid the lottery paradox, and it treated deduction as a form of inference. The last of these is especially serious since it can lead to misguided theories in response to a form of skepticism it encourages; and it can also suggest the construction of inductive or practical logics, having to do with inductive or practical reasoning. These defects are avoided if induction is taken to be inference to the best total explanatory account. We can reject beliefs because the account we come to accept may not contain beliefs we previously accepted. Any inference we are warranted in making is included in the maxi- mal inference we are warranted in making, so the lottery paradox cannot arise. Deductive arguments are not inferences but are explanatory conclusions that can increase the coherence of one's view.
Finally, all this means that principle P must give way to principle P*, which says that we know by inference only if one of our inferences remains warranted and leads to the acceptance only of truths when restricted in premises to the set of things we know ahead of time to be true.

Notes

l. "Mary's friend Mr. Nogot convinces her that he has a Ford. He tells her that he owns a Ford, he shows her his ownership certificate, and he reminds her that she saw him drive up in a Ford. On the basis of this and similar evidence, Mary concludes that Mr. Nogot owns a Ford. From that she infers that one of her friends owns a Ford. . . . However, as it turns out in this case, Mary is wrong about Nogot. His car has just been repossessed and towed away. . . . On the other hand, Mary's friend Mr. Havit does own a Ford, so she is right in thinking that one of her friends owns a Ford. . . but she does not know that one of her friends owns a Ford" (Thought, p. 121).

2. Here and in what follows I am indebted to Saul Kripke, who is, however, not responsible for any faults in my presentation.

3Levi.

4Lehrer.

5. "Words like 'reasoning,' 'argument,' and 'inference' are ambiguous. They may refer to a process of reasoning, argument, or inference, or they may refer to an abstract structure consisting of certain propositions as premises, others as conclusions, perhaps others as intermediate steps.

A functional account of reasoning says how a mental or neurophysiological process can be a process of reasoning by virtue of the way it functions. That is, a functional account says how the functioning of such a process allows it to be correlated with the reasoning, taken to be an abstract inference, which the process instantiates.

To be more precise, the relevant correlation is a mapping F from mental or neurophysiological processes to abstract structures of inference. If x is a process in the

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domain of F, then F(x) is the (abstract) reasoning that x instantiates. Such a mapping F is a reasoning instantiator. . . . To ascribe reasoning r to someone is to presuppose the existence of a reasoning instantiator F and to claim that his belief resulted from a process x such that F(x) = r"; (Thought, pp. 48-49).

6. I have made substantive changes in the discussion of principle Q and in the statement of P* in response to comments by Ernest Sosa.