The first distinction is between two senses of the word “individuation” – one semantic, the other metaphysical. In the semantic sense of the word, to individuate an object is to single it out for reference in language or in thought. By contrast, in the metaphysical sense of the word, the individuation of objects has to do with “what grounds their identity and distinctness.” Sets are often used to illustrate the intended notion of “grounding.” The identity or distinctness of sets is said to be “grounded” in accordance with the principle of extensionality, which says that two sets are identical iff they have precisely the same elements:

SET(x) ∧ SET(y) → [x = y ↔ ∀u(u ∈ x ↔ u ∈ y)]

The metaphysical and semantic senses of individuation are quite different notions, neither of which appears to be reducible to or fully explicable in terms of the other. Since sufficient sense cannot be made of the notion of “grounding of identity” on which the metaphysical notion of individuation is based, focusing on the semantic notion of individuation is an easy way out. This choice of focus means that our investigation is a broadly empirical one drawn on empirical linguistics and psychology.

What is the relation between the semantic notion of individuation and the notion of a criterion of identity? It is by means of criteria of identity that semantic individuation is effected. Singling out an object for reference involves being able to distinguish this object from other possible referents with which one is directly presented. The final distinction is between two types of criteria of identity. A one-level criterion of identity says that two objects of some sort F are identical iff they stand in some relation RF:

Fx ∧ Fy → [x = y ↔ R_{F}(x,y)]

Criteria of this form operate at just one level in the sense that the condition for two objects to be identical is given by a relation on these objects themselves. An example is the set-theoretic principle of extensionality.

A two-level criterion of identity relates the identity of objects of one sort to some condition on entities of another sort. The former sort of objects are typically given as functions of items of the latter sort, in which case the criterion takes the following form:

f(α) = f(β) ↔ α ≈ β

where the variables α and β range over the latter sort of item and ≈ is an equivalence relation on such items. An example is Frege’s famous criterion of identity for directions:

d(l_{1}) = d(l_{2}) ↔ l_{1} || l_{2}

where the variables l_{1} and l_{2} range over lines or other directed items. An analogous two-level criterion relates the identity of geometrical shapes to the congruence of things or figures having the shapes in question. The decision to focus on the semantic notion of individuation makes it natural to focus on two-level criteria. For two-level criteria of identity are much more useful than one-level criteria when we are studying how objects are singled out for reference. A one-level criterion provides little assistance in the task of singling out objects for reference. In order to apply a one-level criterion, one must already be capable of referring to objects of the sort in question. By contrast, a two-level criterion promises a way of singling out an object of one sort in terms of an item of another and less problematic sort. For instance, when * Frege* investigated how directions and other abstract objects “are given to us”, although “we cannot have any ideas or intuitions of them”, he proposed that we relate the identity of two directions to the parallelism of the two lines in terms of which these directions are presented. This would be explanatory progress since reference to lines is less puzzling than reference to directions.