By Jonathan M.S. Pearce, June 2011
Scottish Enlightenment philosopher David Hume has influenced many modern intellectuals with his erudite writings and treatises. One of his most influential works was the 1748 work An Enquiry Concerning Human Understanding, a reworking of his early piece, A Treatise of Human Nature. There is much to ruminate on in An Enquiry, most notable his philosophy concerning epistemological matters, theorising knowledge and fact.
In this essay, I will be looking to explain the terms of the statement above before investigating what is entailed in forming beliefs about the future based on the past. After doing this, I will seek to argue that you can ‘reasonably’ form such beliefs using probability and inductive reasoning. I will also investigate the idea that under metaphysical naturalism, and with perfect knowledge of any given system, we could ‘know’ future events based on past experiences, though on true Pyrrhonism we must doubt almost everything. I will show that this form of reasoning and knowledge is not what Hume had in mind and that he was demanding deductive reasoning to validate the movement from past uniformities to future events. However, I will continue by looking at some attempts to produce deductive arguments using past uniformities before looking briefly at the work of Colin Hoswon who declared that “there are nevertheless demonstrably sound deductive inferences”. Finally, I will question the meaningfulness of raising the problem itself. I will do this whilst also explaining the terms and positions to allow universal understanding of the debate and giving references for further reading when the subject matter becomes too complex or convoluted for the scope of this essay.
Let us first look at how Hume sets out ideas in An Enquiry as this forms some of the fundamental structure to his arguments. Hume differentiates two forms of thought: impressions and ideas. Impressions are sensations, experiences of the senses, whilst ideas are memories or imaginings. Empirically speaking, Hume claims that impressions form the basis for ideas. I would agree with Hume here since every science fiction or fantasy writer would agree that it is impossible to make up a new creature or entity that is not in some way based on entities that have already been experienced in one way or another. In other words, our imagination cannot create ex nihilo. Hume exemplifies the way we do this and categorises them into compounding, transposing, augmenting and diminishing before poking a hole in his argument by declaring that one can imagine a missing colour in a spectrum of blue, hypothetically. I would disagree with his scepticism here by declaring that this example actually fits into the augmenting / diminishing category. Instead of claiming that a giant is the idea of a human but made bigger, we should see this as an increase or decrease in any given characteristic. In the case of blue, it is simply a diminishing or augmenting of the hue of an already existing hue of blue.
This leads on to Hume’s analysis of propositions of relations of ideas and matters of fact. Relations of ideas can be demonstrated, and would include ideas such as geometric, semantic and logical propositions. On the other hand, matters of fact are experiential in nature. This very point here could lead us down several rabbit-holes (language, semantics, logic, conceptualism vs. realism and so on) but this would only serve to throw us off course.
Hume takes the ideas of matters of fact and inquires as follows (Hume p.29):
When it is asked, What is the nature of all our reasonings concerning matter of fact? the proper answer seems to be, that they are founded on the relation of cause and effect. When again it is asked, What is the foundation of all our reasonings and conclusions concerning that relation? it may be replied in one word, Experience. But if we still carry on our sifting humour, and ask, What is the foundation of all conclusions from experience? this implies a new question, which may be of more difficult solution and explication.
I say then, that, even after we have experience of the operations of cause and effect, our conclusions from that experience are not founded on reasoning, or any process of the understanding.
And herein lays the crucial exposition of Hume’s argument. Hume is claiming that there is no reasoning that can lead from experience to matter of fact. A matter of fact, being for example that a pebble landing in a lake (A) always causes a splash (B), is a cause and effect relationship that is derived and understood through experience. There is no a priori reasoning that could lead us, without that experience, to concluding that A would necessitate B. In other words, there is nothing that logically necessitates the causal effect of A to B, such as the definition of the terms and suchlike. We find that B results from A, that a splash is caused by a pebble landing in the lake, through experience.
Now if we threw a pebble into the lake again, and it splashed, we would start to experience a repeated phenomenon. If this happened again, if B consistently resulted from A, then it would be a poor wager to bet against it happening a further time. If, after a thousand pebbles, we were to throw another pebble into the lake, it would be imprudent to bet against there being another splash. This wager that there would be a splash on throwing the pebble becomes a belief that there would be a splash. Hume sees belief as related to fiction but giving a certain feeling of confidence that fiction does not imbue. Hume sums up the belief that a future effect will arise from a cause to be provable only from experience as follows:
…having found, in many instances, that any two kinds of objects–flame and heat, snow and cold–have always been conjoined together; if flame or snow be presented anew to the senses, the mind is carried by custom to expect heat or cold, and to believe that such a quality does exist, and will discover itself upon a nearer approach. This belief is the necessary result of placing the mind in such circumstances.
