Inductive Abstraction - Amateur Logician

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Inductive Abstraction

* “Analytic Judgements” versus “Synthetic Judgements”
* Abstraction
* Inductive Abstraction
* The Four Causes and Uniformity of Nature
* Induction by Enumeration
* Hasty Generalization
* and Unrepresentative Data

The human being is not a disembodied spirit nor a “brain in a vat.” Central to an Aristotelian-Scholastic perspective is the belief that our knowledge, in one way or another, traces back to experience. That is, we learn via experience.

In this, we have both sense powers and intellectual powers. While we have an innate capacity to learn, in addition to certain intuitive insights over basic “axioms” such as the law of contradiction, the vast majority of our knowledge is obtained by extensive observation, experimentation, and testimonies.

Remember, good deduction is totally reliant on good premises.

Sometimes these premises don’t require any extensive investigations or experiments in the world. There’s a sense, however, that all knowledge still traces back to our experiences. For example, by internal reflection.

What are called “analytic judgements” in Scholastic terminology have it that the subject and predicate are somehow logically linked together. Usually this is so such that the predicate is contained in the intention of the subject. We know that a square has four right angles, and we can deduce from the concept of “square” that its angles add up to 360 degrees. Much of mathematics consists of “analytic judgements.” They don’t require experimentation. We can know, e.g., that the whole is greater than its parts. That’s an analytical judgment. Intuitively, we know it is true. And, in praxeology, we can know that human action uses means to obtain ends. We know this through internal reflection.

Other premises are called “synthetic judgments.” These require investigations or experiments in the world. The temperature in which water boils or freezes, for example, is a “synthetic judgement” in Scholastic terminology.

Note, however, that contemporary philosophy defines “analytic judgements” and “synthetic judgments” differently than Scholastic philosophy.

In contemporary philosophy, an “analytic judgment” is a judgment that’s true only by definition. Yet, on the contrary, in Scholastic philosophy, it’s defined more broadly to include things that are not just true by definition. That the angles of an Euclidean triangle add up to 180 degrees is true, though it is not true only by definition.* To reach that conclusion takes a few steps in a proof.

Review the article on the predicables; they relate to this.

*[To be sure, Euclid’s fifth postulate concerning parallelism, mathematicians have discovered, is logically equivalent to a triangle that has its angles add up to 180 degrees. Hence, we can start out an axiomatic system by stating that the definition of a triangle is to have 180 degrees. Even so, we can deduce things from that that are then not true simply by definition.]

Induction and Observations

Induction raises several issues and potential problems. The latter, philosophically speaking, deal with how we can take our experiences of particular things and particular events to then generalize about entire classes of things and events. We have only met a finite number of human beings in our own experiences, for example, but what is it that allows us to make general statements about all human beings. That is, how can we legitimately assert a proposition like “all men are mortal”?

It’s a challenging question, since attempting a thorough answer requires disentangling various sub-issues of both metaphysics and epistemology. Yet, from an Aristotelian-Scholastic perspective, a key component is the human ability to abstract.

It’s our ability to focus-in on what’s essential and to negate what’s not essential.

This is deeply a metaphysical issue!

In order to say that beings A, B, and C are “this kind” of being, it’s because they are similar and, what’s more, this similarity is of essential characteristics. They have the same qualitative pattern or inner intelligible “shape.” It is this—what’s called the “essential form”—that makes a being to be the kind of being it is rather than some other kind of being.

It’s my essential form that makes me a man, not a gorilla.

And we confront a world in which we find several beings belonging to the same “species.” In fact, there are several species to be found, and multiple beings belonging to these different respective species. The members of a species are alike in that they are of the same species, yet unlike in that they are different from each other individually. This is the issue of the one and the many.

By self-reflection and encountering other human beings, we begin to form judgements about what it means to be human. When thinking about any human being, a number of things can be noted. Certain things are specific to an individual person. For example, a person’s skin color or height. Such features are obviously not defining characteristics. They are not commonly shared. What is essential, though, is being an animal and having the potential capability to be rational.

Note that it’s not uniqueness as such that is important in forming a universal concept of the human person. Perhaps, for example, there are other intelligent life-forms in the universe. The issue is instead what best explains what the human person is.

Abstractive induction generalizes.

It takes our experience with a human being (or a number of human beings) and generalizes what a human being is — or at least partially is.

Learning more about the human being — or something else — may take other inductive approaches. Some of those approaches may require far more sophisticated inductive techniques. Learning about human anatomy and medicine, for example, is far more difficult. Sophisticated statistical approaches may be necessary when determining the effectiveness of a medicine. And in physics or chemistry, for example, mathematical models must be developed and tested in their predictive power through the scientific method.

The Four Causes & Uniformity of Nature

Generalizations ultimately concern causality.
There are at least four types: efficient, formal, material, and final.

– The formal cause is what makes a being what it is.
– An efficient cause is what makes something change or be.
– The material cause is what something is made of.
– And the final cause is that which something aims at or how it acts.

Induction can concern any of the four causes. We can induce the formal cause of a man. But then, how do we know it applies to others? Additionally, how do we know it will apply in the future?

The answer is that there is a uniformity of nature.

If the future were totally chaotic, random, and unpredictable, there would then be no relatively reliable means available to us in order to achieve any of our ends. Human action would be impossible since means would have no propitious association to ends. The very study of physics or biology, moreover, would be impossible. Hence the fact that there’s an order to the world makes it intelligible, and thereby theorems are discoverable.

This is not only true of efficient causes, i.e., a cause that brings about something or changes something, but also of formal causes, i.e., that which a thing intelligibly is.

Hence, we can make generalizations. We can claim that e = mc^2. That is true on Earth and on the Moon. And we can claim that men are rational animals. That is true for those in the U.S. – and even those in Canada or Australia.

Induction by Enumeration

One method to infer an attribute of a group of things is to count them up, i.e., to enumerate. Observe individuals of a group one-by-one to then reach a conclusion about that group as a whole.

“Complete” enumerative induction counts up every member of the group.
“Partial” enumerative induction counts only some of the members of the group.

With partial enumerative induction, the premise states that only some members of a group have been observed (i.e., a sample) and the conclusion states an inference about all members of that group (i.e., a population).

We have to be particularly careful with partial enumerative induction.

First, we need a sufficient sample size. Other things being equal, a better induction has a large sample size.

Second, the sample must be representative of the group in question. It shouldn’t be “biased” in favor or against any possible sub-group of the group. A sample that is more varied is therefore better.

Third, we should give a degree of caution to the conclusion based on how large the sample size is and how representative that sample is.

The third point shouldn’t be overlooked. Other things being equal, an inductive argument that has a higher degree of caution is “stronger” than one which does not.

In any case, violating the three rules will lead to fallacious reasoning.

An insufficient sample size leads to the Fallacy of Hasty Generalization. Someone observes a predicate belonging to only a few members of some group and then “hastily” concludes that predicate must belong to every group member.

Alternatively, someone might have a bias sample, which leads to the Fallacy of Unrepresentative Data. Someone wants to do an election poll and then draws up a large sample composed of only those who attended a specific politician’s campaign rally.