Definition

A definition is a semantic statement of the meaning of a term, like a word, phrase, or other set of symbols. It helps us understand what a word really means. There are different ways to give definitions. One way is called an intensional definition, which tries to explain the sense or idea behind a term. Another way is an extensional definition, which lists the things that a term describes. There is also something called an ostensive definition, where you show the meaning by pointing out examples.

Sometimes a word can have many different meanings, so it might need more than one definition to explain all its senses.

In mathematics, definitions are very important. They give a clear and exact meaning to new mathematical ideas by describing exactly what qualifies as that term. Together with axioms, definitions are the foundation that all of modern mathematics is built upon.

Basic terminology

A definition is a way to explain the meaning of a word or group of words. The word or words being defined are called the definiendum, and the words or actions that explain its meaning are called the definiens. For example, in the definition "An elephant is a large gray animal native to Asia and Africa," "elephant" is the definiendum, and the rest of the sentence is the definiens.

Definitions can come in different forms. Some definitions are found in dictionaries and explain common meanings of words. Others might show an example, like pointing to an animal and saying, "This is an Asian elephant." Some definitions make words more specific, such as saying that "large" for female Asian elephants means any individual weighing over 5,500 pounds.

Intensional definitions vs extensional definitions

Main articles: Intension and Extension (semantics)

An intensional definition explains the meaning of a word by describing its key qualities. For example, we can define something by its main features or what it is made of.

An extensional definition lists all the things a word refers to. For example, we can name every object in a group to show what the word means.

These two ways of defining help us understand words in different ways. One tells us about the nature of something, while the other shows us examples of it.

Operational vs. theoretical definitions

A definition can be thought of in two ways: an operational definition or a theoretical definition. These two types help us understand how we use and think about words and ideas in different situations.

Terms with multiple definitions

Homonyms

Main article: Homonym

A homonym is a word that is spelled and pronounced the same as another word but has a different meaning. These words share the same spelling and sound but mean something different. For example, "stalk" can mean a part of a plant or it can mean to follow and watch someone closely. Another example is "left," which can mean the past tense of "leave" or it can mean the opposite of right. Sometimes, these words have different origins, like "skate" meaning to glide on ice and "skate" meaning a type of fish.

Polysemes

Main article: Polysemy

Polysemy is when a word, phrase, or symbol can have several meanings that are usually connected. These meanings are linked by their ideas or concepts. This is different from homonyms, where the meanings are often unrelated. For example, the word "mouth" can refer to the opening of a river or the part of an animal’s face used for eating.

In logic, mathematics and computing

In mathematics, definitions help explain ideas and concepts. Sometimes, the words used might already exist in everyday language, but their meaning in math can be different. For example, a set in math is not exactly the same as what we usually think of as a group of things.

Definitions in math can be used in different ways. Some definitions are created just for a specific discussion, while others try to describe how words are commonly used. There are also definitions that give more details to help with a particular problem.

One special kind of definition is called a recursive definition. This type of definition explains something by using the thing itself in a helpful way. For example, we can define what a natural number is by starting with "0" and then saying that each number has a successor, like "1", "2", and so on. This way, we build up the numbers step by step.

In medicine

In medical dictionaries, guidelines, and other official statements, definitions should be:

• Simple and easy to understand, even for people who are not doctors;
• Useful for doctors and in areas where the definition will be used;
• Specific, so that only the right thing is described;
• Measurable;
• Based on the latest scientific knowledge.

Definitions should clearly show what something is, without being confusing or too broad. They should use words that are well-known and clear. Sometimes, it is hard to define certain things without using words that are not easy to understand.

Some philosophers believe that some important ideas, like "being" or "unity," cannot be defined because there is nothing higher to compare them to. Others think that some words, like the names of people, cannot be defined in the usual way. For example, the word "game" does not have a single, clear definition because games can be very different from each other. Instead, we understand what a game is by looking at many examples and seeing what they have in common.