Occurrences and their counterparts

The details of interpretation of structures created by internal merge, adopting the device of А-operators, differ according to specific choices for the interpretation of expressions consisting of А-terms denoting functions and their arguments. For all those who follow the Carnapian route, taking ^-equivalence to hold irrespective of the presence or absence of modal operators intervening between the variable binding operator and the variable in the body that it binds, the head of the chain is effectively effaced by the interpretation process: taking a structure along the lines of (22), it assigns it an interpretation which defers the interpretation of DP until the level of the copy (translated as variable) is reached.

(22)

Thus, adapting the rule of Fox (2003: 111-112), the interpretation rule for displacement structures under the assumption of free availability of reduction may be stated in a semi-formal way as even ignoring possible occurrences of modal operators intervening between occurrences of a displaced item:

In a structure formed by DP movement, DPn [^ ... DPn ...], where the index n marks occurrences of DP, the derived sister of DP, ф, is interpreted in a structure M with respect to an assignment of values to variables g as a function that maps the interpretation assigned to DP by the interpretation function I and assignment g, to the interpretation assigned to ф[х/п] by the interpretation function I and assignment g which is exactly like g except that it assigns to the variable x the interpretation assigned to DP by the interpretation function I and assignment g, where ф[х/п] is the result of replacing every occurrence of a DP with the index n in ф with the variable x.

On the other hand, if А-operator fixes the level at which the argument DP is assigned an interpretation, it functions as an indicator of the base position being interpreted as ‘scoping out’ all operators intervening between the foot and the head of a chain. Intermediate positions are effectively inert for interpretive purposes, being residues of syntactic constraints, viz. computation by phases, but effects for interpretation have merely the head position and the base one, with an important property that the former is exclusively responsible for providing an object in its entirety—by applying interpretation and assignment functions— and the latter is responsible for the semantic integration of the value assigned to the head with the interpretation of the expression in which it appears. The labour becomes more equally divided once counterpart relations enter the scene—the head position determines now only a starting point for establishing object(s) which either satisfy a predicate or not. Suppose that the C-I component would attempt to establish an interpretive procedure for structures along the lines of (22) availing itself of structures of section 3.2.4, so that F? = {W R, C, D, DO), where D? = {Dn DJ, Dn and D working as in section 3.2.3, D? assigning to each point w a set DI (w) with the requirement that for any of distinct w., w,, Dn (w.) П

Dn(w) = 0; DO assigning to each point w an outer domain of objects, C assigning to each (w, w) a set {(d., d): d. e D(w ) U DO(w.) and d. e D(wj) U DO(wj)}. An interpretation function I, assigning to every point w and every predicate letter P, sets of objects d such that d e DO(w) U D;(w), is additionally added to the structure. Interpretation in a model ML = (F, Г) with respect to an assignment function g and a point w would then apply to an instance of ^Ах.Офх-1 requiring that for such an expression to be true, if О is a possibility operator, ^фхп must be true at some point w' such that (w, w') e R(w) and with respect to some assignment function g [ d:dE{dj:<r.g (TrXa))A >ec(w,w')} ]; other cases follow the same pattern.

Proceeding this way, however, interpretation would neglect the most obvious difference between formal language of modal logic and structures generated by narrow syntax and delivered to the C-I component with regard to possible interpretations of interactions between DP-chains and modal operators, which results from there being distinguished sites along the path of modal operators—links of the chain. Employing А-abstraction to model interpretive effects of consecutive applications of internal merge, the picture is along the lines of (23).

(23)

If narrow syntax guides the C-I component, an interpretive procedure which, once the head of the chain has been taken into account—i. e. once it has been interpreted by the interpretive apparatus as denoting an object of the domain, and coupled with a variable signalled by the A-operator—proceeds uniformly downwards, recording consecutive changes due to crossing modal operators along the way by appropriate modifications of assignments, would not suffice: it does not distinguish interpretive import of the presence of occurrences of DP. A move towards structures richer than above, incorporating multiple counterpart relations of section 3.2.2, seems warranted as a consequence of the presence of chains in their entirety. Let there be a structure F? + = {W, R, Cm, D, DO) as above, except that instead of C there is a family of counterpart relations Cm. In addition to there being a counterpart-based correspondence at each step of interpretation crossing a modal operator, the C-I component follows narrow syntax in distinguishing steps which happen when successive cyclic movement takes place, and requires that there be also a counterpart relation between objects assigned interpretively to such steps such that the composition of modality-induced steps along the path connects the same objects as the counterpart relation linking objects assigned to links of the chain. Counterpart-based interpretive procedures appear to be available without further ado if the fit between syntax and semantics is to be observed; and the fit resulting from their interaction seems most naturally attributable to the influence that narrow syntax exercises over that part of the C-I component which receives directly syntactic objects created during derivations in narrow syntax. Worries about referentialism do not properly even arise in this context; note that the doctrine has its most natural habitat in a setting which strives to connect atomic symbols of a language and external world:

The atomic elements pose deep mysteries. The minimal meaning-bearing elements of human languages—wordlike, but not words—are radically different from anything known in animal communication systems. Their origin is entirely obscure, posing a very serious problem for the evolution of human cognitive capacities, language in particular. (...) Careful examination shows that widely held doctrines about the nature of these elements are untenable: crucially, the widely held referentialist doctrine that words pick out extramental objects. (Berwick and Chomsky 2016: 90-91)

Figure 3.5: Multiple counterpart relations in interpretation of internal merge

Label-theoretic considerations make, however, even the simplest cases of structures involving nominal arguments contain chains thereof. Atomic elements— LI’s—enter the realm of interpretation as parts of structures which, occurring more than once, constitute discontinuous syntactic objects; no issue of direct mapping between LI’s and objects in the outside world ever arises, a possibility being only of establishing links between chains and the extramental world—an entirely different endeavour given, first, the entanglement of chain links and modality and, second, the fact that their properties are established only locally, without there being any element of interpretive machinery which would reify them or which would enable determination of their properties outside the bounds set by the interpreted structure and its properties.

 
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