But still our curiosity will be pardonable, perhaps commendable, if it carry us on to still farther researches, and make us examine more accurately the nature of this belief, and of the customary conjunction, whence it is derived. By this means we may meet with some explications and analogies that will give satisfaction; at least to such as love the abstract sciences, and can be entertained with speculations, which, however accurate, may still retain a degree of doubt and uncertainty.
What Hume is claiming in this very important quote is that what allows the philosopher to arrive at B from A is nothing more than experience and custom. It is only that this effect has taken place with such habit that one can guess, probabilistically, that it will happen again. There is no reason outside of the reliance on custom that can verify that B will always occur after A. Moreover, and crucially, this belief in custom, Hume declares, is open to doubt and scepticism even though it may appear to be accurate. This doubt is born from the difficulty to achieve intellectual certainty in supposing that the sun will continue to rise tomorrow. The certainty that does arise is a different idea of certainty, one that resorts to the plea to uniformity and not to logical verification.
Let us look, then, at what reason is defined as to see what Hume is trying to say and whether he is in a philosophical position to say it. In philosophical terms, reason is a way by which one can connect one idea to a related idea, whether as cause and effect, or by asserting the truth values of a proposition or suchlike. Reason is answerable to the constraints of logic, the axiomatic system of arbitration. Logic, itself, is often split up into two divisions: inductive reasoning and deductive reasoning. Inductive reasoning is a form of argument or logic that allows for the possibility that the conclusion is false, even though the premises are true. For example, we could state:
1. All the penguins we have seen have beaks.
2. All penguins have beaks.
The conclusion to this simple argument may be inferred inductively from our experience but the conclusion does not necessarily follow from the premise. Deductive reasoning would be more concrete:
1. All authors write.
2. Jim is an author.
3. Therefore, Jim writes.
Now the premises may or may not be true, but the conclusion necessarily follows from the premises.
I mention this because it seems to me that Hume is declaring that there is no reasoning that can lead us to make conclusions about the future; that we can only rely on custom. However, it also appears that the use of custom to connect one idea to another is a form of reasoning. Indeed, custom appears to be a form of inductive reasoning. This, with our argument, would be set out as follows:
1. All pebbles that we have observed thrown into a lake cause splashes
2. I will throw a pebble at the lake
3. Therefore, it will cause a splash.
Inductively, we can ‘reason’ that there will be a splash, but the conclusion does not necessarily follow from the premises. The problem with trying to look for an argument that concludes a Uniformity of Nature (UN) is that it must be inductive and have Uniformity of Nature as a premise. This creates an unwanted circularity. As such, the above argument would have to have a hidden a priori premise that upholds UN:
1. All pebbles that we have observed thrown into a lake cause splashes
2. (All consistently observed phenomena will repeat in accordance with the laws of nature)
3. I will throw a pebble at the lake
4. Therefore, it will cause a splash.
Premise 2, or something similar to it, must be there in order for the conclusion to follow necessarily. But that premise is exactly what a critic of Hume would be trying to conclude, so for it to be there makes the entire argument logically fallacious.
When we look at knowledge, epistemologically speaking, we have an interesting situation. Strictly speaking, the only true knowledge that we can hold indubitably is the knowledge that ‘I exist’. If I drop the pebble in question, then I might well think that I know that if I drop the pebble, it will fall downwards. I may think that I have the ‘Law’ of Gravity to define this circumstance. Yet even physical laws are prone to Cartesian doubt. Renee Descartes, the seventeenth century French philosopher, declared that the only statement that does not lay victim to doubt is ‘cogito ergo sum’: I think, therefore I am. We cannot doubt our own existences, in whatever form that may be (I am steering away from a debate about what ‘I’ constitutes here). The Law of Gravity may only be a descriptive law that may not work the next time you drop a stone. There is always an element of doubt. It could be that we are living in a Matrix style of universe where our minds are plugged into a machine and everything else is an illusion. But still ‘we’ exist in some form in order to be able to experience the illusion. Being that we are unable to prove absolutely that this either is, or is not, the case, we are resigned to being at least a little doubtful, or sceptical, about almost everything in the known cosmos.
