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date: 05 April 2020

The Nature of Linguistic Variables

Abstract and Keywords

The concept of a variable is central to a range of explanations in linguistics and the philosophy of language, especially pertaining to quantification and so-called ‘empty categories’. The notion however, is in need of disambiguation. My aim is to remove the ambiguity by first settling on a formal characterization of a variable and then distinguishing between three grades of variable involvement: variables as theoretical artifacts, as features of semantic representations, or as syntactically realized items. I shall suggest that the first grade is relatively innocent, although unfortunately often wedded to a general antirealism about syntax. The second semantic grade, however, is difficult to evaluate on its own terms. The best case for semantic variables is for the third grade to be vindicated, but the evidence for syntactically realized variables is weak and if the notion of a variable is the formal concept, the position is conceptually problematic. The notion of a linguistic variable therefore remains to be properly explained.

Keywords: variables, quantification, empty categories, syntax, semantics

1. Introduction

The concept of a variable is central to a range of explanations across subdisciplines of linguistics (syntax, semantics, and pragmatics) and the philosophy of language. For the purposes of this chapter, I shall assume an essential continuity between the two areas of inquiry. The notion, however, remains unclear or, better, in need of disambiguation. My aim is to remove the ambiguity by first settling on a formal characterization of a variable and then distinguishing between three grades of variable involvement in which a theory may indulge: variables as theoretical artifacts, as features of semantic representations, or as syntactically realized items. I shall raise problems for each ascending level of involvement. In particular, I shall suggest that the first grade is relatively innocent, although unfortunately often wedded to a general antirealism about syntax. The second semantic grade, however, is difficult to evaluate on its own terms for reasons to be explained. Indeed, the best case for semantic variables is for the third grade to be vindicated, but the evidence for syntactically realized variables is weak and if the notion of a variable is the formal concept, the position is anyway conceptually problematic. The notion of a linguistic variable, therefore, remains to be properly explained if we are to appeal to the notion at all. The following may be viewed as prolegomena to that task.

Before we begin in earnest, two qualifications are in order. First, the focus throughout the sequel will be on objectual variables and first-order operations. I shall assume that no issues arise with higher-order variables that would vitiate the arguments on offer. Second, the range of linguistic phenomena that bears on the issue of variables is vast. My selection will not be prejudicial; I discuss what seem to me to be the most clear and neutral cases to exemplify the relevant issues. In particular, I focus on phenomena where there is an accepted syntactic constraint on possible interpretation, the question being: Does the constraint involve a variable in the relevant sense? My attention, therefore, is not on interpretive context-sensitive phenomena, where the existence of a syntactic constraint is very much moot. It is precisely the chief point of the following, as will become clear, that positing a syntactic variable to cater for such phenomena is rashly unwarranted given the conceptual distinctness of variables and indexicals and the general absence of variables in syntactic analysis. Still, arguments specific to such interpretive phenomena along the lines here more narrowly adumbrated are available (Collins, 2007, 2013).

2. The Formal Variable

The developmental of the concept of a variable and its associate, the function, constitutes a long story in the history of mathematics. For our purposes, assume a first-order logical language as applied to a fragment of English. A formal variable, in this context, is what I shall call a term of abstractive generality, a notion that has an obvious resonance with the work of Frege (1879/1967) and subsequent treatments of the variable concept by Quine (1953, 1960a), Curry and Feys (1958), and Dummett (1973/1981), although I am not presently concerned with historical fidelity. Consider a simple sentence such as (1):

  1. (1) Frank loves Ava

To understand the logic of (1) involves understanding the sentence as positioned within a space of entailment relations; in particular, (1) may be viewed as an instantiation of a range of generalizations. For example, on the assumption that Frank and Ava are names of objects, (1) is true given the truth of any of the sentences in (2):

  1. (2)

    1. a Everyone loves Ava

    2. b Frank loves everyone

    3. c Everyone loves everyone

Similarly, if (1) is true, then all of the sentences in (3) are true:

  1. (3)

    1. a Someone loves Ava

    2. b Frank loves someone

    3. c Someone loves someone

If we assume that quantifier phrases express properties of properties (that a property is instantiated by some cardinality of things), then the sentences in (2) and (3) may be viewed as generalizations over three ways of picking out or abstracting a predicate from (1):

  1. (4)

    1. a (_) loves Ava

    2. b Frank loves (_)

    3. c (_) loves (_)

The problem with (4), of course, is that a gap isn’t a something; in particular, the position and distinctness of what we have abstracted over determines the entailment relations exhibited, but a mere gap tells us nothing. We must therefore specify that which has gaps, as it were, in terms of the position and potential distinctness of the items over which we have abstracted. Variables serve such an end. They allow us to specify predicates in abstraction from any linguistic environment, but in a manner that preserves the logical role the predicate has in any such environment. So, substituting variable letters for the gaps in (4), we arrive at (5):

  1. (5)

    1. a x loves Ava

    2. b Frank loves y

    3. c x loves y

The choice of alphabet or symbol is arbitrary (Greek letters or shapes would do as well), but the properties the variables mark are essential. Thus, the number of variables associated with a predicate indicates its adicity, which may be understood either in terms of how many terms we have abstracted away in order to arrive at the predicate or in terms of the number of terms required to turn the predicate into a sentence (a reduction of the predicate’s adicity to zero). Further, the positions of the variables indicate the relation values of the variables hold to the predicate (a lover or a lovee, for example). Finally, the type of the variables indicates whether or not tokens of the same type have been abstracted, or, to put the point in the material mode, whether the predicate may apply to distinct objects.

One difficulty in being clear about the status of a variable is that it leads a triple life. On the one hand, a variable is an arbitrary symbol; all that is important is that we have indefinitely many distinct symbols to serve the variable role. On the other hand, a variable is a logical item that, as indicated, marks adicity, position, and identity. On the third hand, if you will, a variable is valued relative to a domain of objects. The full life of a variable involves all three roles. For example, in (5a) and (5b), the choice of “x” or “y” or some other symbol is arbitrary, for any variable symbol will do because there is only one variable position and a variable is not valued by any object in particular. (5c), though, requires type-distinct variables because we have two positions, the distinctness of which matters to the logical role of the predicate. For instance, a pair of distinct quantifiers may apply to (5c), generalizing over distinct positions, but not (5a) or (5b), and which quantifier applies to which position determines distinct truth conditions. Finally, variables require a domain of objects, if they are to make any semantic contribution to the host formulas.1

So we understand quantifier phrases to express operators that apply to predicates featuring free variables (variables otherwise not associated with or bound by an operator) such that the variable type of the operator binds all occurrences of the same type within the formula. For example:

  1. (6)

    1. ax)[x loves Ava]

    2. b (x)(Ǝy)[x loves y]

Before moving onto the status of the variable in natural language, I shall highlight a chief feature of the formal variable, which will be significant for the sequel.

A ready objection to the abstractionist approach to variables just sketched is that what the variable is supposed to be abstracted from is, at best, arbitrary; no particular complete formula needs to be the source of the variable formula (Fine, 2007). The complaint here is that the abstractionist approach treats free variables as somehow derived from sentences featuring no variables or only bound variables, but the choice of a source for any such derivation appears to be arbitrary.

