5.2.5 Russell’s Postulates for Non-Demonstrable Inference

Among the many ideas Bertrand Russell explored was the problem of showing how we can use non-demonstrable (or non-deductive) inference to draw legitimate conclusions about the world. Take two examples:

If at one moment you see your cat asleep by the fire and later you see it in a doorway, you are confident that it has passed through intermediate positions from the fire to the doorway, although you didn’t see it doing so. Because you were not a witness of the movement, there is no form of deductive logic that would prove that it is the same cat – it could be a completely identical duplicate. Common sense tells us that this is highly unlikely.

Or suppose you are walking along and you notice a shadow following you. You jump and it jumps. You stop and it stops. A reasonable inference is that it is your shadow, but it could equally be a dark spot on the ground with an independent existence that is following you around.

Surprisingly, we must make some assumptions that allow us to trust our inference that the cat walked from the fire to the doorway and that the dark spot is, in fact, only our shadow. The inferences we use in our daily lives and in science are of this sort. But what are the principles underlying this activity? What must the world be like for these non-deductive inferences to be warranted? What grounds do we have for believing that what simply must be true is indeed the case? What extra-logical principles must be true if we are not mistaken in cases like these? Regarding the cat, there must be some principle of endurance or constancy of objects that we assume without any more basic supporting evidence. In the case of the shadow, there must be some concept of causality (our body causes the shadow) in nature on which we can depend.

To provide support for making these types of common sense, but non-deductive inferences from “hard data” (external facts) to “soft data” (derived or inferred interpretations of the facts), Russell provided five postulates. In fact, it was this set of postulates that motivated me to put together this entire blog – the rest grew up around it. They support conclusions we make about our experiences that, although highly likely to be true, cannot be absolutely proven by the use of these or any other postulates. They are self-evident assumptions unaccompanied by proof, which he considered necessary to justify the kind of non-demonstrative inferences about which none of us typically feel any doubt.

Why is something like this needed? Why can’t science prove all its assertions using more basic facts, laws, and theories? If every justifiable belief could be justified only by reference to some more basic belief, there would have to be an infinite chain of such justifications. Because such a chain of proofs cannot reasonably go on forever, the only way to stop it (Russell argued) is to define a set of beliefs that are not proven by any references to more fundamental assumptions. Such are these postulates. They exist a priori and are non-demonstrable, though extremely reasonable. They are foundational postulates and serve as the bedrock upon which all other demonstrable inferences are based.

There is no weakness in resting on postulates. Every branch of mathematics has its first principles that shape the proofs and theorems that arise from them (e.g., in plane geometry: “through any two points, there is exactly one line”). It only makes sense that behind every “proof” is either another set of proofs, or some unproven and unprovable first principles (just pick up any calculus book). It can’t go back to infinity, nor can it loop back on itself or else it becomes tautologous. Postulates do not imply weakness – in fact, they are required. They are not “faith”, but are the required building blocks from which any system of empirical knowledge is constructed.

Although Russell asserted that these postulates were required to keep science from being mere “moonshine” his chief support for them was that they were biologically advantageous. That is, they conveyed survival benefits. He didn’t have great confidence in these precise postulates, but came up with them out of a sense of necessity. Without them, the inductive principle cannot be logically justified:

quasi-permanence - There is a certain kind of persistence in the world, for generally things do not change discontinuously (a kitten becomes a cat, but is still the same entity). Given an event, "A", it is likely that in a neighboring time, and at a neighboring place, there is an event very similar to "A" (pertains to continuity of time, space, and events).

separable causal lines - There is often long term persistence in things and processes. From one or two members of a series of events, we can infer something about the other members of the series. This postulate covers our experience of physical motion. It replaces the concept of a thing changing its position by that of a related series of contiguous events. This principle enables us, from partial knowledge, to make a probable inference. The most obvious examples are such things as sound waves and light waves. It is owing to the permanence of such waves that hearing and sight can give us information about occurrences. It is only on the basis of the idea of causal lines that we can infer distant events from near events.