Therefore, there can only actually be one true, indubitable piece of knowledge, and only the individual experiencing this can have this knowledge; it cannot be transposed onto any other entity, and no other entity can have true knowledge of another entity. So real knowledge is limited to one fact: that you, as the reader, exist. This leads us to Pyrrhonism. Greek philosopher Sextus Empiricus first recorded the use of Pyrrhonian scepticism (named after Pyrrho of Elis) in the late second century CE. This position dictates that we cannot know anything, in a Cartesian manner, and thus we must logically adhere to a strict agnosticism. This is an approach that even goes as far as stating that one must be sceptical over one’s own scepticism to the point that one must remain in perpetual enquiry, making no dogmatic statements about anything. This type of logical sceptic would never, pragmatically speaking, achieve anything in life and so would rely on custom to navigate through everyday life.
Within a naturalistic framework, we take knowledge to be an inductive principle based on probability and overwhelming evidence, or scientific fact. This metaphysical naturalism is a position that entails that the only things that exist are natural phenomena, forces, causes and effect. Supernatural entities or causes are ruled out a priori even though to reach a conclusion of metaphysical naturalism we approach it from an a posteriori (or evidentially based) process. A scientific fact, in the face of Cartesian doubt, is actually redefined as a claim backed up by overwhelming evidence, which seems something rather more subjective and arbitrary than the simple brute ‘scientific fact’. Probability, assuming there is no such thing as true randomness in the universe, becomes the defining principle which allows one to induce a conclusion from previous experiences. Probability is fundamental to inductive reasoning. As Dugald Murdoch said, “not only is this discussion [of probability] essential for a proper understanding of Hume’ philosophy of induction, but it is also of great interest in its own right”. If I have thrown that pebble a thousand times and received a splash each time then probabilistically, on throwing it again, it will cause a splash. In my opinion, naturalism leads to determinism, and these deterministic qualities can lead us to have a much smaller degree of doubt than otherwise. Laplace’s Demon, a theoretical entity that knows every single variable in the cosmos, under a deterministic framework, would have the best chance of knowing everything that might come to pass. Does Laplace’s Demon have good reason, then, to believe claims about the future? I will return to the demon later. Science seems to be a direct result of inductive reasoning since it is based on observation (experience). Thompson (2006) connects inductive reasoning to the scientific method as follows (p. 48):
It is clear that this process of induction, by which a theory is arrived at by the analysis and testing out of observed data, can yield at most only a high degree of probability. There is always the chance that an additional piece of information will show the original hypothesis is wrong, or that it applies only within a limited field.
Here, we look to have proven Hume incorrect in his claims that there is no reasoning that can lead you to conclusions about future events based on past experiences. Inductive reasoning, using probability (especially on a naturalistic foundation which leaves everything to the deterministic machinations of nature) seems to have got us, quite literally, from A to B. However, it is not so easy. And thus we encounter the Problem of Induction: Hume appears to believe that induction is not a form of reasoning, and that deduction is the benchmark for reasoning. Hume is effectively claiming that there is no deductive reason to get a thinker from A to B.
Hume certainly recognises that this inductive reason, this custom and habit, is vital for humans to employ. If we did not induce conclusions from habitual experiences then we would be very different entities indeed:
Without the influence of custom, we should be entirely ignorant of every matter of fact beyond what is immediately present to the memory and senses. We should never know how to adjust means to ends, or to employ our natural powers in the production of any effect. There would be an end at once of all action, as well as of the chief part of speculation. (Hume 2006, p. 36)
But it still remains to be seen whether deduction can get one from A to B. Let us remind ourselves about the pebble argument to weigh it up in a deductive context:
1. All pebbles that we have observed thrown at a lake cause splashes
2. I will throw a pebble at the lake
3. Therefore, it will cause a splash.
This argument is clearly inductive, and one cannot squeeze a deduction out of it. Apples are apples, they are not oranges. But does this mean that there is no form of deductive reasoning that can be employed in defence of the aforementioned argument?
