The Semantics of Reciprocal Expressions in Natural Language

Transcription

1 The Semantics of Reciprocal Expressions in Natural Language Sivan Sabato

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3 The Semantics of Reciprocal Expressions in Natural Language Research Thesis Submitted in partial fulfillment of the requirements for the degree of Master of Science in Computer Science Sivan Sabato Submitted to the Senate of the Technion Israel Institute of Technology Tevet 5766 Haifa January 2006

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5 The research thesis was done under the supervision of Assoc. Prof. Yoad Winter in the department of Computer Science. I would like to express my deep gratitude to Yoad Winter for his continuous encouragement and guidance, and especially for helping me to understand the Big Picture and gain insight to the Meaning of It All. My family and my friends are a part of this work in more ways than one. I am grateful to each of you, for what words cannot express. The generous financial help of the Technion is gratefully acknowledged.

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7 Contents 1 Introduction 9 2 Previous Works Langendoen (1978) Heim et al. (1991) Dalrymple et al. (1998) Beck (2001) Assessment Semantic Restrictions and Reciprocal Meanings Semantic Restrictions Reciprocal Meanings When is a Reciprocal Meaning Attested? Proofs of Propositions in this Chapter The SMH-Based Interpretation of Reciprocals A New Realization of the SMH Applying the SMH-based Interpretation Strong Alternative Reciprocity Some More Complicated Cases Predictions of the System - A Summary of Examples Predicting the Existence of Reciprocal Meanings Characterizing the Congruence Relation Weak Reciprocity Inclusive Alternative Ordering Proofs of Propositions in this Chapter Reciprocals Require Connectivity Partitioned Interpretations of Reciprocal Sentences A New Explanation of Partitioned Interpretations

8 6.3 Fixing the SMH-based Interpretation of the Reciprocal The Strange Case of Heterosexual Relations Conclusion and Directions for Further Research 61

9 List of Figures 4.1 Two settings of bricks Boxes Other boxes The cubes are glued to each other The relation R The men are hitting each other Two settings of schematic pirates Students sitting alongside each other

10 List of Tables 4.1 Reciprocal Meanings Derived from Semantic Restrictions

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12 Abstract The interpretation of reciprocal expressions such as each other and one another exhibits a remarkably wide variation in different contexts. Specifically, the predicate in the scope of the reciprocal expression affects its interpretation. Consider the following examples of two minimally different sentences with reciprocal expressions: (1) Dan, John and Mary saw each other. (2) Dan, John and Mary followed each other. Sentence (1) entails that every person in the specified group saw every other person in the group, while sentence (2) does not entail the analogous claim, that each person followed each of the others. This contrast exemplifies two possibilities for the truth conditions of sentences with reciprocal sentences, out of many more that have been identified in the literature. We propose a new formal system for predicting the interpretation of reciprocal expressions in a given context. This system is based on a principle called the Strongest Meaning Hypothesis (SMH) that was proposed in Dalrymple et al. (1998) (henceforth DKKMP) to account for the effect of contextual factors on the truth conditions of different reciprocal sentences. The SMH as formulated on DKKMP chooses the strongest meaning that is consistent with relevant contextual information, out of an independently defined set of meanings for reciprocals. The system proposed by DKKMP is thus composed of two components: One is the SMH, and the other is the set of meanings the SMH chooses from. The new system we propose is based on the critical observation made in DKKMP about the weakening of the interpretation of a reciprocal sentence by contextual factors. However, unlike DKKMP, we implement the SMH as a mapping from semantic restrictions on the predicate s denotation into the interpretation of the reciprocal, with no independent assumptions about available reciprocal meanings. This new implementation also defines the loose notion of relevant contextual information used by DKKMP, and identifies this information solely with a formal concept of semantic restrictions of predicates. We implement the SMH as a local maximality principle. According to this principle, a reciprocal sentence is consistent with models in which no pairs of non-identical individuals in the antecedent set can be added to the denotation of the predicate, while remaining within its semantic restriction. Our formulation of the SMH allows a systematic analysis of types of predicates 3

13 and the truth conditions they induce on reciprocal sentences that contain them. We analyze the connection between semantic restrictions, interpretations of reciprocal sentences, and several meanings that are proposed in the literature for reciprocal sentences. We show that in our system, the reciprocal meanings Weak Reciprocity (WR) (Langendoen, 1978) and Inclusive Alternative Ordering (IAO) (Kański, 1987) cannot be derived as the strongest meanings of any reciprocal sentence. We also show that the meaning derived by the SMH for a reciprocal sentence should have a lower bound that prevents unlimited weakening of the truth conditions of a simple reciprocal sentence. We argue that partitioned interpretations of reciprocal sentences are always due to partitioning that is independent of the semantics of the reciprocal. 4

