Sunday, January 9, 2011

Advances in Baconian Induction: John Herschel (Part 2 of 3)

(Previous post: Advances in Baconian Induction: John Herschel (Part 1 of 3) )

This essay will focus on the aspects of John Herschel’s Preliminary Discourse on the Study of Natural Philosophy that discuss his ideas on causation and induction. Before presenting his rules of philosophizing, which amounts to his theory of how induction works, John Herschel discusses the characteristics of cause-and-effect.

The Five Characteristics of Cause-and-effect

In Part 1, I mentioned how we study phenomena in order to discover connections among things which we term cause-and-effect. Herschel believes that enumerating the characteristics of cause-and-effect will be necessary for our presentation of general rules, rules designed to guide us in examining the facts we’ve collected, and deciding upon their common cause. Indeed, the rules of induction that Herschel presents are informed by the attributes he lists. There are five characteristics, and they can be abbreviated as (1) invariable antecedence, (2) invariable negation, (3) increase or diminution, (4) proportionality, and (5) reversal:

(1) Invariable antecedence: There’s an unalterable connection between the antecedent, the cause, and the consequent, the effect, unless prevented by some counteracting cause. Herschel cautions that it isn’t always obvious how cause and effect works in a given case. An effect may appear gradually while the cause is still accumulating in intensity (e.g. a slow boil), or the cause and effect happen so instantaneously that the interval cannot be perceived (e.g. the generation of lightning).

(2) Invariable Negation: The effect doesn’t exist whenever the cause is absent, unless there is some other cause that is capable producing the same effect.

(3) Increase or Diminution: The increase or decrease in the magnitude of the effect, corresponding to the increased or diminished intensity of the cause, in such cases as admits of increase and diminution. (Like the increase or decrease in pressure applied to a table by your hand with the increase or decrease in energy and effort you put into pushing your hand into the table.)

(4) Proportionality: The effect is proportional to the cause in cases of direct, unimpeded action.

(5) Reversal: Reversal of the effect with that of the cause—when the cause ceases to exist, so does the effect.

Herschel’s “Rules of Philosophizing”: the Ten Rules of Inductive Reasoning

Following his list of cause-and-effect’s features, he makes ten, “observations, which may be considered as so many propositions readily applicable to particular cases, or rules of philosophizing” (Prelim. Disc., p. 152). He characterizes these methods as an “inductive search for a cause,” and describes them as follows (with titles made up by me):

1. The Method of Exclusion: If in our group of facts there is one in which the sought-after phenomena is wanting or the opposite, then such a peculiarity is not the cause we seek. Herschel postulated that causes precede effects, so if there’s a fact which doesn’t reveal the phenomena, then that fact cannot be the cause we’re looking for.

2. The Method of Agreement: When the facts agree in a certain respect in all cases, then this is the cause in question, if not, it is a collateral effect of the same cause; if there is only one point of agreement, then this becomes a certainty. If there is more than one cause, they may be “concurrent causes.”

3. The Method of Strong Analogy: That we do not deny the existence of a cause when we have many strong analogies to support it, though it may not be apparent how such a cause can produce the effect, or even though it may be difficult to think of its existence under the circumstances—we must appeal to experience rather than decide “a priori” against the cause, and try whether to see if the mystery can be unraveled. (He gives the example of the bright sun which we think to be intensely hot, and the question of how light can produce heat or maintain it, neither of which we knew back in his day. We can’t simply deny either inference just due to our ignorance, however.)

4. The Method of Contrary Facts: Contrary or opposing facts are equally instructive for the discovery of causes as are favorable facts. (He gives the example of an iron vessel, the air of which contains moistened iron filings, and leads to the diminishing of its bulk due to some part of the air being taken out and combining with the iron to produce rust. If you examine the remaining air, you will discover that the air will neither support flames (combustion) nor animal life (respiration). This is a contrary fact (neither an affirmation of combustion nor respiration), but it shows that the cause of the support of flame and animal life is to be seen in the part of the air which iron abstracts, and which rusts it.)

