Semantics & Pragmatics Volume 13, Article 20, 2020 https://doi.org/10.3765/sp.13.20

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Barker, Chris. 2020. The logic of Quantifier Raising. Semantics and Pragmatics 13(20). https://doi.org/10.3765/sp.13.20.

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The logic of Quantifier Raising*

Chris Barker New York University

Abstract Displaced scope is a hallmark of natural language, and Quantifier Raising (QR) has long been the standard tool for analyzing scope. Yet despite the foun- dational importance of QR to theoretical linguistics, as far as I know, there has never been a study of its formal properties. For instance, consider the decidability problem: given an initial syntactic structure, is there an algorithm that will determine whether a semantically coherent QR derivation exists? If at least one such derivation exists, is the number of semantically different analyses always finite? How do we know when we have found them all? Do the answers to these questions depend on imposing scope islands or other constraints on QR, such as forbidding vacuous movement, re-raising, remnant raising, raising of names, repeated type lifting, and so on? I settle these issues by defining QRT (Quantifier Raising with Types), a substructural logic that is a faithful model of QR in the following respect: every semantically coherent QR derivation corresponds to a semantically equivalent proof in QRT, and vice-versa. Since QRT is decidable and has the finite readings property, it follows that a broad class of theories that rely on QR also have these properties, without needing to place any formal constraints on QR. I go on to study the special relationship between type lifting and QR, drawing an analogy with eta reduction in the lambda calculus. Allowing unrestricted type lifting does not compromise decidability. In addition, it turns out that QR with type lifting validates the core type shifting principles of Flexible Montague Grammar, a paradigm example of an in-situ type-shifting approach to scope taking. This suggests that QR is compatible with a local, directly compositional view of scope taking. These results put Quantifier Raising on a reassuringly firm formal footing.

Keywords: quantifier raising, decidability, direct compositionality, scope, substructural logic, type shifting

* Thanks to Simon Charlow, Sandra Chung, Josh Dever, Berit Gehrke, Daniel Lassiter, Richard Moot, Greg Restall, my anonymous referees, and audiences at LENLS 11, ESSLLI 2015, Stanford, and the New York Philosophy of Language Workshop. At LENLS 11, Berit Gehrke pressed me on the relation between Quantifier Raising and a particular logic related to QRT. I gave her a list of reasons why I thought they were deeply different. “So. . . ” said Berit when I had finished, “it’s Quantifier Raising!” This paper explores the sense in which she was right. early access N. osoeilnspa rca oei urneigdcdblt?(o)I it Is (No.) decidability? guaranteeing in role crucial a play islands scope Do (No.) hr a ee enasuyo h omlpoete fQatfirRaising. 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(and address will paper this that questions the of some are Here napatcllvl h nt ednspoet shgl eial.I Rdid QR If desirable. highly is property readings finite the level, practical a On were QR if Theoretically, answer. practical a decidable? and is answer QR theoretical whether a care is we There should Why stake? at what’s So be may There be. could they as favorable as are answers the words, other In 2 hi Barker Chris early access and eta LIFT is special among type IFT L type shifter. 3 LIFT There is a spirit of experimentation and flexibility in Heim and Kratzer’s ap- My strategy will be to consider a substructural logic, QRT (Quantifier Raising I go on to show that these results hold even if we add unrestricted type lifting, a As any experienced semanticist knows, decidability worries don’t arise in simple strictions on Quantifier Raising,to since empirical that may problems foreclose downtakeaway insightful the messages applications of line. this I paperconstraints endorse is whatsoever. that this Quantifier spirit, Raising and requires no one purely of formal the main not give a definition of QuantifierNevertheless, they Raising. characterize (Nor, Quantifier as Raising far clearly asprovide and I a precisely know, solid has enough foundation anyone to for else.) a vast amount of laterproach. work. The spirit is something like this: avoid placing unnecessary formal re- Heim and Kratzer’s 1998Quantifier textbook Raising contains from lengthy both and anserved detailed empirical as and discussions the a of touchstone technicalphilosophers. for point Despite Quantifier of (or Raising view, perhaps and for because has of?) a the generation centrality of of linguists the and topic, they do Since QRT is decidableset and of has coherent the QR finitefoundational derivations — readings seeking does a property, too. deeper it understanding Thusdisciplinary, follows of bringing the Quantifier that to Raising project bear the — the and reported metatheoretical cross- here techniques is of formal both logic. 2 Adding types to Quantifier Raising approach to scope taking. Thiscompositional suggests view that of QR scope-taking. is compatible with a local, directly coherent QR derivation has a semantically equivalent proof in QRT, and vice versa. generalized version of Partee’s 1987 shifters. For instance, there isreduction a deep in and the intriguing lambda analogythe calculus. between core It type turns shifting out principles that of QR Hendriks’ with 1993 type Flexible Montague lifting Grammar, validates However, there are realisticpossible examples analyses, where is not findingonly so any do obvious analysis, the (see results let3 Section givenalgorithm below alonefor settle for a all the computing concrete theoretical all example). issues,example. they Not semantically provide distinct a interpretations practical for an arbitrary situations involving garden-variety generalized quantifiers scoping over clauses. which is a paradigm example of a non-movement, in-situ, directly compositional with Types). QRT is faithful model of QR in the sense that every semantically early access h quantifier The verbatim: a’ uei en ogv iet ulseicto ntecneto general of context the in specification full a to rise give to meant is S) rule (to May’s Q Adjoin QR of (1) rule proposed 1977:18 original May’s quote 1998:210 Kratzer and derivations Heim QR and Raising Quantifier Defining 2.1 aeoyt dont adn ieo n aeoy fasse ihsc radically a any such certainly with then system readings, a finite If with category. decidable any nevertheless any the of is of In site QR expression QR. landing unconstrained an of a allows definition to QR the adjoin that of to assume matter category will a I as to the flexibility, least site, maximizing of at adjunction of identity possible, the spirit as of categorial open-ended identity the categorial as for the left as be prudent well seems as it raised, host begin So expressions a types. and the expression comparatives, cover in adverbs, other adjectives, to analyses of to meant Furthermore, QR QR below). allow is extend insightfully discussed explicitly (1) literature (as Kratzer in DPs and ‘Q’ non-quantificational Heim the of but Likewise, phrases, well. determiner as Kratzer and quantificational VP Heim S, to to adjunction adjunction restricts motivate May although instance, the For for challenge. binder a provide to order in site adjunction is the ‘1’), to trace. 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[ that contains a logical form : Let γ i L | i t : an LF derivation consists of a finite series of logical forms | produces σ LL by replacing | γ W must be chosen fresh, so that it is distinct from any other index in the consisting of an integer index i = : L Quantifier Raising LF derivation in which the initialsubsequent logical logical form form is created contains from no itsof predecessor Quantifier traces by Raising. a or single indexes, application and each tains a logical form applied to replaced by a new logical form whose two daughters are original where the abstraction consists of the index Just to be clear, on the terminology here, a structure that conforms to (3) but that By the way, given that QR is not limited to raising quantificational expressions, With a view towards formalizing Quantifier Raising enough to address the formal cannot target an index. has not undergone any QR operationslogical (that form. is, Likewise, the a proper pre-QRlogical structure) subparts forms, still of with counts a one as complete exception: a logicaldoes an form not index also that by count helps itself as form countalthough an as Quantifier abstraction a Raising structure logical can target form a according trace to for the raising specification (traces in are (3), logical so forms), it Note that the official characterization ofsyntactic a category