spacio-temporal continuity - Denies action at a distance. When there is a causal connection between two events that are not contiguous, there must be intermediate links in the causal chain such that each is contiguous to the next, or (alternatively) such that there is a process that is continuous.

structural postulate - Allows us to infer from structurally similar complex events ranged about a center to an event of similar structure linked by causal lines to each event. That is if you see several similar events arranged about a center, there is something in the center that has causal lines connected to those distributed events.

analogy - Allows us to infer the existence of a causal effect when it is unobservable (where there is smoke, there is fire). If there is reason to believe from previous evidence that A causes B, then when you see A but no B, you can assume that there is a B somewhere hidden. Or if you see B but no A, there is probably an A somewhere hidden.

Paraphrasing Russell, “The inductive principle is incapable of being proved by an appeal to experience. Experience might confirm the inductive principle as regards the cases that have been already examined; but as regards unexamined cases, it is the inductive principle alone that can justify any inference from what has been examined to what has not been examined. All arguments which, on the basis of experience, argue as to the future or the unexperienced parts of the past or present, assume the inductive principle; hence we can never use experience to prove the inductive principle without begging the question. Thus we must either accept the inductive principle on the ground of its intrinsic evidence, or forgo all justification of our expectations about the future. If the principle is unsound, we have no reason to expect the sun to rise tomorrow, to expect bread to be more nourishing than a stone, or to expect that if we throw ourselves off the roof we shall fall. All our conduct is based upon associations which have worked in the past, and which we therefore regard as likely to work in the future; and this likelihood is dependent for its validity upon the inductive principle."

"The general principles of science, such as the belief in the reign of law, and the belief that every event must have a cause, are as completely dependent upon the inductive principle as are the beliefs of daily life. All such general principles are believed because mankind have found innumerable instances of their truth and no instances of their falsehood. But this affords no evidence for their truth in the future, unless the inductive principle is assumed."

Wittgenstein didn’t see it this way. He outlined in Tractatus that you cannot infer one state of affairs (elements of reality) from one another. There is no logical “law” of cause and effect. No “causal nexus” exists in nature. Cause and effect may be useful, but are not provable or even necessary. Similarly, induction (accepting the simplest law that can reconcile our experiences) has no logical justification, but only a psychological one. We would go quite insane without laws of nature. The facts that constitute the world are utterly disconnected. There is no internal, necessary, organic bond between them. He essentially rejected the postulates that Russell proposed. So, as clearly and carefully as Russell laid out his postulates, there is no unanimity of agreement as to their soundness.

David Hume had a similar take on this as Wittgenstein (predating him, of course). Hume believed that all we had was a stream of unconnected impressions from which we formed ideas. We have a psychologically based sense of a "constancy of perception" and the coherence between the unconnected individual perceptions that are the corner stone of the common sense belief in the existence of an external world, and of the continuity of objects over time (that Russell's cat that was in one room is the same cat that crossed over into the other room). There is no way to deduce this - we only infer it, and we believe it to be the case as a matter of habit and convenience. In each case of this type of thing happening, we move from constancy and coherence to it’s being the same object represented. And if it is the same, it must be that the object has existed unperceived through the interval we might not have been watching it, and so it must be something external to the mind.

In closing, here is a little Russell joke -
Q: Why did the kitten cross the road?
A: One must assume the postulate of quasi-permanence to infer that it is the same kitten before and after crossing.

I guess you had to be there. He tells it funnier...

5.2.4 The problem of induction

How shall we deal with the intractable problem of induction? As a means of predicting future events, it can’t be proved either deductively or inductively, which essentially exhausts opportunities for logical justification. Just because something has always occurred a certain way, you can't deduce that it will continue to do so (unless you take as a premise that the future will resemble the past, which just assumes exactly what you are trying to show). And you can't use induction to show that that future events will resemble events observed so far, because that also is what you are trying to prove. Just because induction has worked in the past doesn't mean that it will continue to work. Hume’s solution was to say that we should not refrain from making inductive inferences, but simply realize that we are not being governed by reason when we do so. As Bertrand Russell pointed out, the turkey believes it will be fed every morning until one day when the farmer comes with an axe instead of grain. It is habit, experience, and custom that allows us to rely on inference. We continue to use it because it is the most reliable technique available, despite having no ready proof.