There have been a number of philosophers and mathematicians who have argued that the Problem of Induction can be solved by employing deduction. Hume was of the opinion (though he did not label propositions inductive or deductive as these are more modern terms) that one could not defend inductive reasoning by further inductive reasoning as this would entail circular reasoning, a logical fallacy. In this way, Hume actually showed that if we relied on the probabilistic qualities of induction, such that we could reliably infer that a pebble would cause a splash, then there would be no way to deduce that this was any more ‘reliable’ than any other conclusion. As John Vickers states in the Stanford Encyclopaedia of Philosophy:
Now as concerns inductive inference, it is hardly surprising to be told that the epistemological problem is insoluble; that there can be no formula or recipe, however complex, for ruling out unreliable inductions. But Hume’s arguments, if they are correct, have apparently a much more radical consequence than this: They seem to show that the metaphysical problem for induction is insoluble; that there is no objective difference between reliable and unreliable inductions.
Therefore, it seems necessary to somehow defend inductive reasoning by employing deductive reasoning. As Bertrand Russell complains (Russell 1946;1961, p. 646):
It is therefore important to discover whether there is any answer to Hume within a framework of philosophy that is wholly or mainly empirical. If not, there is no intellectual difference between sanity and insanity. The lunatic who believes he is a poached egg is to be condemned solely on the ground that he is in a minority, or rather – since we must not assume democracy – on the ground that the government does not agree with him. This is a desperate point of view, and it must be hoped that there is some way of escaping from it.
What Russell is therefore begrudgingly accepting is that inductively derived conclusions, on Hume’s grounds, are conclusions that have no more epistemological veracity than any other such knowledge. We must, then, be able to find a way, epistemologically speaking, to use experience to derive knowledge reasonably otherwise the foundations of science fall away, and science itself becomes arbitrary.
Some philosophers have argued that there is a logic to inductive reasoning that gives it a greater credence than Hume gives credit for. The Williams-Stove thesis put forward by D.C. Williams and D.C. Stove in 1947 and then corrected some 40 years later set out to prove that you could get necessary truths from inductive reasoning, thus meaning that there were demonstrable (deductive) qualities to inductive logic. They proposed that relative frequencies of traits in a population are matched in relative frequencies in large samples of that population. In synopsis, “it looks to follow that it is a necessary truth that it is highly probable that the frequency of a trait in a given sample from an inclusive population is close to its frequency in the population.” However, after several criticisms, it seems that the theory (a complex exposition of modal logic too complex for the scope of this essay) only holds probabilistically and in certain limited instances. As John Vickers says:
It began life as the strong and simple modal assertion that it is a necessary truth that inductions of a quite common sort yield their conclusions with high probability. That thesis is seen to be false. What remains are, at best, certain specific instances of it.
D.M Armstrong took a different tack. Rather than trying to prove that an inductive argument itself was intrinsically rational, in the Humean sense, he looked to prove that the use of an inductive argument was rational; that it was necessary. His considerations were the application of the argument, rather than the actual argument itself. As Armstrong (1983, p. 52) himself declares:
…ordinary inference from the observed to the unobserved, is, although invalid, nevertheless a rational form of inference. I add that not merely is it the case that induction is rational, but it is a necessary truth that it is so.
What Armstrong claims is that observed uniformities are best explained by hypothesising strong laws of nature, which are by definition objectively necessary, which then go on to make conclusions about the unobserved (which in our case is the future predicted event of the pebble making a splash). This would then be a testable, verifiable Law. Of course, this existence of a ‘strong law’ goes against the descriptive quality of laws that was mentioned earlier. Again and like the Williams-Stove thesis, Armstrong depends on laws of probability and this is where he, too, is criticised. For the detailed criticisms of his argument, particularly how it falls victim to the vagaries of the laws of probability, see the Stanford Encyclopaedia of Philosophy.
Part of the attractiveness of such theories is that they employ laws of probability and frequency and other mathematical laws. As such laws are themselves mathematical consequences, they are seen as necessary truths. The idea is that the deductive capacity of the maths involved gives the inductive arguments a deductive foundation and thus a reasonable and reliable dimension. In this way, as Vickers states, “Of course the application of these laws in any given empirical situation will require contingent assumptions, but the inductive part of the reasoning certainly depends upon the deductively established laws.” The problem for Hume is that he did not have a sophisticated grasp of probabilistic laws and their deductive qualities. Therefore, he might have been a little premature in his conclusion that there was no ‘reason’ to believe that past uniformities could inform us of future events. One can hardly blame him, of course, because such laws depend upon modern logic and the “axiomatisation of probability”, none of which even existed in the intellectual landscape in which Hume philosophised.