logical labels. form Since in thecoherence, (3) questions does syntactic addressed not categories here provide are concern for not onlyadded directly semantic without relevant, affecting though the they results. could easily be (4) So the derivation diagramed in (2) corresponds(6) to the following LF derivation: [Ann [saw everyone]], [everyone [1 [Ann [saw original logical form. (5) abstraction logical form, what exactly Quantifieras Raising an does LF to derivation. logical forms, and(3) what counts structure consisting of two logical forms, or a trace indexed by an integer However, the term “Quantifier Raising” is firmly embedded in current practice. questions of interest here, it will help to have a more precise idea of what counts as a more highly constrained system will beare as decidable, well of (as course). long as the constraints themselves The index A logical form consists of a word “Quantifier Raising” is not an accurate name. “Scope Taking” would be better. early access ihalo h eiefrflxblt ntewrd hr sagnrlfra constraint formal general a is there world, the in flexibility for desire the of all With htItk ob emadKazrsitnin,a ela ihsadr practice. standard with as well as intentions, Kratzer’s and Heim be to take I what ihtpn judgments: typing with hsmasi atclrta h eatctpso h opnnso h logical the of components the of types semantic the that particular in means This ics eo,tp hfig sdn,tefia oia omms be must form logical we’ll final as the (and, done, modification is predicate end shifting) and the adjoining type below, at and discuss, discuss raising Kratzer all and after day, Heim the as of non-negotiable: is that derivations QR on coherence Type 2.2 (8) 186), 96; (pp. Abstraction semantic Predicate for and rules 49) main (p. two Kratzer’s application and function Heim interpretation, supplementing by types to respect α form logical a If type: a clarity. 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An LF derivation is coherentform with is respect coherent to with types respect iff to its types. final logical Coherent with respect to types types such a way that each internal node of the logical form satisfies the typing With these preliminaries, we can now say what it means for a QR derivation to For instance, the QR derivationlogical in form (6) after is QR coherent is with type-coherent. respect Here is to a types,(10) labeling since that the shows this: be coherent with respect to types: (9) early access tti on,w a tt h anpolm drse nti paper. this in addressed problems main the state can we point, this At t neaan h taeyhr ilb omxmz eiiiy el sueta QR that mismatch. assume type we’ll flexibility: no maximize is to there be where will called here position, often strategy subject is the QR of again, instance, out Once For a quantifiers QR. 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(There | t S · eeet e S [[ | T e [ = : S b. is the number of semanticallyform distinct has derivations on type which the final logical The finite readings problem for QR Like any formal logic, QRT involves formulas, structures built from formulas, The results below provide an algorithm that will find all semantically distinct It may help to have a concrete example to consider: Here’s a restatement of the problem: figure out what to raise to where, figure correspond to a semantically equivalent proof in QRT, and vice versa. and inference rules. Thedefined formulas above of in this (7). logic will be just(14) our set of types Proving decidability and finite readings proceedslogic in that two characterizes steps: the first, class definingthat of a coherent this formal QR logic derivations; is and decidable second,QRT, the showing and logic has of the Quantifier finiteaccurately readings characterize Raising semantic property. with coherence I’ll Types. is call What that the it every logic coherent will QR mean derivation for will QRT to coherent derivations. The essential move willgiven here be into to a translate particular QRical Type and Logical technique the Grammar, pioneered typing and by rules then Gentzen leverage 1934. a metatheoret- 4 QRT, the logic of Quantifier Raising Does a logical form with theon lexical which types it as has indicated type haveis a a coherent hint QR at derivation the end of the Appendix.) found one or more analyses,can how stop do looking we for know more? when we’ve found them all, so we (13) a. 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L A ∆ 10 Axiom · ( i · Γ i · [ t Γ i ]) Γ [ t i [ ] B ` ] ` B A → A a type has A → ` R )floe ya by followed ’) i C yestructure type ∆ hnw can we then , oia form logical · a type has hi Barker Chris A rvdsa provides a type has ∆ ∆ · ∆ and 0 B is e , early access → ‘re- has e ↓ = ( t , et is the type of L t Ax → ↑ → t QR e ` ↑ A t Ann saw everyone = QR ` A . et t L ])] Γ ` i ` t [ Ax → ]] Γ t ∆ t containing a specific occurrence ))) · [ L 1 i ` Γ t ` Γ i ( [ results in a structure that has type · ) · t R Σ → t Γ , ∆ [ t → eet Σ et ( ` · · et from e 11 et ( t ` · B · eet ↓ ( e ` Ax 1 )) · ( ) 1 e · e t QR e is the type of a transitive verb, and · · t ` , A t e ` et eet eet A → ( ( · ])] · e ` i e t Ax e [ ( ]] ’) says that if a structure i · → Γ e rule given here is the standard rule of use for implication, ∆ · R [ 1 e i ` L Γ ( , then abstracting [ · → e = A Σ ‘expansion’. Expansion implements Quantifier Raising (compare → ∆ [ ↑ Σ eet must be distinct from any other index in i has type B is the type of a generalized quantifier. . This rule corresponds to the logical form typing rule T2, and gives the : t is unusual; its relation to the standard rule is discussion below in Section 6.2. A t i In order to check that this proof is valid according to the inference rules just To illustrate, here’s a proof justifying the judgment that The final inference rule (‘QR’) implements Quantifier Raising. This is a structural Although the The right rule (‘ R → → ) t given, we can first read the proof in the direction of deductive inference, that is, I’ve used the traditional abbreviationsa for types, verb phrase, so that (17) Following Barker and Shan 2014 chapterduction’ 17 and and QR Barker 2019, I will call QR type elements arranged as on otherthe side two of kinds the of equation. inferences Hereequivalence: licensed are by schemata this that rule, spell depending out on the direction of the → rule, rather than a logical rule.rather than That characterizing is, the it content imposes of a athat logical constraint matches connective. on one It the side says set that of any of the structure structures, equation can freely be replaced by a structure with the B sequent presentation of the operation Heim andfresh, Kratzer that 1998:96 is, call Predicate of type with (4)); reduction plays a role in a number of discussions below. Abstraction. Just as in every version of Quantifier Raising, the index must be chosen early access (topmost) h second The h e faaye hrceie yQ eiain n yQTpof r the are proofs QRT by and derivations QR by characterized analyses of set The htacas a oss fagnrlzdqatfircmie ihisncerscope. nuclear its with combined quantifier generalized a of consist can clause the a of that instantiation this in of (so instance object direct a first and The instance. axiom the an is with premises begin any must require proof not the does of that branch every inference so only axiom, The bottom. to top from hr sasmnial qiaetQTpof n o vr ai R ro there proof QRT valid every for and derivation proof, QR QRT equivalent coherent semantically semantically a every is for there equivalence: semantic to up same QRT and derivations QR between equivalence The 4.1 13. chapter 2014 Shan and Barker and 2005, the Jäger of Grammars, 1997, discussion Logical them Moortgat For Type brings see thing. to that single applied way as a a specifically of in correspondence aspects chosen Curry-Howard two artfully are be the they can rather, and that alignment; rules things into semantic two Kratzer’s not and are Heim derivations: rules QR typing for rules typing the in namely, proof (10), this in that above corresponding is given the result derivation as content net QR compositional The semantic labeling. same the the on receives effect automatically no have rules structural the and of 1997), application, gives labels Moortgat function the conclusion (e.g., to on correspondence final standard based corresponds the the whole of to a type According as result items. expression lexical the the of of label composition a The semantic receives the label. proof a the as in term Curry-Howard formula lambda the each by correspondence, determined the automatically Under is correspondence. proofs the of strategy. content winning semantic a is this that confirm steps inference remaining The scope. type has is derivation. ask: does QR we is, a conclusion, constructing desired to the approach with normal Starting the matches direction That search. sentence the of types lexical the everyone containing saw structure the that shows