Pragmatic approach
Pragmatists agree with Hume that there is no epistemic justification for induction. Instead, they present a practical explanation of why one is justified in using this method of inference. For one thing, it works better than any alternative, which is a primary selling point for most pragmatic positions. Induction will work if anything will work! Even though the future cannot be known, we can’t avoid having expectations about it. We would be wise to choose a method that would lead to success. One simple way of conceiving of the problem is in a truth table: The world either is uniform or it isn’t. And we can choose to use induction to predict future events or choose not to. Expanding on this exercise, we have six possibilities:

1. Nature really is uniform and regular:
1. induction would be a very reliable method for predicting future events.
2. using some method other than inductive reasoning would be ineffective.


2. or, nature is “somewhat” uniform, and frequently (but not always) evinces a pattern or connection between past and future:
1. induction is of some help, and works as a tool as often as nature chooses to be regular.
2. Non-inductive inference is as reliable as a wild guess.


3. or, nature really is not uniform at all, and there is no significant pattern or connection between past and future:
1. induction is of no help at all.
2. Non-inductive inference is also of no help at all.

Thus, the non-inductive method is useless no matter whether nature is always uniform, somewhat uniform, or chaotic. As for induction, it will certainly be helpful at least in the case when nature is uniform or mostly uniform. Thus, it is rational for us to prefer this method of inference since it is the only one that has any chance at all of being correct.

Avoiding Induction
Karl Popper attempted to resolve the question of induction and inference in science by abandoning the troublesome problem altogether. He had no issue with our inability to conclusively “prove” the validity of the inductive method. If scientific hypotheses (or even less formal everyday hypotheses) are stated in ways that allows them to be falsified, then we can use deductive techniques rather than induction to test them. This technique employs the logical form called “modus tollens”, which was discussed in the section, Moore’s Proof of an External Reality. Knowledge is gradually advanced as tests are made and failures are accounted for. A typical deductive formation of this argument would be along these lines:

• If (some hypothesis is true), then (we will observe some effect)
• We do not (observe some effect)
• Therefore (some hypothesis is true) is proven wrong

Applied to the sunrise, we would say, “If sunrises follow nighttime, the sun will rise tomorrow morning. We do not see the sun rise in the morning. Therefore theory about when the sun rises has been disproved”. As long as we continue to see sunrises after night has passed, we should not reject the theory. When tested many times in many conditions (for example, watch the sunrise from many spots on Earth), and it is never disproved, our confidence in the theory increases (but never becomes 100%). We tentatively and provisionally accept it, barring evidence to the contrary. We should, then continue having confidence in our theory. This is the essence of “falsifiability”. Science, then, could be thought of as a collection of hypotheses that have not been disproved (yet), but none has been conclusively proved, either.

Popper’s seemingly simple argument has not gone unchallenged, though. Among the professional philosophers of science, his view has never been taken as a serious alternative to the consensus theory of probabilistic induction (which takes into account the relevance and weight of evidence, Bayesian probability, and other mathematical representations originated by Carnap and others).

The primary element to consider in Popper’s view is that he believed that focusing on induction, or characterizing our generalizations about the future as exercises in induction was fundamentally mistaken. With finesse worthy of Wittgenstein, he “vanishes” the problem instead of solving it. Statements about how the future will unfold, according to Popper, do not actually employ induction, but instead rely on a technique that only superficially resembles it – the use of tentative hypotheses about future outcomes of our everyday experiments. We conduct an experiment of this sort every morning we look out the window expecting to see the sunrise. When we see it appear over the horizon, we can’t conclusively state that our theory about sunrises is true, but we can say that it has (once again) passed a well-constructed, though informal, test – that our theory can be retained as tentatively valid, useful, and worthy of further testing. We have strong confidence in it because of the countless confirmations of its predictions, and because it is never disproved. It has high verisimilitude, meaning, it correlates strongly with reality.