5. The Method of Degrees or Intensity: Causes become more obvious when we arrange the facts in order of intensity in which some peculiar quality subsists, though not necessarily, since counteracting or modifying causes may be acting at the same time.

6. The Method of Counteracting Causes: That it is the counteracting or modifying causes, operating unperceived, that prevent the effect of the cause we seek in the cases where the resulting phenomena would have been favorable if not for the intervening cause. Exceptions to a proposed general law can often be removed by removing or allowing for such counteracting causes. (Like Galileo’s thoughts on free fall, and how the resistance of different medium affect the rate of fall of objects as opposed to a vacuum. My example.)

7. The Method of Difference: If we can find in nature, or produce by experiment, two instances which agree exactly in all but one particular, and differ in that one, the influence of that difference in producing the phenomenon must be made sensible. If the differing particular is present in one instance but doesn’t exist in the other, the production or non-production will determine if it is or is not the only cause. This is even more evident if we can make the reverse happen: it the differing quality is then absent in the first case but present in the latter, and the effect is reversed from the first case. But if the total presence or absence of this differing aspect only produces a change in the degree or intensity of the phenomenon, we can only conclude that it acts as a concurrent cause or condition with some other cause to be sought elsewhere. In nature an occurrence of two phenomena agreeing in everything except for one respect is rare, but experiment makes this much easier to produce. This is the grand application of experiments of inquiry in physical researches. This quality increases the value of experiments, since it makes the inquiry into nature more pointed, and its answer more decisive.

8. The Method of Concomitant Variation: If we’re trying to discover the influence of a circumstance, and cannot completely wipe it out or oppose it, we must find cases where it varies considerably in degree. If that cannot be done, we may be able to alter its influence one way or another through introducing a new different circumstance, which we think will likely produce this effect, and thus obtain an indirect evidence of its influence. (Think of a catalyst, it would be the new circumstance that starts or makes more powerful a chemical reaction, for instance. We would do well to remember that it is indirect evidence, and that the new circumstance may have a direct influence of its own, or become a modifying one on some other circumstance (like the air reducing the bulk of an iron vessel, which certainly influences the composition of the vessel).

9. The Method of Residues or Subduction: Complicated phenomena have a plurality of causes, which concur, oppose or are independent of each other (like horizontal and vertical motion), and operate at once, and thus produce a compound effect. The phenomena can be simplified by subducting the effect of all known causes either by deductive reasoning or by appeal to experience, the result being a residual phenomenon that requires explanation. This is the process by which an advanced science progresses (or an advanced philosophical theory, explaining minor technical issues; my note). Most of the phenomena of nature are very complicated; when the effects of all known causes are estimated with exactness and thus subducted, the residual facts are constantly appearing in the form of new phenomena, leading to the most important conclusions. (The small discrepancies of predicting the motion of orbiting objects with gravity as the sole cause, lead to the supposition of a resisting medium as the cause of the discrepancy.) (Another example: François Jean Dominique Arago discovered that if you suspend a magnet on a silk thread and vibrate it, and the air resistance, along with the inability of the thread to perpetually move, will cause the magnet and thread to eventually rest. Place a copper plate beneath the magnet and its motion is further retarded, which quickly leads to a whole new relation of facts (copper and magnetic motion.))

10. The Method of Causal Connection: The detection of a possible cause by comparing gathered-up cases must lead to one of two things: (1) the detection of the real cause and its manner of acting, which furnishes the complete explanation of the facts; or (2) the establishment of an abstract law of nature, pointing out two general phenomena as invariably connected—where there is one, the other also appears. The invariable connection is a phenomenon of a higher order than a given particular fact. When many of these are discovered, we can again “classify, combine, and examine them, with a view to the detection of their causes, or the discovery of still more general laws, and so on without end.” (p. 159)

For Herschel, these are the methods by which the process of induction reasons from phenomena to causes.

