Criterial Positions
In the following examples, the phrase [D book] must receive both the interpretive property ‘‘patient of the verb read’’ and the interpretive properties ‘‘interrogative,’’ ‘‘topic,’’ ‘‘focus,’’ respectively:
(1)

Natural languages express this duality of interpretive properties by having the phrase occur in two positions, each assigning one kind of property. In standard terminology, an element is moved from a position of one kind to a position of the other kind. If one adopts the copy theory of traces (Chomsky 1993), according to which ‘‘traces of movement’’ are complete but silent copies of the moved element, it is more appropriate to say that an element can be merged several times in a structure. In other words, occurrences of the element can multiply, so that in (1) [D book] occurs both as complement of read and as left-peripheral focus, topic, or interrogative operator, the unpronounced lower occurrence being represented within angle brackets in (2).
(2)

Departing slightly from Chomsky’s (2001, 2004) terminology, I will call the two kinds of interpretive properties s-selectional and criterial, respectively. An element is first merged in a position in which it is semantically selected, and then it may be merged again in a position dedicated to scope-discourse semantics. The two positions (and other positions hosting occurrences of the same element; see below) form a chain.
Typical s-selectional properties are thematic properties, which I assume to be assigned by lexical items in their immediate structural neighborhood. But the concept should be generalized to whatever interpretive properties license the insertion of a nonargument in the structure: adverbials of various kinds, modifiers, and so on. Following a rather traditional terminology, I call these properties s-selection (semantic selection, as in Grimshaw 1979, Pesetsky 1982, etc.). The case of s-selection for arguments is y-assignment; the analogous case for, say, a temporal adverbial is s-selection by a T node, and similarly for modal and aspectual adverbs, according to much recent work on adverbial syntax, most notably in the systematic approach taken by Cinque (1999).
As for Chomsky’s scope-discourse positions, I will assume that they are determined by a family of principles, the criteria, which require specifier head agreement with respect to features of the relevant class: Q, Top, Foc, R for questions, topic, focus, relatives, and so on (see Rizzi 1996 for an early formulation of this approach). As a first approximation, I will take the format of criteria to be the following:
(3)

That the preposing processes create local specifier-head configurations with dedicated heads is suggested by the fact that such heads are sometimes morphologically expressed. Some plausible cases of morphological expression of Q, Top, Foc, and R, respectively, are the following (the criterial heads are italicized):
(4)

So, a head endowed with the relevant feature attracts a phrase bearing that feature, thereby designating a position dedicated to the relevant type of interpretation. The information is then passed on to the interpretive system (‘‘The phrase in this Spec is to be interpreted as an interrogative operator with scope over this clause, as a Topic, and so on’’). Such dedicated heads are sometimes overt, sometimes silent, but, I will assume, systematically present across languages (as should be expected under Chomsky’s (2001, 2) Uniformity Principle, essentially along the lines commonly assumed for Case theory).
A standard A ¯-chain therefore typically includes two positions, both expressing some interface property and both marked in the structure by the feature content of the local head.
(5)

To explore the form of A ¯-chains in some generality, we should now deter mine what positions are required and what positions are allowed in a chain. Are there other positions in addition to s-selection and criterial positions? This question breaks up into three subcases:
(6)

The answer to (6a) is quite generally assumed to be no: arguments cannot be first-merged in positions lower than their y-position, and then pick up their y-role through movement (but see Kayne 2002 for a different view, in the context of his attempt to derive binding conditions from movement). As for question (6b), the answer is clearly yes: there is rich and diverse evidence for intermediate traces between the s-selection and criterial positions. As for question (6c), it is rarely asked, and trying to address it in some generality is one of the main purposes of this chapter. Let us consider the three subcases in detail.