Like all scientific theories, it cannot ever be completely verified, but it can be quickly falsified. This may appear to share the structure of induction, but it stops short of the end result of induction in that we don’t use the results of this process to construct a general rule from individual outcomes. Instead, we simply can say that, once again, the theory has been corroborated - that it is a very useful and productive theory. We may also construct other theories that have similar logical structure to it regarding moon rises, the rising of Venus, etc. As they are verified, they each help to validate each other and boost our overall confidence in the set of interrelated theories. Popper referred to this as the “Method of conjectures and refutations”. If the sun were to stop rising and not resume its daily circuit across the sky, we would eventually abandon our theory of sunrises as having been falsified. In his view, we can’t accept any theories about the world as absolutely true, regardless of the amount of confirmation they have accumulated. But, those that are consistently corroborated can be made use of because of their eminent practicality and utility, keeping in mind that those same theories may need to be discarded if “eliminative evidence” accrues against them. Although this may seem like a facile manipulation of emphasis, it isn’t – this is exactly how scientists treat all scientific theories, no matter how well established.

Inference to the best explanation
Just as Popper got around the problem of induction by simply dismissing it, others circumvent its difficulties by getting around the issue in other creative ways. One such argument involves “inference to the best explanation” which was first introduced in the chapter on Modern Philosophy of Science. In this view, when one considers all the possibilities for how the future could unfold based on how the past and present events are manifested, the conclusion that the future will resemble the present requires the fewest assumptions, inventions, and stretches – employing Ockham’s Razor – to select the most likely explanation among any of several possible candidates. For example, if one sees a wet sidewalk it is reasonable to assume that it either rained or that the sprinklers were turned on, and less likely that a wave of water suddenly soaked it. Given experience and our knowledge of cause and effect, we can confidently make predictions (using inference) as to past causes of current events and current causes of future events. Drawing other conclusions would require greater leaps of improbability and would stretch credulity. Those who hold this position assert that it is imminently rational to assume that there is order and structure to the universe and that laws of causality actually work, and it would be highly impractical and wildly irrational to assume otherwise. Given the available choices, reliance on induction is the only one that makes any sense.

In fact, inference to the best explanation is used in everyday life far more frequently than either deduction or standard induction. It is how we draw conclusions from partial information. It is how we evaluate social interactions, judge intents of others, and understand potentially ambiguous statements. It is the primary tool of medicine, science, and all other forms of research and discovery. Deduction, although important, is principally a tool used in mathematics, logic, and philosophy.

In making these types of inference, we infer from the fact that a certain hypothesis would explain the evidence, to the truth of that hypothesis. In general, there will be several hypotheses that could potentially explain the evidence, so we systematically consider each one and reject all the least likely ones, leaving one remaining. In this manner, we are able to infer from the premise that a given hypothesis would provide a "better" explanation for the evidence than would any other hypothesis, to the conclusion that the given hypothesis is true. Accepting one of the less probable ones would simply be perverse.

Relax the burden of deductive proof
Another technique for dismissing the inductive problem is to admit and concede that using the strict rules required for deductive logic can’t be applied to induction, which is a fundamentally different and looser form of reasoning. The truth-preserving nature of deductive reasoning doesn’t work when used to justify a reasoning process in which the conclusions are, by definition, not certain. The conclusions of inductive arguments exceed the content of their premises – individual cases when used to construct a general rule necessarily go beyond themselves. However, with deductive arguments the premises contain everything necessary to systematically arrive at a definitive conclusion – the conclusion is inescapable. According to this argument, it is simply inappropriate to impose the tough standards of deduction on the fuzzier process of inductive logic.

As we have seen, inability to disprove a proposition does not render it true. For example:

• Although the Omphalos and Solipsistic positions are immune from disprove, all reasonable people agree they are beneath consideration.
• Russel's celestial teapot and the Flying Spaghetti Monster (blessed be his name) cannot be successfully defeated through argument. But, all satire aside, they are not really out there.
• The many varieties of supernatural mythology all create beings or histories or forces that are beyond the means of science to disprove. This doesn't make them real.
• There is an astronomical number of other incredible claims that bear similar logical structures to the above examples that also are unsusceptible to the power of logic. They are not, therefore, all true.

