Question

You cannot have any sort of inductive argument without having some sort of base set to make an inductive reference from. But choosing elements in that set requires some notion of them being similar to each other. But similar in what respect? Then, you must show that the new object you are making an inductive reference about is similar to that base set. Again, similar in what respect? These questions are clearly fundamental to induction. Of course, the notion of similarity is also fundamental to any argument from analogy for the same reason.

Most of philosophy deals with whether induction is valid, but induction can't even get off the ground without considering answers to these questions. It is almost as if the answers to these questions are presumed. Even the classical inductive case of all swans being black implying the next one being black presumes that the similarity we are talking about is with respect to a property of being a swan, but clearly each black swan in that set would be unique. The property of being a swan involves ignoring differences among those animals and focusing on what is similar between those animals, but similar in what respects?

Induction is often the basis of almost all arguments that are used to figure out the truth about the world. And the truth about the world is often what we care about the most. Why then is there such little philosophical talk about what makes an object similar to another? There is a good amount of psychological research on this subject yet barely any to be found in philosophy apart from the more recent philosophers such as Nelson Goodman. How did such a fundamental concept escape the minds of ancient philosophers?

Such things are commonly discussed by philosophers. The issue goes back at least as far as Plato with his theory of forms and Aristotle with his distinction between essential and accidental properties. In more recent work, the issue is usually discussed in the context of questions about universals and properties, e.g. whether universals are real and correspond to natural kinds, or whether they are just nominal distinctions.

To some extent, we can make use of the fact that we are the product of natural selection, and if we weren't good at making distinctions our species would not have survived. It is vital to be able to recognize and distinguish the similarities between tigers and the similarities between sheep and the differences between tigers and sheep. Humans are good at such things and would not have prospered without that capacity. But such similarities only take us so far.

In the context of inductive reasoning, Nelson Goodman pointed out that projecting properties and relations from observed cases to unobserved ones always involves focusing on some properties and not others. The future always resembles the past in some respects and not others: the trick is to work out which ones matter. Relying on the similarity of appearance can easily be deceptive in this respect. In science, sometimes things may appear similar but be significantly different, or appear different but be similar in some important way. Distinguishing the important similarities from the rest requires a great deal of scientific hard labor.

It is not just inductive reasoning that depends on understanding similarity. Deductive logic does also. One can write a formula such as (∀x)(Fx ⊃ Gx) but what exactly do F and G mean? They may be clear-cut in simple mathematical examples, but as soon as we try to apply a formula to real things and properties, we need to be able to specify what is included and what is not. If we want to use that formula to represent the proposition all people are mortal, then what exactly is a person and what exactly is mortal?

I sometimes wonder whether, in the last analysis, all reasoning isn't just reasoning by analogy. Nearly all of geometry deals with the concepts of similarity. What makes a line segment equal to another? What makes a line similar to another? What makes one triangle equal to another? What makes a triangle similar to another? These relations are expressed as equations. All circles have the same equation. Similar circles have different values for the equation variables. Euclid was doing this as early as 300 BC. What is essential for an inductive argument to occur?
a. Deductive reasoning
b. A base set for inductive reference
c. Mathematical equations
d. None of the above

Answer 1

An inductive argument requires a base set for inductive reference to draw generalizations from specific observations. Philosophers like Nelson Goodman have addressed the issue of relevance in induction, and in both science and logic, distinguishing relevant similarities is a key challenge.

Step-by-step explanation:

The essential element for an inductive argument to occur is (b) A base set for inductive reference. This base set allows us to make inferences by looking at patterns and regularities from specific observations. Induction is the process of reasoning from specific instances to broader generalizations, which is vital in the scientific method and our everyday reasoning.

Even though philosophy has engaged with the problem of induction - a notable example being David Hume - more recent philosophers like Nelson Goodman have considered the problems related to the 'new riddle of induction' and the need to distinguish relevant similarities. Goodman highlighted that in projecting properties and relations from observed cases to unobserved ones, we must decide which properties to focus on, as these properties and relations will be relevant in some respects and not others. This act of categorizing and recognizing relevance is crucial in science as well as in moral reasoning.

In logic and mathematics, deductive reasoning also relies on a notion of similarity when applying logical formulas to real-world situations. The challenge lies in defining the essential properties that make entities similar, which is necessary for both inductive and deductive arguments.