Question

In the context of independence-friendly logic and the order of quantifiers, is there a distinction between for all A, there exists some B... that implies one and the same B for all A versus a reading where there is a separate B for each A? Is this difference tracked by the order of the quantifiers, and is this what quantifier dependence entails?

Answer 1

In the context of independence-friendly logic and the order of quantifiers, there is a distinction between the two readings you mentioned.

we say "for all A, there exists some B," it means that there is a specific B that corresponds to each A. This is different from saying that there is one B that works for all A. The order of the quantifiers helps track this distinction. If the quantifier "for all" comes first, it implies a separate B for each A. If the quantifier "there exists" comes first, it implies one B for all A.

Quantifier dependence refers to the relationship between multiple quantifiers in a statement. In the context of your question, the dependence between the quantifiers determines whether there is one B for all A or a separate B for each A. If the quantifiers are independent, it means there is one B for all A. If they are dependent, it means there is a separate B for each A.