[ Preface | Part One | Part Two | Notes | Bibliography | Cover ]
As a preliminary to identifying Hume's key premise, some explanations are needed. One of them concerns the way the word "inductive" is used throughout this book. It is the more necessary to say something of this, because the word is one which Hume himself never used in print.
A paradigm of "inductive argument", as the phrase is used here, is for example the argument from "All the flames observed in the past have been hot", to "Any flames observed tomorrow will be hot too". (This example is based on one of Hume's). Another paradigm is, the argument from the same premise to "All flames whatever are hot"; another is, the argument from the same premise to "Any flames on Mars at this moment are hot".
Inductive arguments can be of many other forms than these, and of much more complex forms. But these few simple examples will suffice to indicate that "inductive argument" is used here in exactly the same way as it has been generally used by philosophers since Bacon, and in exactly the same way as (for the most part) philosophers still use it. Induction is argument, as the traditional philosophical phrase has it, "from the observed to the unobserved". In an inductive argument, the premises are simply reports of something which has been (or could have been) observed; the conclusion is a contingent proposition about what has not been (and perhaps could not be) observed. In addition, of course, what the conclusion of an inductive argument says about the unobserved is like what the premises say about the observed.
It will be evident from the preceding paragraph, and especially from its last sentence, that the established philosophical concept of inductive argument is not a very well-defined one. Nevertheless, and perhaps surprisingly, philosophers have found the concept well-defined enough for all their purposes. A sufficient proof of this is the fact that in all particular cases---that is, once the premises and conclusion of an argument have been specified---philosophers never have any difficulty in reaching agreement as to whether the argument is an inductive one or not.
There is one aspect of the established sense of "inductive" which nowadays needs to be emphasised, because a sense of the word which is opposite in this respect has grown up in the last forty years. That is that, applied to arguments, it is a purely descriptive epithet. To call an argument or a class of arguments "inductive", is not to evaluate it at all. In particular, it is no part of what is meant by calling an argument "inductive", that its conclusion does not follow from its premises. Inductive arguments are simply a certain class (though indeed, if empiricism is true, a peculiarly important class) of arguments, distinguished from other classes by the fact that their premises and conclusions are propositions which respectively satisfy certain purely descriptive conditions.
Whether all or only some of the members of this class, or none of them, are reasonable arguments; what degree of logical value, if any, such arguments have: this evaluative question can, of course, be asked about inductive arguments, as it can be asked about any others. But no answer to it, or any part of an answer, is implied in simply calling an argument "inductive". Philosophers have, of course, differed deeply in their answers to this evaluative question about induction. But before they can either agree or disagree about the logical value of a certain class of arguments, they need first to have a non-evaluative name for arguments of that class; and "inductive arguments", or "induction", is just such a name.
Now the scepticism of Hume concerning induction is one answer to the evaluative question which has just been mentioned. It is an answer to the question, what reasonableness or logical value, if any, inductive arguments possess; and it is an answer of the most negative kind. The premise of an inductive argument, Hume says, is no reason to believe the conclusion of it; a proposition about the observed is never a reason, however slight, to believe a contingent proposition about the unobserved. Hume, as I have said, does not himself call any arguments "inductive"; but the texts leave no room at all for doubt that, concerning those arguments which we call so, he embraced the thesis just mentioned. And this is his famous scepticism about induction.
Hume's philosophy "of the understanding" includes, however, very many other `scepticisms' beside this one. Two of these require mention here, because there is some danger of their being confused with his scepticism about induction, though in fact they are quite independent of it. It is the latter alone, of course, with which we are concerned.
First, Hume's scepticism about induction must not be confused with what he calls "scepticism with regard to the senses" . This is a scepticism as to whether the senses really give us any access at all to the external world, even to those parts of it closest to us in space and time. There is a certain amount of this kind of scepticism, too, in Hume's own philosophy; but it is quite different from his, or anyone's scepticism about induction. The latter is a denial of the reasonableness, assuming that the deliverances of the senses are to be believed, of believing on their account any conclusion which goes beyond them. It is, in short, a scepticism about arguments from premises of a certain kind, not about whether premises of that kind are ever available to us to begin with.
Second, Hume's scepticism about induction must not be confused with what he called "scepticism with regard to reason" . This is, indeed, a kind of scepticism which is about arguments; and there is some of it, too, in Hume's philosophy. But the grounds Hume gives for his scepticism about induction are entirely independent of those he gives for his `scepticism with regard to reason'; and the latter conclusion is no more about inductive arguments than it is about any other special class of arguments. Hume's `scepticism with regard to reason' is, in fact, a denial of the reasonableness of any kind of argument whatever: logical or illogical, valid or invalid, mathematical or theological, empirical or ethical, philosophical or scientific [...] whatever!
Later philosophers have taken little notice of the part of Hume's Treatise in which he defended this indiscriminating and (it must be admitted) uninteresting kind of scepticism; and of the little notice they have taken of it, most has been unfavorable. Hume himself was apparently willing enough, on more mature reflection, that his `scepticism with regard to reason' should be forgotten. For although he is an exceptionally repetitive writer, and published the substance of the Treatise Book I again in the Abstract, and yet again in the first Enquiry, he never anywhere once mentioned this kind of scepticism again. By contrast, his argument for scepticism about inductive arguments is nothing less than the central thing in the Treatise Book I, the Abstract, and the first Enquiry. And how unwilling later philosophers, at least 20th-century ones, have been to forget this part of Hume's philosophy, we have already seen.
Hume's argued for scepticism about induction (as has just been indicated) in three different books. They are A Treatise of Human Nature ; An Abstract [of A Treatise of Human Nature] ; and An Enquiry concerning Human Understanding . In each of these books Hume gives, for his sceptical conclusion about induction, only one argument. The argument is, however, the same each time. The three different versions of it differ only in conciseness, and in the degree to which the argument is mixed up with extraneous matter. But in these respects the three versions differ widely. In the Treatise, the relevant parts are Book I Part III Sections II-XIV. These Sections, which occupy almost a hundred pages (in the standard edition referred to in the bibliography), contain both several versions of the argument for inductive scepticism, and a great deal of other matter as well. The most concise version of the argument, and overall the best, is that given in the Abstract; where it occupies pp. 11--16 (of the standard edition referred to in the bibliography). In the Enquiry the argument is to be found, in a less concise form than that of the Abstract, but in a far more concise one than that of the Treatise, in Sections IV and V.
The account of this argument which is given below has grown out of an account which I gave in Probability and Hume's Inductive Scepticism . There is only one respect of any importance in which these two accounts of the arguments actually conflict. To this I draw attention below, when I reach the relevant part of Hume's argument. But most of the many differences in detail between the two accounts are simply by way of addition. That is, much of the detail of Hume's argument, which was either left entirely unnoticed or at best suggested by my earlier account, is made explicit here. At the same time, the present account of the argument is intended to be, and I believe is, quite self-contained. In other words, while the reader, in order to judge whether the account given here of Hume's argument is correct and complete, will need familiarity with the parts of Hume's philosophy which were referred to in the preceding paragraph, he will not need anything else.
Hume's argument for inductive scepticism is itself, however, not quite self-contained. His thesis, that the premise of an inductive argument is no reason to believe the conclusion, is not quite the end of the argument in which it occurs. The argument for this thesis is only a part, thought it is indeed by far the greater part, of Hume's argument for a sceptical conclusion which is far more general still. This is the Humean thesis which in the preceding chapter I called "scepticism about the unobserved". It says, there is no reason whatever (as distinct from merely, "no reason from experience") to believe any contingent proposition about the unobserved. It will be worthwhile to extend our account of Hume's argument so as to take in this, its very last, step; even though to do so involves some slight repetition of something which was said in Section 3 of Chapter III above.
The best place to begin is at the end of Hume's argument.
The conclusion of the whole is a general sceptical thesis about whatever has not been observed: that there is no reason (from any source) to believe any contingent proposition about the unobserved. (Call this proposition A). Here it is in some of Hume's own words: "[...] we have no reason to draw any inference concerning any object beyond those of which we have had experience [...]" .
There is no difficulty (as has already been indicated) in determining what Hume's immediate grounds are for this conclusion. For there are two propositions which he constantly asserts and clearly intends to be taken together, and which, when they are taken together, immediately and obviously entail A.
