generality and existence i: quantification and free logic

In this paper, I motivate a cut free sequent calculus for classical logic with first order quantification, allowing for singular terms free of existential import. Along the way, I motivate a criterion for rules designed to answer Prior’s question about what distinguishes rules for logical concepts, like ‘conjunction’ from apparently similar rules for putative concepts like ‘tonk’, and I show that the rules for the quantifiers—and the existence predicate—satisfy that condition. 1 sequents and defining rules Let’s take it for granted for the moment that learning a language involves—at least in part—learning how assertions and denials expressed in that language bear on one another. The basic connection, of course, is that to assert A and to deny A clash. When we learn conjunction, we learn that there is a clash involved in asserting A, asserting B and denying A ∧ B. Similarly, when we learn disjunction, we learn that there is a clash involved in asserting A ∨ B, denying A and denying B. One way to systematically take account of the kinds of clashes involved in these acts of assertion and denial is through the language of the sequent calculus. Given collections Γ and ∆ of sentences from our language L, a sequent Γ ∆ makes the claim that there is a clash involved in asserting each element of Γ and denying each *This work has been in progress for quite some time. I am grateful for audiences at the weekly Logic Seminar and the weekly Philosophy Seminar at the University of Melbourne, as well as presentations in Aberdeen, the Australian National University, Sun Yat-Sen University, St Andrews, LMU Munich, and the Australasian Assocation for Logic Conference for helpful comments on this material. I am grateful to Aldo Antonelli, Conrad Asmus, Bogdan Dicher, Allen Hazen, Lloyd Humberstone, Catarina Dutilh Novaes, Andrew Parisi, Graham Priest, Dave Ripley, Gillian Russell, Jeremy Seligman, Shawn Standefer and Crispin Wright for discussions on these topics, and especially to Bogdan Dicher, Rohan French, Dave Ripley and Shawn Standefer for detailed comments on drafts of this paper. ¶ I dedicate this paper to Aldo Antonelli, whose untimely death leaves us all the poorer. ¶ This research is supported by the Australian Research Council, through Grant dp150103801. ¶ A current draft of this paper is available at http://consequently.org/writing/ generality-and-existence-1/.