Not all logical operators are the same
There is something suspicious about our reliance on simple recursive definitions in logic — specifically, our custom of introducing logical operators into the language via inductive clauses of the form: if F1, F2, … Fn are wffs, then On(F1, F2, … Fn) is also a wff. This formulation tacitly assumes that all logical operators are what I have elsewhere (1994) called "homogeneous": their values belong to the same category as their arguments. In particular, it assumes they are all sentential operators, since both their arguments and their values are sentences. There are undeniable metalogical advantages to conceiving of logical language this way — advantages pressed on me by both Raúl Orayén and David McCarthy, when I was a student. Yet several philosophers of logic, including Frege, Łukasiewicz, Haack, and myself, have challenged this orthodoxy. Frege, for instance, introduced a logical operation from sentences to their contents — a kind of restricted disquotation — that is not...