One of these grounds is empiricism: the thesis that the only reason to believe a contingent proposition about the unobserved is a proposition about what has been observed. (Call this B). In some of Hume's own words: "[...] All the laws of nature, and all the operations of bodies without exception, are known only by experience [...]" .
The other immediate ground of A is Hume's inductive scepticism: the thesis that even propositions about the observed are not a reason to believe any contingent proposition about the unobserved. (Call this C). In some of Hume's own words: "[...] we have no argument to convince us, that objects, which have, in our experience, been frequently conjoined, will likewise, in other instances, be conjoined in the same manner [...]" .
The structure of this last step of Hume's argument is as easily identified as the elements of it. It was as represented in the following diagram.
C } B } -> A
Our object being only to identify Hume's argument, not to evaluate it, the arrows in my `structure-diagrams' are to be understood in a descriptive sense only, not in any evaluative one. Thus "X -> Y", for example, would mean here, not that an argument from X to Y is valid, or that X is a reason to believe Y, or anything of that sort. It would mean that Hume in fact gave X as a reason to believe Y, and it would mean nothing else.
At the same time, it is quite obvious that Hume intended his argument to be a valid one, and thought that it was. I too believe that the argument he intended is in fact valid. Of course Hume sometimes left unexpressed certain premises which are necessary to make his arguments valid, as every arguer must do if he is not to be tedious. But when he does, it is almost always easy, for a reader familiar with his philosophy "of the understanding", to supply the additional premise which Hume intended, and which is needed to make his argument valid. There is in fact only a single step in the entire argument of which this is not true. I intend to proceed, therefore, by assuming at each step that the argument is valid, and attributing to Hume the additional premise necessary and sufficient premise to make it so. Hence an arrow in my structure-diagrams, although it will not signify a valid step, will always represent a step which was (I believe) made by Hume, and which is (I believe) valid as well.
It will be obvious that Hume's scepticism about the unobserved (A) does follow from the conjunction of empiricism (B) with inductive scepticism (C). It will be equally obvious that A does not follow from B alone; even though many philosophers have thought, to the contrary, that Hume's scepticism about the unobserved is an inevitable consequence simply of his empiricism. It is not so obvious, but it is true and of some importance, that A does not follow from inductive scepticism C alone, either.
If you want to reach a certain place, it is no fatal news to be informed that the route via X will not get you there. This will be fatal news if and only if it is conjoined with the information that no route other than the one via X will get you there either. Just so, if you want to reach knowledge or reasonable belief about the unobserved, it is no fatal news to be told that the inductive route (the route via the observed), will not get you there. Yet that is all that inductive scepticism C says. This will be fatal news if and only if it is conjoined with the information that no route other than the inductive one will get you there. Just that, however, is what empiricism B asserts. Hence scepticism about induction will not commit you to scepticism about the unobserved, unless you also subscribe to empiricism. Someone who held that there are non-inductive routes to knowledge or reasonable belief about the unobserved---routes via pure reason, say, or revelation---could with perfect consistency admit C and yet deny A: that is, be a sceptic about induction without being at all sceptical about the unobserved.
Neither B nor C was a premise of Hume's argument. Inductive scepticism C is, of course, so irrationalist a thesis that it could hardly be a starting-point of any argument advanced by a sane person (at any rate before about 1950); certainly Hume had to argue for it. But neither was empiricism B a starting-point of Hume's argument. For it, too, he argues. Hume's argument for B was sometimes perfunctory, it is true, as well as being usually short and elliptical. The historical reason for this is obvious: empiricism was a commonplace with Hume and with his readers. Hence B, quite unlike inductive scepticism C, was something which require little defense. Still, Hume does have an argument for empiricism. What was it?
Hume's main ground for empiricism, in the sense that it is the ground which he usually gives as though it were a sufficient one, is this: that propositions which are necessarily true are not a reason to believe any contingent propositions. (Call this D). Unlike necessary truths, "matters of fact are not ascertained", Hume says, "by the mere operation of thought" , by "demonstrative arguments" or "abstract reasoning a priori". (When Hume speaks of "demonstrative arguments", he does not mean, as we might mean, just valid arguments; he means, valid arguments from necessarily true premises ).
It may appear from this that Hume begged the question in favor of empiricism. For the ground D just mentioned may seem scarcely indistinguishable from the empiricism B for which it is supposed to be a ground. Well, it would not have been surprising, nor would it have mattered much at the time, if Hume's argument had been question-begging here, the reason being historical circumstance mentioned a moment ago: that with Hume's contemporary readers empiricism was virtually a datum anyway. And since, as it happens, empiricism is virtually a datum with most of Hume's readers now, too, it would not matter much now, either, if his argument here had been circular. In fact, however, it is not.
Empiricism B says that if there is any reason to believe a contingent proposition about the unobserved, it is a proposition about the observed. Hume gives, as though it were sufficient to establish this, the ground D, that necessary truths are no reason to believe a contingent proposition about the unobserved (or any other contingent proposition). Now this is just like some one saying that the murder, if it was murder, was committed by the gardener, and giving, as though it were sufficient reason to establish this, the ground that at any rate the butler did not do it. Such a person is clearly assuming that the murderer, if there is one, is either the gardener or the butler. Equally clearly, Hume is assuming that if anything is a reason to believe a contingent proposition about the unobserved, it is either a necessary truth or a proposition about the observed.
This assumption, or rather the even stronger one, that any reason to believe any proposition is either a necessary truth or a proposition about the observed, is one which, once it is stated, will be acknowledged by every student of Hume to have been absolutely central to his thought. No account of his philosophy of the understanding can possibly be adequate if it does not make this assumption explicit and prominent. Without it, for example, it is quite impossible to explain Hume's special affinity with the empiricists of the present century: an affinity which (as was implied in Chapter III above), is no less obvious than it is deep. And the deficiencies of my own earlier-published account of the present argument, I may observe, stem almost entirely from my having failed to make explicit the part played in the argument by this assumption.
The assumption has two parts, and it is helpful to separate them. Consider the class, at first sight the oddly disjunctive class, of propositions which are either necessary truths or propositions about the observed. What is common and peculiar to the members of this class? Or rather, what did Hume think is common and peculiar to them, and what gives them the special status that they enjoy in his philosophy? These statements are not hard to answer. Hume thinks of necessary truths and propositions about the observed as being propositions, and the only propositions, which can be known or reasonably believed, without having to be inferred from other propositions known or reasonably believed: as being propositions, and the only propositions, which are (as we may say) directly accessible to knowledge or reasonable belief. This is one half of the assumption. (Call it E).
The other is the very natural assumption, about one proposition's being a reason to believe another, which is almost inevitably expressed (and is expressed by Hume) by means of a comparison with a chain or a ladder. P's being a reason to believe Q is like a ladder which reaches, whether by few rungs or many, from P to Q; and Hume's assumption is the exceedingly plausible one that such a ladder, no matter how safe and climbable it may be, will be no help at all to us for reaching Q, if we cannot reach P. In order, then, for P to be a reason, however remote or indirect, to believe Q, P must be directly accessible to knowledge or reasonable belief. Otherwise, as Hume says, "all our reasonings would be merely hypothetical; and however the particular links might be connected with one another, the whole chain of inferences would have nothing to support it [...]" . (The same condition is necessary, evidently, in order for P to be a member of a conjunction, P-and-R, which is a reason to believe Q). (Call this second part of the assumption F).
It is easy now to understand why Hume regularly proceeds as though
D: No necessary truth is a reason to believe any contingent proposition,
is sufficient to establish
B: Any reason to believe a contingent proposition about the unobserved is a proposition about the observed.
It is because he was assuming both
F: If P is a reason or part of a reason to believe Q, then P is directly accessible to knowledge or reasonable belief,
E: A proposition is directly accessible to knowledge or reasonable belief if and only if it is either a necessary truth or a proposition about the observed.
The structure, then, of Hume's argument for empiricism, was:
D } E } -> B F }
I have implied that Hume's argument for empiricism B comes into his argument for scepticism about the unobserved A, only near the very end. So it does, logically speaking, since what it supplies is one of the immediate grounds of the ultimate conclusion. In the actual order of Hume's presentation, however, the opposite is true. He always completes the argument for empiricism B first, before he even begins the argument for inductive scepticism C. Moreover, when he does complete the latter, its conclusion is always a deliberate echo of a premise of the earlier argument for empiricism. C, the thesis that even after experience we have no reason to believe anything about the unobserved, echoes D, the thesis that we have no such reason before experience, or a priori. And Hume had good literary and historical motives for adopting this order of presentation of the parts of his argument, and in particular for adopting this echo-device.