First off, the objection is perfectly apt against the sketch just offered, for the open formulae or predicates presented in (5) are clearly not essentially associated with (1). In most general terms, however, we do understand a free variable relative to a domain and so relative to an operator defined over the domain:

  1. (7)

    1. a λx. x D. Φ(x)

    2. b {x D: Φ(x)}

Both formulas here are complex terms: (7a) designates the function from the members of D to those members of D that are Φ, or the property of being Φ, and (7b) designates the subset of D whose members are Φ. Both characterisations are available, and so, without further ado, the choice is arbitrary. Yet the need for some such closed specification of an open formula is not gratuitous. So, the abstractionist account of variables isn’t required to single out some particular source from which the abstraction to variables proceeds; rather, the basic claim is that the free variable is not explanatorily prior to closed formulae. On the contrary, the free variable is always understood relative to the valuation of its host predicate over a domain, and this can either be left implicit or made explicit as in (7).

Crucially, however, it does not follow that there really are no free variables. A free variable is simply a variable that is not explicitly bound by an operator within its host formula. A variable, though, is essentially such as to be bound in the sense that it marks a position within a predicate open to operators defined over the predicate. It is in this sense that (7) makes explicit how we should understand a free variable as open to valuation relative to a domain such that it must always be available to be bound by an explicit operator. As we shall see, this essential duality of a variable as being either free or available to be bound plays a central role in the evaluation of the status of variables in natural language.

There is much else to be said about variables, but the detail is better presented in the context of thinking about natural language, to which I shall now turn.

3. Variables in Natural Language

A formal language is invented, and has its properties by agreed stipulation. A natural language is a phenomenon whose properties are to be discovered. It is not an a priori truth, therefore, that natural language should feature anything approximate to a variable. Still, the formal languages that feature variables clearly capture aspects of natural language, as we shall see, and if variables are essential for that achievement, then natural language must, it seems, feature variables. Consider the following cases as examples of how both overt and covert variables appear to be involved in the interpretational possibilities of sentences:

  1. (8)

    1. a Every philosopher thinks he is a genius

    2. b (every philosopher x)[x thinks x is a genius]

    3. c (every philosopher x)[x thinks y is a genius]

  2. (9)

    1. a The duck is ready to eat

    2. b (the duck x)(some y)[y ready (y to eat x)]

    3. c (the duck x)(some y)[x ready (x to eat y)]

There are indefinitely many constructions that would have served as well, some of which we shall meet below. These cases simply strike me as being relatively clear in their readings. (8a) is ambiguous between readings that treat the pronoun he as bound by every philosopher or as free, contextually referring to some salient person (Plato or Kant, say). (9a) exhibits a case where there are no explicit pronouns, but “gaps” that require filling for a coherent interpretation to be available. In particular, eat requires an eater and a thing eaten, but both are missing in (9a). On the face of it, therefore, natural language features items that can be bound or free and be overt or covert; indeed, the presence of overt variable-like items offers a prima facie case for such items being covert too, which appears to be realized. At any rate, that is how things appear to be. In the sequel, I shall consider three ways of understanding the role of variables in relation to the kind of data exhibited in (8–9).

4. The First Grade of Involvement: Variables as Descriptive Artifacts

A useful way of investigating the status of variables in natural language is to ask what the relation is between an invented formalism featuring variables and natural language. A deflationary attitude to the question is, in effect, to say that there is no real distinction here. Consider the following remarks from Quine (1960b, p. 159):

The artificial notation of logic is itself explained, of course, in ordinary language. The explanations amount to the implicit specification of simple mechanical operations whereby any sentence in logical notation can be directly expanded, if not into quite ordinary language, at least into semi-ordinary language … [T]o paraphrase a sentence of ordinary language into logical symbols is virtually to paraphrase it into a special part still of ordinary or semi-ordinary language; for the shapes of the individual characters are unimportant.

The thought here is that the formal language and its special variable and operator symbols are just a “regimented” paraphrase of the source sentences. “Shape” doesn’t matter: either x or he, or every or“ ∀ will do. All that is crucial is that the paraphrase serves some useful end of the theorist who deploys the regimented structures. Thus, Quine (1972, p. 451) says the following on the question of the “reality” of a particular logical analysis:

I find the phrase “logical analysis” misleading, in its suggestion that we are exposing a logical structure that lay hidden in the sentence all along. This conception I find both obscure and idle. When we move from verbal sentences to logical formulas we are merely retreating to a notation that has certain technical advantages, algorithmic and conceptual … [D]eep structure loses its objectivity … [T]he grammarian’s deep structure is similar in a way to logical structure. Both are paraphrases of sentences of ordinary language.

These remarks anticipate some of the issues to follow, but let us, pro tem, consider Quine’s position on its own terms. Quine’s chief point is that logical analysis, and with it the variable, is a descriptive artifact, which does not explain the linguistic phenomena but merely renders it in a way amenable to various ends the theorist might be independently pursuing. Such a descriptive view cannot be faulted on grounds of immodesty; after all, some descriptions are better than others along various dimensions, and we do expect logical/variable paraphrase to give way to a natural language formulation, albeit a tortuous one. Still, the position is so weak that it cannot be occupied in good faith in the face of the complexity of the relevant phenomena. A theorist, I suggest, is obliged to explain why only some but not other paraphrases are acceptable, regardless of what end the theorist might have in mind for being interested in a logical analysis in the first place. After all, no one thinks that just any analysis/paraphrase is supported by the language.

Let us grant that each English sentence has a variable-involving paraphrase of the kind indicated in (8–10); further grant that these paraphrases in turn can be rerendered into English. The question remains, as just suggested, why such paraphrases are available. The point of this query is simply that a sentence’s having one set of potential paraphrases rather than another, or its submitting to one analysis rather than another, is itself a linguistic phenomenon that is not explained by any particular class of paraphrases or choice of analysis. One wants to say: there is some property of the language, independent of any analysis, which explains why one analysis rather than another is admissible. Of course, one may seek to offload such an explanatory demand onto some nonlinguistic field of inquiry—language is the way it is because of the history of a practice or the dispositions of speakers or psychology or whatever. Yet the demand I have in mind is not so much for a metaphysical grounding of an acceptable analysis, which may be an “obscure,” if not “idle” request, but for clarity on the constraints apparently inherent to language itself that admit and preclude certain analyses. The point is pressing, for whatever the basis for the permissible logical paraphrases is, a description of the language itself needs to cleave to the constraints that obtain, which are not evident, let alone explained, by the mere availability of an invented formal language whose formulas submit to natural paraphrases. More specifically, the constraints on permissible paraphrases are intriguingly misaligned with what should be available from a formal perspective (i.e., formally sanctioned paraphrases are not linguistically admissible).

On the first and general point, reflect on why it is that (10b) is acceptable, if (10a) is:

  1. (10)

    1. a Bill loves Mary

    2. b Everyone loves someone

The query here is not skeptical, as if (10b) only seems to be acceptable, but why it should be acceptable given (10a). In prosaic terms, love is a relation between persons (or persons and things), but the subject and object of (10b) do not name particular persons. The explanation, therefore, cannot be straightforwardly semantic, for quantifier phrases such as everyone and someone do not name entities at all; or at any rate, if they do, then the truth conditions of (10b) go haywire, for (10b) does not claim that the abstract object or set that includes every person stands in an emotional relation to an equally abstract entity that includes some or other number of persons. The explanation is not straightforwardly grammatical either. Not any old noun phrase (NP) can be an argument of love. Witness:

  1. (11)

    1. a *Bill’s leaving loves Mary

    2. b *Easiness loves Mary

Moreover, (10b) is subject to a familiar scopal ambiguity in a way (10a) isn’t.

