H. P. Grice, "A Three-Year-Old's Guide to Russell's Theory of Types"

Grice’s three-year-old’s guide to Russell’s theory of types. Grice put forward the empirical hypothesis that a three-year old CAN understand Russell’s theory of types. “In more than one way.” This brought confusion in the household, with some members saying they could not – “And I trust few of your tutees do!” Russell’s influential solution to the problem of logical paradoxes. The theory was developed in particular to overcome Russell’s paradox, which seemed to destroy the possibility of Frege’s logicist program of deriving mathematics from logic. Suppose we ask whether the set of all sets which are not members of themselves is a member of itself. If it is, then it is not, but if it is not, then it is. The theory of types suggests classifying objects, properties, relations, and sets into a hierarchy of types. For example, a class of type 0 has members that are ordinary objects; type 1 has members that are properties of objects of type 0; type 2 has members that are properties of the properties in type 1; and so on. What can be true or false of items of one type can not significantly be said about those of another type and is simply nonsense. If we observe the prohibitions against classes containing members of different types, Russell’s paradox and similar paradoxes can be avoided. The theory of types has two variants. The simple theory of types classifies different objects and properties, while the ramified theory of types further sorts types into levels and adds a hierarchy of levels to that of types. By restricting predicates to those that relate to items of lower types or lower levels within their own type, predicates giving rise to paradox are excluded. The simple theory of types is sufficient for solving logical paradoxes, while the ramified theory of type is introduced to solve semantic paradoxes, that is, paradoxes depending on notions such as reference and truth. “Any expression containing an apparent variable is of higher type than that variable. This is the fundamental principles of the doctrines of types.” Russell, Logic and Knowledge. Grice’s commentary in “In defense of a dogma,” The H. P. Grice Papers, BANC.