One standard characterization of quantifiers is that they are bits of speech that follow certain rules of inference (e.g., universal instantiation, existential generalization, etc.). This characterization is surely incorrect.
Let p be any complicated logical truth. Consider the symbols ∃* and ∀* such that ∃*xF and ∀*xF are abbreviations for p ∧ ∃xF and p ∧ ∀xF. Then ∃* and ∀* satisfy exactly the same rules of inference as ∃ and ∀, but they are not quantifiers. The sentence ∃*Fx expresses a conjunctive proposition rather than a quantified one.
In other words, the characterization of quantifiers by logical rules misses the hyperintensionality at issue. The same is true of the characterization of any other logical connectives by logical rules.