Likewise, no one can disprove this claim: "Reliance on induction is unwarranted". That does not automatically render this proposition true. If we can't disprove that "induction is groundless", reliance on induction is not, therefore, groundless. In fact, it is "probably true".

So, as with all assertions about the future (which are what both scientific theories and the inductive process concern themselves), they are beyond positive proof, though their conclusions (when supported by much confirming evidence) are well worth relying on. Interesting.

Probabilistic approach
For all of human history and as far back in time as we can collect evidence, the laws of cause and effect and the uniformity of nature have existed, unchanged. It is completely true that we can't use that past run to make conclusive statements about the future continuation of this consistent track record. However, it would be a gigantic leap of faith to assume that all of this will suddenly change as soon as I finish typing this sentence... See, nothing changed! If these laws were going to change at some time, and that time has not occurred in the last several billion years, there is not a shred of evidence that indicates that it is going to occur in the next few seconds, years, or centuries. From a purely probabilistic framework, the odds of everything being turned topsy-turvy exactly right now are very very very slim when measured against all of the opportunities for change that came and went in the past. For this reason it would be rational to assume the present trend is likely to continue, and highly irrational to assume it will not. For all practical purposes, for all of us, for the rest of our lives and the rest of humanity's existence, the chances of something like this that has never ever occurred and shows no sign of occurring now, are not likely to suddenly happen. Although we can't prove that the continuity of past/present/future will persist, a betting man could reliably count on it.

Even Deduction Cannot Be Proved
The problem that Induction has is that it cannot be proved deductively, and using induction to prove it would be circular. However, the same type of attack could be made on deduction, even though no one serious contemplates abandoning that reasoning technique. Lewis Carroll, the author of Alice in Wonderland, wrote in one of his stories that reaching conclusions by deduction can only be justified by an appeal to deductive inference, yet that doesn't dissuade us from believing that it is a valid methodology. We still consider it to be a rational approach to problem solving, so why would we have a higher standard for induction? If you were to try to convince a person of

• if p then q
• p
• therefore q

and they rejected it how would you respond? They might agree to "if p then q" and also agree to "p", but still not believe "q", because they don't accept the rules of deductive logic. The only response is to tell them that they are not being logical, that you can deductively show them their error, that they are not following the rules of deduction! But that is the issue - they don't accept the rules of deduction, and the only response you can give them is that they really ought to. So, we see that induction can be inductively justified after all, because even deduction can only be given a circular (in other words, deductive) justification.

People still debate the many ways of viewing the inductive process and its legitimacy. There are complex mathematical and probabilistic arguments too intricate to try to try to explain here. However, it is fairly clear that there is no clear cut and unambiguously convincing logical argument in its favor. We are left with one of several responses – to agree with Hume that there is no legitimate rule of inference as induction, and rely either on habit and experience. Or we can agree with Popper that we are mistaken in calling what we do "induction". Or with those who argue that requiring a proof of the validity of induction is not needed, requiring a less rigorous proof of a process that itself is less than purely rigorous. Also there is the argument that we don't require a non-deductive proof of deduction - we think of the rules of logic and deduction as fundamental and intrinsic to our concept of rationality. There is nothing more fundamental that we can use to demonstrate that deduction is justified. The same might also be said of induction - it is just fundamental to what it means to be "rational". It remains an interesting, and still unsolved, problem.

See chapter 2 of James Ladyman's Understanding Philosophy of Science, called "The problem of induction and other problems with inductivism" here for several more responses to the Problem of Induction. I just wish I would have read that before writing this chapter, because he does a much better job than I!

5.2.3 David Hume and Induction

The study of how induction and inference were involved in the acquisition of knowledge was a cornerstone of Hume's epistemological research. He believed that reliance on induction was fundamental to to making determinations about things when they go “beyond the present testimony of the senses, and the records of our memory”.