It may help to consider five of these methods to be kinds of inferences, and the other five as precautions or tips for the inductive investigator to keep in mind as he searches for a cause. I’ll now restate them as “kinds of inference” and as “causal tips”:

On the five kinds of inference:

(1) The first method (Exclusion) is a kind of eliminative reasoning, in which you reason that some thing or circumstance isn’t a necessary or sufficient cause of an effect or phenomena being studied, based on either observation or experiment.

(2) The second (Agreement) is a more sophisticated form of enumerative induction, in which from the instances it is reasoned that the facts share the same cause in the quality that they have in common, or it’s reasoned that this common quality is a collateral effect of whatever causes the facts being studied; if there is more than one cause, then the induction would conclude that there may be two or more causes which act at the same time to produce the effect (“concurrent causes”).

(3) The seventh method (Difference) is a form of causal reasoning, in which we note how the one factor that a group of facts do not share plays into the production or non-production of the phenomena being studied, adding that it may or may not be the cause, or only a concurrent cause.

(4) The eighth method (Concomitant Variation) is another kind of casual reasoning in which we infer the influence of a circumstance on a phenomena by finding instances in which the circumstance varies in degree, or failing this, introduce a new circumstance which affects the first, giving us indirect evidence of its influence.

(5) The ninth method (Residuals/Subduction) involves either deductive reasoning or inference from an appeal to experience, by which we break down a complex phenomenon of compound effects with multiple causes, correlating known causes with their effects, in order to wind up with a simpler, residual phenomena that requires a new inductive investigation for its cause.

(There is one other method that Herschel considers important for induction: analogical reasoning. I will discuss this more in-depth in Part three.)

On the causal tips:

(1) The third method (Strong Analogy) cautions us to avoid denying that something is the cause of a phenomenon “a priori,” without experience, when that thing has many strong analogies in support of it being the cause. Even if it’s hard to conceive of how the cause could produce the effect, we can’t allow our reason to be sole arbiter over what can be the appropriate cause, or ignore relevant analogical evidence.

(2) The fourth method (Contrary Facts) instructs us that contradictory instances, like air not supporting flame or life (as opposed to the positive instances that normally apply in life), is informative in our search for a cause. This is similar to Bacon’s account of induction, which considers positive and negative (or “contradictory”) instances of the phenomena being investigated.

(3) The fifth method (Degrees) suggests that we arrange the phenomena being studied according to the range of intensity belonging to some quality of them, which may make the discovery of the cause easier. The arranging of phenomena by their intensity in order to discover the cause was first used in Bacon’s theory of induction, in his Table of Degrees or Comparison.

(4) The sixth method (Counteracting Causes) warns us that in cases which don’t convey the effects of the cause we’re investigating, it is due to counteracting or modifying causes, maybe acting in a manner that cannot be directly perceived.

(5) The final one (Causal Connections) remarks that the detection of a possible cause must lead to either the detection of a real cause and manner of acting which explains the facts, or the formation of an abstract law of nature, which describes two general phenomena as being invariably connected.

It’s important to note that Herschel believes that these inductive rules are more like general guidelines than a strict methodology, which is contrary to Mill’s inductive method. (But Mill’s theory of induction is a topic for another time.) This also means that he doesn’t believe that scientific induction must discover certain causes in some necessary order (like heat, then radiation, then molecules, etc.). In his presentation of how to induce a theory of dew-formation (following the theory of Dr. William Well, Essay on Dew (1818)), prior knowledge that heat radiates from objects was crucial for understanding the cause of dew, but Herschel remarks that, even with no knowledge of heat radiation, our induction of dew would nevertheless had made the fact of radiation known to us. Following this idea, he says:
In the study of nature, we must not, therefore, be scrupulous as to how we reach to a knowledge of such general facts: provided only we verify them carefully when once detected, we must be content to seize them wherever they are to be found. (Preliminary Discourse, p. 164, Aphorism 170)
Herschel then proceeds to list the ways in which we can verify the inductive conclusions we reach.

