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(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 ∀<sub>''A''</sub>, which can be viewed as set-theoretic inclusion, satisfies all of the above except [symmetry]. Clearly [symmetry] holds for ∃<sub>''A''</sub> 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]].
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