So where does this leave us now? Well, it seems that there can be specific cases made for the deductive justification of the inductive method. Metaphysically speaking, though, there is still the seemingly insoluble problem. Why should we trust induction? It is here that I want to concentrate the rest of the essay. I would hope that by answering that question “because it is reasonable” should give Hume pause to think.
Colin Howson, British philosopher and Professor of Philosophy at the University of Toronto, dealt with this very idea in his book Hume’s Problem: Induction and the Justification of Belief. He claims that (p. 2):
The resolution of the paradox is that inductive inference arises as a necessary feature of consistent reasoning, given the sorts of initial plausibility assumptions scientists habitually make.
What Howson is adamant in saying is that Hume’s argument does not say that we, in relying on the scientific method of induction, are “misguided” in any way. This method is not wrong because it is not deductive. Howson claims that there is a sound logic to be employed in inductive inferences. He combines the work of Hume himself, and F.P. Ramsey. From Hume he takes the notion that “inductive conclusions may be soundly inferred from inductive premises” (Merrill 2003). Ramsey’s influence for Howson is that since evidence-based reasoning is dependent upon probability (as we saw earlier) then it is “nothing but the application of logical principles of consistency” (p. 4). Essentially, Howson’s argument is that by using Bayesian probability theory, induction is reliable. Moreover, it appears that the basic axioms of probability (something that Hume would not have known) produce this reliability. Induction, based upon valid assumptions (likely prior probabilities) results in real knowledge. Thus, Howson claims, this model of induction, using Bayesian theories, has the same qualities as deductive reasoning. As Lipton (2002, p. 579) puts it:
Howson makes a great deal of the analogy between deductive and probabilistic argument. The principles of deductive logic place constraints on proper deductive reasoning even though they do not specify the premises. Similarly, the principles of probability place constraints on proper inductive reasoning, even though they do not specify the prior probabilities…
Howson gives more credence to his argument by highlighting the probabilistic arguments’ deductive characteristics. I will now move away from the often confusing world of modern logic and into the world of intuition. It seems to me, at any rate, that inductive reasoning yields results that are pragmatically very useful. It is in this way that the reasoning appears sound. Though it may (or may not) have the necessary conviction of a deductive logic, there is soundness in achieving a goal. And the more knowledge we have of all the variables, the more accurate the predictive the probabilistic calculations for the inductive argument will be. Reverting to Laplace’s Demon, we would surely have an entity, with its perfect knowledge of all the variables in the cosmos, all the forces and natural laws, which could not just probabilistically calculate future events, but which could have complete certitude over future events. Thus, it appears that Laplace’s Demon could circumvent Hume’s argument, given the adherence of everything to natural laws. This is a position held by some objectivists who claim the behaviour of entities in causal relationships are ‘entity-based’ which recognises that causation is a necessary relationship between an entity and its actions.
With these ideas in mind, could Hume argue that this entity (Laplace’s Demon) is not inducing knowledge form experience, and is not using only past experiences to inform future predictions? However, this objection would surely apply to humanity. With each passing year, our collective intellect and knowledge base grows hugely. Humanity, together with human designed computing systems, is approaching the zenith of Laplace’s Demon. Though we may never reach those heady heights of knowledge, we seem to be getting forever closer. And thus our calculations are becoming ever more accurate. Our probabilities for calculating future events, future effects from their causes are meaning that we are more consistently arriving at the predicted truth. Are we not becoming more sound in our predictions? In this way, I would argue that theoretically speaking, Laplace’s Demon would have deductive reason to form beliefs about the future based on past uniformities (given the assumption of determinism). It is only humanity’s imperfect knowledge of our system that requires us, from Hume’s perspective, to rely on custom.
Statements of knowledge, the common parlance (non-Cartesian) knowledge, should be understood in the light of experience and probability. They are empirically meaningful. On a daily basis they have meaning to all of us, even to Hume, by his own admission. Induction is very much an integral part of human nature. It does not matter, surely, at whether our conclusions which turn out to be reliable and correct are deductively derived. What matter is that we have sound mechanisms to predict future events using data and observations (experience) garnered from past events. This is the scientific method, and it is inductive. But it works. This is not too far from Hume’s own position of accepting the usefulness of everyday pragmatism, though he did concern himself greatly with notion of the Problem of Induction.
