Definition:Definition

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Definition

A definition lays down the meaning of a concept.

It is a statement which tells the reader what something is.

It can be understood as an equation in (usually) natural language.


Some authors distinguish between particular types of definition, particularly of symbols:


Stipulative Definition

A stipulative definition is a definition which defines how to interpret the meaning of a symbol.

It stipulates, or lays down, the meaning of a symbol in terms of previously defined symbols or concepts.


The symbol used for a stipulative definition is:

$\text {(the symbol being defined)} := \text {(the meaning of that symbol)}$


This can be written the other way round:

$\text {(a concept being assigned a symbol)} =: \text {(the symbol for it)}$

when it is necessary to emphasise that the symbol has been crafted to abbreviate the notation for the concept.


Ostensive Definition

An ostensive definition is a definition which shows what a symbol is, rather than use words to explain what it is or what it does.


As an example of an ostensive definition, we offer up:

The symbol used for a stipulative definition is "$:=$", as in:
$\text {(the symbol being defined)} := \text {(the meaning of that symbol)}$


Also defined as

In the words of 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica:

A definition is a declaration that a certain newly-introduced symbol or combination of symbols is to mean the same as a certain other combination of symbols of which the meaning is already known.


Note

When a definiens contains "if", the intent is generally "if and only if".

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the standard style is to use "if and only if".


Also see

You can't get much more circular than defining the definition of definition.


Sources