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I have finally finished my major essay on Induction which incorporate many of my ideas on the topic in a much better written essay, which I present below. 06/07/2014

[Revised 10/09/2022 after some criticism of it by Brett Hall at: ]



by Thomas M. Miovas, Jr.


After giving it considerable thought, I think it is possible to incorporate concept formation, generalizations, scientific induction, and philosophical induction under one general term of “Induction.” I think there are enough similarities between the mental processes -- including observation to abstraction, omitting measurements , the unit perspective, and that each leads to a product that is open ended within a range -- to say the similarities are sufficient to warrant having one general concept of induction. In fact, what got me thinking along these lines is the following passage from Ayn Rand's “Introduction to Objectivist Epistemology”:

“Thus the process of forming and applying concepts contains the essential pattern of two fundamental methods of cognition: induction and deduction. “The process of observing the facts of reality and of integrating them into concepts is, in essence, a process of induction. The process of subsuming new instances under a known concept is, in essence, a process of deduction.”

[Now, what is meant by creating an abstraction? It means mentally isolating out certain aspects of that which we observe, and retaining it in our minds, until we can omit the measurements and integrate those aspects to form a concept.]

Aristotle defined “induction” to be “the process of reasoning from the observation of concretes or individuals to a general or universal conclusion." I think this is a good working definition and I think that type of ability to abstract from particulars to the more general conclusion stems from the human mind's ability to omit measurements. While this has been explicitly identified as an aspect of concept formation by Ayn Rand, I think it is implicit in the types of inductions that Dr. Peikoff and David Harriman discuss, and the purpose of this essay is to draw out those similarities to make the case that there are four types of inductions and that these have enough similarities that they can be integrated together into one global conception of “induction.” One doesn't always think of forming a concept as the same thing as drawing a reasoned conclusion from the facts, but I think it is clear that sometimes a great deal of thought and effort must be put into concluding if there are enough similarities between known things to incorporate them into one concept. As a pointer to that type of reasoning stemming from observations in an inductive process leading to a conclusion and then to a concept see how Miss Rand handled the concept of “justice” and how much effort she put into making the case very clear that some things are similar while others are not in her discussion of this term.

[For those times in which one makes further and further inductive generalizations (see below) I call this "reiterative induction," meaning that as one makes observations as to what something is doing and identify the various causes of the actions observed, one can organize one's thinking in a refined manner. An example would be making observations of cats and then after some further investigation into the causes of cats conclude that cats have different DNA than, say, an elephant or a human. This sort of process is not circular thinking, but rather refined observations and integrations according to the various causes.]

Ayn Rand's “Introduction to Objectivist Epistemology” (or how the human mind works) with Dr. Peikoff's course on “Induction in Physics and Philosophy” and David Harriman's book “The Logical Leap” answer the question of how can we go from observation to abstract understanding, which is a unique approach based on the philosophy of Objectivism to the issues of universals, how they are formed, and what rules and guidance can be given to those trying to form proper and valid concepts, generalizations, and scientific laws of nature, and philosophical inductions.

Historically, induction has been taken to mean going from the observation of particulars to abstract thinking, which I think is possible within Objectivism only if the measurements are omitted. So part of my approach will be to show that measurement omission is involved in all four types of induction: concept formation, generalizations, scientific induction, and philosophical induction.

For concept formation, one observes the facts of reality, notices similarities of various things observed within a context or an abstraction from the background (which requires a type of selective mental focus), unites the various observations together by omitting measurements while retaining the differentiated characteristics, names the concept (i.e. “cat”) and then defines this concept either by being able to point to instances of those things which are subsumed under the concept or in terms of concepts already created so as to start building a conceptual hierarchy. The concept “cat” can be understood by pointing to various cats observed or by referencing to other concepts, like “A cat is a four legged [observed thing] with claws and fur and meows as a means of communication.” Once one has a series of first-level concepts that are generally identified by referencing that which can be observed in reality (the referent and meaning of the concept), one can then use these concepts together to make sentences (such as the above definition of “cat”), or to build concepts from concepts (abstractions from abstractions) to get higher-level concepts stemming from a noticed similarity between things that have already been conceptualized (i.e. “animal” from “”bird,” “cat,” “snake;” or “running” which is an abstraction from the conceptualized animals that can run; or “jumping”, or “clawing”, or “communication”). Technically speaking the above definition of “cat” is formed from both first-level concepts and higher-level concepts, and would not be the definition given by a young child, who may not yet have the concepts of “animal” or “communication.” At the early stages of concept formation or using concepts and defining them, a young child with a small vocabulary would point to the household cat and say, “Cat!” which is an ostensive definition in this context.