The apparent distinctness of names from quantifiers can be made clear by way of standard categorical grammar type theory. Assume that <e> is the type of a name and <t> the type of a sentence. Let the transitive verb love be of the type <e <e, t>> (i.e., a function from an entity e to a function from another entity e to t). Thus, providing love with an object argument produces a monadic predicate (<e, t>) that in turn takes a subject argument (<e>) to produce a sentence (<t>) that is truth-evaluable. So much makes sense of (10a), but quantifier phrases are not type <e>, but rather type <<e, t>, t>, i.e., they express functions from monadic properties (<e, t>) to truth values, in line with the intuitive thought that quantifiers express properties of properties. Suffice it to say that the type-mismatch here has generated much debate, which is ongoing (Szabolcsi, 2010).

The crucial thing to note, however, is that the mismatch becomes much less puzzling, if we assume that the kind of logical form quantifier phrases exhibit in a formal language is in fact realised in natural language, for in the formal language, quantifiers always scope out of the predicate. Problems persist, but the salient point is to recognize the mismatch as an explanatory burden, which the mere availability of a paraphrase does not resolve (cf. Heim and Kratzer, 1998).

The general point here is made all the more intriguing because of the expressive misalignment between the target language and the typical formal languages in which the logical analysis/paraphrase is couched. The principal phenomenon is that many operator-variable relations are not realised in natural language, even though they are formally and semantically permissible. Consider the following cases:

  1. (12)

    1. a If every philosopher comes to the party, the host will be upset

    2. b A host asked whether he could invite every philosopher

    3. c A host organised a party that every philosopher shunned

These sentences exhibit so-called island effects: the DP every philosophy is restricted to narrow scope. Note, however, that there is nothing the least bit semantically bizarre with the readings in which every philosophy wide scopes, and, of course, a Quinese paraphrase can easily be cooked up. Still, the sentences in (12) lack wide-scope readings of the universal DPs. Why this should be so is not my present concern. The crucial point is that viewing quantifier-variable relations as merely matters of description fails to provide the resources to explain the phenomena, or even predict it.

A broader concern for being content with the first grade of variable involvement is that it is known that variable-free systems of logic/syntax are (at least) formally equivalent to systems that employ variables. Let us dub such systems combinatory categorical grammars (CCGs). The precise character of this class of formal systems is not a matter we need dwell on here.2 Suffice it to note that they constitute pure function systems, where all relations are realised by function application. There are, therefore, no dependencies between items beyond adjacent interpretation, and so there are no operator-variable relations. None of this, of course, was lost on Quine. Quine (1960a) readily endorsed the logical power of CCGs, and saw it as a way of “explaining variables away.” Hence it might have been that he saw one formal system as having mere technical advantage over another. At any rate, notwithstanding his regard for all things first-order, Quine happily recognized that CCGs can do everything first-order logic can. Yet far from militating for a descriptive attitude to natural language form, this situation obliges us to search for a resolution. If we wish to explain linguistic phenomena and not merely describe them, then that the phenomena pattern one way rather than another is what is to be explained (inter alia). It might be, therefore, that variable-free and variable-involving accounts are not equally up to such an explanatory demand; or it might be that neither are. In short, the right description will simply be the description that embodies the theory that explains the phenomena. The overdetermination Quine readily embraced can only be indulged if one has forsaken ambitions to explain the phenomena, or if one thinks that there is no fact of the matter at all. As just noted above, however, there are clear and very curious facts to language that require explanation.

The appropriate conclusion here is not necessarily that we must move beyond the first grade of variable involvement. What is certainly correct is that a general descriptivist attitude to syntax and logical from is ruinous of the prospects of any explanatory ambition. It might be, however, that variables specifically do not find a fundamental place in linguistic explanation. If so, then they are perfectly legitimate as first-grade entities, from which it does not follow that the whole of syntax is fated to attract mere first-grade commitment. So, as things stand, variables are kosher as first-grade entities, but if they are to be linguistically real, we must find a place for them beyond contributing to a description of linguistic facts.

5. The Second Grade of Variable Involvement: Variables as Semantic Features

A second and higher grade of variable involvement is to take variables themselves to be semantic kinds that constitute or contribute to the meaning of natural language expressions. The picture is what Jacobson (1999) refers to as the ‘standard view.” Here, variables are not merely part of a notation employed to describe semantic phenomena, but part of the phenomena themselves. In the “formal mode,” this claim amounts to the thought that any language adequate for semantic theorising will contain variables. Take a semantic theory to be an assignment function that maps elements from a domain onto expressions of the language, where the values of such assignments compose to constitute the semantic properties of complex expressions. A variable, on this model, enters into the theory in case the assignment for an expression is nonconstant (i.e., it varies relative to a determinate range of factors). Personal pronouns, demonstratives, and temporal and spatial adverbs are the paradigm, but the model can and has been extended to a whole range of other linguistic phenomena that appear to exhibit nonconstant interpretation, such as quantifier noun phrases, gradable adjectives, relational nouns, weather reports, and many other kinds of construction. Crucially, these apparent variables may be free or bound, and so appear to be much like the variables that occur in formal languages. Also, the variables may be covert in the sense of possessing no morphophonemic signature, a point to be returned to at length.

I shall leave mostly unmolested the idea that a proper semantics must deal with variable assignments. This is for two interrelated reasons. First, it might well be that items, such as pronouns in cases such as (8a), are interpreted as variables, at least in some constructions. If so, then an interpretive mechanism is required to convert a syntactic item into an element that may be construed as free or bound. We shall consider a system of indexation in Section 6.1 as a syntactic-semantic model of variables. On the other hand, if the variable-free semantics approach is correct, variable construal will be a theoretical or descriptive option with no essential “linguistic reality” even in syntax or semantics. The defender of semantic variables, though, may retreat to a neutral position that retains a variable semantics but doesn’t treat it as an interpretation of syntactic structure; perhaps the semantics describes a level of language-independent thought, or the variable interpretations are an aspect of lexical understanding that does not project into the syntax. The point, put simply, is that there is a lot to argue about even if one dispenses with the thought that syntax contains variable items. The general plausibility of these positions is mostly a matter for another day.

Second, understanding variables to be semantic features is a somewhat unstable position. Arguments can always be had that the variable effect on interpretation is a pragmatic feature, rather than linguistic proper (cf., Recanati, 2004, 2010). Such a complaint would be stymied, if the variable feature had an independent basis in syntax (Stanley, 2000; Glanzberg, 2007; Schaffer, 2011). Indeed, we might say that the best argument for the second grade of variable involvement would be a vindication of syntactic variables of the required sort, which will be our topic for the remainder.

6. The Third Grade of Variable Involvement: Syntactic Projection

The third and most committing grade of variable involvement considers variables to be interpretable syntactic items. Of course, such a position is equivalent to the semantic variable position only if syntax is viewed as a projection from semantics, (i.e., semantic types and their composition determines syntactic types and their composition into “grammatical” forms). If, though, we assume that syntax is autonomous of semantics (i.e., a distinct system of structures with their own principles of organization), then the position becomes much stronger than our second grade of variable involvement. This is because the presence of variables in syntax would be an independent parameter, which, when in accord with a semantic account, would provide an independent source of confirmation for some given analysis of the relevant linguistic phenomenon, such as a locative variable for weather reports or a comparison class variable for gradable adjectives. The problem facing a defender of such an account, however, is to provide syntactic evidence or rationale for the presence of the variables in syntax. If no such support is forthcoming, then the syntax will not provide independent evidence for the account at hand; indeed, it would appear that the putative syntax is just a species of semantics and the account would collapse into the weaker second grade involvement.

