Reductions and occurrences
The presence or absence of A-operators in translations of structures involving A-movement may seem an irrelevant technical detail due to the widely assumed availability of substitution of the argument term for the variable bound by the operator—free availability of ^-reduction, in short. This assumption has a noble pedigree, going back to Carnap:
If a sentence consists of an abstraction expression followed by an individual constant, it says that the individual has the property in question. Therefore, ‘(Ax)(... x ...)a means the same as ‘... a... ' that is, the sentence formed from . x... ’ by substituting a’ for ‘x’. The rules of our system will permit the transformation of ‘(Ax)(... x ...)a’ into ‘.a.’ and vice versa; these transformations are called conversions. (Carnap 1947: 3)
In such an environment, the displacement of, say, EA to the EPP position would be an entirely syntax-internal affair, required by syntactic principles and/or filters (however tightly they might be connected with the interfaces, as the labeling requirement is thought to be), but without interpretive consequences—a byproduct of the way narrow syntax operates, easily eliminable by an application of the reduction rule. It is, of course, significant that Carnap takes expressions with individual constants into consideration—their semantic properties are indeed quite unique and make it easier to accept unrestricted ^-conversions; by the same token, it immediately calls into question applicability of such a solution for semantics of even such simple structures as EA to Spec-TP raising, as in
(16) would be entirely nondistinct—as far as the interpretive relationship of EA with the structure is concerned—from (17), with EA ‘undergoing full reconstruction at LF’ (‘Tr(T')[Tr(EA)/x)]’ indicating the result of substituting the term resulting from the translation of EA for the variable in the body of the A-term being the translation of T').
This ‘LF reconstruction’ of EA would be an instance of ^-reduction indeed—a syntactic operation substituting an expression for a variable in an appropriate configuration, with the effect interpretively identical to leaving EA in situ. All syntactic requirements, be it Case-valuation, ^-valuation, labeling etc., are on this picture strictly confined to narrow syntax without leaving any effect once the C-I component enter into play. This is indeed supposed to be so: as Matush- ansky (2006) notes, explaining lack of semantic effects of head movement via в-reduction, ‘Movement of DPs denoting individuals, of which kinds are a special case, has no LF effect for the same reason: they have the same semantic type e as their traces, so it is unsurprising that head movement will have no semantic effect in this case either’ (Matushansky 2006: 103), and the reasoning might be generalized: ‘[EA] is interpreted in exactly the same way in its moved position as in its first-merged position, and, given the fact that [it] is of type (e), denoting a set of individuals (in this case, a singleton set), it has no scope effects. Hence this is an instance of DP-movement that lacks semantic effects,’ Roberts (2010: 25) concludes. Such understanding of displacement is clearly put forward e. g. in Fox (2003), who proposes that structures obtained by the Trace Conversion Rule are subject to the interpretation according to the following recipe:
In a structure formed by DP movement, DPn [^ ... DPn ...], the derived sister of DP, ф, is interpreted as a function that maps an individual, x, to the meaning of Ф[х/п]. Ф[х/п] is the result of substituting every constituent with the index n in ф with him, a pronoun that denotes the individual x. (Fox 2003: 110)
Fox (2003: 111-112) contemplates ‘the possibility that Trace Conversion is an artifact of the semantic rule that interprets the derived sister of a moved constituent’ and does not exclude having an interpretive rule which would interpret structures created by internal merge directly as an alternative:
In a structure formed by DP movement, DPn [ф ... DPn .], the derived sister of DP, ф, is interpreted as a function that maps an individual, x, to the meaning of Ф[х/п]. Ф[х/п] is the result of replacing the head of every constituent with the index n in ф with the head thex, whose interpretation, [[thex]], is, AP. [[the]](PnAy.y = x). (Fox 2003: 111-112)
Not everyone succumbed to the charm of Carnap’s siren call; already Feys (1963) raises a warning flag, discussing the fate of identity in modal contexts:
But this argumentation accepts as granted that both sentences ‘DA/ and y has the predicate “to be an x, which is necessarily an A,’” i. e., ‘(XxOAx)y, are logically equivalent.
But this is not the case. ‘OAy is a sentence expressing a necessary proposition, whereas ‘(AxDAx)y’ is a modally ambiguous statement attributing a necessary predicate to y. Abstract modal predicates and abstract modal (necessary) individual concepts seem to lie at the root of the (modal) name-relation paradoxes. If now the legitimacy of this abstraction be restricted—in the sense that a modal abstract applied to something may not be reduced to a usual sentence as ‘OAy—we must recognize that modal logic represents a greater departure from “logical common sense” than had been supposed hitherto. A modal logic is commonly considered as adding the consideration of modalities to the consideration of facts (of factual propositions); but it goes much farther indeed if the assertion of a fact becomes ambiguous and hence may no more be handled simply as “hard fact”. (Feys 1963: 297)
Carnap’s kind disagreement notwithstanding (Carnap 1963: 908), the abstraction operator was subsequently employed as a device to distinguish expressions of the kind Feys discusses in systems rejecting unrestricted availability of fi- reduction (Stalnaker and Thomason (1968); Thomason and Stalnaker (1968), Thomason (1969), Stalnaker (1977), Fitting (1991, 2002, 2004, 2006), Fitting and Mendelsohn (1998)). It might be noted in passing that parts of the stance taken by logicians adopting abstraction operators to indicate interactions between the interpretation of terms and modal operators was already taken by their medieval predecessors: the theory of ampliatio, although embedded in a very different framework of assumptions about semantics and comprising various phenomena which would be today distinguished as being due to different causes (as e. g. resulting also from the internal complexity of terms discussed in this connection), not to mention differences of theoretical machinery, involves crucially attempts to analyze differences in supposition of subject terms as contrasting with the behaviour of predicates and taking place in the context of modal verbs, tenses, and verbs indicating ‘acts of soul’ (see e. g. Uckelman (2013), Parsons (1995, 2008, 2014) for recent discussions and copious references).