Conditional quantifier

In logic, a conditional quantifier is a kind of Lindström quantifier (or generalized quantifier) $Q_{A}$ that, relative to a classical model $A$, satisfies some or all of the following conditions ('X' and 'Y' range over arbitrary formulas in one free variable):

• $Q_{A}XX$ [reflexivity]
• $Q_{A}XY\Rightarrow Q_{A}X(Y\land X)$ [right conservativity]
• $Q_{A}X(Y\land X)\Rightarrow Q_{A}XY$ [left conservativity]
• $Q_{A}XY\Rightarrow Q_{A}X(Y\lor Z)$ [positive confirmation]
• $Q_{A}X(Y\land Z)\Rightarrow Q_{A}(X\land Y)Z$
• $Q_{A}XY\Rightarrow Q_{A}(X\lor Z)(Y\lor Z)$ [postiive and negative confirmation]
• $Q_{A}XY\Rightarrow Q_{A}\lnot X\lnot Y$ [contraposition]
• $Q_{A}XY\land Q_{A}YZ\Rightarrow Q_{A}XZ$ [transitivity]
• $Q_{A}XY\Rightarrow Q_{A}(X\land Z)Y$ [weakening]
• $Q_{A}XY\land Q_{A}XZ\Rightarrow Q_{A}X(Y\land Z)$ [conjunction]
• $Q_{A}XZ\land Q_{A}YZ\Rightarrow Q_{A}(X\lor Y)Z$ [disjunction]
• $Q_{A}XY\Rightarrow Q_{A}YX$ [symmetry].

(The implication arrow denotes material implication in the metalanguage.) The minimal conditional logic M is characterized by the first six properties, and stronger conditional logics include some of the other ones. For example, the quantifier $\forall _{A}$, which can be viewed as set-theoretic inclusion, satisfies all of the above except [symmetry]. Clearly [symmetry] holds for $\exists _{A}$ while e.g. [contraposition] fails.

A semantic interpretation of conditional quantifiers involves a relation between sets of subsets of a given structure—i.e. a relation between properties defined on the structure. Some of the details can be found in the article Lindström quantifier.

References

Serge Lapierre. Conditionals and Quantifiers, in Quantifiers, Logic, and Language, Stanford University, pp. 237-253, 1995.