It must also be emphasized, and I shall return to this point at length, that sanctioning quantifier-variable relations in syntax, or relations interpreted in that way, does not, ipso facto, involve conceding that free variables occur in syntax. In a formal setting, a free variable is invariant in its interpretation over bound and free readings, and so is what a quantifier binds in the sense that the variable makes a compositional contribution to the value of the host structure. A quantifier-variable relation, however, need not involve a variable that is free; it suffices merely that there is a dependence relation of interpretation (cf., Chomsky, 1980a, pp. 163–165). Thus, Chomsky (1981, p. 102), for instance, characterizes a variable as follows: “α is a variable if and only if α = [NP e] in S bound by an operator.” This definition stipulates that all variables are bound. A theorist is free to stipulate however they please. The interesting issue is whether in fact natural language features items that are variables in being bound or free, regardless of what anyone calls them. Within generative syntactic theory, various items, both overt and covert, appear to be akin to variables in precisely the sense that they can be free or bound. I think appearances are misleading. First, though, let us consider why the idea is tempting.

As noted in Section 3, many overt pronouns certainly appear to be variable-like by way of their Quinese paraphrases. Accepting such appearances at face value, it now seems as if there is a license to posit covert variables, free and bound, for the overt cases demonstrate that variables occur in language. Before we get to the putative covert cases, it is worthwhile to undermine this idea.

6.1 Against Variable Indexicalism

First off, pronouns are clearly not like x, y, z, etc. for they are morphologically invariant relative to binding (they differ as regards case, but that is irrelevant). Strictly speaking, then, no pronoun can be a variable, because there is nothing to the item as a morphological overt unit that differentiates its being free or bound. This is obvious enough in the case of (8a), where the overt form he is precisely ambiguous because of a lack of overt marking between the free and bound construals. What does distinguish the cases is not at all obvious; indeed, the answer to that devolves onto the fate of binding theory as a branch of linguistics, which is currently in some turmoil (more on which shortly). What is clear, however, is that what makes a pronoun variable-like is covert, not overt. In that case, one cannot immediately conclude from the behaviour of overt pronouns that there is a general license for variables in syntax. The problem here is not a mere skeptical one; rather, it remains unclear if variable is so much as a syntactic kind, because it remains unclear what the covert mechanism is that allows pronouns to be differentially construed. Assume, however, as was done traditionally, that pronouns have affixed indices where identity of index means joint construal, and distinctness means disjoint construal (Chomsky, 1981). “Classic” binding theory does not constitutively involve indexation as presented in Chomsky (Chomsky, 1981, p. 188).

  • (CBT) A: A reflexive is locally bound3

  • B: A pronoun is locally free

  • C: An R-expression is unbound

Indexes enter into the picture as an explanation of binding:

  1. (B) α binds β iff

    1. (i) α c-commands β and

    2. (ii) α and β are co-indexed.

Thus, indexes mark when (intended) reference is the same (coreference) by a matched pair of indexes or disjoint by distinct indexes, but the binding conditions themselves do not independently tell us which is which apart from in the case of Principle A, which renders indexes there redundant (i.e., because binding is necessary, indexes could only be matched so do not carry any information). As first introduced by Chomsky (1977a, 1980b), indexes were explicitly understood as part of a theory of construal of syntax, not an aspect of syntax itself. This way of slicing the phenomena tallied with the view of Chomsky (1976), Lasnik (1976), and Reinhart (1976), according to which optional relations of coreference are not syntactically marked.4 Something like this old view is correct, I think: Syntax only marks that which is obligatory (a problem with the view as initially presented is casting binding relations as if they were ones of coreference). Regardless of the syntactic ins and outs, however, arming pronouns with indexes definitely invites treating the pronouns akin to Tarski’s (1936/1983) ordinal-indexed variables. The invitation to conflate variables with indexicals has been accepted by many but remains erroneous from both sides, as it were: Tarski and natural language are misrepresented equally.

Tarski treats each variable of a defined object-language as having an ordinal index such that the variable may simply be understood as the index. Each index can be thought of as designating a position in every (infinite) sequence of objects, where each sequence in turn may be thought of simply as an ordinal of type ω. A formula with a free variable, ψ(xi), is satisfied by every sequence Σ iff σi (the ith member of Σ) satisfies the predicate hosting the variable—that is, iff ψ(σi). The method generalizes, of course, such that, for any sequence of indexed free variables in a formula, ψ(xi, …, xn,” the formula is satisfied by a sequence whose corresponding ordinal positions are occupied by objects such that ψ(σi, …, σn). Crucially, satisfaction is understood in relation to every sequence, never a designated sequence; this has to be so in order to capture the generality of the variable. For instance, if we took prime(x2), say, to be evaluated relative to some given sequence, then, absurdly, we would be saying that only the second object of the sequence is prime. Rather, we take the formula to be evaluated relative to every sequence and take the formula to be satisfied by every sequence whose second object is prime. In each case, a formula will be satisfied by infinitely many sequences, even if only one object possesses the property the formula expresses. For example, there are infinitely many sequences with 2 as their first member, and all of them satisfy even-prime(x1). We may say that a variable is a name for the objects that occupy the position of the sequences that satisfy the formula that hosts the variable. That way of talking, though, is cute, and part of Tarski’s point, but is misleading. Sequences are defined such that, for every position i in a sequence Σ, there is a sequence Σ´ that differs from Σ in at most position i. So for every object there is a sequence whose ith position is occupied by the object. Since formulas are not satisfied by unique sequences, if a variable is a name, it is a universal name for every object. The indexation is a way of distinguishing positions within a sequence of variables, and is arbitrary vis-à-vis satisfaction.

The moral here is that if a free (/unbound) variable is restricted in its indexation such that it is satisfied by a unique object, then it is not a variable. It is tempting, for instance, to view natural language indexical items as if they were indexed free variables that designate definite objects relative to a sequence (cf. Fiengo and May, 1994; Heim and Kratzer, 1998; Landman, 2006; Larson and Segal, 1995). Call this view variable indexicalism. The view is one traditional semantic approach to pronouns, for sure, but it has not been without its critics, for indexicals also appear to be constants in being unbindable in some cases, such as I and here, although the data is somewhat complex (see Kratzer, 2009; Partee, 1989). Also, the view need not be taken to imply that pronouns are of type <e>. A general approach is to view pronouns as having a determiner rather than singular term structure, whether syntactically (e.g., Kayne, 2004; Longobardi, 1994; Postal, 1966) and/or semantically by way of some form of a definite description construal (e.g., Elbourne, 2005; Heim, 1990; Heim and Kratzer, 1998; Neale, 1990). If pronouns are definite descriptions or have their semantics fixed by way of such descriptions (D- or E-type), then they may still be analyzed, and typically are, as involving an indexed variable, as an argument of a function or as attached to the (covert) complement of the determiner (see Elbourne, 2005).5 It is confusion, however, to identify free variables with indexicals.

Heim and Kratzer (1998, p. 92) state that “[a] variable denotes an individual, but only relative to a choice of an assignment of a value.” Heim and Kratzer, in fact, identify pronouns, variables, and traces with the following rule:

  1. (V) If α is a pronoun or a trace, a is a variable assignment, and i ∊ dom(a), then [[αi]]a = a(i).