Everyone dislikes a sudden loud noise, but it is worse still if you are half-asleep at the time. Now D, the thesis that we have no reason prior to experience to believe anything about the unobserved, is a proposition which I have elsewhere called "Bacon's bell" , in reference to Bacon's famous boast: that he had "rung the bell that called the wits together", by insisting that all contingent propositions be subjected to the test of experience and to no other. But of course by the time Hume wrote, this empiricist maxim D, once so revolutionary, had become almost as much a part of the British constitution as a church by law established, and almost as soporific. So Hume, by sounding Bacon's bell early in his argument, as he always does, artfully creates in his readers a sense of security. Its familiar note assures them that this author is a decent British empiricist, a Bacon-and-Newton man like the rest of us: he will not disturb our Royal Society slumbers. How much the more appalling, then, when at the end of his argument he sounds what I have called "Hume's bell", with its ghastly parody of this familiar note: the thesis of inductive scepticism C, that we have no reason for any beliefs about the unobserved, after experience either!
To the main part of Hume's argument, his argument for this staggering conclusion, we now turn. Here most of his premises are easily identifiable, and it is best to go straight to them.
One premise is, that the conclusion of an inductive argument does not follow from its premise, except in the presence of an additional premise, or assumption, that the unobserved is like the observed. In some of Hume's own words: "All inferences from experience suppose, as their foundation, that the future will resemble the past" . Again: "All probable arguments are built on the supposition that there is [...] conformity between the future and the past [...]" . Yet again: "[...] probability is founded on the presumption of a resemblance, between those objects, of which we have had experience, and those, of which we have had none [...]" .
Let us be sure we understand just what Hume is saying in these passages. "Inferences from experience", "probable arguments", and "probability", are simply some of the many names which Hume uses for what we call inductive arguments: those arguments from the observed to the unobserved, of which the argument from "All the many flames observed in the past have been hot", to "Any flames observed tomorrow will be hot", may serve as a paradigm. And what Hume is pointing out is simply that this argument, for example, is invalid as it stands, (the conclusion does not follow from the premise), and that in order to turn it into a valid argument, you would need to add to it a premise which asserts at least that tomorrow's flames resemble the past observed ones.
Let us call a proposition which asserts that there is a resemblance between the observed and the unobserved, a "Resemblance Thesis". Then this first premise of Hume's argument for inductive scepticism is:
G: Any inductive argument is invalid, and the weakest addition to its premises sufficient to turn it into a valid argument is a Resemblance Thesis.
Hume's next premise is also easily identified. It is a proposition about the nature of the Resemblance Thesis; namely
H: A Resemblance Thesis is a contingent proposition about the unobserved.
In some of Hume's own words: "[...] that there is this conformity between the future and the past, [...] is a matter of fact [...]" .
Now Hume concludes, from this characterization of Resemblance Theses, something about the nature of possible evidence for them; namely
I: A Resemblance Thesis is not deducible from necessary truths.
In some of his own words: a Resemblance Thesis "can never be proved [...] by any demonstrative argument or abstract reasoning a priori" .
It will be obvious that I does not follow from H alone, but equally obvious what Hume's tacit premise was here. It is the maxim, which he is never tired of repeating, that "there can be no demonstrative arguments for a matter of fact and existence". That is, it was
J: No contingent proposition is deducible from necessary truths.
This part of Hume's argument for C had, then, the structure:
H } J } -> I
Hume next considers the possibility of observational proof of a Resemblance Thesis. But the result, he finds, is as negative as in the case of a priori proof. That is, he concludes, in an obvious parallel to J,
K: A Resemblance Thesis is not deducible from propositions about the observed.
On what grounds? Well, recall H: that a Resemblance Thesis is a contingent proposition about the unobserved. Any argument to a Resemblance Thesis from the observed will thus be an inductive argument, and, in view of G, therefore, it will be invalid unless to the observational premises is added a Resemblance Thesis. That, however, is the very proposition which we are trying to prove! Any argument from experience for a Resemblance Thesis, therefore, will be invalid unless it is circular. Or in some of Hume's words, "To endeavor [...] the proof of [a Resemblance Thesis] by probable arguments, or arguments regarding existence, must evidently be going in a circle, and taking that for granted, which is the very point in question" .
Hume's grounds for K, then, are H and G. For they together entail
L: A Resemblance Thesis is deducible from propositions about the observed, only when to the latter is conjoined a Resemblance Thesis;
and L in turn (remembering that we already have H as a premise), is Hume's warrant for concluding that K.
Here, then, the structure of the argument was:
H } G } -> L -> K
So far, then, Hume's argument for C has been as represented in the diagram:
H } J } ------> I H } -> L -> K G }
The only premises have been G, H, and J.
But now, what is represented above is, as far as I can discover, the whole of Hume's explicit argument for inductive scepticism C.
This assertion may be found surprising. Proof of it would certainly be desirable. Unfortunately, however, it is impossible. An old logical saw says that one cannot prove a negative, and certainly one cannot prove an exegetical negative such as this. I am obliged, therefore, to rely entirely here on what elsewhere Hume called "the method of challenge", and to invite anyone who thinks there is something explicit in Hume's argument for C, which is omitted in the above account of it, to point it out: either another premise, or a result drawn by Hume from premises already presented here.
If, as I believe, this cannot be done, then it must be admitted that Hume's argument, while it is admirably explicit as far as it goes, stopped a good deal short of the conclusion C which it was intended to prove. For C says that the premise of an inductive argument is not a reason to believe its conclusion; yet so far we have not got anything like that. In the premises G, H, and J (the only ones so far), there is nothing whatsoever, for example, about what is required for one proposition to be a reason to believe another.
Well, what have we got? Or rather, since it is inductive arguments and no others that are the subject of C, we should ask how much, at the best, has so far been established about inductive arguments? The answer is plain. At the best (that is, assuming Hume's premises true as well as his steps valid), the most that follows from Hume's premises, about inductive arguments, is the conjunction of G, I, and K. All that this conjunction says is this: that any inductive argument is invalid, and that the weakest additional premise sufficient to turn it into a valid argument is a proposition which is not deducible either from necessary truths or from propositions about the observed.
We need to make this result less unwieldy. First, let us call any additional premise which is sufficient to turn a given invalid argument into a valid one, "a validator" of it; and let us call the weakest of all the validators of a given argument, the validator of it. Second, let us make use of the fact that, necessarily, a proposition is deducible from a necessary truth or a proposition about the observed, if and only if it is itself a necessary truth or a proposition about the observed. Because of this, instead of saying that the validator of an inductive argument is not deducible from necessary truths or observation-statements, we can say, without losing logical equivalence, that that validator is itself neither a necessary truth nor an observation-statement. With the aid of these two abbreviations, what the conjunction of G, I, and K says about inductive arguments can be expressed as
M: Any inductive argument is invalid, and the validator of it is neither a necessary truth nor a proposition about the observed.
Obviously, M does not entail inductive scepticism C. Indeed, since it is only an abbreviated logical equivalent of what G, I, and K say about inductive arguments, M cannot bring us any closer to C than the conjunction of G, I, and K did; which is, as I said, not very close. Many philosophers nowadays would go much further than this, and say that while C, though false, is at least important, M, though true, is unimportant, because its truth is obvious.
I wish that some of these philosophers would tell us an important and true result that does follow from the premises of Hume's argument: this famous argument which all of us (and not only those who accept its sceptical conclusion C) admire so much. In fact, however, the result M is not only true and original, but is of profound importance. Indeed it is Hume's central insight concerning induction, and is what separates his philosophy of induction, and the best of ours, from the slipshod philosophy of Bacon before and of Mill after him, and of most empiricists even now. Not only is M important in itself. In conjunction with some of the premises of Hume's empiricism (and ours), it entails, as will be shown later, a further result which is still more important, and one which most empiricists even now are far from having fully absorbed.
These are large claims to be made for the not-very-pregnant-looking result M; but I think I can establish them. To do so, however, it is necessary to take a step back from Hume's argument for a while.
