Definitions - Amateur Logician

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Definitions

Definition = definiendum + definiens
The definiendum is that which is to be defined.
The definiens are the words used to do the defining.

What good is an argument without clearly defined terms (or names)?

Good logic necessitates clearly defined terms.

A definition of a term should — ideally — summarize what it is referring to essentially. Here we are concerning ourselves with a universal term which is applicable to many things of a common nature.

This is what is called a “real definition” or a “logical definition.” It is to reveal what is the common essence. It implicitly invokes the predicables. This is done through classifying the term to be defined as a species that falls under a specified genus that is different from other species of that specified genus through a specific difference.

“Man is a rational animal” is a real or logical definition. It tells us why man is what he is. A man is a man because he is a rational animal. Man falls under the genus animal and has the specific difference of rationality.

Often we fall short of that. Most definitions will be nominal.

A “nominal definition” gives us the meaning of a word, not necessarily the essence of what it is referring to. Usually we have to settle for a nominal definition. But a good nominal definition should thrive at providing at least a good description of what it is denoting.

Example of a universal term’s *intension* and *extension*.

Remember, the intension of a term concerns attributes and the extension concerns all of the members that are concretely referred to by the term.

For example, a “dog” is a domesticated carnivorous animal with a long muzzle, a fur coat, a long tail, etc. These are attributes of what we call a “dog.” They are accordingly the term’s intension. Its extension includes everything that is in fact a dog. This includes the famous dog Lassie, Rin Tin Tin, Toto from The Wizard of Oz, etc.

A good definition will have the same extension as what’s being defined. It will thereby cover each and every member of the term without leaving any member excluded. It will also be “intensional” in the sense of it reaching for the real meaning of the thing in question. It should give a sufficient description, if not a summary of its essence, of what the thing really is.

How do we arrive at a good definition? Generally, we start from experience. This means we look at individual examples. There are things in the world that share a common nature. We should try to find out what makes these things similar. What is their common nature? What are the common attributes? That will guide us as we form a definition. Our definition should neither be too narrow nor too wide. We want to only refer to the individuals in question with our definition.

Five Rules for Defining. . .

Rule One: It should be adequate such that the definition is not too broad and not too narrow. That is, the definition should have an equal extension as the term’s.

Rule Two: It should reveal the intension of the thing being defined. That is, the definition should express the correct meaning of the term. The essential or important attributes of the term should be expressed.

Rule Three: It should be clear, not obscure. No definition should be ambiguous or vague. Figurative language should be avoided.

Rule Four: It should not be tautological or circular. The subject term and the predicate term shouldn’t be identical or synonymous, since then no new information would be conveyed.

Rule Five: It should be expressed, if possible, in a positive (not negative) manner. Don’t define something by what it is not, but by what it is.

Limits in Defining. . .

Individuals cannot be strictly defined. Rather, we can only describe an individual. Definitions are of the universal. There’s no definition of “Socrates.”

Also, that which is of the highest genera cannot be strictly defined.
These don’t fall under any genus as a species with a specific difference.

Examples include “being” and “unity.”

Being is “transcendental” because it transcends, or moves beyond or above, all the different classifications, groupings, and divisions between and within beings. It’s totally all-inclusive. Being transcends over all of the differences of being. Metaphysics is the study being.

Some Types of Definitions: Lexical, Theoretical, Stipulative, & Causal

A “lexical definition” — a.k.a. “dictionary definition” — only reports how a particular word is being commonly used in a particular language community. It’s descriptive, not strictly prescriptive. Thus we can challenge a lexical definition if it’s flawed or not precise enough.

Decreasing the vagueness of a word lexically defined to become more precise is a “precising definition.” It tends to decrease the extension and increase the intention. For example, what exactly is “small”? Context matters! A precising definition of the term can make it more exacting to fit a particular context.

A “theoretical definition” evokes a theory to scientifically or philosophically elucidate a term. It both explains the meaning of the term and provides a scientific or philosophical explanation of it. A physics textbook defining “work,” unlike an ordinary dictionary, will likely mention theories of energy and Newtonian or Hamiltonian equations.

A definition assigned by fiat, that is, without explanation or justification, is a “stipulative definition.” A first attempt to define something new or unknown tentatively is a stipulative definition.

Moreover, consider that in mathematics a whole system of theories can be derived from various axioms and stipulative definitions. These terms might even be “undefined.” In geometry we have “points,” “lines,” and “planes.” Axioms relate these “undefined” terms.

Then we have “causal definitions.” Here the definition concerns a thing’s cause(s).
There are four types of causes: efficient, final, formal, and material.

A Brief Tour of the Four Causes . . .

A cause provides the reasons or grounds for something’s existence, essence, or its various and changeable modes of being. They contribute, in some way, to a thing being what it is presently. Thus causes play an ontological role, i.e., they are real features about or of being.

And this grounds the possibility for things to be intelligible to us.

Efficient Cause: is that which brings an entity into existence or changes the manner in which it exists.

It answers questions like…
— Who or what made it?
— What caused it to change?

Final Cause: is that for the sake of which the entity exists or is made for.

It answers questions like…
— Why was this made?
— What purpose or goal does it have?
— What kind of effects does it tend to produce on other entities?

Formal Cause: is that which makes an entity be a particular kind of thing versus another kind of thing.

It answers questions like…
— What is it?
— What is its essence?
— What shape does it have?

Material Cause: is that which an entity is made of.

It answers questions like…
— What is it made out of?

Even More Types of Definitions: Extensional & Intensional

Another viewpoint makes a distinction between “extensional definitions” and “intensional definitions.” A definition can be expressed via a term’s extension or intension.

An “ostensive definition” is formed by pointing. Hence it’s extensional. A “tree” is this object, as you point to a tree. It gives no more explicit meaning than that. So, is it strictly speaking a definition? Perhaps not.

But then, wouldn’t all “extensional definitions” be disqualified? After all, they don’t explicitly give a term’s intension. A traditional definition is always intensional. Yet it’s possible to give non-traditional definitions. They have a use!

To be sure, we should not ignore the mind’s power of abstraction. Implicitly a very rough intenstional definition will be formed. A kid given an ostensive definition of tree will easily be able to identify most other trees in the world. That’s by rough abstraction, i.e., by noticing characteristic notes. Hence there’s implicit intensional content.

Along with ostensive definitions, another extensional definition is the “enumerative definition.” It has no explicit intensional content as it merely lists members of the term. For example, a “scientist” is “a person like Galileo Galilei, Johannes Kepler, Sir Isaac Newton, and Albert Einstein.”

Intensional definitions include “real definitions” and “nominal definitions.”

It also includes “synonymous definitions.” It defines a word by another word having the same meaning. However, doesn’t this violate Rule Four of a good definition? Still, there are literary reasons to resort to this sort of definition.

Similarly, there’s “etymological definitions.” These define a word in terms of its origins as a word.

Another intensional definition is an “operational definition.” Here the meaning of a word is connected to an experimental test to ascertain whether a thing in question falls under this word. It’s related to a “theoretical definition.” Something is such-and-such “if and only if” it passes the test XYZ.