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Let us assume that syntax realizes relations between covert items in argument positions and operators in higher positions with the relations being interpreted as quantifier-variable relations. From the perspective of the interpretation, we can thus designate the covert item as a free variable relative to its predicate position—that is,whatever head projection dominates the covert item v P, say.

It does not follow, however, that the covert item is a free variable relative to any constituent part of the syntactic structure, either the completed structure or the immediately dominating head projection; it only gets to be interpreted as a free variable relative to the interpretation we give to the whole structure.

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The crucial difference between free variables and traces is that traces are formed via movement, and so there is no trace in the lower projection independently of the higher operator, i. This raises the question of why there are traces at all, which we shall broach presently, but for the moment, the issue is the status of traces as variables, on the assumption that there are traces.

In first-order languages where an n -place predicate is understood to contain n variable positions independent of whether any of these positions, given a host formula, are bound or quantified into; this is because the open predicate enters into syntactic composition with the quantifier. In natural language syntax, on the other hand, a predicate does not contain any traces or gaps independently of the moved item interpreted into the trace-marked position.

It is true that a trace syntactically saturates a predicate position, but that does not make it a free variable semantically speaking, for its interpretation is never free, but always bound by the item of which it is a trace. In effect, then, to treat traces as free variables is to read a semantic interpretation onto a compositionally distinct syntactic structure: the trace only appears free given a global semantic interpretation of the structure that does not track the syntactic formation of the trace.

This point holds generally, but the situation is most clear with wh -items, whose movement is overt. Wh -movement may be understood as based upon a rule that targets a wh -item in an argument or adjunct position and moves the item to a higher nonargument position that scopes over the original position now occupied by a trace. Thus, we go from 23a to 23b :. Crucially, the predicate love t of 23b does not occur in 23a at any level of analysis. Such a predicate is the result of the movement operation and its constituent trace is necessarily bound.


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On the other hand, in the language of first-order logic, variables are understood to be constituents of the predicate, bound or not. It might now be thought, and I suspect Higginbotham has this idea in mind, that the derivational history of 23b is irrelevant.

As submitted to interpretation, the lower predicate contains something equivalent to a free variable. Thus, Bach , p. Even this more modest position is misstated, however. If at any stage of interpretation the predicate with a trace could be genuinely open, then there would appear to be no ban against vacuous quantification of the kind witnessed in 23c as a possible interpretation of Who does Mary love?

Taking the variable in object position of love to be free appears to allow for vacuous quantification, which somehow must be excluded. Stipulating against the quantification or treating it as merely semantic nonsense is not a happy position. Vacuous quantification is formally and semantically permissible true, even , albeit absent from natural language. Partee et al. As it is, the trace must be interpreted relative to the higher operator precisely because its trace is formed via movement; it is not independently interpreted, even given just the final structure. It must be emphasized that my concern here is for syntactic vacuous quantification involving movement.

It seems to me obvious that there is no syntactic bar against semantic vacuity as evidenced by 24 :.

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With wh -movement cases, on the other hand, the relevant readings are not available. As we saw, 23c is just not a contextually-salient possible reading of 23a , even if 23c is otherwise OK, which I think it certainly is not.


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In the wh -cases at least, therefore, it appears as if vacuous quantification is a syntactic matter and cannot be explained away as mere semantic oddity. A ban on vacuous quantification is often assumed as a principle of the syntax-semantics interface without explanation e. May , p. Similarly, Koopman and Sportiche propose a Bijection Principle that puts operators and variables in correspondence, from which it follows that an operator cannot be vacuous. As we shall see, I think vacuous quantification can be banned on principled grounds, partly involving a ban on free variables.

Strikingly, Kratzer , pp. If the absence of vacuous quantification should be syntactically explained, then the introduction of free variables appears to be inconsistent with such a desideratum. Vacuous quantification should be avoided without stipulation, but the introduction of free variables appears to invite such quantification in the absence of other considerations. Leaving aside the issue of vacuous quantification, we should observe that the fundamental point here against free variables is crucial to the very idea of LF as a syntactic level where scope relations are accounted for by way of otherwise available syntactic movement operations.