All philosophers and logicians are interested in evaluating arguments. The evaluation of arguments is a complex matter, requiring many different distinctions to be made. For example, in some arguments the premises cannot be a reason to believe the conclusion, while it other arguments they are, and hence can be. Then, where the premises are a reason to believe the conclusion, there is the distinction between arguments in which the premises are an absolutely conclusive reason to believe the conclusion, and those in which they are not; that is, between valid and invalid arguments from P to Q, where P is a reason to believe Q. Then there is the distinction, entirely independent of the two just mentioned, between arguments in which the premises are all true, and those in which they are not. And so on. In short, two arguments can differ in value along a number of different, and even independent, dimensions.
The ordinary or `deductive' logician, however, is interested, ex officio at least, in only one dimension of the value of arguments: namely in the distinction between validity and invalidity. Most philosophers, on the other hand, regard the distinction between valid and invalid arguments as a silly thing to have as an exclusive object of interest. They are right. For one distinction which the evaluation of arguments requires to be made is, as I have said, that between arguments of which the premises are or could be a reason to believe the conclusion, and arguments in which they cannot; while that distinction is largely, if not entirely, independent of the distinction between the valid and the invalid. At any rate, it is certainly not enough to make P a reason to believe Q, that the argument from P to Q is valid.
If it were enough then no one, however irrational, need ever lack a reason, and even an absolutely conclusion reason, to believe any and every proposition whatever. For you can always turn an invalid argument into a valid one, merely by making a suitable addition to the premises. Let your argument from P to Q be invalid; let it even be as atrocious as a piece of reasoning can be; still, you can always turn it into a valid argument, by the trifling expedient of adding the premise that P is false or Q is true, or some other premise which entails that one. Nothing could be easier. And if the conclusions following from the premises were enough to make those premises a reason to believe it, then nothing could be more important than this stratagem, since it would enable us all to ensure that whatever we believe, we believe reasonably.
In fact, of course, as is obvious, nothing could be more trivial. That the premises of an argument entail the conclusion is not enough to make them a reason to believe it. And if the premises of an argument are to succeed in being a reason to believe the conclusion, not every validator R of the argument from P to Q is available to every arguer as an additional premise. Such an R, to be available to an arguer as an additional premise, must at least be such that it can be part of a reason to believe Q.
Very often, of course, such a validator is available to an arguer. My companion may disagree with my identification of a bird which we are both looking at, and argue "The bird on that post is no raven, since all ravens are black"; omitting, just in order not to be tedious, the premise, which we have both just learnt from experience, that the bird on the post is not black. This proposition, which is of course a validator, in fact the validator, of his argument, is available to him as an additional premise, presumably. At any rate it certainly satisfies a necessary condition of such availability; that of being a proposition which can be part of a reason to believe his conclusion.
But it is not so in every case, that is, for every invalid argument; and there are some validators which are never in any case available to arguers as an additional premise. If I am to succeed in giving a reason to believe a contingent proposition Q, but my argument to Q from P is invalid, I cannot add a premise R which is, for example, self-contradictory. A self-contradictory additional premise is indeed a validator of every invalid argument. But such a validator is not available to any arguer to Q, because a self-contradiction cannot be part of a reason to believe a contingent proposition. Again, if I aim to give a reason to believe Q, but my argument to Q from P is invalid, I may not add as a premise the very proposition Q which I am trying to give a reason to believe. The conclusion of any invalid argument is indeed a validator of it, but Q is not available to me or any arguer to Q as an additional premise, because Q cannot be part of a reason to believe Q.
Given an argument from P to Q which is invalid, then, a validator of it, R, may be available as an additional premise to an arguer whose object is to give a reason to believe Q. For R may be, in addition to whatever else is required for availability, a proposition that can be part of a reason to believe Q. In such a case, for example the argument about the bird on the post, the invalidity of the original argument is an unimportant defect of it, because a cure for the defect is available to the arguer. But not every validator R of a given invalid argument is available as an additional premise to every arguer whose object is to give a reason to believe Q. For a validator R may be a proposition which cannot be part of a reason to believe Q; or it may be a proposition which is unavailable on some other ground.
Hence for philosophers, who must distinguish, as deductive-logicians need not, between arguments in which the premises are or at least could be a reason to believe the conclusion, and arguments in which they cannot, an important general question arises. In just what cases is a cure for the invalidity of our arguments available to us, consistently with our premises remaining a reason to believe our conclusion? What propositions are, and what are not, available validators of our invalid arguments?
This question, in its general form, is not one which Hume ever explicitly considered. Still, we know well enough what his answer to it was, even in its general form. His answer to it is given by his premises E and F above. But Hume did of course consider, and most explicitly, the special case of this general question in which the arguments from P to Q are inductive ones. That is, he did consider the question whether, when we argue from the observed P to the unobserved Q, any validator R is available to us. Indeed Hume never considered any question, concerning the evaluation of induction, except this one. His answer to it is, of course, that no validator is available for inductive arguments. His argument for part of this answer is that which I have set out above.
What Hume did was to consider two classes of candidates for the position of available validators of induction. The first class consists of necessary truths. These were obvious candidates for consideration. Propositions which cannot be false are, presumably, always available as additional premises, to any arguer. At any rate they certainly satisfy the necessary condition of availability, that they can be part of a reason to believe the conclusion of the arguments now under discussion. When the argument from P to Q is inductive, a necessary truth R can be part of a reason, P-and-R, to believe Q.
Alas, where the argument from P and Q is inductive, a necessarily-true additional premise R, although available, will never satisfy the other requirement of the position we are seeking to fill; it will not be a validator. The conclusion of any inductive argument is a contingent proposition. Where R is a necessary truth, the conjunction P-and-R is logically equivalent just to P itself. And two arguments with the same contingent conclusion, and logically equivalent premises, cannot differ in value along any dimension (except perhaps an economic or an aesthetic one). At any rate they cannot differ in that one of them is valid and the other invalid. So where R is a necessary truth, an argument from P-and-R to contingent Q would be valid only if the argument to Q from P alone were valid to begin with; which, in the case of inductive arguments (as Hume's premise G says), it is not. Trying to turn inductive arguments into valid ones by adding necessarily true premises, is like trying to increase a boat's displacement by taking on weightless ballast.
Hume then considers a second class of candidates for the position of available validators of induction: propositions about the observed. These, too, were natural candidates for consideration. When we argue from the observed P to the unobserved Q, another proposition R about the observed is, presumably, very often available to us. Certainly such an R can be part of a reason, P-and-R, to believe a conclusion Q about the unobserved.
But alas, these candidates too are unequal to the task of turning inductive arguments into valid ones. By adding a proposition R about the observed, to the original premise P about the observed, the best you can get, that is at the same time a proposition which can be a reason to believe Q, is just another proposition about the observed; a stronger one, indeed, than that with which you began, but still a proposition about the observed. But the conclusion Q is still a proposition about the unobserved. So, even with the premise P-and-R, our argument to Q is an inductive one still. And all inductive arguments (as Hume's premise G says) are invalid. As far as turning inductive arguments into valid ones goes, therefore, propositions about the observed behave, as additional premises, in exactly the same way as necessary truths. At least this much is true, then, in the famous sceptical passage in which Hume writes: "Now where is that reasoning, which, from one instance, draws a conclusion, so different from that which it infers from a hundred instances, that are nowise different from that single one? [...] I cannot find, I cannot imagine any such reasoning" .
No proposition, then, which is either a necessary truth or a proposition about the observed, is sufficient as an additional premise to turn an inductive argument into a valid one. A fortiriori no necessary truth or proposition about the observed is the weakest of all the validators (that is, is the validator) of any inductive argument. That is, Hume's result M is true.
(It is, I hope, unnecessary to say that the argument just given for M was simply a modernized version of the argument, set out above, which Hume himself gave for it; and little enough modernized at that).
I need not contend here for originality of this result M. It is, in fact, as original as anything in philosophy ever is. What I do need to contend for is its importance. For, as I indicated earlier, many philosophers nowadays suppose that the truth of M is obvious, and even that it always was so, at least to philosophers. Some go as far as to suggest that M is an analytic truth of common English: that what it says about inductive arguments is as trivial, and as well-known to normal English-speakers, as what "A father is a male parent" says about fathers. These suppositions are not only false, but grotesque, and the exact opposite of the truth. The simplest way to prove this is to show that we have ample testimony, from authorities too numerous, recent, high, and even in a sense irresistible, to the falsity of M.
