THE LOGIC NOTES

Identity Glossary

Definition

Identity is the relation which everything has to itself and to nothing else. In logic, as in mathematics, it is represented by the symbol '='.

Comments

We standardly write the identity or equality sign in infix position (between two terms rather than in prefix, just because '2+2 = 4' is more familiar than '=(+(2,2),4)'.

The addition of identity to the formal vocabulary of logic makes the formal language more expressive. In partiular, it makes it possible to express numerical quantifiers.

In the natural deduction calculus, the introduction rule for identity says that for any term t, the self-identity formula t = t may be introduced at any point, with no input lines and depending on no assumptions. The elimination rule says that for any terms t and u and for any formula A containing a name m, Aumis an immediate consequence of Atmand t = u.

Examples

  1. Tasmania is [the same thing as] Van Diemen's Land.
  2. George Orwell is [the same person as] Eric Blair.
  3. 7+5 is [the same number as] 4×3.

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