Vacuous quantification and free variables: historical and speculative reflections on LF

1. Vacuous quantification and free variables: historical and speculative reflections on LF john.collins@uea.ac.uk

2. Epigram • ‘If possession of variables is a syntactic matter, then it is doubtful that natural language quantification has variables in any interesting sense, even if pronouns have sometimes been thought to be such’ • Max Cresswell, 1990, p. 61.

3. The Frege Question • Frege bequeathed us a polyadic theory of quantifier-variable relations of many orders. The insight this offers for natural language is not to be doubted, but its reflecting the design of NL can be doubted. • Bach, Higginbotham: The insight of LF was the recovery of Frege’s insight of operators and open formulae. But how deep does the insight go? • Three grades of variable involvement in natural language: • (i) Mere theoretical artefacts • (ii) Semantic construal • (iii) Syntactic projection • Reasons to doubt that the third grade is realised.

4. Three distinctions concerning ‘variable’ • Notation: Alphabetic types recursively specified. • Logical form: A variable marks a position of generality such that the predicate/function that has the variable as its independent argument is an invariance over operations defined on the variable. (A variable is a way of talking about functions = Frege, Curry & Feys.) • Semantics: A variable is interpreted as a term of indefinite generality over a domain. • Logical form mediates between notation and semantics.

5. An illustration (1)a x > y b x < y • Both (1a) and (1b) may be satisfied. ’x’ and ‘y’ do not refer to any particular numbers at all, since no number is both greater than and less than some given number. On the other hand, being well-formed, we should be able to conjoin the relations in (1) to form a coherent complex relation: • (2) x > y x < y • Thus, the notational type of a variable is a semantically significant intrinsic property as regards its logical form. The property can affect the semantics of the formulae in which it occurs. Pace Kit Fine, variables can have intrinsic properties, but not ones that are always semantically relevant. • Simple version: the function/property of being greater than is distinct from the property of being less than, so generalisations over one are not ipso facto the same as over the other, so this must be marked by distinct variables where both properties occur in the scope of a potential operator.

6. Free and bound variables • Free variables can only make an indefinite semantic contribution to formulae, but do compose with quantifiers: • () [SAT[, ‘(xi)’] ↔(i)]. • () [SAT[, ‘(x)(xi)’] ↔(´i )(SAT[´, (xi)]] • Note: The free variable ‘xi’ is not an indexical; it is a general term of indefinite reference such that it can compose with quantifiers. It ‘refers’ to a unique position in every sequence (given an arbitrary assignment of an index), but for every object , there is a sequence that has  in its ith position. • So, a variable has a different significance between the free and bound cases.

7. Duality endorsed • ‘I don’t know of any cases of implicit arguments which can be interpreted only as bound variables or only as indexicals’ (Partee, 1984, p. 171). • ‘For variables 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’ (Stanley, 2000, p. 417). • ‘I postulate a phonologically null variable in the syntax of sentences with rain and eat, and that variable can be either free or bound’ (Martí, 2006, p. 141). • But free variables are not indexicals!

8. To be argued • As it happens, there are no variables in natural language that are either free or bound or bound or indexical. • Variable-like elements are either bound or have a fixed interpretation. • Variable indexicalism is a mistake, but it is not the crucial mistake. • The absence of variables is explicable via locality (hopefully, mother-daughter).

9. 1st Grade: Theoretical artefact • 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 (Quine,1960, p. 159)

10. 1st Grade: Theoretical artefact • I am happy to admit that much of the interest of logical form comes from an interest in logical geography [what sentences a given sentence entails and is entailed by]... The location must be given relative to a specific deductive theory; so logical form itself is relative to a theory. The relativity does not stop here, either, since even given a theory of deduction there may be more than one total scheme for interpreting the sentences we are interested in and that preserves the pattern of entailments... Seen in this light, to call the paraphrase of a sentence into some standard first-order quantificational form the logical form of the sentence seems arbitrary indeed(Davidson,1980, p. 140).

11. 2nd Grade: Semantic content • Variables are not merely part of a notation employed to describe or explain semantic phenomena, but part of the phenomenon itself. 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 account for complex expressions. A variable, on this model, enters into the theory in case the assignment for an expression is non-constant, i.e., it varies relative to a determinate range of factors. • I shall leave this second grade mostly unmolested, for the best argument for it is that the 3rd grade holds, which entails the 2nd grade. Much more about this later, though.

