 Each departure from classical quantificational logic we have considered originated from an objection to either axioms of pure quantificational logic or the Tarskian definition of satisfaction in a model by an assignment of objects to the variables of the language. We now take both for granted and look at proposed extensions of classical quantificational logic with new styles of quantification. Each extension will generally require us to add new axioms for the new styles of quantification and to expand the Tarskian definition of truth in a model in terms of satisfaction.