So, one makes observations, mentally abstracts out observed similarities, takes any of the observed existents as a unit to compare to the other similar existents, omits the measurements and unites the units in order to make the abstraction a specific mental entity (the concept) and this concept is open ended in the sense that the same concept would apply to any future observations of other things that are similar to the first observed things that were conceptualized, such that those newly observed similar existents can be mentally incorporated into the previously formed concept (i.e. the household cat is then seen as being similar to all other animals that have the characteristics that the household cat has, such as those he sees at neighbor's houses or on TV or on the Internet).

As it turns out, there are also first-level inductive generalizations that work similar to first-level concepts in that one can simply point to those aspects of reality, and then state the causative generalization – i.e. “Flipping the light switch turns on the lights”, or “Pushing on a ball gets it to roll,” or “Typing on a keyboard displays alpha-numeric characters on a computer screen.” A first-level generalization is one in which the causal sequence or relationship is given in observation. In other words, for those causal events observed causality is given in observation and requires a minimal effort to abstract out the relationship.

This is a form of induction as presented by Peikoff and Harriman in their works on induction and it is important to point out that the statement must be a statement of a causal relationship and not just a generalized observation that is not connected to causation – i.e. “All swans are white” is not a proper or valid causative generalization, and so it is not an inductive statement according to Peikoff and Harriman, because it is only by speaking in terms of identified causes that one can be assured that the conceptualized abstract inductive generalization will apply to all members of that identified causative class. “All swans are white” is only an identification of the swans one has observed in the past, but since it is not causative in nature, one has no assurance that any future swans seen will be white (in fact, historically, they were all thought to be white in Europe, until some black swans were found in other parts of the world). In other words, it is by identifying causes that one comes to understand the world abstractly in terms of scientific principles, when other types of historically considered inductive statements would not discover what the cause of the effect is and would not provide any guidance as to whether to be able to expect more of the same relationship identified or not.

At the stage of first-level causative inductive generalizations, usually the concepts already exist (one has formed them previously) and one is simply relating concepts to other concepts in a causative manner without any further types of measurement omission involved because the measurement were already omitted when one formed the original concepts (i.e. “switch”, “light” “ball”, “rolling”, “typing” and “computer screen” were already formed following the process of concept formation stated above). From time to time new concepts may have to be formed to fully conceptualize the inductive generalization, such as when one observes that a changing electric circuit current leads to a fluctuating magnet or compassed placed near it, but in this case, the concept for the causative relationship would have to be formed (i.e. “electro-magnetism”). So concept formation can definitely be involved in identifying a given causal inductive generalization, but does not have to be involved if the concepts were already formed before making the new observations. [With the electric current running through the wire, one can observe by turning it off and on again that the compass needle moves and is therefor caused by the electric current doing something to the magnet. After further investigation, we find that a changing electric current induces a magnetic field which influences the compass needle.]

With a generalization one can have something like, "Flipping the light switch turns on the lights." This would be a first-level generalization of the type spoken about in Peikoff's course and in The Logical leap. But notice what can be done with this. Because it is conceptualized, it covers all types of electrical switches and all types of lights being turned on by them. It can be the usual switch in a house (the flip kind with a little throw switch) used to turn on all sorts of lights one can have in one's house; it covers even much bigger switches turning on more powerful lights, like at a sports stadium; it covers even things like motion sensor switches for many offices these days; it covers laser switches that were popular to indicate someone has walked in your store and might turn the lights on in a certain section; etc. This means that inductive generalizations are open ended in much the same way that concepts are open ended – they relate to many different instances of the conceptualized inductive generalization.

But notice that just as one has a unit that serves as a standard in concept formation (i.e. any given dog can be used as a standard for the concept of "dog"), one can keep the whole causal sequence in mind (of turning on lights) such that any given means of turning on light can be used as a unit that serves as a standard to make further integrated observations, such that when one comes across some unique way of switching on lights, one doesn't have to start from scratch and re-conceptualize the causal sequence.