We all act as if we believe the world behaves in a consistent and regular manner; that past patterns of behavior will persist into the future, and into the unobserved present. This persistence of regularities is sometimes called the Principle of the Uniformity of Nature, which is discussed later in this document.

Hume wrote that we could not conclusively prove the principle of uniformity in nature, because justification comes in only two varieties, and both of these are inadequate. These two types of reasoning are commonly called deductive (or a priori) and inferential/inductive (or a posteriori). The uniformity principle cannot be deduced because past regularity in nature is no guarantee of future regularity, no matter how probable we may think it is. There are no general principles inherent in past events that compel belief in the orderly progression future events. It is conceivable that nature might stop being regular at any time, as it has on rare instances in the past (consider the occasional, uncommon meteor strike, supernova, or earthquake). We can’t logically maintain that nature will continue to be uniform because it always has been up to now, because this way of reasoning uses induction to prove that induction is valid. This is circular reasoning (discussed in the Infinite Regress Problem section earlier in this document). Thus no form of logical justification will rationally warrant our inductive inferences. Yet we still believe in them.

Hume’s solution to this problem was to say that natural instinct, rather than reason, explains our ability to make inductive inferences. It is our natural instinct that allows us to connect this intuitive series of propositions together:

• In our past, the sun has risen every day
• Based on what we know about how sunrises work, there is no evidence to suggest that this will not continue to be the case
• Therefore the sun will rise tomorrow

Our expectations about such things depend on the relation of cause and effect. It is our common sense about this relation that tells us that depending on tomorrow's sunrise is a reasonable expectation. However, if all matters of fact are based on similar types of causal relations, and if all of these causal relationships depend upon induction, then we must somehow demonstrate that induction is valid.

Hume uses the fact that induction assumes a valid connection between a proposition like "today the sunrise followed a long period of darkness" and the proposition "tonight's darkness will be followed by a similar sunrise." We connects these two propositions not by reason, but by induction.

Probably the first modern philosopher to exhaustively study the problem of induction, Hume was followed by many who tried to address the problems he posed. But they still continue to puzzle us. He argued that it is just as possible to conceive of a contrary proposition to the sun rising:

"That the sun will not rise tomorrow is no less intelligible a proposition, and implies no more contradiction, than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply contradiction, and could be distinctly conceived by the mind."

In other words, just as we can’t prove the sun will rise, we can’t prove it won’t rise. We can conceive of either course, and neither would contradict any firmly accepted premises. As Hume said elsewhere, existence (or non-existence) cannot be proved through a priori reasoning unless one or the other would cause a contradiction. Of course, everything we know about how orbiting objects work tells us that it would be quite an unlikely feat to stop this well understood phenomenon from happening. But how do we really know that these same physical laws will persist, that the future will resemble the past? This reasoning has to be either a priori or a posteriori. Hume contended that it couldn’t be a priori (deductive). If we were to see the sun rising for the first time we would never discover from that event alone what produced it, just as a child can't use reason to stop himself from touching a flame for the first time. Only after having experienced the pain does the child learn the relation. Knowledge of such causal relations must come only through experience of the relations between objects – therefore our reasoning must be a posteriori (inferential) and require the collection of evidence, not the exercise of pure logic.

Our expectations of the future matters of fact lies in the relation of cause and effect, say both Hume and common sense. "By means of that relation alone, we can go beyond the evidence of our memory and senses". The only way we could obtain knowledge of causality would be to infer it from our past observations of regularities. Our prediction of future events based on the past observations is not a rational activity, but just a matter of habit and an intuitive sense of probability – the odds of the sun not rising are infinitesimal. When we project findings about these relations into the future, we must use an intermediate premise, the uniformity of nature, which is risky, because it can change at any time and be proven false. The chicken thinks that the human will always bring it grain until the day he comes with a hatchet. According to Hume:

"It is impossible, therefore, that any arguments from experience can prove this resemblance of the past to the future; since all these arguments are founded on the supposition of that resemblance. All inferences from experience, therefore, are effects of custom, not of reasoning."