12. 3rd Grade: Syntactic Projection • [A] logic with variables and a logic without variables have the same expressive power. But if logical form is derived step by step, it turns out that a logic with variables is required to express certain general principles which explain facts of language... [A logic without variables] do[es] not furnish the types of representation appropriate for formulating rules that relate the surface structure to logical form in the most general way... As far as I can see, certain significant generalizations require a classical logic containing variables, where at times the variables reflect the presence of a trace in surface structure (Chomsky, 1979, 166-7). • The crucial issue: Does syntax contain variables that may be bound or free?

13. CCG, etc. • CCG also does without variables, but works off ‘surface form’. • No reason to favour surface form – all linguistic description is as artefactual as empty categories. • If we define variables as items that can occur free or bound, then generative theory has never sanctioned variables. • Treating variables as gaps elides this fundamental feature in favour of a needless fixation on surface form. • Still, CCG suggests that syntax might be a constrained CCG at each point of merge (forget surface form). Maybe this allows us not to stipulate against vacuous quantification.

14. Overt ‘variables’ • (3)a Every philosopher loves himself/herself b Every philosopher thinks he/she is a genius • (3a) is unambiguous with the single Quinese paraphrase: • (4) Every philosopher x is such that x loves x • (3b) is ambiguous between a bound and deictic construal of the pronoun: • (5)a Every philosopher x is such that x thinks x is a genius b Every philosopher x is such that x thinks y is a genius

15. Problems • But overt variables have no overt feature that marks them as variables: the indexes are covert. • Further, matched or disjoint indexation is a global property, which cannot be locally determined. • Co-indexation is symmetric, but the semantic relation is asymmetric. • An indexical cannot be bound by a quantifier without the indexation being cancelled, in which case the Q-v relation is not established, or the indexes are matched, which is a global stipulation.

16. Traces/copies: quantification without open formulae • (6)a Mary loves who • b Whoi does Mary love ti • The trace here is necessarily bound. 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 (6b), by itself, 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 case assignment and object selection, and thematic marking without any special assumptions whatsoever.

17. Quantifiers and QR: no open formulae • Same as above… If quantifiers scope on the basis of QR, then they scope by creating antecedent-trace relations. • One might think that derivational history is irrelevant. As submitted to interpretation, the lower predicates contain something equivalent to a free variable. That is not so. • The trace must be interpreted relative to the higher operator precisely because it is its trace formed via movement; it is not independently interpreted, even given just the final structure (the object of interpretation is the chain). • 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).

18. No free variables • [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. (Chomsky, 1977, pp. 10-1) • In other words: the grammar does not generate open formulae. • This is good, because free variables invite vacuous quantification.

19. Vacuous quantification • (7)a Who does Bill love? b [CP Who does [TP Bill love <who>]] c (which person x)[Bill loves x] d *(which person x)[Bill loves y] • (8) a *All some men left b *Who does Bill love Sam c *The man who John saw Bill

20. An assumption • The ban on VQ is nigh-on invariably taken to be axiomatic (May, Kennedy, Kratzer, etc.). • May (1977, p. 22/46-7) proposes the condition of quantifier binding: ‘Every quantifier phrase must properly bind a variable’, i.e., it must bind a variable it c-commands. • May (ibid., pp. 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 formulae like ‘x(2 + 2)’’.

21. Vacuous quantification and free variables • [T]he barrier to vacuous quantification should be supplemented by the requirement that variables at LF must either be assigned a range [by an operator] or a value by an antecedent and in this sense cannot be “free”… [T]hese are not principles of logic, but rather empirical principles concerning the syntax of LF (Chomsky, 1982, 32). • [L]anguage does not permit free variables: the strong binding property [i.e., that ‘variables’ either have antecedents or fixed interpretations or are traces/copies of movement] determines the curious semantic properties of these constructions. We might think of this condition as a specific application of the UG condition of FI (Chomsky, 1995, p. 153)

22. The ban • A mere ban on non-movement scoping will not exclude vacuous quantification: • (9)a [who does [<who> Bill love] b *(which person x)[Bill loves y] We get the ban by excluding free variables and treating scoping as via movement. This should all fall under a minimalist/CCG/TAG-like demand for locality, without appeal to FI (pace Potts, Asudeh).

23. Besides… • The bound variable proper occurs as part of the interpretation of the structure. • (10)a Whose book did Mary read t b (which person x)[Mary read x’s book] • 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.