conclusion with equation) structural the structure the replacing by QR the Finally, nfc,w’earayse h ur-oadcrepnec twr above work at correspondence Curry-Howard the seen already we’ve fact, In compositional the that is Grammar Logical Type of properties pleasant the of One proof of direction the in up, bottom the from proof the read also can we But n a everyone saw Ann t → etyQatfirRiigtedrc bet donn tt t nuclear its to it adjoining object, direct the Raising Quantifier try We ? → → L a type has R ↑ neec asta luecncnito ujc n predicate. a and subject a of consist can clause a that says inference L i neec rp h eeaie unie noissraeposition surface its into quantifier generalized the drops inference btat h ietojc.Tetid(oet ntneof instance (lowest) third The object. direct the abstracts neec asta ebprs a oss fatastv verb transitive a of consist can phrase verb a that says inference t . et,t aeasmnial oeetQ eiaino hc it which on derivation QR coherent semantically a have e · ( · eet ( 1 · ( · e et,t → · ( 12 eet R ) i everyone mthn h ethn ie.Tefinal The side). hand left the (matching orsod oPeiaeAbstraction, Predicate to corresponds · t 1 ))) e · ( mthn h ih adsd of side hand right the (matching eet → ( λ L · x et rule, . saw , t ) x Σ ` [ Ann A t = ] hoe?That theorem? a ) . e · hi Barker Chris [ et → ] L .The ). → says Ann L early access e : 2 t et eet t loves : ). e ]]]]]] e 2 t : et and 1 t )) ; nodes licensed by typing rule T2 e e L [loves )( 1 t 1 : → xx t λ 13 (( , of course, corresponds to Quantifier Raising. ↑ , where 2 et,t : σ , 1 [loves everyone]]]] σ 1 : t et , 2 ; and QR σ σ ↓ someone : e ; certain mother-daughter pairs that are licensed by typing rules i t 2 : R [everyone [2 [someone [1 [ [someone [1 [ [someone [loves everyone]] = = → t = , 2 1 σ σ σ et Someone loves everyone f. Type labeling of b. QR derivation = e. c. d. Here’s an additional example involving a QR derivation of the inverse scope Specifying how to turn an LF derivation into a QRT proof involves building (18) a. Someone loves everyone. everyone : Full details are given in thehere. Appendix, For and instance, several the illustrating QR examplesproof derivation are summarized in given in (17), (10) and corresponds vice to versa. the QRT reading of arbitrary sequence of QR operationsbetween exactly. The the second internal phase nodes of islicensed a a labeled correspondence by logical typing form rule and T1correspond logical to correspond inferences: nodes to a QRT proof in twoderivations phases. and The sequences first of phase expansion is inferences.the a The QR essential operation one and similarity to the between one QR mapping structural between rule QR allows a QRT proof to recapitulate an is a semantically equivalent QR derivation.having Here, the ‘semantically same equivalent’ compositional means semanticbeta value, where reduction two count lambda as terms the related same by (e.g., T1 and T2 correspond to QR early access n h o?Q prtsjs ienra,adi ses ofidalbln that labeling a find to easy is it and normal, like just operates QR not? why And clear: this make will example An around. way h xaso neecstakteQ eiaineaty oeseicly h final the specifically, More exactly. derivation QR the track inferences expansion The eei h orsodn R ro eiee yteagrtmi h Appendix: the in algorithm the (19) by delivered proof QRT corresponding the is Here Ro ae,a hw ee sepiil loe yHi n rte 1998:210. Kratzer and Heim by allowed explicitly is here, shown as names, of QR left. Ann other the (20) or argument, its as scope nuclear (QR the takes rule structural structure QR raised the the of whether instance on to reduction corresponds a sometimes to derivation sometimes QR QR and a of in exception Abstraction the Predicate with nodes. one, two to one is inferences logical types. σ form logical form logical to corresponds sequent 2 h eann neecsjsiytecamthat claim the justify inferences remaining The . smnind h orsodnebtenndsi aee oia omand form logical labeled a in nodes between correspondence the mentioned, As n : Ann e σ 1 : 1 t n h hr eun rmtebto orsod olgclform logical to corresponds bottom the from sequent third the and , e 1 2 e · · et t ( e ( et 1 ` et t et 1 · : e ( , · , e t eet t , ( t eet e · et · t · ( ( ` et: left ( 1 2 · , 1 e t · e t e · · · · ( ( )) et e · t ( et et ( 1 )) t et 1 1 ` , · t , · · ( ` ` t σ t ( eet ( · eet t · tt et t ( h eutmt oia omcrepnsto corresponds form logical penultimate the , 1 ( eet ` · 1 → 14 ( · t → · eet · e ( t L → · )))) t 2 1 et e R )))) · i · e · L ( , ( et t eet ` ` 1 ) ` e e · , t t e ` t ( ` · · tt et t ))) · et et 1 t σ t t 2 · 2 ))))) → et → ` ` ` schrn ihrsetto respect with coherent is t t t )) R L ` ↓ i QR ` hc orsod to corresponds which , ` t t t → ↑ ` QR QR QR t L → ↑ ↓ ↑ ↓ L ,depending ), hi Barker Chris → R i , early access subformula ) guarantees that whenever a 15 cut elimination , which beta reduces to the semantic content ann ) 1 x left . . 1 x λ ( left ann The argument that QRT is decidable and has the finite readings property follows Within those constraints, the correspondence is as close as it could be: QRT The relationship between QR derivations and QRT proofs cannot be one to one. Thus the QR derivation wastes semantic effort. Like the Duke of York, QR limits the possible premises that need to be considered to a finite number. Further- Gentzen’s 1934 Hauptsatz. In any logic,set of proving premises a from conclusion which depends theof on conclusion presenting proving follows. the a One logic of in theproperty: sequent distinctive each form advantages formula is in that a the premise logical appears rules (exactly have once) the in the conclusion. This there is a coherent QRof derivation for the every algorithm QRT into proof, a wesemantically set can distinct of translate QR QR the derivations, derivations. results since We know thereproof that is for we a every haven’t QR semantically missed derivation equivalent — any QRT semantically and distinct since QRT the proofs, proof we search can algorithm be delivers sure all we got them all. a semantically coherent QR derivationlogical exists form, the is number decidable; of andHere’s such for a derivations synopsis: that any given are particular that semantically QRT distinctthere is is is decidable finite. an and algorithm has that the will finite deliver readings all property, semantically distinct QRT proofs. Since executed, which can be irrelevant to the final result (seecovers the the Appendix full for space examples). of semantic analyses generated by QR, adding nothing5 extra. QR is decidable and has the finite readings property For one thing, type structures ignore linear order,of so siblings for within each a QRT logicalthere are proof, form. many For there distinct another, QRT as proofsthat that usual QRT are proofs with semantically differ substructural equivalent. according logics, The to reason the is order in which the logical inferences are marches the subject into scope-takingicate position, Abstraction only beta-reduce to the have the subjectresults semantics back proven of into in Pred- argument the position. Appendix OneQRT (namely, of proof the contains such a semanticallyequivalent proof useless without excursion, there the is excursion. a semantically after a reduction has a null effect, of course, andthe if QR we derivation eliminate is both QRof inferences, the QRT proof, satisfies the typing rules. Thealgorithm corresponding in QRT the proof Appendix, in is20), ( likewise valid. delivered Performing by an the expansion immediately will be many semantically equivalent QR derivations that differ only in the order we have an equally valid proof with the same semantics. The semantic content of We can now answer our two main questions affirmatively. Yes, finding out whether early access ihoeo h oia ue;teol pini R is QRT in option only the rules; logical the of one with ,snetetoQ neecsaeadjacent are inferences QR two the since (20), in proof the for true obviously is This ieto fsbttto eaaey h QR The separately. substitution of direction the in present connectives logical of of conclusion. number number final the logical the that Since exceed follows rule. cannot it logical inferences eliminated, the logical be by never created can appear formula not introduced, the does once namely, connectives, that premises, conclusion the the of in any appearing in formula one (exactly) is there more, eatcefc.I re o Rt aeannnl eatcefc,i utinteract must it effect, semantic non-null non-trivial a a have have to key must QR The QR for operations. of order QR instance In of each effect. number that semantic the requirement with to the coherent limit is be no decidability must be to form this, would logical see there To final types, system. the to logical that respect the requirement in the elements abandoned non-trivial other we accomplishes the if QR with concert which in in work way semantic the from follows decidability Rather, every remove safely can we Since semantically