In other words, I am expanding on the idea of a unit that serves as a standard and applying this to generalizations, scientific induction, and philosophical induction, each of which deals with particular causal sequences of specific types. Instead of just retaining one item at a time like one does with concept formation and having one thing as the unit the serves as the standard, the entirety of a causal sequence can be retained and it can be used as a unit that serves as a standard for any further understanding of similar causal sequences. In this sense, measurement omission is involved in a bit wider range than just omitting the measurements from particular things observed to form a concept. Each instance of the observed causal sequence is like one thing observed by the individual and instead of forming a new concept, concepts are united via a language base to cover all similar types of causal events or causal sequences. So, the measurement omission and the open endedness of generalizations stemming from observed causes is like a step up from simple concept formation. The entire causal sequence is retained in an abstract manner, and anything sufficiently similar to that particular causal sequence is covered by the generalization formed.

Once one has these first-level inductive generalizations, one can, in effect, form second- and third-level generalizations from these conceptualized generalizations by integrating these separate observations into other newer such generalizations. For example, let's say one has the generalization “Flipping the switch turns on the lights” and “Typing on a keyboard makes alpha-numeric characters on the computer screen” and with some further mental work, one can unite these two together to get “Pressing or throwing buttons or switches connects an electric circuit to do work electronically.” Or, in the case of electro-magnetism, the scientific principles or inductive generalizations were well known for electricity and others for magnetism, but with a bit more observations it was found that one could integrate these two together into a broader generalization that a changing electric field leads to a magnetic field and conversely a changing magnetic field leads to an electric field.

Notice that the aspects of concept formation are involved in this process. One makes observations, mentally isolates out from the observations (forms an abstraction), and then one can use the various instances of the causal relationship as a type of unit (a dynamic unit) such that any one of the observed causal relationships being observed can be seen as similar other such observations, one forms new concepts as needed, and the final product, the inductive generalization, is open ended in that it applies to any future observations of similar causal sequences.

As a further thought, it might be wondered that if the generalization has units that can serve as a standard, then why don't we form new concepts to identify these identified causal connections? I think sometimes we do this, but it isn't really necessary because these causal phrases just are not that difficult to retain in one's mind as a sentence, but also if we did this, I think it would leads top a great deal of confusions if each and every type of generalization was given a concept of its own. In other words, naming each of the types of generalizations one can identify would be violating the crow epistemology and would be multiplying concepts more than is necessary, so we don't do that as a general principle.

For scientific induction of the types that leads to natural laws, many of the same principles are involved. When one first forms the first-level or higher-level generalization, an extra step or two is involved to move from a simple generalization towards a principle of natural law. Instead of just using ordinal numbers of less or more of the standard, one relates the causal components together in precisely identified mathematical units of cardinal measurements requiring a systematized standard of units of measurement (that is an integrated system of mathematical measurements). This requires re-introducing the measurements into the conceptual framework that one omitted from forming the concepts in the first place.

For example, to get the relationship that force is equal to mass times acceleration (F=ma), one would have to measure the force exerted on the object, the mass of the object, and the acceleration of the object, for each instance of one's observations. There would be many such measurement data points, each with a different mathematical unit value, and these could be placed into a chart or some type so that one would have a series of observations tabulated out. That is, one would have F1, m1, a1; F2, m2, a2; F3, m3, a3; etc. each with their own quantity of measurements. From this data, one could abstract out the exact mathematical relationship between force, mass, and acceleration after one omits the measurements while retaining the causal relationship. [The cause, in this simple case is the force acting on a body and getting it to change its direction of motion. There are more complicated relationships for this equation, but I am not going into that here.] So, for scientific induction, one does wind up with measurements between the components that one then has to omit to get the precise mathematical relationship between the components stated abstractly, F=ma. In other words, measurement omission is explicit in the formation of a scientifically induced law of nature stemming from the fact that the measurements are re-introduced, and then have to be re-omitted in order to come up with the abstract form of the natural law. And note that these laws of nature stated abstractly are open ended within a range, just as the case for concepts and generalizations. And any given example of a natural law observation can be used as a unit the serves as a standard. That is, any particular instance of a force acting on a body and getting it to accelerate can be taken as a unit of the conceptualization of F=ma.

