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...