Abstract

Neo-logicists have claimed that Hume's Principle (HP) may be taken as a stipulative definition of cardinal number. This claim is threatened by the fact that HP is not conservative over pure second-order logic. I argue that the dominant neo-logicist response to the conservativeness objection is not satisfactory. Then I propose a novel version of neo-logicism, based on Heck's Two-sorted Hume's Principle (2HP), which does meet the conservativeness objection—provided that conservativeness is understood semantically and not deductively. I also argue that on my proposal, the Bad Company problem is solved by conservativeness.