Every generalized quantifier has an arity, which is the number of formulas it takes as arguments, and a type, which for an -ary quantifier is a tuple of length . The tuple represents the number of quantified variables for each argument.
The most common quantifiers are those of type , including and . If is a quantifier of type , is the universe of a model, and is the relation associated with in that model, then .
So , since the quantified formula is only true when all elements satisfy it. On the other hand .
In general, the monadic quantifiers are those of type and if is an -ary monadic quantifier then . Härtig’s quantifier, for instance, is , and .
A quantifier is polyadic if it is of type where each . Then:
These can get quite elaborate; is a quantifier where is a well-ordering. That is, it is true if the set of pairs making true is a well-ordering.
|Date of creation||2013-03-22 12:59:57|
|Last modified on||2013-03-22 12:59:57|
|Last modified by||Henry (455)|