THE LOGIC NOTES

Restricted/unrestricted quantifier Glossary


Definition

A quantifier is restricted if it ranges over things of a specified sort. It is unrestricted if it ranges over the entire domain.

Comments

In the formal language of logic, we write the restriction or sort indicator after the variable(s) inside parentheses. That is, to restrict to things satisfying a description A, we write ALL(x: A) B or SOME(x: A) B. For unrestricted quantifiers, the restriction is omitted, and of course the parentheses are not needed, so we write ALLx B or SOMEx B.

For the universal and existential quantifiers (and for numerical quantifiers generally) it is possible to translate between the two notations. ALL(x: Fx) Gx is equivalent to ALLx(Fx IMP Gx), and SOME(x: Fx) Gx is equivalent to SOMEx(Fx AND Gx). Conversely, ALLx Fx is equivalent to ALL(x: Fx IMP Fx) Fx or more briefly to ALL(x: NOTFx) Fx, while SOMEx Fx is equivalent to SOME(x: Fx IMP Fx) Fx or more briefly to SOME(x: Fx) Fx.

Examples

  1. To say 'Every goat is hairy' with a restricted quantifier, write ALL(x: Gx) Hx. To say the same thing with an unrestricted quantifier, write ALLx(Gx IMP Hx).
  2. To say 'Some footballer is hairy' with a restricted quantifier, write SOME(x: Fx) Hx. To say the same thing with an unrestricted quantifier, write SOMEx(Fx AND Hx).
  3. To say 'Every footballer who owns a goat kicks it' with restricted quantifiers, write ALL(x: Fx AND SOME(y: Gy) Oxy) Kxy. To say the same thing with unrestricted quantifiers, write ALLx((Fx AND SOMEy(Gy AND Oxy)) IMP Kxy).

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