In the first place everyone, in his bones, nerves, and muscles, believes that M is false. Strike a match and look at the flame. Then try not to believe that you would feel heat if you held you hand an inch over it. You cannot do it. You cannot even be less confident, about this future thermal phenomenon, than you are about the present visual phenomenon of the flame. This is an example, of course, of Hume's favorite kind of inductive inference, and the kind in relation to which his entire argument for scepticism was in fact conducted: what he calls "the inference from an impression to an (associated) idea" , after we have had "a long course of uniform experience"  of the conjunction of the two properties, such as being a flame and being hot. The corresponding inference before experience, or (as Hume likes to say) in Adam's situation, is course the inference just from "This is a flame" to "This will be hot". Now no one, as Hume is always saying, takes that premise as a reason to believe that conclusion, and still less would anyone mistake it for an absolutely conclusive reason to believe it. But then, as Hume is also always saying, once experience has supplied us with the additional premise that all the many flames observed so far have been hot, we do draw the conclusion that a flame as yet untested will likewise be hot; and draw it, with a degree of confidence which is introspectively indistinguishable from that with which we conclude, given that all men are mortal and Socrates is a man, that Socrates is mortal. In other words, we all do believe that, contrary to what M says, the observation-statement about past flames is sufficient, as an additional premise, to turn the argument from "This is a flame" to "This will be hot" into a valid one. At least, our bones, nerves and muscles believe so.
These authorities against M will be admitted to be numerous and recent, and even in a sense irresistible. But they may be thought to be rather low. So let us turn to Bacon and Mill, who are sufficiently high authorities on induction. And let us ask what they would have thought of the following inductive argument. "The canary was alive and well when we left the room an hour ago; but it is dead now. Gas from the oven was leaking into the room during that time. So, if nothing else caused the canary's death, the gas did".
That is, of course, a homely example of the very kind of inductive argument with which Bacon and Mill were especially occupied: "eliminative induction", as Mill aptly called it. This argument is invalid, just as Hume's premise G requires. The validator of it is the proposition that something caused the canary's death. This proposition is indeed, just as Hume's result M requires, neither a necessary truth nor a proposition about the observed. But the question is, was this fact obvious to Bacon and Mill? It would take a very bold man, or a very ignorant one, to say so.
From what Bacon wrote about inductive arguments of essentially this kind, it seems never to have crossed his mind what kind of proposition its validator might be: for the simple reason that he seems to have thought such an argument valid as it stands. Mill at least knew better than that, and accordingly he tried for a while, in Book III of his Logic, to show that the validator of such an argument is, after all, known from experience; or rather (with his characteristic rigor) that anyway it nearly is; or that it is known from experience, at any rate with "all the assurance we require for the guidance of our conduct"  (`Conduct'?!) But this apparent modesty Mill was unable to sustain: his real confidence in what he called the Law of Universal Causation was too deep. To suppose that the deterministic assumption (that the canary's death had a cause) was not available to inductive reasoners, in 1843, evidently seemed to Mill merely a solemn farce, and he could not keep it up. So in the end he simply throws up in impatience the question of the validity, or the curable invalidity, of eliminative induction. By doing so he seems, to 20th-century philosophers, as he would have seemed to Hume, to have left his philosophy of science in ruins. In his own century, however, there were very few who were of that opinion.
But leave even these mighty dead out of it. Consider the argument about the canary, and let us ask ourselves this. What experimental scientist, now, would have any more patience than Mill had, with someone who tormented him with reminders that the additional premise, which this argument needs to be valid, is not known to be true either a priori or from experience? Come to that, how many experimental scientists would be conscious, any more than Bacon was, that the argument is not valid as it stands? These questions answer themselves.
So very wide of the truth, then, is the belief that M is a truth which has always been obvious, at least to philosophers. As for the suggestion which some philosophers have made in recent decades, that M is known to every competent English-speaker, like "A father is a male parent" [...] I blush for my profession! Quite the contrary to all this, to bring to light the truth M about inductive arguments, required the peculiarly fixed, strong, and passionless gaze by which Hume was distinguished in mind as he was in body. After Hume, of course, you do not need to be a genius to know that M is true; but that is a little different.
It is not only dead philosophers or living scientists, however, who have not fully taken in the truth of M, or have not perceived the full extent of its consequences for empiricist philosophy of science. The same is true of most empiricist philosophers now.
The grounds which Hume explicitly gave for C amounted; as we saw, only to M. Yet C is his shocking conclusion about induction; while M is so far from being shocking, at least to philosophers now, that the difficulty with it is rather, as we have seen, to secure recognition of its importance. Hume proceeded, in other words, as though his premises yield a result which is even stronger and more important than M.
They do, too. What this result is, will become clear if we ask ourselves the following natural question. Why did Hume consider, as candidates for the position of available validators of induction, necessary truths, observation-statements, and no others? The answer is obvious. To be available to inductive reasoners, a validator of their arguments must at least be such that it can be part of a reason to believe the conclusions of induction; and Hume thinks that only necessary truths and observation-statements can be part of a reason to believe anything. In other words, Hume was here drawing again on two of the premises of his earlier arguments for empiricism. He was taking M not on its own, but in conjunction with E and F. And when that it done, a result which is even more important than M does follow.
It follows at once from M, E, and F, that the validator R of an inductive argument from P to Q is not a reason or part of a reason to believe the conclusion Q. And necessarily, if even the weakest validator R of an argument from P to Q is not a reason or part of a reason to believe Q, then a fortiriori any stronger validator of the argument cannot be a reason or part of a reason to believe Q either. For any stronger validator than R will be logically equivalent to R-and-S for some S, and if R cannot be even part of a reason to believe Q, then evidently no proposition logically equivalent to R-and-S can be so either. That is, no validator of an inductive argument can be even part of a reason to believe the conclusion of that argument. Taken with E and F, then, M entails:
N: An inductive argument is invalid, and any validator of it is not a reason or part of a reason to believe its conclusion.
This is an enormously important result of Hume's argument. I call it the thesis of the incurable invalidity of induction. Some invalid arguments, we have seen (for example, the one about the bird on the post) are only curably invalid; a validator of them is available to the arguer, at least in the sense that such a validator can be part of a reason to believe the conclusion of the argument. What N says is that inductive arguments are not like that; for their invalidity, no cure is available. Any additional premise, if it is sufficient to make the conclusion of an inductive argument logically follow, is not a reason or part of a reason to believe the conclusion. In other words, the fallibility or invalidity of inductive arguments (the possibility of their having a false conclusion even though their premises be true) is a feature absolutely inseparable from them.
Whereas many philosophers now need to be reminded of the importance of M, the importance of the present result N is obvious to them all. Indeed many philosophers, beginning with Hume himself, believe that, or at least proceed as though, N is so devastating a result about induction that the sceptical conclusion C follows from it at once. If induction really is, as N says it is, not only invalid but incurably so, does it not follow that induction is unreasonable, as C says it is?
Good philosophers have a very exacting standard of what constitutes a reasonable argument; and other things being equal, one philosopher is better than another, the more exacting his standard of reasonable argument is. The highest possible such standard would say, that the premise of an argument is not a reason to believe the conclusion, unless the argument is actually valid; and it is, accordingly to such a standard as this that all good philosophers more or less incline. They have the deepest reluctance, consequently, to admit that an argument can be a reasonable one, if it is not only invalid, but cannot be turned into a valid argument by any additional premise that can form part of a reason to believe its conclusions. Hence the admission of N, that inductive arguments are all in this position, is bound to impose at least some strain on any good philosopher's belief in the reasonableness of induction. It is natural, therefore, for a good philosopher to think that C follows from N. He will even, other things being equal move from N to C the more easily, the better philosopher he is. There is nothing at all surprising, then, but quite the reverse, in Hume and many other philosophers having proceeded as though N entails C.
Nevertheless, some other philosophers (of whom I am one) resist this step from N to C. We have a less exacting standard of reasonable argument than most philosophers incline to. We say that an argument can be invalid, and even incurably so, but still its premise can be a reason to believe its conclusion. It is so, we say, with some inductive arguments in particular. Hume's result N we accept, and we admire it, as a profound truth about induction which his argument brought to light. But the sceptical conclusion C which Hume drew from N does not follow, we say, and is false.