The rule interprets an indexed pronoun according to a variable assignment function as the value of its index under the function.6 Similarly, Dowty et al. (1981, p. 203), explaining a Montogovian treatment, claim that “subscripted pronouns he0, he1, he2, … [ … ] serve roughly as free variables in English syntax.”7 Larson and Segal (1995, pp. 202–203) index pronouns in direct analogy to the Tarski-style treatment of variables (“Tarski … [made] use of something like the pointing idea” constitutive of the notion of a pronoun (Larson and Segal, 1995, p. 202). Thus, “x is the value of the pronoun hei with respect to sequence σ if and only if x is the ith element of σ” (Larson and Segal, 1995, p. 203). Similarly, Fiengo and May (1994, pp. 6–7) “interpret NPi … as picking out σ(i), the ith position of σ. This NP will refer to the individual assigned to that position.” Fiengo and May correctly distinguish positions from objects but then immediately run the two together. More generally, consider the following reasoning from Stanley (2000, p. 53): “[V]ariables can either be bound or free. An account involving variables, therefore, predicts there to be two readings, one in which the value of the relevant variable is supplied by context, and one in which it is bound.” In all these cases, the semantic properties of variables are identified with those of indexicals. There are two fundamental problems with this conflation.

First, a free variable is not variable name, as it were, as if diverse tokens of the type may refer to diverse objects, here codified by the notion of different “variable assignments.” That conception effectively turns a free variable into an arbitrary name, as if it referred to some definite object upon a tokening, but it does not refer to any definite object as a type. As explained, indexation of free variables is not a referential device for picking out objects, for no unique object occupies an ith position across sequences. What an index refers to, qua ordinal, is a position in each sequence. It is simply no part of Tarski’s (1936/1983) formal method to treat variables as uniquely referring relative to an assignment. Hence, a variable is general, not because it can refer to many different objects relative to an interpretation, but because it refers indefinitely to every object. On Tarski’s method, quantifying over a variable effectively cancels its index such that the variable imposes no restriction on what sequence satisfies the host formula. The generality of the variable, therefore, is unaffected; the quantifier, rather, imposes a restriction on the satisfying sequences to be (some, every, … ) i-variants of sequences that do satisfy the open formula with the variable free. The method is not compositional at the level of sequences, for all true formulas are satisfied by every sequence. The method is compositional, rather, in defining a unique relation of satisfaction that recursively applies to every wff of a specified language. It is difficult to understand how an indexical could be quantified over if understood as an arbitrary name that, on an occasion, refers to a particular object. The quantifier could cancel the index, but then there would be no establishment of the relation between the quantifier and the variable. Alternatively, the indexes could be matched, but this would amount to a stipulation, which would, anyway, fundamentally change the semantic significance of the indexed item. In general, one would, it seems, be treating the item as effectively ambiguous depending on whether it is bound or not, an ambiguity that is globally resolved. A variable, though, just is an item that can be free or bound, and so may compositionally enter into the determination of the value of a formula that contains it as a bound item. So indexicality is not a property of formal variables. My quarry, however, is the very distinction between free and bound readings of variables in natural language. If that distinction does not hold, then indexicalism is sunk, even if it were to be a plausible model of free variables, which it certainly isn’t.

Second, as baldly presented, treating variables and pronouns alike as if they both referred to unique objects relative to a sequence seems to get pronouns wrong. A deictic token of a pronoun refers to a particular object as a function of what the speaker intends (or perhaps as encoded in the item’s character, such as with I). We precisely don’t want sequences to enter into the picture here, for the token refers to a definite object, not whatever occupies a position of a sequence. In other words, the definiteness of deictic pronouns appears to be at odds with the general variability of variables, which is just what we expect given that variables, qua intrinsically general, are quite unlike pronouns, qua definite as deictically tokened. In other words, free variables are not free pronouns.

Larson and Segal (1995, pp. 207–208) propose a solution to this problem. They posit a “selection relation” Σ that holds between utterances and sequences such that “[i]f a speaker uses αi [a deictic pronoun] in u to refer to x, then Σ(u, σ) only if σ(i) = x.” The idea is that the use of an indexed pronoun (αi) will select a sequence that satisfies the host utterance (u) such that it has the intended referent of αi in its ith position. The interpretation rule will issue in conditionalized statements of truth conditions, where the consequent may be detached given the provision of extralinguistic information about the relevant contextual factors that bear upon the interpretation of u (who the speaker is, what time it is, etc.). An obscurity in the proposal poses a dilemma.

Assume that Bill is the intended referent of a token of himi and that Σ selects one sequence σ such that σ(i) = Bill. The choice of sequence is thus entirely arbitrary save that its ith member is Bill. Yet this is very odd, for, presumably, whatever fixes the sequence as a relatum of Σ is not arbitrary at all; the intended referent is just Bill, not sometimes <Chomsky, Paul McCartney, … Bill, … > and some other times <Montague, John Lennon, … Bill, … >. Of course, the other members of the sequence are irrelevant, so the arbitrariness of the sequence selected might seem to be inoffensive. This just raises the question, though, of why any sequence at all should be selected, given that no answer can be provided why one sequence rather than another is selected, whereas it is clear that one shared member of all the admissible sequences is selected. The arbitrariness is removed by assuming that Σ selects every sequence σ such that σ(i) = Bill. The puzzle is not resolved, though, for an equally pressing question is pending: Why should intending to pick out Bill with him be gratuitously conditioned by infinitely many sequences? The problem for both options, note, is not so much that no speaker using a deictic pronoun has intentions towards infinitely many sequences, although that is patently true. The problem, rather, is that the intended referent is just Bill, so no sequence at all is semantically essential. The selection of one sequence is arbitrary, and the selection of every sequence is gratuitous. The appeal to sequences only seems to be a good idea because of an illicit conflation. Sequences are required for variables precisely because they are interpreted always generally, which is not the same feature at all as the deictic freedom of the pronoun. The fundamental difference, which is what indexation to sequence positions in fact enshrines, is that variables never refer to any definite thing, but are valued only relative to positions; hence, they are intrinsically general as regards the objects that may be their values. On the other hand, pronouns precisely do refer to definite things, which indexation relative to sequences fails to capture. Only relativization to a definite sequence can secure a definite value, but that is not how variable indexation is understood, and the appeal to sequences here can only be arbitrary or gratuitous anyway.8

The apparent mismatch between the generality of the free variable and the definiteness of the indexical is also noted by Fiengo and May (1994, pp. 64–66). Their solution is, in effect, to dispense with sequences for nonquantificational constructions. Instead, a context is “constructed” in which definite values are assigned to the indexical items of a structure as determined by the lexical content of the items plus the intentions of the speaker. Of course, one may retain talk of sequences as “partial,” as Fiengo and May do, but any relation between indexes and variables is here forgone. Indexed items, as noted above, are simply treated like variable names, names relative to context, whereas variables are always general.

My aim here has just been to reject the idea that free variables are like indexical pronouns in taking definite values via an indexation (relative to an assignment). It might still be that pronouns are associated with syntactically represented indexes, albeit ones unlike Tarski-style indexes. I shall presently reject that idea. My only claim based upon the above reasoning is that formal variables are not deictic pronouns, regardless of whether such pronouns have indexes as part of their syntactic structure. As advertised, variable indexicalism is sunk anyway, for syntax contains no free variables independently of the fact that pronouns and indexed variables are quite distinct.

