Quantifier Glossary
Definition
A quantifier is an expression which applies to a predicate to form a predicate of lower arity. Where the arity is reduced to zero, it forms a sentence stating what quantity of things of some sort satisfy the predicate. In first order logic, it has three parts: the sort indicator saying which things it ranges over, the quantity indicator saying what quantity of them satisfy the description, and one or more variables which it binds.
Comments
It is quite common to omit the sort indicator, meaning that the quantifier ranges over the entire domain. Such quantifiers are said to be unrestricted.
Examples
- In English, quantity indicators include all, some, most, at least two, exactly five, few, many, etc. These can be used to form sentences like 'Most people have exactly two feet', in which the 'most' quantifier ranges ofver people and the 'exactly two' quantifier ranges over feet.
- The only quantifiers we take as primitive in ordinary formal logic are the universal one, written 'ALL', and the existential one, written 'SOME'.
-
All:
ALLxFx means
that everything in the domain of discourse
is F. ALL(x: Gx) Fx
means that everything in the set of Gs
(i.e. every G) is F. More generally, any
complex description can go in place of F
and G. So for instance,
ALLx((Hx AND Fx) IMP NOTGx)
might formalise "All hairy footballers are non-goats". -
Some:
SOMExFx means
that at least one thing in the domain of discourse
is F. SOME(x: Gx) Fx
means that something in the set of Gs
(i.e. a G) is F. More generally, any
complex description can go in place of F
and G. So for instance,
SOMEx((Hx AND Fx) AND NOTGx)
might formalise "Some hairy footballer is not a goat".