The Three Methods of Verifying Inductions


It is a tendency of the human mind to speculate, to leap forward, on the basis of a sketchy analogy between some phenomena, to a cause or law. Because of this, many of our most important inductions must be considered as conclusions drawn from few cases, and “verified by trial on many.”

“Whenever,” Herschel states,
therefore, we think we have been led by induction to the knowledge of the proximate cause of a phenomenon or of a law of nature, our next business is to examine deliberately and seriatim [that is, points taken one at a time] all the cases we have collected of its occurrence, in order to satisfy ourselves that they are explicable by our cause, or fairly included in the expression of our law… (ibid., p. 165, Aphorism 172, words in brackets mine)
An induction has to be able to account for and explain all the cases and phenomena from which it was generated. Herschel advises the inductive reasoner to examine all the cases he has gathered up to determine if each and every one of them can be explained by the induction, and that they all are properly represented in the inductive law or generalization that has been formulated.

What Herschel decides to do with “contradictory instances,” with instances that seem to disconfirm the induction, is well worth noting. He remarks that any exceptions of the induction must be “carefully noted and set aside for re-examination at a more advanced period,” where at this more advanced stage of knowledge, the cause of the exception might become known, and afterwards the exception may turn out to be an affirmation of the induction in a way that was never thought to be so.

In verifying our induction, the steps needed for the verification will differ depending on whether the cause or law we’ve reached is already known and generally recognized as a general causal law, of which the phenomenon we were studying is merely one additional effect of this general cause, or if it is less general, less known, or altogether new. If it is less known, less general, or new, then our verification, examining all the known cases and noting that they all agree with the induction, will suffice, because it shows that induction really does fit with the facts that are known. But if it is generally known and recognized as a more general cause, “the process of verification is of a much more severe and definite kind,” Herschel imparts. This more severe kind of verification traces the relevant causal actions with more precision, and this precision is reached by modifying the circumstances of each case gathered in the induction; we test the circumstances and estimate the effects of our tests, in order to show, “that nothing unexplained remains behind.” He modifies this point by stating that we’re only concerned with explaining this known cause or causes, not with unknown modifying causes.

If unknown modifying causes occur, we’ll first discover their existence by the presence of residual phenomena that occur in our evidence for our induction. If an induction is really valid and a comprehensive one, then any unexplained phenomena that remain after comparing the inductive conclusion with its cases, in all their circumstances, must be a residual one (instead of a necessary indicator that the induction is false). The residual phenomena then become the subject of another train of inductive reasoning to discover what their cause or law(s) is. This is how inductions become more general and more specific and how new sciences rise up:
It is thus that we may be said to witness facts with the eyes of reason; and it is thus that we are continually attaining a knowledge of new phenomena and new laws which lie beneath the surface of things, and give rise to the creation of fresh branches of science more and more remote from common observation. (ibid., p. 166, aphorism 174)
An example of an induction leading to residual phenomena that Herschel discusses is the gravitational law--that planets are kept in their solar orbits, and moons in their planetary orbits, by an attractive force which decreases in strength as the squared distance increases—which historically led to discrepancies in explaining the motions of the planets, and larger ones in the cases of the moons. These two became “residual phenomena,” were studied in subsequent inquiries, and determined to be cause by the same gravitational law, but now applied to the mutual attractive force of the planets on each other (answering the issues of measuring solar orbits) and the force by which the sun influences the motions of the moons (answering the issues of planetary orbits).


The second verification of inductions is the induction’s capability to predict new phenomena that are analogous to the ones that were originally considered in the formation of the induction. For an inductive law of nature to be general enough to serve as a foundation upon which greater inductions may be built, the law of nature must be universal in its applications. And we won’t know if it will be general enough to apply to more than what instances were used to form it, unless we’ve already experienced the law’s ability to do that very thing—to allow us, before trial or experiment, to state what will happen in cases similar to the ones already included under the induction.

