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Is a definition either intensional or extensional?

  1. Can a definition be neither?

  2. Can a definition be both?

    How about this definition? when there is only one object that satisfies a definition, e.g.

    define $A:= 1$

    I think this definition of $A$ is both intensional and extensional.

    Intensional because being $1$ is a necessary and sufficient condition for something being $A$.

    an intensional definition gives the meaning of a term by specifying all the properties required to come to that definition, that is, the necessary and sufficient conditions for belonging to the set being defined.

    Here we have "the set" being singleton i.e. $\{1\}$.

    Extensional, because we list all objects that satisfies the definition, and there is only one object $1$.

    An extensional definition of a concept or term formulates its meaning by specifying its extension, that is, every object that falls under the definition of the concept or term in question.

Thanks.

Tim
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  • You want "intensional." – Qiaochu Yuan Jul 19 '14 at 00:39
  • That definition of $A$ is extensional, by definition. – Git Gud Jul 19 '14 at 00:42
  • Is it intensional? @Git – Tim Jul 19 '14 at 00:43
  • @Tim No, an intentional definition would be $A:={x\in \mathbb N\colon 1\leq x\leq 3}$. – Git Gud Jul 19 '14 at 00:45
  • The issue of whether a definition is intensional or extensional is not cleanly tied to whether there is only one object that satisfies the definition. In many cases we would complain that a definition was incomplete if more than one object satisfied the formulation. Here many Readers would understand your definition of $A$ to be extensional since it depends on the relation of $A$ to other objects ($1,2,3$ as members, everything else as nonmembers). – hardmath Jul 19 '14 at 00:45
  • @GitGud: I think both your and my ways of defining $A$ are both intensional and extensional. See my edit – Tim Jul 19 '14 at 00:51
  • @Tim My way certainly isn't extensional as I didn't list the elements. Yours isn't intensional because you didn't define $A$ as "the set whose elements are $1,2$ and $3$". You actually listed them. You didn't define it by any property. – Git Gud Jul 19 '14 at 00:52
  • @hardmath: I am defining $A$, not if an object is its element. So I think the definition of $A$ is both intensional and extensional. – Tim Jul 19 '14 at 00:53
  • @GitGud Here I am defining A, not if an object is its element. So I think either of your and my definitions is both intensional and extensional.. – Tim Jul 19 '14 at 00:54
  • @Tim For it to be intensional, you need to define it with properties, you haven't done this. And I don't understand why you say that $A:={x\in \mathbb N\colon 1\leq x\leq 3}$ is an extensional definition. – Git Gud Jul 19 '14 at 00:55
  • @Tim: I'm relating how most Readers would understand your "definition". Absent the conventional interpretation of listing the elements of a set, you are free to assert an idiosyncratic semantic. But you should share this, if the meaning is different from what the Readers expect. – hardmath Jul 19 '14 at 00:56
  • @GitGud: I simplified the example, see the edit. I hope you can understand what I meant earlier with the original example. Both examples are similar in that there is only one object that satisfies their definitions. – Tim Jul 19 '14 at 01:02
  • @hardmath: I simplified the example, see the edit. I hope you can understand what I meant earlier with the original example. Both examples are similar in that there is only one object that satisfies their definitions. – Tim Jul 19 '14 at 01:02
  • @QiaochuYuan: do you happen to have some idea? – Tim Jul 19 '14 at 01:05
  • It's not intensional because you're not defining it with a property. "Intensional because being $1$ is a necessary and sufficient condition for something being $A$." You didn't define it as "the only object which equals $1$". Nowhere in the sequence of symbols "$A:=1$" did you wrote a property. – Git Gud Jul 19 '14 at 01:07

1 Answers1

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Extensional: $A:=\{1\}$

Intensional: $A:=\{x\in\mathbb{R}:x=1\}$

This is due to the agreement in naive (and probably also in ZF) set theory that sets can be specified in either listing its elements or specifying some rules (in the context of a universe) - so precisely by giving an extensional or intensional definition.

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