It should also be said, as intimated above, that the status of binding and indexes has always been somewhat contentious. As first introduced by Chomsky (1977a, 1980b), indexes were explicitly understood as part of a theory of construal of syntax, not an aspect of syntax itself. Syntactic relations appear to be indifferent to the identity or distinctness of a sequence of indexes; moreover, identity being symmetrical, a coindexing relation all by itself will not establish a dependence relation, which is asymmetrical (Higginbotham, 1983, 1985).9 Furthermore, Reinhart (1983, 2006) pointed out that indexes appear redundant for establishing coreference as opposed to variable binding, not least because coreference can take place in violation of binding conditions.10 Still, indexes became an integral part of theoretical description in binding theory, and had a crucial role in conditions on movement. As noted above, “classic” binding theory does not constitutively involve indexation as presented in Chomsky (1981, p. 188). Still, Chomsky (ibid., p. 187) assumes that “each lexical element is inserted in D-structure with an index (including PRO), indices being copied by Move-α.” This is an assumption simply required by the assumption of indexation in the first place. Indexation cannot be formed by composition or movement, for not all referential relations are due to movement (apparently), and indexation could only be formed by composition by way of an extrastructural principle of construal (indexation disambiguates between structural equivalents). Yet it is utterly arbitrary what indexation a lexical item should have, and whether two or more items should be the same or disjoint in their indexation can only be decided by stipulation or chance, for if items have their indexes inherently, then there is no reason why they should match or not beyond independent syntactic conditions such as movement that cannot by themselves establish indexation.

Recent years have witnessed a growing skepticism about indexes concomitant with the thought that binding phenomena are an interaction effect of a range of linguistic principles interfacing with extralinguistic systems of interpretation, an idea that harks back to Reinhart’s pioneering work. If indexes are supposed to lend a unity to binding relations, then they have a dubious status precisely because they are features that mark a global construal of a structure rather than being inherent to lexical items or generated by syntactic composition (Chomsky, 1995, p. 228).11 As Rooryck and Wyngaerd (2011, p. 11) put it, features that indicate whether an item is bound or not appear to be “little more than a shorthand for what needs to be explained.”12 For our purposes, one apparent moral is that overt binding is not a syntactic matter and so cannot be appealed to as a general rationale for the syntactic realization of free variables. That thought is not quite right. The fundamental point is that something like a system of indexes appears not to be syntactic in character, but it might well be that much if not all of classic binding theory can be reconstructed in an index-free syntax. An alternative, as remarked upon above, is to treat bound pronouns as not variable-like at all, but more like functional items, such as definite descriptions, with binding realized by a matching of lexical content, perhaps under ellipsis (Elbourne, 2005; Heim, 1990, 2008; Sauerland, 2007).13

If such trends are on the right lines, and there is no system of indexes, and so no general constraints on their matching or not in a structure, then the “conclusion is unavoidable: there is no ‘binding theory,’ and the superficial constraints on anaphoric dependencies tell us very little in isolation of other properties of a language” (Reuland, 2011, p. xv).

Whatever the ultimate truth might be here, there is no general argument from overt binding to a syntactic principle that will license free variables generally (overt and covert) as part of a mechanism for binding, for none of the accounts advertised endorse syntactic items that are free or bound (i.e., at the relevant level of linguistic representation, the bound and unbound items, if marked at all, will be locally distinct from each other independent of any higher operator) (see Section 7).

There is an irony to all of this. Tarski-style indexation is a semantic notion, but a free variable is evaluated relative to a position (arbitrarily selected) in every sequence, not any definite object that occupies the positions. Syntactic indexation, on the other hand, whether ultimately to be accepted or not, is a purely formal mechanism, determining whether items are referentially dependent or independent. To conflate the two notions, as if variables and pronouns both refer to definite objects by way of an assignment to an index is to get both notions wrong. So, what the two notions do, in fact, share is that neither kind of index is a mechanism for reference to definite objects, but a mechanism for determining whether two or more items make a joint or disjoint contribution to the semantic properties of the host construction.

Let us consider the attraction of covert variables independently of the dubious charms of overt would-be free variables.

6.2 Covert Variables?

Syntax looks as if it trades in covert variables quite freely; that is, variable-like items that have no morphophonemic signature but are essential to the construal of the host sentence (so-called ‘empty categories’). Consider (13):

  1. (13) Who does Mary want to succeed?

(13) is ambiguous between a reading where the subject of succeed is being questioned, with succeed construed intransitively, and a reading where the object of succeed is being questioned, with succeed construed transitively, Mary being the understood subject. We may take the structures in (14) to be partial analyses of the two readings:

  1. (14)

    1. a Whoi does Mary want ti to succeed

    2. b Whoj does Maryi want PROi to succeed tj

Traces (t) of movement may be understood to be like bound variables coordinated with their index-mate, in this case who, which functions like an operator (per the above admonishments, treat the indexes as merely notational). A trace is the information remaining after an item has been displaced to a higher position in the structure. PRO is a covert pronominal item that serves as the syntactic correlate for the understood subject of infinitival clauses, inter alia (see below). PRO also appears to be variable-like in, again, being referentially dependent on an antecedent, in this case Mary. Rendered into Quinese, the structures in (14) become

  1. (15)

    1. a Which person (x) is such that Mary wants him (x) to succeed

    2. b Which person (x) is such that Mary wants herself to succeed him (x)

As a second class of examples, consider the constructions in (16):

  1. (16)

    1. a Bill seems to be happy

    2. b Critique of Pure Reason is hard to read

    3. c Mary is happy to read

(16a) has the paraphrase in (17):

  1. (17) It seems that Bill is happy

This indicates that Bill is not the “semantic subject” of (16a), for the subject of seem in (17) is an expletiveBill doesn’t seem, whatever that might mean. We take seem, therefore, to be a predicate of the state of Bill’s being happy. Put into Quinese, then, (16a) becomes something like (18a), with (18b) depicting the relation of Bill to its trace:

  1. (18)

    1. a Bill is such that it seems he is happy

    2. b Billi seems ti to be happy

In both cases an element (pronoun or trace) referentially dependent on Bill is explicitly the subject of the monadic predicate happy.

  1. (16b) is somewhat different. “Critique of Pure Reason” is understood as both the subject of the predicate is hard to read and the object of read, which requires a subject as well. This gives us the paraphrase in (19):

  2. (19) Critique of Pure Reason is such that it is hard for one to read it

Syntactically, a standard approach to such so-called “tough-constructions” is to posit a covert operator, OP, whose trace may serve as the object of the embedded verb and whose landing position serves to fuse the complex predicate in its scope so that the matrix subject may serve as its subject (arbitrary PRO serves as the embedded subject):

  1. (20) Critique of Pure Reasoni is [OPi hard PRO to read ti]

  2. (16c), constituting a minimal pair with (16b), is quite different again. In particular, (21) is patently not a paraphrase:

  3. (21) Mary is such that it is happy for one to read Mary

Indeed, (21) is gibberish unless we take it to be referential, whereas in (17) it must be an expletive. (16c) is simply a “control structure” with the subject of the embedded infinitive PRO indexed to matrix Mary:

  1. (22) Maryi is happy [PROi to read]

7. Against Syntactic Variables

My aim is to show in the space permitting that traces and PRO are not to be understood as variables in the sense characterized above.

A natural question to explore, assuming I am right, is why natural language should be adverse to variables as items that can be free or bound. The suggestion I favor is that a genuine variable must have both a free and a bound reading, but that just means that the status of a putative variable cannot be settled in a local way (i.e., one has to specify the full structure to see if a lower variable is bound or not). Space precludes my exploring this idea fully, but the next section should make the core idea clear.