14 Abbreviations and Notations SMH The Strongest Meaning Hypothesis DKKMP Dalrymple et al. (1998) HLM Heim et al. (1991) I The Identity relation. I def = {(x, x) x E} R For R a binary relation, the symmetric closure of R: R def = R R 1 R For R a binary relation, the transitive closure of R: R def = i N R i R A For R a binary relation and A a set, R A is R restricted to A: def R A = R A 2 R A For R a binary relation and A a set, R A is R restricted to A def and disregarding identities: R A = R A \ I (A) For a set A, (A) def = {S S A} min(α) For α a set of relations, min(α) def = {R α : S α [S R S = R]} D(x, y) For D a binary relation, D(x, y) (x, y) D D(x) For D a binary relation, D X Y, the image of x X under D: D(x) def = {y Y D(x, y)} T A S If T and S are binary relations, then T A S T A S A If T and S are sets of binary relations, then T A S {R A R T } {R A R S} R = A S, R A S, etc. are defined similarly. 5

15 Θ P For P a binary predicate, the semantic restriction of P: Θ (E 2 ) recip Θ The reciprocal interpretation domain of Θ: def recip Θ = ( (E) Θ) R Θ For Θ a semantic restriction, R Θ recip Θ is the SMH-based interpretation of the reciprocal, defined as follows: A E, R Θ [R Θ (A, R) R Θ [(R A R ) (R = A R )]] 6

16 Reciprocal Meanings In the following definitions of reciprocal meanings, A is the antecedent set of the reciprocal, and R is a binary relation denoted by the predicate in the scope of the relation. For each reciprocal meaning, we give the formula that represents its truth conditions. SR Strong Reciprocity: x, y A [y x R(x, y)] PSR Partitioned Strong Reciprocity: P (A) [( P = A) P P [ x, y P [y x R(x, y)]]] SAR Strong Alternative Reciprocity; SmR Symmetric Reciprocity: x, y A [y x R(x, y) R(y, x)] x A [ y A [y x R(x, y) R(y, x)]] IR Intermediate Reciprocity: x, y A [y x R (x, y)] Where R is the transitive closure of R. IAR Intermediate Alternative Reciprocity: x, y A [y x (R ) (x, y)] Where (R ) is the symmetric and transitive closure of R. IAO Inclusive Alternative Ordering: WR Weak Reciprocity: x A [ y A [y x (R(x, y) R(y, x))]] x A [ y A [y x R(x, y)] y A [y x R(y, x)]] OWR One-way Weak Reciprocity: x A [ y A [y x R(x, y)]] 7

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18 Chapter 1 Introduction The interpretation of reciprocal expressions such as each other and one another exhibits a remarkably wide variation in different contexts. Specifically, the predicate in the scope of the reciprocal expression affects the interpretation of the reciprocal expression. Consider the following examples of two minimally different sentences with reciprocal expressions: (1) Dan, John and Mary saw each other. (2) Dan, John and Mary followed each other. We henceforth refer to sentences with reciprocal expressions as reciprocal sentences. Sentence (1) entails that every person in the specified group saw every other person in the group, while sentence (2) does not entail the analogous claim: Obviously, it is not possible that each person followed each of the others. For instance, if Dan followed John and John followed Mary, sentence (2) is evaluated as true. This contrast exemplifies two possibilities for the truth conditions of sentences with reciprocal sentences, out of many more that have been identified in the literature. The aim of this thesis is to propose a new formal system for predicting the interpretation of reciprocal expressions in a given context, using as few assumptions as possible. In Dalrymple et al. (1998), a principle called the Strongest Meaning Hypothesis (SMH) is proposed to account for the effect of contextual factors on the truth conditions of different reciprocal sentences. We develop a system that is based on a new implementation of the SMH. In this new system, the interpretation of a reciprocal expression is derived directly from the properties of the predicate in its scope. Our system incorporates less assumptions than the one proposed by Dalrymple et al., and it correctly predicts the interpretations of reciprocal expressions in many cases. It also allows us to analyze suggested meanings of reciprocal expressions and 9

19 to predict the properties a predicate must have in order to generate these meanings. Earlier versions of part of this work appear in Sabato & Winter (2005a) and in Sabato & Winter (2005b). The structure of this thesis is as follows: In chapter 2 we survey related works on the semantics of reciprocal expressions. In chapter 3 the formal framework and concepts we use are introduced and defined. Chapter 4 introduces our new formulation of the SMH and its implications. In Chapter 5 we analyze within our system two previously suggested meanings of reciprocal expressions. In chapter 6 we refine our formulation of the SMH based on additional empiric evidence. Chapter 7 concludes the thesis. 10

