Four Types of Quantification

Four Types of Quantification

It has been said that there are really two types of quantification not one: “objectual” and “substitutional”. The objectual type may be paraphrased as follows: “There is an object x such that x satisfies F, where F is a predicate” (and similarly for the universal quantifier). The substitutional type may be paraphrased: “There is singular term t such that when t is substituted into the appended open sentence it produces a truth” (similarly for universal quantification). The intent of the distinction is clear enough, though the terminology leaves something to be desired. We might better speak of metalinguistic quantification versus objectual quantification, and admit that the former is a special case of the latter, since bits of language are also “objects”. Also, why is inserting a term into an open sentence a type of “substitution”—isn’t it just occupying an argument place (a blank)? Still, there is no very good terminology to fall back on, so we can stick with the standard formulation, so long as we are clear about what is involved. The question is whether the quantifier is “over” objects or “over” words, i.e., what the “domain” includes.

I think this two-way distinction is too limited. There are (at least) four distinguishable types of quantification; we need to add what I will call conceptual quantification and intentional quantification. Again, the terminology is not pellucid, but the distinctions are real. Intuitively, a conceptual quantifier says something like this: “There is a concept C such that when C occurs in a certain proposition the proposition is true”. For example, “There exists a man” is equivalent to “There is a concept C such that when C is combined with the concept man we get a truth”, where C might be the concept Socrates. This is really the twin of metalinguistic quantification except that we “substitute” concepts into propositions not words into sentences. It proceeds at the level of sense not of reference or signs. What I am calling an intentional quantifier (read “intentional object quantifier”) ranges over existent and nonexistent objects: thus, we can say “Some men fly” meaning to include men like Superman. We no longer require that our range of quantification includes only existent objects; we also quantify “over” nonexistent objects. Obviously, this type of quantifier belongs alongside the usual “objectual” kind. Even if natural languages (or logic texts) don’t include such quantifiers, we can stipulate them: they range over objects of thought as well as objects that exist. Ontologically, there are four kinds of entity to work with—existing objects, objects of thought, concepts, and words—so we have four types of quantification to go with them.

We can construct languages with each type of quantifier and explore their properties. One type may be suitable for some purposes but not for others. If we are discussing fiction, we might want the intentional quantifier (“All of Shakespeare’s tragic figures are flawed”); whereas in the sciences we will limit ourselves to existing things. If we are discussing quantificational thoughts, we might want to keep language out of it. Nonsensical language might invite the substitutional quantifier (“Some mome raths are greedy”). Quantifiers are not logically univocal or ontologically parsimonious. They are flexible and pragmatic, up for anything. They are willing to “range over” anything we can think up. They are not semantically austere or judgmental. They are certainly not the measure of what we take seriously and literally to exist in the world of actual objects.

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