On a generalization of quantifiers

TLDR

This chapter explores operators that generalize logical quantifiers, formulates analogous first‑order logic problems, and questions whether a formal calculus can prove all true propositions involving them, noting that such calculi are not the sole aim of symbolic logic. The authors aim to investigate certain generalized quantifiers further and consider including them in systematic symbolic logic expositions despite the negative result. While some problems are solved, the partial results indicate that a formal calculus proving all true propositions with these quantifiers is essentially impossible.

Abstract

Publisher Summary This chapter deals with operators that represent a natural generalization of the logical quantifiers and formulates, for the generalized quantifiers, problems that correspond to the classical problems of the first-order logic. Some of these problems are solved in this chapter. Most of the discussion centers on the problem—whether it is possible to set up a formal calculus that would enable to prove all true propositions involving the new quantifiers. Although this problem is not solved in its full generality, it is clear from the partial results that are discussed in the chapter that the answer to the problem is essentially negative. Despite this negative result, it is believed that some of the generalized quantifiers deserve a closer study and some deserve even to be included into systematic expositions of symbolic logic. This belief is based on the conviction that the construction of formal calculi is not the unique and even not the most important goal of symbolic logic.