Definition:Definition
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.
Warning: If or Iff
It is a standard convention, when making a definition in mathematics, to use if to introduce the definiens, when in fact the intent is generally iff, that is: if and only if.
This convention is specifically not followed on $\mathsf{Pr} \infty \mathsf{fWiki}$, where the mandatory style is to use if and only if.
Also see
- Definition:Definiendum
- Definition:Definiens
- Definition:Undefined Term
- Definition:Recursive Definition
You can't get much more circular than defining the definition of definition.
Sources
- 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume $\text { 1 }$ ... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.11$: The formulation of definitions
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $4$ The Biconditional