Therefore, we have ended up in a position that is known as reliabilism which contends with Hume’s claim that inductive reasoning cannot justify inductive reasoning – induction cannot be circular. Whether reliabilism, with its claims of reliable predictions of future events, and scientific realism with its claims of correlating predictions with truth, are able to escape the whirlpool of inductive circular reasoning (and I don’t think they can) is to me neither here nor there. When it comes down to it, pragmatism and reliability are far more useful, and therefore meaningful, qualities to a person who acts in the real world of experience and data than whether something can be justified deductively or not. Being able to accurately predict, using gathered data from experience and observation, whether or not a certain experiment will result in disaster and death is much more relevant and no less important because it is only ‘justifiable’ inductively.
Inductive reasoning also has self-correcting methods so that when it is unreliable, these observations and calculations get taken into consideration for future calculations. The scientific inductive method is always improving. In this way, it is more likely to bring us closer to the truth than not. It is this view that Hume himself saw as giving the scientific approach as being preferable to one of superstition.
To conclude, it seems pertinent to question the question. Hume’s point is potentially devastating and yet, whilst it can theoretically undermine the entire rubric of scientific academy, it can also be (and is by so many scientists) ignored. The question that really remains is: does Hume’s problem matter?
“We are all convinced by inductive arguments”, says Ramsey, “and our conviction is reasonable because the world is so constituted that inductive arguments lead on the whole to true opinions. We are not, therefore, able to help trusting induction, nor, if we could help it do we see any reason why we should” (Ramsey 1931, 197). We can, however, trust selectively and reflectively; we can winnow out the ephemera of experience to find what is fundamental and enduring.
 I will call his work An Enquiry from hereon in.
 Howson (2000) p.2
 The text I will be using is the Project Gutenberg ebook of An Enquiry Concerning Human Understanding by David Hume. It is a public domain copy released in January 2006, first posted on October 14, 2003.
 Thompson (2006), p. 49, states “In common parlance, ‘law’ is taken to be something which is imposed, a rule that is to be obeyed. But it would be wrong to assume that a scientific law can dictate how things behave. The law simply describes that behaviour, it does not control it (as Hume argued).”
 I am intuitively a follower of the narrow frequency interpretation of probability which asserts that no event can be said to have probability, since there is only one universal outcome (when I am not being entirely Pyrrhonian!).
Dugald Murdoch, Induction, Hume and Probability, The Journal of Philosophy, Vol. 99, No. 4, April 2002, p.186
 As I have set out in detail in my book Free Will? An investigation into whether we have free will or whether I was always going to write this book (2010), Ginger Prince Publications.
 http://plato.stanford.edu/entries/induction-problem/#CanIndJus retrieved 02/06/2011
 http://plato.stanford.edu/entries/induction-problem/#VerConParInd retrieved 4/6/2011
 http://plato.stanford.edu/entries/induction-problem/#VerConParInd retrieved 4/6/2011
 http://plato.stanford.edu/entries/induction-problem/#VerConParInd retrieved 04/06/2011
 http://plato.stanford.edu/entries/induction-problem/ retrieved 04/06/2011
Blackburn, S. (1994;2008), ‘Oxford Dictionary of Philosophy’, Oxford ; Oxford University Press
Howson, C. (2000), ‘Hume’s Problem: Induction and the Justification of Belief’, Oxford University Press ; Oxford
Lipton, P. (2002), ‘Review: Colin Howson – Hume’s Problem: Induction and the Justification of Belief’, British Journal For The Philosophy of Science (2002) 53 (4): 579-583
Merrill, K. (2003), ‘Review of COLIN HOWSON. Hume’s Problem: Induction and the Justification of Belief.’ Hume Studies, Volume XXVIX, Number 1 (April, 2003) 155-162.
Murdoch, D. (2002), ‘Induction, Hume and Probability,’ The Journal of Philosophy, Vol. 99, No. 4, April 2002, 185-199
Russell, B. (1961, 2nd edition), ‘The History Of Western Philosophy’, London ; Routledge
Thompson, M. (1995;2006), ‘Teach Yourself Philosophy’, London ; Hachette Livre UK
The Problem of Induction (Princeton) – http://www.princeton.edu/~grosen/puc/phi203/induction.html
The Problem of Induction, First published Wed Nov 15, 2006; substantive revision Mon Jun 21, 2010, by John Vickers – http://plato.stanford.edu/entries/induction-problem/