To verify the induction in this way, we must extend its application to cases not originally part of the induction; this means carefully varying the circumstances under which the causes act in our induction, in order to determine if its effects are truly universal. This includes applying the inductive law to extreme cases. Herschel illustrates the importance of the extreme case with Galileo’s inductive conclusion that gravity’s acceleration was the same on all bodies, great and small, remarking that Galileo couldn’t prove this with extremely light objects like feathers or cotton due to the counteracting resistance of the air during their fall. The invention of the air pump, however, allowed this law of acceleration to be tested by an extreme case: Isaac Newton exhausted the air from a glass using the pump, and dropped a guinea (a British coin) and a downy feather at the same time, the result being that they struck the bottom at the same time. Of course, in air the coin would strike first, followed by the feather as it slowly floats to the ground. After giving this example, Herschel announces, “[let] any one make the trial in the air, and he will perceive the force of an extreme case.”

(For some cool videos demonstrating the validity of Galileo’s induction that resistance is what prevents objects of different shapes and weights from falling at the same rate of acceleration, see the famous “Apollo 15 Hammer and Feather Experiment” and a new version of the “Guinea and Feather” experiment made by a person using his personally-designed apparatus.)

A further requirement is to be applied to inductive laws whose expression is quantitative: its universal validity must be established not only by subjecting it to trial in all manners of varying circumstance, but every trial made must be of precise measurement. The means for subjecting the quantitative law to trial should also be designed such that the trial can be repeated many times in order to make any deviations from the law apparent.


Consilience is the third, and the best, verification criterion for an induction. And it consists of completely unsuspected verifications of the induction arising from areas of study least expected. Herschel focuses on the psychological aspects of consilience, in that an induction is consilient when unsuspected or unknown cases or even groups of facts actually verify the induction’s truth when they weren’t expected to, especially cases that were at first considered hostile to the induction’s validity. “Evidence of this kind,” Herschel remarks, “is irresistible, and compels assent with a weight which scarcely any other possesses” (ibid., p. 170, Aph. 180).

He states that this is often the case with “residual phenomena”:
Unexpected and peculiarly striking confirmations of inductive laws frequently occur in the form of residual phenomena, in the course of investigations of a widely different nature from those which gave rise to the inductions themselves. (ibid., p. 171, Aph. 181)
The term “consilience” was coined by Herschel’s good friend and fellow scientist, William Whewell, who also worked out his own theory of induction. As I’ll discuss in my future essays on Whewell, consilience had this psychological aspect for Whewell too, but also a causal, logical element: a “jumping together of inductions” in the form of causal unification of different event or process kinds into more general kinds (a more general induction), whose members all share a common cause or property. (The last sentence is paraphrased from Dr. Laura Snyder’s “Reforming Philosophy,” p. 182)

Science as a Process of Induction and Deduction

The lower stage of induction that’s been discussed is how we reach proximate causes, laws of nature which apply to vast amounts of phenomena of certain kinds. Deduction is the method by which we trace out these laws into their farthest reaching consequences and effects. Indeed, “it is very important to observe, that the successful process of scientific enquiry demands continually the alternate use of both the inductive and deductive method.” (Prelim. Disc. p. 175, Aph. 184)

Speaking further about the relation between induction and deduction, he states:
The path by which we rise to knowledge must be made smooth and beaten in its lower steps, and often ascended and descended, before we can scale our way to any eminence, much less climb to the summit. The achievement is too great for a single effort; stations must be established, and communications kept open with all below. To quit metaphor ; there is nothing so instructive, or so likely to lead to the acquisition of general views, as this pursuit of the consequences of a law once arrived at into every subject where it may seem likely to have an influence. (ibid.)
This is also how greater inductions (hypotheses and theories) are verified, by deductions made from the inductive conclusions using specific facts, and testing to see if the theory fits with the results. I’ll cover this relationship more in Part 3.

Part 3 will cover Herschel’s views on analogy, the greater inductions called “theories,” the role of hypothesis, and the three ways for discovering the general laws which are the foundation for these theories.

(Next post: Advances in Baconian Induction: John Herschel (Part 3 of 3) )

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