ABSTRACT
I. In two very in teresting recen t p apers,2 Boolos has tried to tame second-order logic completely. Second-order logic p er se, he now claims, does no t commit one to sets, classes, F regean concepts, o r anything else. The gist o f his argum ent is this: 1. We need no t posit classes o r collections in o rd e r to rend e r
second-order sentences intelligible. We can simply translate them in to ord inary language using plural quantifiers. (For example, the second-order version o f the least-num ber principle, “ For any F, if there are num bers th a t F, then th e re is a least such” may be trans lated as “ It is false tha t there are some num bers such th a t no one o f them is the least.” ) 2. Using plural quantifiers does no t commit one to classes o r collec tions. Indeed , it does no t commit one to anything th a t one is no t already comm itted to by means o f o n e ’s use o f singular quantifiers. 3. Thus, the use o f second-order logic need no t commit one to collections o r sets. Q uine is wrong; second-order logic is no t class theory in disguise. I f this is co rrect, Boolos has made a significant con tribu tion to the
philosophy o f mathematics. H e will have vindicated those who have held tha t conflating second-order logic with set theory has perverted F rege’s logicism, and he will have streng thened the ir version o f logi-
cism.3 H e will also have helped H artry Field ou t o f a pitfall he fell in to while trying to nom inalize New tonian mechanics.4 H e will have shown structuralists, such as Stewart Shapiro and me, how we can use second-order characterizations o f set-theoretic h ierarch ies with ou t fear o f circularity .5 Finally, he will have shown set theorists how they can em brace second-order axioms w ithout abandon ing the idea tha t the first-o rder universe o f sets contains all the setlike entities tha t there are. (Shapiro tells me tha t this has been the principal motive fo r Boolos’s work on second-o rder logic.) A lthough one m ight see Boolos’s transla tion as a “ nom inalization”
o f second-order logic, it does no t significantly affect the central ontological p rob lem in the philosophy o f m athematics. Unless we adjoin it to a p rog ram such as F ield’s, which has o th e r problem s besides its use o f second-order logic, Boolos’s m ethods do no t dis pense w ith sets and o th e r abstract objects. A t most, his technique elim inates the use o f abstract entities from several specific (but im portan t) contexts. Despite the attractiveness o f Boolos’s view fo r ph ilosophers o f
mathematics, and fo r me in particu lar, I am no t convinced tha t it is an advance over the s tandard view o f second-order logic. H ere are my reservations. First, my logical/linguistic in tu itions differ sharply from Boolos’s. As I see it, some plural quantifications do commit us
SECOND-ORDER LOGIC STILL WILD 77
to collections, sets, o r some o th e r so rt o f second-order entity, and I find fault with Boolos’s argum ents against my position. I will expand on this in section 11. But there is m ore to deciding m atters o f ontic comm itm en t th an b ran d ish in g logical and lingu istic in tu itio n s . Boolos may have had this in m ind when he constructed an alternative semantics fo r second -o rder logic acco rd ing to which no secondo rd e r variable has a second-order entity as one o f its values. In section h i, I argue that, a lthough this semantics is necessary fo r the consistency o f Boolos’s position, it fails to dem onstra te tha t secondo rd e r logic is no t comm itted to collections. In section iv, I argue tha t ne ither the intelligibility o f plural quantifications n o r th e ir p rim a facie lack o f comm itm ent to collections is sufficient to dem onstra te that they never commit us to collections. D eterm ining w hether they do involves rep resen ting them in an acceptable and suitably in te r p re ted logical no tation . Thus, those o f us who find p lural quantifi cation in need o f logical analysis will no t be en ligh tened by formal explications such as Boolos’s, which presuppose p lural quantifica tion fo r the ir in te rp re ta tion .