7.1 Traces and Open Sentences

In a highly insightful paper, Higginbotham (1990) argues that covert items in argument positions precisely function like free variables (he calls them that) (Higginbotham, 1990, pp. 156–157), which syntactically saturate a predicate position and are open to generalization operators, such as quantifier DPs and adverbials (often, seldom, etc.). In effect, Higginbotham endorses the position that Jacobson (1996, p. 102) views as the generative “crucial assumption”: that traces form open sentences that are quantified over at a “later” stage by the scoped out item. Heim and Kratzer (1998, p. 92/241) do indeed interpret traces, as if free variables, relative to an assignment of values, as if a trace were independent of the item of which it is the trace. There is, however, an ambiguity here. 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 (vP, 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. 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.e., a trace is a consequence of movement, so is necessarily bound by whatever does move. 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):

  1. (23)

    1. a Mary loves who

    2. b Whoi does Mary love ti

    3. c *Who does Mary love Bill

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. 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 (1990) 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 (1977, p. 137), commenting on Chomsky’s (1977a) landmark paper “On wh-movement,” praises Chomsky for recapitulating Frege’s “insight” as regards the essential role of open sentences in any adequate understanding of quantification: “the concept [of an open sentence] comes in the interpretive rules that give parts of logical form as open sentences.” Bach’s point is that the predicate from which a wh-item has moved is interpreted as an open sentence, even if no free variable actually occurs in the syntax. 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?, where Bill is contextually salient, or the interpretation of (23b) with the object of love being construed generally (“Which person is such that Mary loves someone or other”). 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. (1990, p. 140) rightly note that, in the context of first-order logic, banning vacuous quantification results in a more complex syntax. Thus, “[w]e prefer to have a simpler syntax and render vacuous quantification meaningless or harmless in the semantics.” Similarly, Dowty et al. (1981, pp. 101–102) commend vacuous λ-abstraction.14 Still, vacuous quantification is absent in natural language. 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. In a phrase: syntax doesn’t forget its derivational history, and so traces can never be free, either syntactically or semantically (if construed local to their head projections).

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):

  1. (24) Everything is such that 2 + 2 = 4

The problem with (24) is simply that the quantification adds nothing to the sentence’s truth conditions. 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. 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.15

A ban on vacuous quantification is often assumed as a principle of the syntax-semantics interface without explanation (e.g., Chomsky, 1982; Fox, 2000; Heim and Kratzer, 1998; Kennedy, 1997; May, 1985). May (1977, p. 22/46–47) did initially propose the condition of quantifier binding, which says that “Every quantifier phrase must properly bind a variable,” i.e., it must bind a variable it c-commands (May, 1977, p. 22). May (1977, p. 46) notes that this condition just amounts to the assertion that “in natural languages, there is nothing comparable to the sort of “vacuous quantification” exhibited by logical formulas like “x(2 = 2).” May’s point here is that all syntactic scope-taking is derived from movement, so syntactic vacuous quantification is impossible (cases such as [24] are merely semantically odd, for their syntax involves movement). Similarly, Koopman and Sportiche (1982) propose a Bijection Principle that puts operators and variables in 1:1 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 (1995, pp. 130–131), following Kamp (1981) and Heim (1982), takes indefinites to introduce free variables, but then stipulates against vacuous quantification. 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 (1977b, pp. 10–11) says:

[A] trace is analogous to a bound variable, and an NP node with no lexical representation and not bound by the trace relation might be thought of as analogous to a free variable. 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.

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, 1977, pp. 432/449).16

In recent years, suspicion has fallen on traces precisely because they appear to be items whose significance is spread over the completed structure. 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, 1995). 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. The type/token distinction is essential to any conception of language, whether in terms of use or cognitive representation, insofar as the one lexical item type can be tokened indefinitely many times within a given sentence. Traces, on the other hand, must be posited as items over and above the lexical items (i.e., a trace is not an instance of a general type/token distinction). 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. 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. So, the trace in (23b) fails to meet any local condition pertaining to the verb phrase, for love t is simply not a constituent independent of the prefixed/moved wh-item. 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.17


Next consider PRO. ‘PRO’ is the term given to the understood covert subject of infinitival clauses, gerunds, and potentially other constructions. 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. For example:

  1. (25) Sam tried [PRO to win]

Here, we have the one obligatory reading, where Sam tried to bring it about that Sam himself won. It is impossible to read PRO as unbound, where (25) might express the thought that Sam tried to bring it about that someone or other won (witness, *To win, Sam tried, where the position of PRO is not available to be bound by Sam). 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:

  1. (26)

    1. a PRO Smoking is antisocial

    2. b Jill thought [that PRO to talk so freely was asking for trouble]

The occurrences of PRO here are “free” or “arbitrary,” but they are not construed as ranging over a domain or being indexical; rather, they are necessarily read as agentively indefinite or generic. (26a) does not have a reading where PRO designates a salient individual, even though one may use it precisely to talk of some individual. 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. For (26b) to be true in the imagined context, it suffices that Jill would hold the same thought about anyone who spoke in Harry’s manner. Note that the attempt to force a bound reading will always fail.

  1. (27) Jill thought that [PRO to talk so freely was damaging to him/Harry]

We might here construe who is talking to be Harry or the value of deictic him, but such a construal is dependent on the speaker’s intention, not on the syntactic position of PRO. 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. Whoever is talking freely determines Jill’s thought to be true or false, but the thought would remain true or false whoever is speaking so long as they were speaking freely in the relevant manner. Arbitrary PRO is indefinite in such cases much like the use of one by Queen Elizabeth II.18

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):

  1. (28) Everyone thinks that PRO smoking is bad habit19

  2. (8a) Every philosopher thinks he is a genius

(28) cannot express the proposition that everyone thinks that his or her smoking is a bad habit, only that smoking is, whoever does it. 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 (1984) and Moltmann (2006). 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.20

It also bears emphasis that PRO is necessarily agentive, whatever the construction. This feature alone signals that PRO is not a genuine free variable insofar as its interpretation is fixed over agents, regardless of whether the predicate itself is selective of an agent in the position of PRO. More than that, though, it tells us something important about control. Consider the embedded TP of (25), we may more elaborately depict as [TP PRO to [VP <PRO> win]]. One might think that PRO here is genuinely free to be controlled when embedded as the complement of a verb, or left free as an infinitive nominal. This is not so. The agentive character of PRO remains fixed. If the infinitive is in a nominal position, PRO is not free at all, but must refer indefinitely or generically to an agent (one), but exactly the same interpretation holds when it is controlled (subject or object). The difference is that when controlled, the explicit agent controller relates by way of the matrix verb to the activity generically specified by way of PRO. PRO cannot but be controlled, even though it retains its generic agentive form, because the structure involves a double predication to the one explicit agent. This is evident with the cleft of (25): To win is what Sam tried for. PRO here is not in a control syntactic configuration with Sam, but the cleft cannot establish a non-control construal, as if someone other than Sam tried to win, even though PRO is generically construed.

8. Conclusion

I have neglected many other constructions and hypothesized items that fall under a variable treatment within linguistic theory. What has been discussed, however, is of foundational significance and at the very least a challenge to theorists who would happily use the variable notion.

