Quantification

Medieval authors approached the semantic phenomenon now known as quantification essentially by means of the concept of supposition, more specifically the different modes of personal supposition. The modes of personal supposition were meant to codify the quantificational behavior of what we now refer to as quantifier expressions, and what the medievals referred to as syncategorematic terms. Perhaps the best way to understand the medieval approach to these quantifier expressions by means of the notion of supposition is as a two-step procedure that explicates their meaning and semantic behavior. First, the syntactical structure of the proposition, that is, the presence and order of its syncategorematic terms, determines the kind of personal supposition that each categorematic term has. Then, the semantic definitions of each mode of personal supposition determine the effect of quantifying syncategoremata over the quantity of objects involved in the assertion of a proposition. This entry discusses both groups of rules, and the contrasting thirteenth and fourteenth century approaches. The former is based on the verification of propositions and focuses on the semantics of quantifier expressions taken individually; the latter focuses on the inferential relations of ascent and descent between propositions with quantifying syncategorematic terms and singular propositions of the form “This a is b,” and on the study of the global quantificational effect of syncategorematic terms in wider propositional contexts.

The phrase “medieval theories of quantification” is, properly speaking, an anachronism; medieval authors never used the term “quantification” in this sense, and even though they did treat semantic phenomena similar to what we now refer to as quantification, their theories differ from modern theories of quantification in significant aspects – to the point that this approximation may even be unwarranted (Matthews 1973). Nevertheless, their treatments of such phenomena are often insightful and sophisticated, justifying thus that we consider them from the viewpoint of modern theories of quantification, but provided that the term “quantification” be understood very broadly.

Broadly understood, quantification can be defined as a construct or procedure by means of which one specifies the quantity of individuals of the domain of discourse that apply to or verify a given statement. Typical quantifier expressions are “Some,” “All,” “None,” and they usually determine the quantity of individuals involved in an assertion. Medieval authors approached quantification and quantifier expressions essentially by means of the concept of supposition, more specifically the different modes of personal supposition.

Besides supposition, they also treated quantificational phenomena from the vantage point of their theories of syllogisms, following the traditional Aristotelian approach. However, it is widely acknowledged that medieval authors did not contribute much to the development of Aristotle’s theory of syllogisms for assertoric propositions, and that their main contributions concern modal syllogisms. Therefore, the innovations proposed by medieval authors with respect to quantification are not to be found in their theories of syllogisms, but rather in this typical medieval development, theories of supposition.

The different modes of personal supposition are indeed the closest medieval counterpart of our theories of quantification. The modes of personal supposition were meant to codify the quantificational behavior of what we now refer to as quantifier expressions, and what the medievals referred to as syncategorematic terms. Such analyses can be found in virtually every later medieval textbook in logic, but for reasons of space I shall focus on three representative texts: William of Sherwood’s Introduction to Logic, William of Ockham’s Sum of Logic (part I), and Buridan’s Treatise on Supposition.

Perhaps the best way to understand the medieval approach to these quantifier expressions by means of the notion of supposition is as a two-step procedure that explicates their meaning and semantic behavior. First, the syntactic structure of the proposition, that is, the presence and order of its syncategorematic terms, determines the kind of personal supposition that each categorematic term has. Then, the semantic definitions of each mode of personal supposition determine the effect of quantifying syncategoremata over the quantity of objects involved in the assertion of a proposition.

In other words, the various theories of supposition presented by medieval authors typically have two groups of rules for the modes of personal supposition: the syntactic rules mapping terms in the propositional contexts created by quantifier expressions into modes of personal supposition; and the semantic rules mapping modes of personal supposition into specific semantic behaviors (see Ashworth 1978). To illustrate this, let us first discuss the four Aristotelian classes of categorical propositions: universal affirmative (A), particular affirmative (I), universal negative (E), and particular negative (O); and provide the two kinds of rules for these propositional forms. (Notice that, even at early stages of its development, supposition theory already recognized a wide variety of quantifier expressions – unlike modern quantification theory, which started out with the existential and universal quantifiers and only later developed into a theory of generalized quantifiers. Notice also that, for medieval logicians, following Aristotle, all affirmative propositions have existential import, existential and universal propositions alike.)

(A) Every a is b.

(E) No a is b. 

(I) Some a is b.