This kind of philosopher, the `inductive probabilist' as he may be called, does not think of the invalidity of inductive arguments as a mere surface blemish of them. He knows better than that, for he has taken Hume's result N to heart, and therefore holds that the invalidity of induction is incurable. Still less will he join those philosophers who search for a validator of induction. Invalidity which cannot be cured, he considers, had better be endured. On the other hand he still maintains that some inductive arguments are reasonable, in the sense in which C says that none are; that is, he maintains that their premises are a reason to believe their conclusions. Of course he does not regard the reasonableness of those arguments as an extrinsic feature of them: as consisting in the fact that some other argument, which has the same conclusion and augmented premises, is actually valid. On the contrary, he regards the reasonableness of those inductive arguments which are reasonable as an intrinsic logical feature of them; just as, for example, their invalidity is. So while he admits that inductive arguments have an incurable infirmity, in that it is possible for their premises to be true and conclusion false, he does not obsessively concentrate on this logical feature of them to the exclusion of every other .
But most philosophers, it must be admitted, consider the inductive probabilist's position a feeble evasion, and one impossible to maintain. Many suspect that the inductive probabilist, despite the lip-service he pays to Hume, has never really taken in the full force of his argument. Some even suspect that he is engaged, most embarrassingly, in defending a position about induction Hume himself had already shown, in the course of the very argument we are discussing, to be indefensible.
`Consider' (these critics say) `an inductive argument, for example that from P, "All the many flames observed in the past have been hot", to Q "Tomorrow's flames will be hot". There is no connection whatever between the premise and the conclusion. P and Q are propositions entirely logically independent of one another. Nevertheless, you tell us, P is a reason to believe Q. Now, is it not obvious that, if this is so, it is because there is some connection between past flames and tomorrow's flames, or between being a flame and being hot, or between the observed and the unobserved? To make P a reason to believe Q, there must be some ground in nature, some fact about the cosmos, some "cement in the universe" (in Hume's phrase) which, taken along with P, logically connects that premise with the conclusion Q; that is, turns the original inductive argument into a valid one. Yet you reject Hume's C while you accept his N. There is no proposition, you say, which is at once part of a reason to believe Q, and sufficient to make the argument from P to Q valid. If so, then a fortiriori there is no true proposition of that kind. And what is this but to say that there is in the nature of things no ground for inferring Q from P, or that P is not a reason to believe Q? Forbear these evasions, then, and admit at any rate the truth of Hume's conditional, that if N is true, C is: that if induction is really incurably invalid, then it is unreasonable. Or, if you persist in affirming N and denying C, at least tell us what it is, according to you, which makes it true that (for example) P is a reason to believe Q. It cannot be, that the argument from P to Q is valid, or only curably invalid; for you accept N, and insist that the argument is not so. What is it, then, that makes P a reason to believe Q? If you tell me that it is just an ultimate fact of inductive logic, or of the theory of probability, that P is a reason to believe Q, then I will know what to think of your so-called inductive logic: that it is simply speculative metaphysics in disguise'.
Criticism of this kind has often been thought to be fatal to inductive probabilism. Its plausibility must have been felt, at least at times, even by the inductive probabilist himself. It can fairly be summed up thus: "If the universe were not connected or cemented in some way (that is, if there were no true validator of at least some inductive arguments), then Hume's scepticism about induction would be true".
Some of these critics of inductive probabilism have an anti-sceptical intent. They intend to go on to say that, since Hume's inductive scepticism C is plainly false, the universe must in fact be cemented in some way. (As to the nature of the cement, they may and do differ. Some of them say it is causation; others that it is a certain connection that exists between properties; others again that it is the providence of God; etc.). A second group of the critics of inductive probabilism have a sceptical intent. They mean to go on to say that, since the universe is in fact not cemented or connected, Hume's inductive scepticism C is true. Of this second group, a recent example is Popper, in The Logic of Scientific Discovery . Of the first, a recent example is D.M.Armstrong .
For our purposes, however, the difference between these two groups of critics does not matter. What matters is what they agree on. For this can be shown to be a complete mistake.
What the critics of inductive probabilism unite in believing is, that Hume's inductive scepticism would be true if the universe is not cemented or connected in some way; or what is equivalent, that his inductive scepticism would be false only if the universe were cemented or connected. That the universe is cemented or connected, whatever exactly it means, is a proposition which is, as an additional premise, sufficient to turn at least some inductive arguments into valid ones. Inductive arguments, however, as well as being invalid, all have contingent conclusions; and any additional premise, which is sufficient to turn an invalid argument with a contingent conclusion into a valid one, must be contingent itself. It is therefore a contingent proposition that the universe is cemented or connected. Since the negation of any contingent proposition is itself contingent, it is also a contingent proposition that the universe is not cemented or connected. Our critics therefore all imply that Hume's scepticism about induction would be true if a certain contingent proposition ("no cement") is true, false only if a certain contingent proposition ("cement") is true. But a proposition which is true on a certain contingent condition and false otherwise, is contingent itself. All these critics therefore imply that Hume's inductive scepticism is a contingent proposition.
But this is so extreme a misconception of the nature of Hume's C that no one, I believe, will venture to defend it, once it is thus explicitly stated.
On its very face it is most implausible. Recall the proposition C, or any of Hume's own words for it: for example, "even after the observation of the frequent or constant conjunction of objects, we have no reason to draw any inference concerning any object beyond those of which we have had experience" . This certainly does not appear to be a contingent claim about the overall character of the actual universe. It appears to be, rather, a logical thesis of some kind: a proposition about whether certain propositions are or are not a reason to believe certain other propositions.
If C is contingent, then Hume's argument for it must either have been invalid, or have had at least one contingent premise. For remember, `there can be no demonstrative arguments for a matter of fact and existence'! That Hume's argument for C was valid, we are assuming. Where, then, is its contingent premise? The reader has only to recall E, F, G, H, and J, to see that there is no contingent premise in Hume's argument for C, according to my account of it above. Nor is there a contingent premise in Hume's argument for C, according to any other account of it which is worthy of consideration in other respects. Such a premise, then, is not easy to find in Hume's argument for inductive scepticism. Yet if it were there at all, it must be a huge one; since it has to entail a conclusion, C, which on the present hypothesis is nothing less than a contingent proposition about the entire universe. We must ask to have this elusive giant---this Yeti, as it were, among Hume's premises---pointed out to us.
But it is not really necessary to rely here on the method of challenge. Any hope, or fear, that the challenge just issued might be met, can be very easily extinguished. For suppose that Hume's inductive scepticism C were contingent. Then, since what is says, it says about any inductive argument, it would be a universal proposition, as well as a contingent one. Contingent universal propositions, however, are a species of contingent propositions about the unobserved; and we know what Hume here implies about all of them. His empiricism B says that only propositions about the observed can be reason to believe them; and his inductive scepticism C says that even propositions about the observed are no reason to believe them. On the hypothesis that C itself is contingent, then, Hume's scepticism A about the unobserved, which follows from B and C, would imply, concerning inductive scepticism C itself, that there is no reason to believe it; any more than to believe, say, that tomorrow's flames will be hot.
Well! Philosophically, of course, no result could possibly be more welcome than this, to any empiricist who denies inductive scepticism: if only, as a matter of Hume-exegesis, he could believe it! `Hume's inductive scepticism, given his empiricism, entails that there is no reason to believe his inductive scepticism': what an optimum result! Alas, as a matter of exegesis of Hume, even the inductive probabilist finds this far too good to be true. Still less will any other student of Hume be able to believe it. Hume's inductive scepticism C, however it is to be refuted, if indeed it can be refuted at all, is certainly not a proposition which makes its own refutation unnecessary, by committing suicide at birth in this obliging and even graceful manner. It would be a proposition of exactly that kind, however, if it were contingent. Therefore it is not contingent.
It is not hard to see how the anti-sceptical critics of inductive probabilism have been led into the exegetical absurdity just noticed. They have not fully taken in Hume's result N. No one would search for a cement of the universe which would validate inductive arguments, if he were once fully persuaded that anything which was equal to that task would not be part of any reason to believe the conclusions of induction. The position of the sceptical critics of inductive probabilism, such as Popper, is much less intelligible. They are empiricists, and even inductive sceptics, yet somehow they have had revealed to them what they imply is a natural law: that the cement-content of the universe is constant zero. Criticism is superfluous in such a case.
