Arbitrary objects in a bilateral setting

Arbitrary objects play an important role in both ordinary and mathematical reasoning. The reason for that is their distinct behavior: an arbitrary object presents those properties common to all individual objects in its range - what, following Kit Fine, has become known as the "Principle of Generic Attribution", or simply "PGA". Many philosophers have argued this precise quality to be contradictory, therefore rejecting arbitrary objects altogether - of which the most famous is George Berkeley's argument against Locke's general ideas. However, all the arguments that have been offered assume rejecting properties of an arbitrary object to be expressible in terms of asserting something of it - a unilateral setting. This paper has two main purposes: first, to show how, in the propositional bilateral system of Weak Rejectivist Logic, introduced by Luca Incurvati and Julian Schloder (2017), the PGA does not fall prey to the counter arguments which rise in tension with the nature of the logical connectives; second, to argue understanding sentences about arbitrary objects in terms of assertions which preserve (or not) generality better frames our reasoning about them, and is reflected by the Weak Rejectivist Logic. Then, following the relation between this propositional system and arbitrary objects, a further goal of this paper is to offer a characterization result for the Weak Rejectivist Logic with respect to first-order logic.