conclusion. QR is final of proof instance same modified the the has structural removed labeling, and Since two semantic equivalent order. the same the the of affect in exception don’t remains, the inferences inference With other effect. every net inferences, QR same QR of the it instances with brief: matching instantiated In the be remove adjustments. for simply minor 2019 we with Barker if QR here in that given over show to are carry possible argument and is the logic, of related details closely Full a effect. null removed. a been have have inferences and both which in proof equivalent an is there occurs, QR of instance QR of instances coindexed of number the of instances of of instance number an the by introduced was inference the the to limits are there conclusion. that the proofs. show in distinct must present we not QR decidable, are of is that QRT number trace that argue a to the and order than index connectives In an structural as more well contains as it conclusion: conclusion, its than simpler not is connectives structural conclusion. fewer the strictly than contains premise its and property, subformula ↑ omc o h oia ue.A o h tutrlrl,w utteteach treat must we rule, structural the for As rules. logical the for much So steeapriua etr fQ hti epnil o eiaiiy o exactly. Not decidability? for responsible is that QR of feature particular a there Is QR by eliminated index the when is case second The by eliminated index the when is case first The consider. to cases two are There QR the for As vr neec nbtenteroiia oiin ntepofcnb still be can proof the in positions original their between in inference every , ↑ ↑ ↓ ntescn ae nytefis aeremains. case first the only case, second the in neecsta r eddi re ocntutalsemantically all construct to order in needed are that inferences .Hwvr ttrsotta hnvrti configuration this whenever that out turns it However, (20 ). in as ↑ neec,atog thstesboml rpry t premise its property, subformula the has it although inference, → R i slmtdb h opeiyo h nlconclusion, final the of complexity the by limited is 16 ↑ slmtda well. as limited is ↓ neec ses,snei a the has it since easy, is inference → R i ic earayko that know already we Since . → R ↑ i ic ahexpansion each Since . a nrdcdb an by introduced was hi Barker Chris ↓ and early access . It also LIFT turns it into a e . LIFT algorithm. There are many inferences is limited means that i . Since any expression can poten- R efficient — after all, a given index can only be → i et,t R bears a special relationship to QR. → 17 to QR is parallel to adding eta reduction to LIFT LIFT , a logic that is closely related to QRT (as we’ll see in the next λ interacts with scope taking. Lifting a name of type LIFT ? (Yes!) Second, we’ll see that adding I’ll mention three reasons why this is worthwhile in the context of this paper. It is important to say that having an algorithm (i.e., a method that is guaranteed Since the proof search space is finite, we notFor only the guarantee sake decidability, of but concreteness, here is one especially simple algorithm for finding LIFT the lambda calculus. This provideshelps a explain new why conceptual it perspective is on that such within a the natural class and of indispensable type type shifters, shifter, and suggests First, scope-seeking generalized quantifier of type tially undergo the lift operation —threatens including decidability a and previously finite lifted readings. expression — Is lifting QR still decidable in the presence of Raising, no more, no less.supplementing This QR section with pulls the back familiar and type takes shifting a operation slightly wider view, by section). I make no claimsthe here simple about procedure computational just complexity, except sketched to can say be that vastly improved upon froma the theory point that of includes QR is not even well posed unless QR derivations are decidable 6 Type shifting and direct compositionality to terminate) is not thestudies same giving thing bounds as on having the an timeFor complexity example, of Kuhlmann various et grammatical formalisms. natory al. Categorial 2018 Grammar, discusses a formalism the withLogical parsing some Grammar. complexity For important of an similarities especially to Combi- relevantcomplexity Type investigation, of Moot NL 2020 considers the of constructing premises from whichrules. the Recursively sequent repeat follows the search by forsequent one each in of premise. which the Abandon the inference trying numberconnectives. to of prove abstraction any indexes exceeds the number of logical the number corresponding expansion inferences is also limited. finite readings as well. all semantically distinct analyses: given a sequent to be proven, try all possible ways can interact with at mosteliminated one once instance — of the fact that the number of view of time cost. Note that the question of the computational complexity of parsing This paper is about Quantifier Raising, and QRT delivers exactly standard Quantifier with finite readings. early access ilcl RT(unie asn ihSit ye) ic RTi rgetof fragment a is QRST Since Types). Shifty with Raising (Quantifier QRST call will ietycmoiinl u ahrtentr fters fteifrnilsystem. inferential the non of it rest renders the that of or nature theory, presence the QR the rather a but not in is compositional, movement it directly forces that that shows Raising This Quantifier approach. shifting of type in-situ 2012). an Jacobson and e.g., taking (see, taking scope of account So a in-situ is an Grammar of Montague example Flexible Grammar. paradigm Montague Flexible 1993 Hendriks’ of omdfo h rvosoeb igeapiaino ihrQ or QR either is of sequence application the single in a form by derivations logical one LF each previous of which the set in from our forms formed enlarge logical to of need sequences however, include do, We to LFs. for rules typing the of result the that Note (21) Raising: Quantifier a of here rule suggest the will as I (see others). category among treating syntactic discussion, approach, its for novel 2007 affecting Winter without and expression 1993 an Hendriks of denotation the and presence the in decidable remains QR that of whether extent consider the To should cost’.” we ‘no near-universal, or or cost’ ‘low “at language particular type of expressions lift B to generalizes it that observes (e.g., all. quantifiers at generalized shifting with coordinate type type have can any to names allows proper that as of such of advantages expressions aware compelling more I’m the system of every One by types form the adjusting some for in technique a as expressions. shifting linguistic type of pioneered 1983 Rooth and Partee L 6.1 readings. finite with decidable also is lifting unrestricted I with that QR logic a to equivalent is NL eta + QRT that show I QRT. to (“eta”) rule structural h pcltsta lhuhi a o euiesl tsol eaalbei any in available be should it universal, be not may it although that speculates She . LIFT λ LIFT hr,i un u htQ with QR that out turns it Third, yesitr r sal rmda ietsmni prtr htajs h type the adjust that operators semantic silent as framed usually are shifters Type of says 1987 Partee adding side, logic the On Bre 09,i sdcdbewt nt edns tflosta unrestricted that follows It readings. finite with decidable is it 2019), (Barker . Then IFT LIFT hrceie h ifrnebtenapr oeetapoc oscope to approach movement pure a between difference the characterizes seaexpansion eta as Let : LIFT σ ple to applied [ = LIFT ... LIFT L LIFT sawl-omdlgclfr,s hr sn edt adjust to need no is there so form, logical well-formed a is IFT γ ... σ sarl flgclfr,wt xcl h aestatus same the exactly with form, logical of rule a as LIFT saogtertp hfes(.38,adi included is and 378), (p. shifters type their among is hti flsdrcl u ftetp hoy”and theory,” type the of out directly “falls it that ] produces ealgclfr htcnan oia form logical a contains that form logical a be oQ eiain orsod oadn new a adding to corresponds derivations QR to LIFT 18 [ aiae h oetp-hfigprinciples type-shifting core the validates ... LIFT [ e i [ ttesm ieterlfe versions lifted their time same the at , t i sta talw individual-denoting allows it that is γ ]] n n vr student every and Ann ... A ] . otype to AB LIFT , B o any for LIFT suniversal, is ). hi Barker Chris . A and γ . early access , a — the exact ) , which means ann ( P . There are terms in . ) P g λ = f ( . The addition of eta reduction type shifter. f beta equivalent → ) just in case they deliver the same LIFT ga , since for any choice of argument f = , in comparison with all of the logical f a , guarantees (Barendregt 1984:63) that any . f 19 a LIFT ∀ η (

f x . x λ is not beta equivalent to e f x . QR, then, is the inverse operation, taking a subexpression, . extensionally equivalent x . (The fact that the argument appears to the left of its functor ... λ Ann: ]] a t ... Ann . But et β : is extensionally equivalent to 1 t 1 f a [ t

expansion 1 f x et,t [ a β . , ) x