Both groups of critics suppose, as I said earlier, that they adhere more rigorously than the inductive probabilist does to the truths about induction which Hume's argument can teach us. In fact, as we have now seen, the boot is on the other foot. The inductive probabilist has taken in, far better than his anti-sceptical critics, the truth of Hume's result N. And he has taken in, far better than any of his critics, the non-contingent character of Hume's conclusion C. As for the belief that the inductive probabilist is trying to revive a position refuted in advance by Hume himself, this is a mere myth. Its only foundation is ignorance of the texts; ignorance, in particular, of what Hume meant by the phrase "probable arguments". Far from having refuted inductive probabilism, Hume never so much as considered it. He scarcely could have done so, because inductive probabilism came into being, in the modern period of philosophy, or at least assumed a definite form, only in response to his sceptical attack on induction . Hume's only question about induction was, as I said earlier: what can validate it (while being also part of a reason to believe its conclusions)? Finding, N, that nothing can, he forthwith concluded C, that induction is unreasonable. That inductive arguments, or that any arguments, might be reasonable although incurably invalid, is a position which Hume nowhere attempted to exclude .
The inductive probabilist can easily show, too, that his critics' implied philosophy of logic is no more satisfactory than their implied criticism of Hume.
`You challenged me to say' (he might reply to his critics) `what makes it true that "All the many flames observed in the past have been hot", is a reason to believe "Tomorrow's flames will be hot"; or to say, in general, what makes it true that Hume's inductive scepticism C is false. And in accordance with your misconception of the nature of C, you were then demanding a contingent truth-maker for this assertion of mine. But inductive scepticism C is not a contingent proposition. No more, then, is my denial of it contingent. The negation of C, like C itself, a logical thesis in a broad sense, and whichever one of the two is true, that proposition does not require a contingent truth-maker, any more than other propositions of the same kind do'.
`When we say' (he might continue), `as we all do say, and as Hume's D implies, that a tautology, for example, is not a reason to believe, "Tomorrow's flames will be hot", does this assertion of ours require a ground in nature to make it true? When we say, as all philosophers do, that "All men are mortal and Socrates is mortal" is not an absolutely conclusive reason to believe "Socrates is a man", does this assertion depend for its truth on some cosmic contingency? If it does, what is that contingent feature of the universe which makes undistributed middle a fallacy? Is it an unfortunate local deficiency of cement, perhaps, or a vein of actual anti-cement which runs through our fallen world? When philosophers say, as almost all of them do say, that "All men are mortal and Socrates is a man" is an absolutely conclusive reason to believe "Socrates is mortal", does their assertion require a contingent truth-maker? If it does, then deductive logic too, no less than non-deductive logic, will be `speculative metaphysics'. To every one of these questions, the answer is obviously "no". And no more does my assertion, when I say that some propositions about the observed are a reason to believe some contingent propositions about the unobserved, require any contingent fact to make it true'.
There is one compromise, however, which the inductive probabilist can and should offer to his critics. He should undertake to reveal what the contingent fact is, which makes the premises of some inductive arguments a reason to believe their conclusions, on the very day that his critics reveal what the contingent fact is, which makes the premise of every inductive argument not an absolutely conclusive reason to believe its conclusion. Mutual disclosure of all contingent assets is a fair principle. Let the ground in nature of the reasonableness of some induction be disclosed, then, in return for disclosure of the ground in nature of the invalidity of all of them.
Nor is the inductive probabilist obliged to maintain, wherever the premises of an induction is a reason to believe the conclusion, that this is an ultimate logical feature of those arguments. It may sometimes be possible to show that it is a derivative one. It may be possible, that is, to derive the conclusion that certain inductive arguments are reasonable, from premises about the reasonableness of certain non-inductive arguments. Indeed, it is already known (thanks originally to Bernoulli and Laplace ) that this can in certain cases be done. But any premise of such a derivation will be, like the conclusion of it, a proposition not of a contingent but of a logical kind. And even where such a derivation is possible, the reasonableness of the inductive arguments in question remains an intrinsic feature of them, even though not an ultimate one.
The philosophical dispute between inductive probabilism and its critics, as I have presented it, arose from a historical dispute, about the interpretation of Hume's argument for inductive scepticism. The question was whether, supposing induction to be incurably invalid, as N says it is, it follows that induction is unreasonable, as C says. The inductive probabilist believes it does not follow. Hume, and the critics of inductive probabilism, believe that it does. But now, there is nothing to prevent us from condensing this whole cloud of philosophy and of Hume-exegesis into a single drop of elementary logic. Does C follow from N, or does it not?
Well, C says this: that the premise of an inductive argument is not a reason to believe its conclusion. N says this and only this: that any inductive argument is invalid, and that no validator of it is a reason or part of a reason to believe its conclusions. But evidently, from the fact that no validator R, of an inductive argument from P to Q, is a reason or part of a reason to believe Q, it does not follow that the premise P itself is not a reason to believe Q. Yet this is what C says. So C does not follow from N. The incurable invalidity of induction is no proof of its unreasonableness.
There is, therefore, a gap in Hume's argument for inductive scepticism C. Proceeding, as we are, on the assumption that the argument which Hume intended was valid, we therefore have no alternative but to suppose that his argument had some premise which he did not state: some tacit assumption which, when it is added to his other premises, is sufficient (and of course no more than is necessary) to turn his argument for C into a valid one. For our purposes it is necessary to identify all the premises of his argument. We need, therefore, to identify this suppressed premise: the validator (for it is nothing less) of Hume's own argument for scepticism about induction.
The general nature of this missing premise is obvious enough. One who believes that C follows from N regards it as necessary, in order for the premise of an argument to be a reason to believe the conclusion, that the argument be valid, or at least not incurably invalid. One who denies that C follows from N, denies that this is a necessary condition for an argument to be reasonable. The former philosophers therefore (as was said earlier) have a higher or more exacting standard than the latter, of what it is for the premise of an argument to be a reason to believe its conclusion. As we called the latter, with obvious propriety, `inductive probabilists', so we may call the former `deductivists'. For their standard of a reasonable argument, whatever exactly it may be, is one which demands that, if P is a reason to believe Q, then Q is deducible either from P itself, or from P along with such limited additional premises as can be themselves part of a reason to believe Q. Hume, since he believes that C follows from N, is one of these deductivist philosophers. We are therefore entitled to call the unexpressed assumption, which enabled him to mistake the argument from N to C for a valid one, Hume's deductivist premise; even though it is yet to be determined exactly what it says.
The presence of such an assumption in Hume's argument, as it is obvious, has often been noticed. Many writers have detected, not only in his argument for inductive scepticism, but elsewhere in Hume's philosophy, the influence of an inexplicit standard, of a `high' or `deductivist' or `rationalist' kind, as to what constitutes a reasonable argument. Thus one writer says, for example, that in Hume's argument for inductive scepticism, "the tacit assumption [was that] all rational inference is deductive" . Another says that Hume's assumption was, that "arguments are deductive or defective" . Many other writers could easily be cited to the same effect.
The last-quoted version of Hume's deductivism is too vague to be of any use to us; for the writer does not explain, and it is not obvious, what he means by "defective". The previously-quoted one is identical with the version of deductivism which, in my earlier-published account of this argument, I myself attributed to Hume. For I there concluded that Hume's unstated premise was, that an argument is reasonable only if it is valid; or in other words, that P is a reason to believe Q, only if Q is deducible from P .
But it is easy to see, in the light of the more detailed account of Hume's argument which has been given here, that this identification is wrong. This simple version of deductivism makes Hume assume both too much and too little. Too little, because it takes no notice at all of the distinction between arguments which are (simply) invalid, and arguments which are incurably so. Too much, because it is, evidently, stronger than the validator of the argument from N to C. (To attribute it to Hume is therefore to attribute to him more than is needed to make his argument valid: a serious fault in exegesis). And both of these defects, it will be obvious, arise from the same source: namely, that this version of deductivism `engages' only with the first clause of N (that inductive arguments are invalid). Hence on this identification of Hume's deductivism, the second clause of N, which adds that the invalidity of induction is incurable, plays no essential part in his argument at all.
