THE LOGIC NOTES

Numerical quantifier Glossary

Definition

A numerically indefinite quantifier is one which gives an upper or lower bound on the number of things satisfying a description. A numerically definite quantifier gives the exact number.

For each natural number n there are two numerically indefinite quantifiers: for at least n things... and for at at most n things... The numerically definite quantifier is essentially the conjunction of these two.

Comments

In first order logic with identity, 'There are at least n Fs' can be written

SOMEx1...SOMExn((Fx1 AND ... AND Fxn) AND (x1 ≠ x2 AND ... xi ≠ xj AND ... ))

or more briefly

ALLx1...ALLxn-1SOMEy(Fy AND (x1 ≠ y AND ... AND xn ≠ y))

'There are at most n Fs' means that there are not at least n+1, so it can be written

SOMEx1...SOMExnALLy(Fy IMP (x1 = y OR ... OR xn = y))

Note that on these definitions, the familiar existential quantifier is the numerically indefinite 'at least one', and the universal is 'at most zero...not', so those two count as numerically indefinite quantifiers. The universal one is also numerically definite, of course, since there cannot be fewer than zero of anything.

Restricted numerical quantifiers can be emulated using unrestricted ones and the connectives AND and IMP as in the case of ALL and SOME, since n As are B iff n things are A and B.

Examples

  1. At least two Fs are G:   SOMExSOMEy(Fx AND Fy AND x ≠ y)
  2. At most two Fs are G:   SOMExSOMEyALLz(Fz IMP (x=z OR y=z))
  3. Exactly two Fs are G:   SOMExSOMEy(x ≠ y AND ALLz(Fz IFF (x=z OR y=z)))

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