24. Controlled PRO • When in a position to be bound, PRO is necessarily bound. For example: • (11) 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 (11) might express the thought that Sam tried to bring it about that someone or other won or that a specific discursively salient person is who Sam tried to have win.

25. Arbitrary PRO • (12)a PRO Smoking is anti-social • b Jill thought [that PRO to talk so freely was asking for trouble] • c Everyone thinks PRO smoking is anti-social • (12a) does not have a reading where PRO designates a salient individual, even though one may use (12a) precisely to talk of some individual. PRO here is interpreted akin to one’s. • Similarly, the PRO in (12b) is necessarily indefinite. • (12c) doesn’t express the thought that everyone thinks their smoking is anti-social.

26. No mention of functions in syntax • Heim and Kratzer (1998, pp. 178-88) pose the familiar problem of quantifier phrases in object position: • (15) Bill offended every linguist • There is a type-mismatch. Assume that transitive offend is of type <e, <e, t>> and that the DP every linguist is of type <<e, t>, t> . If these are the semantic types, then offend can’t semantically compose with every linguist, for every linguist would have to be of type <e>, not type <<e, t>, t>. The solution Heim and Kratzer propose is that the QR-movement of every linguist from object position creates a variable, which serves to satisfy the type requirements explained above.

27. A syntactic solution? • Thus: • (12) [TP [DP Every linguist]1[α 1 [TPBill offended t1]]] • The type mismatches are now resolved. Offend may now take the trace (t1) as its argument, which is of type <e>. The DP every linguist requires an argument of type <e, t>. Since TPs are type <t>, a projection is required to render the TP as type <e, t>. The abstraction operator serves this end, presenting the TP as a function from the set of individuals to Truth just in case Bill offended the individual.

28. Problems: bullshit • But the projection is really just a λ-abstraction term (a difference of Greek letter is irrelevant), which is doubly syncategorematic, i.e., the sisters are non-type-driven interpreted and the projection is headless, as all λs are. • (i) The operator is not a lexical item, so can’t project. • (ii) It is syntactically unlicensed. • (iii) The term designates a function, but can only compose syncategorematically, so is not a syntactically composed item – semantic interpretation is read into the syntax. NB: H&K’s operator is quite distinct from the familiar ‘empty operator’ employed in the analysis of tough constructions and is not a relative pronoun. • Moral: Don’t mix talking about functions, which variables are for, with functions.

29. Local and global • If syntax is strictly local, then the significance of an item as encoded in syntax must be settled as the item is merged, not at the level of a completed structure (deep insight of TAG). • Hence, free variables are ruled out, because whether they are free or bound is a matter of whether the relevant higher operators occur. Likewise, free variables cannot be indexicals, because indexicals cannot be bound. • Chomsky (1995, pp. 152-3) on tough constructions. Consider (8): • (14) John is too clever to catch • John is so clever that an arbitrary person/thing will not catch him. • *John is too clever to catch him/it/someone, etc. • The acceptable reading: (15) John is [OP too clever PRO to catch <OP>] • The posited empty operator (OP) is necessarily bound by the matrix subject, John. Chomsky (ibid., p, 153) concludes that ‘language does not permit free variables’.

30. Locality of traces/copies and PRO • Assuming strict cyclicity, all traces will be locally related to the correlated lexical item. • Copies carry their inherent features. • PRO, where arbitrary, is obligatorily indefinite/generic. • PRO where controlled, patterns with copies, if MTC is true. • Assuming a traditional control model, PRO must be bound either because the predicate is some kind of λ-abstraction or a proper clause thematically linked to the matrix subject. • Any putative syntactic variable, therefore, must have its status settled locally, unlike (local) free variables whose ultimate status is globally determined.

31. Conclusions • My aim is not to overthrow any particular way of pursuing syntactic or semantic theory, but rather to cast doubt on a way of conceiving of the relation between the two. • The relation, I think, is not as intimate as many philosophers of language imagine. As Partee puts it, some matters of interpretation appear to be ‘rather holistic’ bearing on the sentence as a whole, but syntactic matters seem to be local (witness generics). One signature of this is the absence of variables proper in syntax. • The deep problem is the interface between semantics and syntax, a problem, if I am right, that is not ameliorated, let alone resolved, by positing free variables in syntax; for that is to conflate semantics or, at a higher level, propositional content with syntax.