A further observation of the unit perspective for such scientific laws written in mathematical terms (equations) is that each valid scientific equation reduces to 1=1 (the unit perspective in mathematics) when one takes only the units of measurement into account but not their specific values in relations to one another. That is, taking only the units into account in the equation, and doing mathematics on their mathematical relationship, each legitimate equation reduces to 1=1. A simple verification of this is that since each equation is equal across the equal sign, then dividing both sides of the equation by either side of the equation results in 1=1. F=ma; F/F = ma/F; F/F = F/F; 1=1.

Other examples of natural law written in mathematical form using the above process are the universal gravitational equation: Fg=Gm1m2/d^2, kinetic energy: Ek=1/2(mv^2), conservation of momentum: m1v1 + m2v2 (before interaction) = m1v1 +m2v2 (after interaction), and many more. I'm not going to derive them all here, just pointing out that each one of these would follow the principles stated above of having a variety of specific measurements that would be tabulated in some form, then the measurements would have to be omitted to arrive at the abstract form of the relationship as an induced natural law based upon observed causal relationships, and in each case any particular individual observation would serve as the unit for that class of observations.

In scientific induction one has the observation of similarities, abstracting out from a context to get the generalized causative relationship, any of the observed causal sequences can serve as a unit of comparison for future reference, re-introduce the measurements to get the exacting relationship, omit these measurements once one discovers the abstract causative law, which leads to that law being stated abstractly in a mathematical form, and this mathematicalized law of nature is open ended in the sense that it would apply to all future observations of similar causative events one observes, just as what occurs with concept formation and generalizations.

For philosophical induction, the broadest type of induction possible because the focus is on causal events that show the fundamental nature of reality or the relationship between man and reality, the same basic principles stated above for concept formation, generalizations, and scientific induction are involved, except that due to fact that conceptualizations of philosophical principles induced from observations concern concepts of consciousness, exacting cardinal numbers are not involved; but one can use ordinal numbers of less and more and then form the causative generalizations and principles in philosophical terms. One would have various concepts already at hand, make one's observations, state these in terms of concepts (or forming new ones as necessary), and identify the causes of that which is observed, and then state the principle in the broadest means possible for that class of observations and their causative connections.

As an example, one might observe throughout history or what is directly available to you at work or home that rewarding a man for the values he presents to you leads to him being more productive. This is a causal relationship and can probably be stated more succinctly as: “Rewarding a man for his productive efforts generally leads to him becoming even more productive in the future.” Other types of causative philosophical inductions include: “A romantic relationship is based upon a mutual sense of life as a response to the character of another,” “Thinking works best if one organizes one's mind according to similarities,” “Since a man has a volitional consciousness, he must choose his values wisely according to what he is and according to what benefits him,” “Historically, societies thrive best who follow reason instead of Faith,” and “An entity acts according to its nature.” Of course, there are many more inductive principles that one can arrive at by making observations according to philosophical causes, but these will give the reader the general idea.

Note that while a philosophical induction is not stated in terms of explicit cardinal numbers, like a scientific induction, it is based upon an observed causal relationship. And like concept formation, it does have units that can serve as a standard in the fact that any one observed particular instance of the philosophical induction – i.e. a particular romantic relationship – can be used as a model or a standard for all other types of similar causative relationships. And like concept formation, the measurements are omitted (simply because concepts are being used); though in a sense, one can say that further measurement omission is involve in that one states the principle not in terms of less or more (the ordinal measurements), but rather as a causative statement that includes the range of possibilities within the statements. For example, one can say that one loves one's wife more than one's mother-in-law, or that rewarding a man a little or a lot will influence his greater production to a lesser or greater degree, or note that societies that are a mixture of reason and Faith do not thrive as well as a fully rational culture, etc. In other words, the idea of the statement holding true for a range of observations or being open ended within a range is true for philosophical inductions.

So, a philosophical induction based upon observations leading to a philosophical principle has measurement omission (in the concepts and further in that it includes less or more of the cause involved), a unit that serves as a standard for identifications of future instances of the principle, and it is open ended within a range due to its statement in an abstract form.

