A History of Quantification

Aristotle (384–322 BC), the founder of the discipline of logic, also founded the study of quantification. Normally, Aristotle begins a topic by reviewing the common opinions, including the opinions of his chief predecessors. In logic, however, he could not adopt the same strategy; before him, he reports, “there was nothing at all” (Sophistical Refutations 183b34–36). Aristotle’s theory dominated logical approaches to quantification until the nineteenth century. That is not to say that others did not make important contributions. Medieval logicians elaborated Aristotle’s theory, structuring it in the form familiar to us today. They also contemplated a series of problems the theory generated, devising increasingly complex theories of semantic relations to account for them. Textbook treatments of quantification in the seventeenth and nineteenth centuries made important contributions while also advancing some peculiar theories based on medieval contributions. Modern quantification theory emerged from mathematical insights in the middle and late nineteenth century, displacing Aristotelian logic as the dominant theory of quantifiers for roughly a century. It has become common to see the history of logic as little more than a prelude to what we now call classical first-order logic, the logic of Frege, Peirce, and their successors. Aristotle’s theory of quantification is nevertheless in some respects more powerful than its modern replacement. Aristotle’s theory combines a relational conception of quantifiers with a monadic conception of terms. The modern theory combines a monadic conception of quantifiers with a relational theory of terms. Only recently have logicians combined relational conceptions of quantifiers and terms to devise a theory of generalized quantifiers capable of combining the strengths of the Aristotelian and modern approaches. There is no theory-neutral way of defining quantification, or even of delineating the class of quantifiers. Some logicians treat determiners such as ‘all,’ ‘’every,’ ‘most,’ ‘no,’ ‘some,’ and the like as quantifiers; others think of them as denoting quantifiers. Still others think of quantifiers as noun phrases containing such determiners (‘all men,’ ‘every book,’ etc.). Some include other noun phrases (‘Aristotle,’ ‘Peter, Paul, and John,’ etc.). Some define quantifiers as variable-binding expressions; others lack the concept of a variable. My sketch of the history of our understanding of quantification thus traces the development of understandings of what is to be explained as much as how it is to be explained.