et λ ... a x 1: ) f x The lambda calculus is a theory of functional equivalence. If one term can be What, if anything, is special about Here is a simple derivation in which a proper name is lifted into a generalized . .... x x λ λ instance, to the system, which says that two extensionally equivalent terms (that havein normal that forms) are they provably can equivalent, renders be the reduced lambda calculus to complete the with same respect to beta-eta extensionality. normal form. So eta reduction derived from another viaone beta can reduction, be they substituted for are functions the are other said without to affecting be theresult computational for result. every Two choice ofthe argument: lambda calculus that are extensionally equivalent but not beta equivalent. For replacing it with a distinguished variable,is, and QR creating is an abstraction beta structure —in that logical form is not important.) form rules we couldcorrespondence have between added QR to derivationsthat QR? and Quantifier An Raising the closely intriguing lambda resembles answer calculus.In beta comes equivalence It the in from is the lambda lambda the obvious calculus,replacing calculus. occurrences combining of a the distinguished lambda variable term in the with body of an the argument lambda term results in Function Application, so it translatesgeneralized into quantifier the delivered lambda by term the standard quantifier: (22) Ann (23) This logical form contains one instance of Predicate Abstraction and one instance of with a copy of the argument. This substitution operation is known as beta reduction: And here is a type labeling for the final LF: ( ( early access t.Hwvr ecngi diinlisgtb nidrc prah ’lshow I’ll approach. indirect an by insight additional gain can we However, eta. + Adding Then ye) ic RTi rgeto NL of fragment a is QRST Since Types). form the of structure any that says rule This orL hoyt ul hrceiesnatcetninleuvlne o need you equivalence, extensional syntactic characterize fully to theory LF your httolgclfrsaeetninlyeuvln uti aete eie the deliver [ sibling: they of case choice say in every We’ll just for result. equivalent result syntactic extensionally syntactic the are same affecting forms without logical other two the that for are substituted they be Raising, can Quantifier via another from (25) the has and decidable is eta + QRT property. that readings follows finite Shifty it with readings, Raising finite (Quantifier with QRST decidable call I logic a to equivalent is eta + QRT that by (24) QRT: QRST = eta + QRT 6.2 So way. calculus. intimate lambda and the natural to uniquely reduction a eta in adding as natural as is and want Raising you Quantifier if both Likewise, reduction. eta and reduction beta both need you equivalence, syntactic same the from QR via derived namely, be structure, both can they since QR-equivalent, are of readings scope namely distinct semantically two the instance, equivalence. extensional to respect with complete they and LF is, QR that via equivalent, derived provably be are can forms logical equivalent extensionally two any of addition The equivalent. α [ ∆ 1 Fi hoyof theory a is LF RThstesm e ftpsa R,adtesm structures. same the and QRT, as types of set same the has QRST QRT to extended be can QRT for decidability justified that reasoning same The extensional functional characterize fully to calculus lambda your want you If of theory a is LF that Note [ nteconclusion. the in t [ 1 1 h neec ue fQRST: of rules inference The i [ left [ t LIFT someone ∗ 1 ( left ]]] t i ∗ ]] oQ eiain orsod oadn eodsrcua ueto rule structural second a adding to corresponds derivations QR to and ∆ and ) [ ⇒ 1 [ [ someone α [ left everyone ∆ left syntactic r xesoal qiaet ic o n hieo sibling, of choice any for since equivalent, extensionally are ] r Reuvln.But QR-equivalent. are LIFT [ LIFT LIFT a everyone saw [ 2 syntactic [ t rmtesm trn oia om So form. logical staring same the from 1 qiaec.I n oia omcnb derived be can form logical one If equivalence. hssget htadding that suggests This . hc asthat says which , [ saw t 2 ]]]]]] qiaec,ntsmni qiaec.For equivalence. semantic not equivalence, 20 ]] λ . n akr21 rvsta NL that proves 2019 Barker and , i and ∗ ( t QR-equivalent [ i everyone ∗ [ ∆ [ ( i 1 ∀ [ ) t [ i α t napeiecnb replaced be can premise a in 1 γ . ]] left [ β α

[ oen a everyone saw Someone 2 ]] LIFT [ LIFT [ = ] someone and LIFT hc en one means which , γ oQ derivations QR to γ α urnesthat guarantees , left srltdt QR to related is ]) [ LIFT 1 r o QR- not are → [ hi Barker Chris t 1 [ ( saw β renders = (eta) λ t 2 γ ]]]]]] is ) , . , early access Ri → ↓ R R → QR → A A R A ` A → R A → ]) → ` i B ` t → A [ B Γ ] ` A · Γ B A ] ` · [ i → ` A B t i [ Γ Γ ( B Γ → ` Γ · · · ` B Γ B i B · Γ ` is a combination of the standard B ]) i in QRST: Γ i ≡ t R [ R Γ in a QRT proof can be reproduced in · ≡ i → has some structural reasoning baked in. i i → R i ( R i · L Axiom R 21 R → ∆ → A → i → A → eta ` R C ] = A A C → ∆ A ` → [ ] ` B Γ ` → A A ] A ] [ ` A A B B ] Σ . That is, [ i → → ` A t ↓ ` [ Γ → B B Γ ) ` Γ · B ] · Γ ` ` · i · B B ] i B [ i Γ t ∆ t [ [ Γ ( ` · Σ Γ i · ∆ i in Gentzen’s 1934 system LJ, which is a sequent presentation of → It’s easy to see that every QRST proof can be reproduced by QRT + eta, thanks Because of this conflation of logical and structural inference, the QRT version is able rule with an instance of QR In the other direction, anyQRST instance using of the following inferences: (28) to the following equivalence: (27) intuitionistic logic). This same rule of proof for implication is used throughout the right rule for Comparison with (16) reveals that QRT is identical to QRST except that the (QR) rule in QRT is replaced with a simpler rule (26) This equivalence reveals that the QRT rule Type Logical Grammar literature. The original rule given above for QRT is on the left. The modified rule is the standard early access ecnadthe add can we bsdshvn tnadrl fpoffripiain sta hr sasudand sound a is there that is implication) for proof of rule standard a having (besides h ur-oadlbln o hspofgvseatytegnrlzdquantifier generalized the exactly gives proof this for labeling Curry-Howard The oeta hr r nytowy htasrcueo h form the of structure First, a inference. that eta ways an two contains faithfully only that be are can proof there inferences eta that eta + note QRT contain a that Consider eta QRST. + in QRT simulated in proofs that show to only added. nothing with Raising QR by created of availability the limit to opeeitrrtto o RT(e akr2019). Barker (see QRST QRST for with interpretation going complete of rule favor standard in the consideration has additional which QRST, One else implication. or for added, proof reduction of eta of rule structural side, derivation a QR the On derivations. QR of properties formal the compromising property. readings logic finite modified the the has results, and decidable these also change is not here does Exchange adding and NL property, Since added. Exchange rule structural type to specific Partee’s of semantics versa. vice and QRST, in proof equivalent semantically a it has until eta proof + QRT the or in in QR proof higher either moved of be instance can It the reduction conclusion. meets eta final immediately of the inference affecting instance an without any of exchanged that order follows be the can cases, reduction other eta all by an In followed If (27). omitted. in safely shown be as the can inference, of inferences result of net pair the the of of reduction, and instance instance eta change, an by no by followed is or immediately inferences rule, is two structural QR QR of the instance of an instance If an by conclusion: a in nodrt sals h qiaec ewe R t n RT tremains it QRST, and eta + QRT between equivalence the establish to order In hsmasta ecnfel lo netitdgnrlzdtp itn without lifting type generalized unrestricted allow freely can we that means This NL of fragment a is QRST mentioned, As QRST: in straightforward is lifting Generalized → ↑ R LIFT e speet hsrsrcini rcsl htlmt R oQuantifier to QRT limits what precisely is restriction This present. is i n types any — simdaeyfloe yeardcin hsi xcl an exactly is this reduction, eta by followed immediately is uegvnaoe ntelgcsd,w a ihrueQTwith QRT use either can we side, logic the On above. given rule LIFT → yesitr h esnn eei ul eea,adnot and general, fully is here reasoning The shifter. type rgtifrne ostain nwiha abstraction an which in situations to inferences -right A e and ` et e → t e ` B · R e et,t i a esbtttdhr for here substituted be can ` htitoue h eeatidx oevery So index. relevant the introduced that 22 λ t ` sdcdbeadhstefiiereadings finite the has and decidable is → t λ R → dsusdi akr21)wt the with 2019) Barker in (discussed L i · ( t i · Γ e ) and a ecreated be can hi Barker Chris t . → → R R i . early access saw in (10) , on the saw saw ). On such an Ann saw everyone only if every syntactic constituent has a that will conveniently make salient the 23 Ann saw everyone, and Bill did too Someone