While I cannot make the case that due to the similarities between concept formation and the types of inductions identified by Peikoff and Harriman are at the same level, due to causation being central to three of them but not to concept formation, I think one can say there are enough similarities to incorporate them into one higher-level concept of “Induction.” In other words, I think a proper conception of induction would include concept formation, generalizations, scientific inductions, and philosophical inductions since I have shown that each of these has several aspects that are similar to each other.

And a friend of mine just pointed out something that is interesting regarding whether causation is the specific differentia between concept formation (which supposedly is not based on causal considerations) versus the types of induction Peikoff and Harriman talk about. He said that if causation stems from the nature of an entity, and if the concept of an entity contains everything you know about that entity, then for concepts of entities they do have a causal connection in that what an entity can do or might do is contained in the concept of that entity. For example, the concept of “dog” does contain what a dog does, like barking, running, drooling, eating, wagging its tail, etc. So, I'm not even sure causation can be ruled out for concept formation, especially when one considers that concepts of actions are abstraction from abstractions – i.e. “running” stems from an entity that has legs and is capable of running, as an abstraction from an abstraction. We do not form concepts of actions apart from those things conceptualized that can act or behave certain ways. But this does not hold true for all concepts, and so I cannot state definitively that causation of the type that an entity acts according to its nature is involved in all types of the process of forming concepts, though it is an interesting observation.

Proper concept formation requires items to be integrated according to similarities, and for higher-level concepts that incorporate previously formed concepts, there can be differences at the lower level that are abstracted out at the higher level. For example, we can have the concepts of “bird,” “turtle,” “snake,” and “dog” that are different from one another due to, in part, their means of locomotion and the types of skin coverings or skin protections that they have. However, when one integrates these concepts together into the higher-level concept of “animal” these differences are abstracted out, as if they were measurements that are omitted at the higher level. The definition and concept of animals would not require having the concepts of, say, feather, shell, fur, or skin as these would be differences abstracted out in forming the concept of “animal” which is more focused on, say, a means of locomotion and awareness of existence by some sensory means.

Likewise, what I am attempting to do is form a higher-level concept of “induction” from the similarities pointed out between concept formation, generalizations, scientific induction, and philosophical induction whereby their differences are measurements that are omitted on that higher-level concept of “induction” (in its most general form). [By the way, it is not circular reasoning to say that some new scientific inductions require new concepts which are themselves inductive, because at some point one must get to the observed facts of reality and it is reiterative induction that leads to certainty for scientific and philosophic understandings.]

Aristotle defined induction as the process of reasoning from the observation of concretes or individuals to a general or universal conclusion. I've tried to show that there are definitely times when one has to put considerable thought into deciding if two or more observed things belong in the same concept or not. I think I have shown that concept formation, generalizations, scientific induction, and philosophical induction all have enough components that are similar that these can be integrated into one concept of “Induction.” Each starts with observing existence and organizing one's mind according to observed similarities between the units observed, omitting the measurements so as to form an abstraction of those units so as to be able to integrate those units together, having a unit the serves as a standard, and leading to a final product that is open ended within a range. In this case, the fact that generalizations and scientific induction and philosophical induction are based on causative elements whereas concept formation is not always, is abstracted out to retain a very general and abstract understanding of existence based upon observed facts of reality; and the cardinal nature of forming a scientific law of nature is abstracted out retaining the similarities of the mental processes of forming concepts, generalizations, and philosophical inductions.

In other words, I am attempting to do what one does when one has the concepts of “snake,” “bird,” “cat,” and “turtle” whereas on one level of abstraction, these are seen as different, but on a higher level of abstraction, we come to conclude that these can be integrated into one concept of “animal” by retaining the similarities while dealing with the differences as measurements omitted at the higher level of abstraction. There are significant differences between concept formation, generalizations, scientific induction, and philosophical induction on those levels of abstraction, but looked at more globally, there are enough similarities that these can be integrated together into one concept at the higher level of abstraction.

Graphed out, concept formation would be lower than the special inductions, then the special inductions would be at a slightly higher level, with a final cap on all four under the concept of “Induction.” See the chart below.

The following hierarchical chart is induction going from bottom to top, reduction going from top to bottom, and is deduction insofar as one comes across a new instance of either a concept or a causal sequence when one tries to incorporate that new instance observed into something already conceptualized:



Scientific Induction, Philosophical Induction


Concept formation

The observed facts of reality, including introspection