On Definitions

When two people converse, each necessarily assumes the other is using words like themself. Yet frequently, neither person could define the very same words they are using.

If you show a picture to a three-year-old and ask if there is a tree in it, you will likely get the correct answer. If you ask a thirty-year-old what the definition of a tree is, you will likely get an inconclusive answer. We didn’t learn what a tree is by studying the mathematical definition of trees. We learned it by looking at trees. Y. S. Abu-Mostafa, Learning From Data

It is difficult to define words because they are contextual and recursive.

When a veterinarian asks whether a recently sedated dog is conscious, we know they are asking if the dog is a awake. When a contemplative person asks whether a dog is conscious, we know they are asking if the dog’s mind is somehow like a human’s mind.

When a botanist is told by their friend to “turn right after the fourth tree,” they know when to turn, despite the second and third plants being shrubs.

Words can have several meanings. The word “conscious” can mean awake, or self-aware. We infer the appropriate use from context.

The more fundamental problem with definitions is that they are recursive.

A dictionary contains these two definitions

If definitions are recursive, then how do we know what they mean? Consider this set of definitions:

Clearly, we don’t know the meaning of “A”, “B”, or “C.” If our language is also recursive, where does meaning come from in our language? I think that the meaning of abstract words like consciousness is built up from more concrete words in a huge web of definitions. At the lowest level, the meaning of words comes directly from association with our sensory inputs and motor outputs. Thus, meaning bubbles up from the concrete uses at the bottom, to their more abstract uses at the top.

Recursive definitions are especially problematic during abstract philosophical discussions. I have had heated conversations resolve when both sides put aside the debate for a moment to define their words, only then to realize they had been agreeing all along!

I think this is one of the reasons why many philosophical treatises begin by defining their terms. Philosophical definitions are different than the definitions we find in a dictionary. A formal definition is an intended equivalence between a word and other words.

The definition of the word “definition” is seen to have two uses. The first is the standard definition—“what people usually mean.” The second is a formal, almost mathematical, equivalence between a word and other words.

A standard definition is wrong when it is, in fact, not what people usually mean. For example, the word “book” does not mean “yellow fruit.” On the other hand, a formal definition can not be wrong—it can only be useless. We can formally define the word “book” to mean “yellow fruit” if we like, but it is not very useful. This distinction between standard definitions and formal definitions is often obscured because, to aid our memory, we often provide formal definitions for words that are very similar to that same word’s standard definition.

Words that are too broad or too narrow are not useful for making distinctions. Everyday words make distinctions that are useful in everyday life. Specialists develop lingo to make finer distinctions.