 In order to assess the Logicality Thesis in a more principled way, more will have to be said about what it might mean for a theory to be “purely logical”. So I will now survey some of the features commonly thought to play a role in such a definition. Although people are free to use the word “logic” as they please, it is important to get clear on what different usages entail; in particular, theories that qualify as purely logical are often assumed to enjoy a variety of desirable philosophical properties such as epistemic and ontological innocence. In the next section, where various applications of plural quantification will be discussed, I will carefully note which strains of the notion of logicality our theories PFO and PFO+ must possess for the various applications of them to succeed.