Formal languages and their devices are very fruitful as a means of understanding the properties of natural language, but great care must be taken if such devices are not to be unthinkingly read into the phenomena. In particular, the structure of language should not be immediately identified with the structure of thought. So even if the latter is apparently captured by a formal structure, it does not follow that the former must realize such structure―one has to establish that it does, if at all, by paying attention to language itself.21


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(1) The standard approach, after Tarski (1936/1983), is to take first-order formulas to be valued by sequences of objects, such that an i-indexed variable is valued by ith members of sequences (σi). Thus, schematically, an open formula has the following satisfaction conditions:



Quantifying over a variable has the effect of cancelling the index, on condition that an i-variant sequence satisfies the open formula depending on the character of the quantifier:



(2) The development of CCG goes back to Ajdukiewicz (1935/1967) and Bar-Hillel (1953). See Steedman (2000) for an excellent overview of CCG in relation to natural language.

(3) Chomsky (1981) and much subsequent work construed the relevant notion of locality in terms of “governing category,” but that notion is not essential to the core principles that structurally distinguish the relevant items in terms of possible antecedents.

(4) Such a proposed eschewal of certain relations of construal was a response to the traditional “transformational” view that conceived of a bound item as a “surface” substitution of a copy of the binder at “deep structure.” Such an account is inadequate, as was quickly noted. For example, although (i) is perfectly coherent, it does not offer a reading of (ii):

  1. ((i)) Every boy loves his mother

  2. ((ii)) Every boy loves every boy’s mother

(5) A Montague-style treatment is somewhat different again but still retains the indexed variable approach (see note 7).

(6) Here I sideline the question of the semantics of the φ-features of pronouns (Heim, 2008).

(7) The correspondence is only rough, for where variables are of type <e>, the standard Montague-style semantics treats names as sets of properties (or unary generalized quantifiers of type <<e, t>, t>), which applies to pronouns as well. Thus:

  1. ((i)) hen translates as λP[P{xn}]

The semantic value of the indexed pronoun is the set of properties instantiated by the indexed object—i.e., all the properties that have the property of being instantiated by the indexed object. Unlike binary every, say, which is of type <<e, t>, <<e, t>, t>>, names/pronouns have their restrictor set built into their semantics, as it were, so are type-equivalent to full quantifier DPs. My criticisms of the standard approach hold for this approach too.

(8) Independent of the problems about indexation, one might think of a conditional rule from contextual information of a specific kind to a semantic value as an aspect of the character of indexical items, a part of their lexical entries (Borg, 2004; Higginbotham, 1994). For present purposes, I have no complaint against such a position, for it does not involve the notion of indexation.

(9) Another problem for indexation in classic binding theory is that it can only express relations of joint or disjoint reference, which do not include “split antecedents”:

  1. ((i)) Bill asked Jane whether they should leave

Bill, Jane, and they cannot all be coindexed without Bill and Jane being rendered coreferential. Higginbotham’s (1983) “linking theory” is designed to resolve such problems, but its directed arrows going from the bindee to the binder have as much (or as little) independent syntactic plausibility as indexes; indeed, less, if anything, for linking is a matter of global construal, not locally determinable in terms of syntactic relations or lexical properties. The idea of each lexical nominal having an index as a feature might be unmotivated, but it is at least a local feature.

(10) The phenomena here can be quite complex. Imagine the topic of conversation being the politician Bill, who had called for a vote of confidence in his performance, a vote which turned out bad for him. One might utter (i) and intend coreference:

  1. ((i)) Only he/Bill voted for Bill/him

Either formulation here violates a principle of classic binding theory, which takes names to be always unbound and pronouns to be free in their clause (Chomsky, 1981; Reinhart, 1983).

(11) An interesting issue is the status of variables/indexes in sign language. Assuming that sign languages are expressions of UG on an equal footing to spoken languages, if I am right, then sign languages will lack variables/indexes too as syntactic items. Lillo-Martin and Klima (1990) and Sandler and Lillio-Martin (2006) argue, however, that various sign languages contain indexes as overt items. The basic idea is that the loci of pointing gestures in the “sign space” are overt indexes. This appears to be confirmed by two facts. First, the gestures cohere with classic binding conditions. Second, the gestures and their dependencies (Principle B‒like effects) are unbounded, which rules out the gestures being a mere morphological effect. All of this, though, is perfectly consistent with syntax being index-free. If we antecedently think that syntax contains indexes, then the sign phenomena should be expected (on the assumption that there is a common UG structure). If we think that syntax does not contain indexes, and binding is brought about by other means, then we will understand the sign phenomena as semantic or pragmatic, or perhaps partly a function of the peculiar iconicity of signing (cf. Schlenker et al., 2013). Such an approach is quite in accord with the classic binding conditions and the dependencies being unbounded, i.e., the theoretical options are no different from those that pertain to binding phenomena in spoken language.

(13) A further option is to seek to account for binding in a wholly semantic manner (Jacobson, 1999, 2000; Schlenker, 2005). For present purposes, given our focus on the status of variables as syntactic items, we may let this option go in peace.

(14) See Quine (1954) for discussion of vacuous quantification in different systems of formal logic.

(15) It is difficult to construct a case that evidences a ban on syntactically vacuous quantification involving quantifier DPs because their movement, unlike wh-items, is covert, so the would-be interpretation of the launch-site position of the DP would be competing with an overt DP. For example, if the relevant vacuous quantification were possible, then (ia) would have a possible reading as in (ib):

  1. ((i))

    1. (a) Everyone loves someone

    2. (b) Someone is such that everyone loves Bill

The impossibility of this reading, however, is already accounted for by the lack of competition for the interpretation of overt positions.

(16) It is worthwhile to note that even the notion of a bound variable does not accurately capture the notion of a trace. As Chomsky (1977a, p. 83) remarks, a trace is ‘non-terminal’ as it retains the category status of the moved item, i.e., an NP. The bound variable proper occurs as part of the interpretation of the structure. This is evident with cases like (i):

  1. ((i))

    1. (a) [Whose book]i did Mary read ti

    2. (b) (which person x)[Mary read x’s book]

That is, the interpretation must introduce the terminal symbol ‘x’ as part of the interpretation that expands the NP trace relative to the higher operator. If ‘t’ were just a variable then its interpretation would just pick out an object rather than the specific relational structure that is quantified into, i.e., the object that possess a book.

(17) A crucial issue for copies, however, is how they may be structurally distinguished from mere repetitions and so how lower copies may be interpreted as discontinuous items dependent upon their antecedents. This is a complex issue (Fox, 2000; Safir, 2004b; Chomsky, 2013). It suffices for present purposes to note that, whatever the right story is, a lower copy itself cannot be a variable as it cannot be free or bound; qua discontinuous, it must be bound.

(18) See Landau (2013) for a survey of control phenomena and the options for their explanation.

(19) Likewise, the attempt to bind arbitrary PRO with a wh-item results in a vacuous quantification:

  1. ((i)) *Who is it illegal [PRO to park the car in the morning]

(20) The interpretation of PRO in a surface form may be ambiguous, as is witnessed by the familiar duality between material and psychological construals of ready that lead to ‘tough’ or control structures:

  1. ((i)) The duck is ready [PRO to eat]

  2. ((i)) may mean that the duck has been prepared to be eaten (‘tough’) or that the duck is in a state of preparedness to eat something or other (control). The interpretation of the subject and object of eat differ between the cases. On neither construal is PRO free.

(21) For conversations on the nature of variables, my thanks go to Francois Recanati, Paul Pietroski, Michael Glanzberg, Jody Azzouni, Barbara Partee, Ora Matushansky, Chris Fox, Adele Mercier, and Noam Chomsky.