saw everyone and nothing else. is established as a constituent: there is no type fails to have a semantic interpretation until it saw everyone everyone directly compositional and saw everyone is a syntactic constituent, then a directly compositional at the point in the derivation in which it is first formed by saw everyone saw are constituents. For instance, this particular verb phrase can serve as the analysis, there is no semanticeveryone value to assign to thethe verb syntax phrase (or at anydirectly other compositional point in analysis, the by derivation, contrast,semantic for is value that forced for matter). to any A expression provide thatin a is the recognized syntax. as a Thuseveryone constituent if there areanalysis good must reasons provide to it with believe a that meaning. [In the standard analysissuch using as Quantifier Raising,] ahas verb been phrase embedded within ato large raise enough structure and for take the scope quantifier (e.g., Quantifier Raising contrasts in this regard with Hendriks’ 1993 Flexible Mon- There are good reasons, of course, to suppose that verb phrases such as strategically deployed type shifting, withoutconfiguration. any The movement two or main other type structuralarbitrary shifters re- are argument Argument of Raising, a whichpredicate, allows predicate an and to Value take Raising, which scopesuitable lifts over the for the result other taking type arguments scope. ofdicto/de of a (There re predicate that ambiguities is into that a a I type third will not type discuss shifting here.) Flexible rule Montague that Grammar deals is with de system that Barker andcomputed Jacobson for the have structure insemantic mind, value captured there by will the ellipsis. be a semantic value tague Grammar. In that system, expressions take displaced scope entirely through contribution of everyone antecedent of verb phrase ellipsis,interpretation as in on which Bill saw everyone. In the kind of directly compositional or the corresponding QRT proofcorresponding in to (17), there isassociated no with stage that at particular whichcorresponding substructure QRT the either proof, structure and in no the Curry-Howard QR labeling derivation that contains or the in semantic the about Quantifier Raising: 6.3 In-situ scope and direct compositionality And indeed, if we examine either the QR derivation of A semantic theory is well-defined semantic value. Here is what Barker and Jacobson 2007:2 say (p. 2) early access ecnpoeta h ebphrase verb the that prove can we h ur-oadlbln is labeling Curry-Howard The ebi question. in verb take to QR undergoes argument quantifier generalized The arguments. its and verb (29) unie ietojc (type an object of direct argument quantifier first the (type to verb applied transitive Raising extensional Argument illustrate, To QRST. in rems scope to approach compositional directly a 2012). of Jacobson example (e.g., paradigm taking a as up held often eiain nwihsm ytci tutr osntrcieasmni au,as value, semantic a receive not does structure syntactic some which on desired. derivations as Bill), (or Ann by seen was everyone of that denotation entails the to applied is meaning (30) that instance, proof For a it. gives is Grammar here Montague Flexible that denotations) corresponding shifted the as pro- is, same that proof the Grammar, Q the exactly Montague of is Flexible rest labeling in the Curry-Howard given The and meaning (19). verb, e.g., saturated in, the as by ceeds created clause the extensional over the con- scope of the consisting of structure types argument a building the turnstyle, of each the pushes across proof result the clusion up, bottom the from Reading sagnrlzdqatfirand quantifier generalized a is ttrsotta l ntne fAgmn asn n au asn r theo- are Raising Value and Raising Argument of instances all that out turns It RTi o a not is QRST that means Raising Value and Raising Argument validates QRST that fact The strictly a everyone saw ietycmoiinlter,sneteeaepet of plenty are there since theory, compositional directly et,t et,t (et,t)et λ eet e e sapeiaeo type of predicate a is · · x saw eet · · eet eet ( ( . a everyone saw ( ( 1 1 everyone rae hfe ebta ae generalized a takes that verb shifted a creates ) eet eet · · ( ( ` stemaigo h xesoa transitive extensional the of meaning the is · · ): e e Ann et,t et,t 24 (et,t)et · · · · et,t et,t ( ( · · · · · · eet eet (or ( ` ` λ ) ) · · Bill et et y t t ` ` . 1 1 saw ))) ))) a aho h ye adtheir (and types the of each has t t ,tersl sapooiinthat proposition a is result the ), → → → ` ` x y λ R et t t R R Q ) : QR QR hnti ebphrase verb this When . λ x . ↑ ↑ Q ( λ y . saw x y hi Barker Chris ) where , early access with the i is a clause, R → that characterizes R B ∧ : given any theorem B ∧ (QRT with conjunction) B ` ∧ ∧ Γ a type . That is, given any specific A ∆ A ` is not the only way to embed ` Ann (saw everyone) Γ ] A · B [ ` Σ LIFT Σ Σ . Then QRT (equivalently, replacing B and B 25 and LIFT L ` A ∧ ∆ C C ` ` ] ] B . The construction given by the interpolation theorem B t · ∧ A such that ` A [ [ for all types ) Σ B Σ t , B ∧ et · A eet ( ), we have in-situ analyses and direct compositionality on demand. · R e → , there is a type A ` For instance, the proof in (17) establishes that Without going into complete detail, here’s how it works. First, we expand the set One way to see this is to note that adding What should we make of this situation? There is no difference between QRT By the way, the fact that QRST is decidable with finite readings, along with ] ∆ [ its role in that proof. that is, that Σ proof, it is possible to assign an arbitrary substructure routinely include a conjunction, starting with Lambek 1958 (see Moortgatof 1997). types to include is QRT plus the following two standard logical inference rules: (31) taking analysis is embedded. QRT in a directly compositionala system. conjunction For to instance, the anotherone logic. strategy conjunction After is in all, to addition many to add logics implication. In (to particular, put Type it Logical mildly) grammars have at least rule). Yet in bare QRT, scopecompositional; taking but always with requires the movement and additionstandard is of not directly directly compositional theory resides notscope-taking, in but the in conception the or nature the of implementation the of larger inferential system in which the scope a QRST proof means thatMontague (the Grammar Argument is Raising/Value Raising also decidable corethe with of) first finite Flexible time readings. that As fact far has as been I established. know, this is and QRST with respect to the structural rule that encodes scope-taking (the QR on Demand’. In the sametions spirit, delivered we by can Flexible reconstruct Montague the Grammarconsider same whenever QRST type-shifted desired, to interpreta- so provide we in-situ can scope also taking on demand. the fact that every Flexible Montague Grammar derivation can be reproduced by The interpolation theorem of Barker 2019 holds for QRT Apparently, the difference between a pure movement theory of scope and an in-situ, we have seen. Nevertheless, we canwhenever compute desired. the Barker missing 2007b semantic calls interpretations this kind of situation ‘Direct Compositionality early access (32) proposes oeetoeainsc sQ,btdpnsisedo rpriso h ytmas system the of properties on long-distance instead a depends whole. allows but a it QR, whether as interpretation on such an depend operation and not movement type does — a structure provides syntactic it grammar every whether a for is, whether that that — means compositional This directly demand. is on compositionality direct supply can demand. QRT So QRT QRT. of plain theorems of the are theorems to that contrast sequents In (conjunction-free) no 2014:70). does, are Shan QRT and that Barker analyses by allow of noted to addition (as fails scope it parasitic lifting), as as (such such allow not Grammar does Montague QRT Flexible that does analyses only Not Grammar. Montague Flexible than that derivation. labeling the Curry-Howard of a version and non-local these type allows the grammar a tracks given the accurately be to to conjunction of points addition stopping The intermediate hops. local strictly of series is to conclusion final pair the ordered left Curry- for the (the label is standard phrase proof) verb the the the Using of for branch labeling inferences. hand semantic cut the of conjunction, for discussion semantics a Howard for A.1 Appendix See everyone naycs,QS n QRT and QRST case, any In closely more QR of power expressive the matches approach conjunction The a into movement QR arbitrary an up breaking allows theorem interpolation The et et et LIFT , , ( , t t t λ ` eet · ∧ y ( . et 1 digacnucint R scnevtv,i h es htthere that sense the in conservative, is QRT to conjunction a adding , saw eet · · , ( et t eet , y stetp ftevr phrase. verb the of type the as t ann · 1 ` t e · 1 et ( )) ` eet eet ) , ` et e t susual. as ∧ et · · e t eet everyone , 1 ` t ) ∧ ∧ ∧ ` e et ` hwta w ifrn aso xedn QRT extending of ways different two that show · eet spr Rbtwt ietcmoiinlt on compositionality direct with but QR pure is et eet ( eet → QR ∧ → · 26 ( L R et λ R i y , h t . everyone ( ) 2 λ ` · x ( t . e e et saw · · ( ( , e t t 2 · e · x e eet · , ( · · · · ) eet 2 λ · ( et y et · x ) ( ann . e )) , ` , saw t t · ∧ t ` ∧ ( · t eet ) eet htwr o already not were that tt et 2 hc eareduces beta which , x · i eet → ) hc en the means which , ` ` R t ))) t i cut ∧ ` L hi Barker Chris t ` QR t → L early access (Jacobs is able to λ kein to create unbound traces ↑ 27 (with Exchange added), a substructural logic first proposed in λ equivalent. Likewise, on the logical side, the bi-directional structural account not only for in-situ scope-taking, but syntactic movement as well. Because If computing covert scope analyses isis decidable, a what fragment about of overt NL movement?Barker QRST 2007a. As shown in Barker and Shan 2014 and in Barker 2019, NL all, since the Curry-Howard correspondencethat ignores displaced structural scope inferences. is It an follows other essentially, purely grammatical syntactic phenomena phenomenon, that on correspond to a structural par rules, with such as scrambling coordination (corresponding to the structural rule of associativity). 