Variables, on this model, are not projections of predicates independent of the operators that bind them, so they cannot be free or bound, depending on the presence or not of a higher operator. Thus, Chomsky b , pp. To say that an NP must be either bound, by the trace relation, or lexically specified, is in effect to say that open sentences are not sentences.

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In other words, predicates with at least one argument position occupied by an item that is either unbound or lexically empty is just not generated by the grammar, although it can be, as it were, specified as an open sentence given the full structure where the item in fact occurs as a trace of an operator moved outside of the predicate. In short, there is a ban on free variables at LF Chomsky and Lasnik, , pp. In recent years, suspicion has fallen on traces precisely because they appear to be items whose significance is spread over the completed structure.

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Traces, in this sense, appear to be syncategorematic in terms of their interpretation—that is, they are uninterpretable by themselves; they are only interpretable relative to the moved item of which they are the trace. This is why the trace cannot be thought of a free variable in the ways explained above.

Of course traces are not syncategorematic from the perspective of syntactic composition: They occupy a terminal node and have intrinsic categorical features inherited from the moved item. Still, the significance of traces is not strictly local, which is evident from there being products of a derivation, rather than lexical items. In the place of traces, copies are now widely accepted Chomsky, Under copy theory, there is nothing resembling a variable at all.

Copies come for free, as it were, for copies are just tokens of a single type within different structural positions of a phrase. Traces, on the other hand, must be posited as items over and above the lexical items i. In effect, therefore, a trace looks as if it is simply a theory-internal item to record a bound reading of a sentence rather than an item that must be a part of the language itself. Alternatively put, traces are posited to meet global interpretive conditions on structures, properties whole structures possess.

1 Introduction: the problem of indefiniteness in Inuktitut

Copies are essentially local in the sense that they meet conditions that hold between structurally adjacent lexical items, simply because they are lexical items, not unique items designed to express sentence-level internal relations. On a copy construal, on the other hand, who in object position meets the local conditions of object selection and thematic marking without any special assumptions whatsoever.

Whether one goes for copies or traces, though, it remains clear that the items do not possess a variable-like duality of being free or bound.

Next consider PRO. For present purposes, we may sideline the various disputes about PRO, in order to consider just its status as a variable. When in a position to be bound, PRO is necessarily bound. Here, we have the one obligatory reading, where Sam tried to bring it about that Sam himself won. Likewise, PRO cannot be construed indexically such that a specific discursively salient person is who Sam tried to have win.

PRO may occur unbound, but not as the subject of a subordinate infinitive:. PRO here is interpreted akin to one. Similarly, the PRO in 26b is necessarily agentively indefinite. Imagine that the topic of discourse is Harry, who always talks in the most impolite terms about senior colleagues to junior colleagues. In such a context, one may utter 26b and intend to be understood as making a comment about Harry, but PRO does not designate Harry, for the thought attributed to Jill is not about Harry as such but his kind of talk whether spoken by Harry or not. Note that the attempt to force a bound reading will always fail.

For the structure to be coherent, it suffices that the talker is indefinite, and so PRO has no bound or contextual value as a matter of linguistic license. It might be thought, however, that arbitrary PRO is as close to a free variable as to make no difference, for its interpretation is free, nonspecific. There is a crucial difference, though, as noted with regard to 27 : arbitrary PRO cannot be bound. Compare 28 with 8a :. Thus, PRO as it occurs in 28 is not a free variable that can be bound. This fact might be due to PRO always being covertly bound, when not controlled, as suggested in different ways by Epstein and Moltmann I shall leave discussion of this possibility aside.

It suffices to note that when arbitrary, PRO cannot be contextually valued or bound by an overt operator.