But on the contrary, it is the second clause of N which Hume's entire argument had been directed to establishing; not the first clause of it, the mere fact that inductive arguments are invalid. That was assumed from the outset, in Hume's premise G. What Hume argued for, and argued successfully for, was I and K: that the validator of inductive arguments is neither necessarily true nor observational. It was this result, combined with his assumption, which follows from E and F, that it is only necessary truths and observation-statements which can be even part of a reason to believe another proposition, from which Hume validly inferred that the invalidity of induction cannot be cured at all; that is, N. And some deductivist assumption, the exact nature of which we wish to identify, conjoined with N and perhaps with some other premises of his argument, then carried Hume to inductive scepticism C.
With this recapitulation of Hume's argument before us, then, let us ask afresh, what is the assumption, of a deductivist kind, which is implicit in the argument?
The conclusion to be reached, C, says that the premise of an inductive argument is not a reason to believe its conclusion. We ought, therefore, in trying to identify the missing deductivist premise of the argument for this conclusion, to take account not only of N, but of anything which the already-identified premises of the argument say, about what is required in order for one proposition to be a reason to believe another. E and F contain everything there is of that kind, in the premises of the argument which have been already identified. We ought therefore, in trying to identify the deductivist assumption which supervened between N and C, to take N not on its own, but in conjunction with E and F.
Once this is done, the missing premise stands out clearly. The most that is entailed about inductive arguments by the conjunction of E and F, is:
M+: Any inductive argument is invalid, and any validator of it is neither a necessary truth not a proposition about the observed.
(I call this M+ because it says, about any validator of inductive arguments, just what M says about the weakest one). The missing deductivist premise is therefore the validator of the argument from M+ to C. That proposition is evidently:
O: P is a reason to believe Q only if the argument from P to Q is valid, or there is a validator of it which is either a necessary truth or a proposition about the observed.
This proposition, therefore, is Hume's deductivist premise.
In order to satisfy ourselves of this, it is sufficient to cast our minds back over Hume's argument, and ask ourselves the following simple question. What, after all, did Hume have against inductive arguments? What is it, at bottom, about inductive arguments, which elicits from him the dire verdict C against them? Well, certainly not the mere fact that they are not valid. That feature of them was taken for granted by Hume (in premise G) from the start, and is, besides, a feature, unimportant in itself, which inductive arguments share with many arguments that neither Hume or anyone else condemns; for example, the argument mentioned earlier, about the bird on the post. No, what makes inductive arguments unreasonable in Hume's eyes is that, precisely unlike the argument about the bird on the post, their invalidity cannot be cured by any additional premise which might be supplied either by a priori knowledge or by experience. That was what Hume has against induction. Which is to say that he assumed that, if an argument is a reasonable one, it is either valid, or can be made so by an additional premise which is either necessarily true or observational. Which is to say that his tacit assumption of a deductivist kind was O; and that what that premise engaged with was M+.
In the presence of E and F, N entails M+; but equally, in the presence of E and F, M+ entails N. In other words, given Hume's assumptions, that a reason to believe another proposition must be directly accessible to knowledge or reasonable belief, and that all and only observation-statements and necessary truths are so directly accessible, M+ and N are logically equivalent. From a logical point of view, therefore, it is immaterial whether we regard deductivist O as engaging with N or with M+; that is, whether we regard the last step of Hume's argument for C as having the structure:
E } F } N } -> C O }
or the structure:
E } F } -> M+ } N } O } -> C
The structure, then, of Hume's argument for inductive scepticism was the following:
E } H } E } F } -> M+ } J } ------> I } F } -> N } O } -> C } -> M } H } -> L -> K } G }
It is important to realize, as was pointed out earlier, that it is possible consistently to be an inductive sceptic without being a sceptic about the unobserved; that is, that A does not follow from C alone, but only from C conjoined with empiricism B. In the same way it is important to realize that it is possible consistently to be a deductivist, without being an inductive sceptic; that is, that C does not follow from O alone, but only from O conjoined with M+, which says that no validator of induction is either necessarily true or observational.
It ought to be obvious that C does not follow from O. For deductivism O says nothing about inductive arguments, or indeed about any particular class of arguments, at all. All it does is to allege that a certain condition is necessary in order for an argument to be a reasonable one. Clearly, this on its own cannot entail that some particular class of arguments satisfies, or fails to satisfy, that condition. It is only when O is conjoined with M+, which is about inductive arguments, and says of them that they fail to satisfy the condition demanded by O for reasonable arguments, that C, the unreasonableness of induction, follows.
It ought to be equally obvious that C does not entail O, either. C says only that inductive arguments are unreasonable. From such a proposition as that, it is evidently impossible to deduce any positive condition which arguments in general must satisfy in order to be reasonable. But such a condition is precisely what O lays down.
Deductivism O and inductive scepticism C are, then, neither of them deducible from the other. These facts are important. For these two theses, although in fact independent, are nowadays often, or rather, usually, inextricably confused with one another. The historical reason for this confusion is, of course, just the fact that the two theses are closely connected in the context of Hume's argument about induction, while that argument is nowadays vividly, though confusedly, present in the minds of most philosophers. Hence nowadays the deductivist believes himself obliged to be a sceptic about induction; the friend of induction believes himself bound to reject deductivism; the inductive sceptic imagines himself bound to be a deductivist; the enemy of deductivism considers himself safe from inductive scepticism; and every single one of these beliefs is false. It is scarcely possible, in fact, to overestimate the damage which has been done, in the way of positive error but even more in the way of mere confusion, by the failure to recognize that deductivism, and scepticism about induction, are separate theses. And this damage has been inflicted, not only on our ability to understand Hume's argument about induction, but also, and even more importantly, on what is based on that argument: 20th century philosophy of science.
It is the mistaken belief that O entails C, rather than the converse belief, which has been the more productive of error and confusion. This belief has had the effect, among others, of making unintelligible to most philosophers nowadays a philosophy of science which is as recent, as influential, and as intelligible, as that of J.S.Mill. For the kind of position which was pointed out (three paragraphs back) as a logically possible one, namely deductivism without inductive scepticism, was in fact that of Mill. What Mill really believed, as was indicated earlier, is that M+ (and even M) is false. That is, he thought that the deterministic validator of eliminative induction (such as the argument about the dead canary) was observational. That is why he could consistently be, what he was, a deductivist and yet no sceptic about induction.
Now, among the premises of Hume's argument for inductive scepticism C, which one is the key to the scepticism or irrationalism of that conclusion? What is that premise, without which this argument would have neither C nor any other irrationalist thesis as a consequence?
Well, the only premises of the argument for C are G, H, and J, E and F, and O.
The first five of these premises, however, entail nothing of a sceptical or irrationalist kind about induction. They entail, indeed, M+, that the invalidity of induction is incurable by any observation-statement or necessary truth. And they entail, what is equivalent to M+ in the presence of E and F, N, that the invalidity of any inductive argument is incurable by any additional premise which is even part of a reason to believe its conclusion. But these results M+ and N are the most that these five premises entail about induction. And these is nothing sceptical or irrationalist about either of them. They do not entail C. (They say, indeed, no more than is acknowledged nowadays by almost all philosophers, and not just by inductive sceptics: that induction `cannot be turned into deduction').
Thus before the sole remaining premise, deductivism O, comes into the argument, there is nothing to necessitate C or any other irrationalist conclusion about induction. Once that assumption does come in, however, it engages with M+, and scepticism C concerning induction is an inevitable result. It is, therefore, deductivism which is the key premise of Hume's argument for inductive scepticism.
Nothing fatal to empiricist philosophy of science, in other words, follows from the admission that arguments from the observed to the unobserved are not the best; unless this assumption was combined, as it was with Hume, with the fatal assumption that only the best will do.
Finally, it is worthwhile, for the sake of getting an overall view of Hume's argument for scepticism about the unobserved A, to put together the two parts of its structure-diagram: the argument for B, and the argument for C. This is done below. For convenience of reference, all the elements of the argument are also listen below: and I have here given each of them a summary title, which may be found helpful.
E } H } E } F } -> M+ } J } ------> I } F } -> N } O } -> C } } -> M } } -> A H } -> L -> K } D } } G } E } -> B } F }
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