7.4 The logic of movement? in-situ in its surface positionsyntactically and the corresponding logical formequation created in by QRT QR that are characterizes QuantifierPut Raising another way, is recall a that structural the inference QR rule. structural rule does not affect semantic labeling at sion of the details andRaising the analysis. trade-offs of having higher-order traces in a Quantifier 7.3 Quantifier Raising is syntactic 2017), and cumulative readings (Charlow 2020,traces Charlow are to perfectly appear). compatible Higher-order with the system here. See Charlow 2020 for a discus- special action here, since anybe QR derivation semantically that coherent. creates Likewise, an in unbound QRT, trace using will QR not 7.2 Higher-order traces can happen when material containingraises the higher trace of than a theQuantifier previously Raising raised lambda is scope that to taker simply binds prohibit the unbound traces. trace. There The is no traditional need solution to take for any 7 Four additional issues 7.1 Unbound traces There are many analyses that rely on higher-order traces: semantic reconstruction (corresponding to the structural rule of Exchange) or so-called non-constituent (e.g., Cresti 1995, Barker and Shan 2014), split-scope analyses (German As discussed in Section 6.1, LF is a theory of syntactic equivalence: a quantifier will never lead to a complete and valid proof. As May noticed, unconstrained Quantifier Raising can create unbound traces. This 1980), donkey anaphora (Barker and Shan 2014), Haddock sentences (Bumford early access Euvln’hr en aefia eun,adeuvln ur-oadsemantic Curry-Howard equivalent and sequent, final same means here ‘Equivalent’ elbhvd hruhepoaino h oi foetmvmn ln h lines the along movement overt of logic the of exploration thorough A well-behaved. tnadifrnerl o nestesoy the story, the enters now rule inference standard A hsApni rvscteiiainfrQT n ie eal ftemappings the of details gives and QRT, for elimination cut proves Appendix This h u uesy hti yestructure type a if that says rule cut The (Well, deduction. of transitivity the expresses and logic, every in valid is rule cut The (33) Conclusion 8 occasion. another for wait to have will computationally paper is this that of grammar unified single a in scope-taking and movement NL aeigu obt reduction. beta to up labeling cut. (34) using without proven be can rule cut is the rule cut the that proving proof. larger cut.) of of occurrence version restricted a endorse logics these even Yet paradoxes. certain address almost elimination Cut problem A.1 the for hint a gives also It again. (13). back in and posed QRT to derivations QR from Appendix A scope. analyzing for Raising technique Quantifier well-behaved in formally confidence and full coherent justify a of together as presence taken the results in These of even lifting. number expression, given type finite any strictly for a interpretations provides for semantic and method distinct decidable, explicit is an it in with compatibility, scope combined type displaced is checking analyzing Raising for Quantifier tool When standard language. the natural been long has Raising Quantifier λ suulfrdcdblt rosi oi,tedcdblt fQTwl ig on hinge will QRT of decidability the logic, in proofs decidability for usual As sas eial ihfiieraig,i hw o ocmiesyntactic combine to how shows it readings, finite with decidable also is ∆ htde o oti n u inferences. cut any contain not does that elimination Cut vr oi e er21 o icsino o-rniielgc that logics non-transitive of discussion a for 2015 Weir see — logic every ` A Σ [ A ∆ ] nsm te ro ihtemtra in material the with proof other some in ` Σ [ B A ] ` ie nabtayQTpof hr sa qiaetproof equivalent an is there proof, QRT arbitrary an Given . B admissible cut rdnat:ta n hoe rvbeusing provable theorem any that (redundant): 28 ∆ a type has cut A rule. ecnsfl elc any replace safely can we , ∆ ihu fetn the affecting without hi Barker Chris early access in the A inference i R is the formula → L → cut C C cut ` ] ` cut formula C ] A [ ` A cut Σ ] ↓ A → C [ Σ ` B QR · ]] C B ∆ C B [ ` [ rule in effect incorporates an instance ` Σ ` Γ i A [ ∆ C ])] 29 R i Σ ` ])] t of a logical inference is the formula created ). ` i [ ] i t [ Γ → ]] B R · [ Γ ∆ i · [ Γ ( i → Γ · ( [ · ∆ A Σ [ ∆ inference. The reason is that the refactoring eliminates [ Σ ↓ → Σ A B principal cut B , which is the active formula for both the ` ` ] A active formula ` B ] ∆ [ i t → , and since the [ Γ i Γ B R · i → inference. This cut can be transformed into a pair of smaller cuts using L → The logical rules have the subformula property, which means that every formula The only remaining case to consider is when the cut formula is the active formula The structural rules do not impede pushing cut inferences higher in the proof. As in all cut elimination proofs, the general strategy here is to push each cut The proof here is almost completely standard (see especially Gentzen 1934, reduction inference in the replacement proof fragment. in the premises; see, e.g.,non-standard wrinkle Jäger in 2005:43 the for proof arequires is more the that addition the detailed of transformed definition. a reasoning The QR an with only smaller instance cuts of of reduction (as discussed in Section ),6.2 it is necessary to add a compensating Here, ‘smaller’ refers to the total number of base types and logical connectives and the the same initial premises and arriving at the same final conclusion. in both of the premises (a the cut can always be swapped. in the conclusion appears inactive (exactly) formula. one As of a the result, premises,one with whenever of the the the exception cut premises of formula of the is a not cut the inference, active it formula is in possible to push the cut upwards. an axiom, the other premisesafely must eliminated. be identical to the conclusion, and the cut canSince be structural rules do not addin or the subtract premise formulas, of the the cut structural formula inference, will so be the present order of the structural inference and shared by the twoschema premises above), of and the the cutin inference the conclusion (the that formula is matching not present in any ofinference the higher premises. in the proof until it reaches an axiom. When one premise of a cut is Restall 2000, Jäger 2005:41). Some vocabulary: the The cut formula is early access with instances the tracks phase first The phases. two into divides proof the Building wards. derivation QR coherent semantically a have we Assume orsodn to corresponding type proof QRT equivalent an has derivation QR Every A.2 ananta smrhs.Te hr r he ae,crepnigt h three the to will corresponding construction cases, the three rules: of are typing application there recursive Then each isomorphism. and that siblings, maintain of order the to up structure type form a logical and labeled a parameters: two on prove to needed rules inference instance derivation. corresponding form the logical matches the exactly from that QR inference of expansion an by sequent it (lowest) below initial The derivation. QR the is constitute that Raising Quantifier of Σ T1. T0. T2. ebidtepofsatn rmtebtoms ocuinadwrigup- working and conclusion bottommost the from starting proof the build We h eodpaeue h yelbln of labeling type the uses phase second The ` π A A u oli obidaQTpofof proof QRT a build to is goal Our . = ev o eue proving reduced now We’ve ol)lxclie in item lexical (only) ae ttero of root the at label the instantiate π with once twice: algorithm construction proving namely, follows: as upwards proof the that Because π rule. inference form π π h etis next the , ossso w daughters, two of consists = sa btato ihform with abstraction an is ossso igewr,i hc aetesqett epoe a the has proven be to sequent the case which in word, single a of consists σ n γ γ and , P a type has and 0 ` π σ P Π Π aifistetpn ue,w a suethat assume can we rules, typing the satisfies . where , Σ = Π = 1 ` → hc stepr of part the is which , B Σ Γ A n → . ∆ R oeta h nta ausfor values initial the that Note . Γ n oo pto up on so and , π i P ` A ` : uet xedtepofuwrs where upwards, proof the extend to rule 0 π B o ievra ihsalcagsblw.W extend We below). changes small with versa, vice (or B sasnl ye yconstruction, By type. single a is So . Σ → n proving and n ` A δ P A i ∆ and ∆ γ h osrcinpoed eusvl based recursively proceeds construction The . = [ · · t 30 i Γ Γ P ] γ and , 0 Σ ` ` π hr ete sa ne,and index, an is neither where , n ehv nisac fteaxiom the of instance an have we and , ∆ Σ n ∆ A A hc sapr ftelgclform logical the of part a is which ` n ` Γ · A B orsodn to corresponding B Σ Π notosrcl mle problems, smaller strictly two into A ` σ π ahsqeti eae oteone the to related is sequent Each . → ` n = B = ogieteisataino the of instantiation the guide to A i A → where , · δ Γ ` A and ` [ σ A t ercrieycl the call recursively We . i A ] , π where , cut σ Π → 1 and ,..., Σ = L P satp structure type a is π Π ∆ δ stelblo the of label the is i σ hs begins 2 Phase . n gi with again and , r isomorphic are sa ne.We index. an is a type has n where B → hi Barker Chris Π A = σ B sthe is n ∆ and has σ · Γ n , . early access , i t L = Π → when and A ] i t [ γ = . If the expansion π A has type γ will guide construction γ i instead of R A → is of finite complexity, we will → A Σ B · is a single type, so every subproblem → A ∆ Γ B ` ] or ` 31 B ] ∆ [ i t [ Γ Γ · . i A ` ] . inference, there are two subcases: the raised element is B i [ B R Γ → , so we can be sure that the labeling of instead of B , or not. If not, the expansion can be delayed, in which case the i t A : whenever the conclusion of a QR instance is a premise of a logical . The typing rule for abstractions guarantees that ] B [ Fact inference, the order of theproof. inferences can be reversed without affecting the Γ has type of a valid proof of We now recursively call the construction algorithm with Here is an illustration of the last subcase: The official inference rules for QRT given in (16) do not contain cut, and the , or not. If not, the expansion can clearly be delayed. If so, the delayed expansion affects the premise of a B raises the trace sequent. For instance, if the expansionthe affects expansion the can first safely premise be ofthe delayed an second till instance premise, after of the there logical areoccurrence rule. two If subcases: of the the expansion raised affects elementdelayed is expansion the simply distinguished raises the structure an expansion inference is a premisethe of order a of following the logical inferences inference. can In be each reversed case, without affecting the initial or the final QRT places no restrictionsthe on correspondence between when QRT a proofs andstructural QR QR inferences inference derivations lower depends in can on the pushing occur, proof. so establishing (35) proof construction just given doesSection containA.1 cut;explains but of how to course arrive the at method an described equivalent in cut-free proof. A.3 Pushing structural inferences lower building into strictly smallereventually problems. reach a Since point at which either To see why, consider first all possible configurations in which the conclusion of These three cases show that we can always divide up the second phase of proof will terminate in axiom instances. early access hc l QR all which h eiaiiyrsl n h nt ednsrsl ol otnet apply. to continue would result readings finite the and result decidability The are left the on inferences unswapped the for sequents ending and beginning The dnia otebgnigadedn eunsfrtesapdinferences. swapped the for sequents ending and beginning the to identical ruetpoed yidcino h tutr of structure the on induction Let by inferences. proceeds expansion argument final the including not but of of nodes structure corresponding the the onto with labels labels the map associate to to isomorphism convenient be will it which on and how show must corresponding the sequence with the step in lock structures in the QR relate of that application inferences an expansion by one previous the to lated σ final QR the all adjustments, these After inferences). QR each which replacing in by cut-free, begin is we QRT, for result ability operations. Raising Quantifier initial of the number which some in undergone forms already logical has to the of form need maintain sequences logical also include to would to we order derivations derivations, QR in QR generalize but and relaxed, proofs Quantifier QRT any be between undergone can yet correspondence restriction not has This that operations. form Raising logical a with begin derivations QR use to is derivation goal Our structures. abstraction any derivation QR equivalent Let an has proof QRT Every A.4 ences. , σ iia ruethlsfrsapn euto neecswt oia infer- logical with inferences reduction swapping for holds argument similar A osdrteprinof portion the Consider we derivation, QR coherent semantically a is this that demonstrate to order In decid- the and Appendix the of sections previous the in results the on Relying of sequent final the Requiring p 1 ,..., eayQTpofwoefia ocuinis conclusion final whose proof QRT any be Σ i · [ B Σ n Σ [ σ · σ Π neecsof inferences ( [ n Π , [ j t uhta each that such σ ↓ · σ i [ ]] B Π 1 n neecshv eneiiae aogwt hi onee QR coindexed their with (along eliminated been have inferences ,..., ]] ` [ p a type has t 0 ` j B ])] eemnsatp aeigfor labeling type a determines A → σ ` n A uhthat such A p A → 0 QR hnteQR the Then . Because . Σ p R i ↑ 0 i ↑ satp tutr for structure type a is htproves that neec olw vr oia neec,adin and inference, logical every follows inference Σ p ob btato-rei aallt eurn that requiring to parallel is abstraction-free be to satp tutr for structure type a is ≡ σ n and 32 ↑ neecsin inferences ↑ Σ i Σ neecswl egtee oehras together gathered be will inferences n · p n Σ ` ofidasmnial oeetQR coherent semantically a find to r smrhc(pt iln order), sibling to (up isomorphic are p Σ [ i t A · i [ iha qiaetproof equivalent an with B Σ · σ hti,teprinof portion the is, that , Σ ( [ p · n Π σ j ` ( 00 htstse h yigrules typing the satisfies that · i p . [ j n ahlgclfr sre- is form logical each and , A Π t 00 · i ]] where , Π [ p eti oto of portion this be t σ 0 ` j [ ])] t nueaQ derivation QR a induce and j B ])] ` Σ → ` , B σ σ Σ Σ Σ A A n n n → 1 osntcontain not does eyn nthe on relying , . ,..., a type has A QR → Σ hi Barker Chris n R . p ↑ p i 0 A p 0 pto up The . 0 . that ↑ early access ] i t [ Γ . By . We . We · . Any 00 ↓ B i p A has type ` ]) i A t [ Γ both have type , and QR · i has label by virtue of the ] i ( R ∆ ∆ A [ · Γ ∆ → → , L B → be the label at the root of B . ↓ L i C has type R , by virtue of the typing rule T1, → QR ] A i t → C [ C C Γ , the newly created structure ` A ` , and let · A ] ] ` i C , the labeling satisfies the typing rules, ] ∆ A → n ])] [ [ A ` i A t σ Γ Σ [ B . 33 ]] ` , we can assume that we have a labeling for Γ A ] ) has type → ∆ ` B · [ B A ] i [ i B Γ according to the typing rules. Then the newly- ` ( t [ · [ has type Γ · B → Σ ∆ B Γ ] ∆ ∆ [ · [ ` B B i [ Σ · Σ to the newly-created substructure. By the inductive ∆ has type Γ consists of an axiom inference of the form ∆ n A 00 σ p is copied onto . Since all expansion inferences have already been pushed n by virtue of the typing rule T2. We label the new structure A Σ A ` n → ∆ B . n that satisfies the typing rules such that the root of . σ be the label at the root of ∆ on which it has type C A ∆ . Then the newly created structure , by virtue of the typing rule T1., We the label recursive the assumption guarantees newly that created the structures newly-extended labeling has A accordingly. Since the larger newly createdA structure and type Since the labeling of has type accordingly. Let ∆ typing rule T2, and so the larger newly created structure By the inductive assumption applied toof the left premise, wecreated have a substructure labeling ( so we add the label assumption applied to the rightthe premise, conclusion there sequent is on now which a it complete has labeling type of : : : i ↓ L For the inductive case of the argument, consider the final inference in The base case is when R → QR → and justifies the claim that need to prove that lower in the proof, therereason are as three follows: subcases to consider: a labeling for the recursive assumption, we can assumepremise. that That we is, have if a the suitablethe premise labeling structure is for each logical form labeled with the left hand side (trivially) has the result type, so we have When the labeling on early access References Raising. 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