20 Chapter 2 Previous Works In this chapter we introduce previous works that investigate the semantics of reciprocals. First we introduce Langendoen (1978), where a single interpretation is attributed to all simple sentences with reciprocal expressions. This interpretation is claimed to be compositionally derived from the interpretation of similar sentences that contain plurals instead of reciprocal expressions. The next work we introduce is Heim et al. (1991). In this work reciprocals are analyzed using syntactic mechanisms to explain some anaphora and scope puzzles with reciprocals. This work assumes a single straight-forward interpretation for reciprocal expressions. A different course of investigation is undertaken in Dalrymple et al. (1998) (henceforth DKKMP). DKKMP observe the rich diversity of interpretations of reciprocal expressions in different contexts and propose a system based on a principle they term the Strongest Meaning Hypothesis (SMH) to account for the various interpretations. In this system the interpretation of the reciprocal expression in a specific context is chosen out of a set of available meanings. DKKMP s system is specific for reciprocal expressions and has no immediate bearing on the semantics of plurals. Beck (2001) attempts to combine the insights of Heim et al. (1991) and Langendoen (1978) to show that the multiple interpretations of reciprocal expressions follow from general mechanisms of plural predication. This work derives also from Sternefeld (1998), who proposes a formal account for the parallels between plurals and reciprocals that was observed by Langendoen. Beck leaves room for the SMH principle suggested by Dalrymple et al. as a choice mechanism over the set of interpretations derived in her system. We summarize this chapter by presenting the limitations of the above proposals, and laying out the points our work proposes to improve upon these works. 11

21 2.1 Langendoen (1978) Langendoen (1978) investigates elementary reciprocal sentences (ERSs) of the form in (3), in which A denotes a set of at least two individuals, R is a binary relation on A A, and r is a reciprocal expression such as each other or one another. (3) A R r Langendoen analyzes a reciprocal expression as a lexical and semantic unit that has one inherent interpretation that is manifested in all ERSs. Out of the interpretations that had been suggested in the literature for reciprocal sentences, Langendoen concludes that the correct interpretation of reciprocal sentences is Weak Reciprocity (WR), defined by: x A[ y A [y x R(x, y)] y A [y x R(y, x)]]] In words, WR requires that each member of the set A participate in the relation R both as the first and as the second argument. WR is justified empirically by Langendoen by eliminating other possible candidates for truth conditions of ERSs. For example, consider sentence (4): (4) They scratched one another s backs. The truth conditions of this sentence are weaker than those required by Symmetric Reciprocity (SmR), defined as follows: x A [ y A [y x R(x, y) R(y, x)]] Therefore Langendoen concludes that SmR cannot be the correct interpretation. The interpretation of reciprocal expressions as WR is claimed to be consistent in all but special cases of temporal and spatial predicates, as in the following sentence: (5) The plates are stacked on top of each other. Langendoen attributes the inconsistency of this sentence and others of this sort with WR to extra-grammatical reasons that are not explored in full in his work. Langendoen further justifies the choice of WR on theoretical grounds, by showing that the truth conditions of simple reciprocal sentences can be derived from the truth conditions of elementary plural relational sentences (EPRSs), of the form in (6), in which A and B denote sets of at least two individuals, and R is a binary relation on A B: 12

22 (6) A R B This form is exemplified in the following sentence: (7) The women released the prisoners. The truth conditions of EPRSs are claimed to match the following formula: (8) x A [ y B [R(x, y)]] y B [ x A [R(x, y)]] WR as the interpretation of ERSs can then be arrived at by replacing B with A, and adding the condition that x and y be distinct elements of A. The exact process by which the interpretation of ERSs is derived from the interpretation of EPRSs is, however, left unspecified. Langendoen does not account for the observed variation in the interpretation of reciprocal sentences, and specifically does not account for the fact that WR is weaker than the actual truth conditions of many sentences. One example is sentence (1) in chapter 1, which requires that each of the named individuals saw each of the other ones, a requirement that is not captured by WR. 2.2 Heim et al. (1991) Heim et al. (1991) (henceforth HLM) focus on anaphora and scope phenomena of reciprocal sentences, and propose a representation of the syntax and semantics of reciprocal expressions at Logical Form (LF) that explains those phenomena. Their analysis of reciprocal expressions is built upon a more general claim they make as to the representation of plurals at LF. HLM s work focuses on cases where the subject NP denotes exactly two individuals, hence they do not address the variety of meanings that reciprocal sentences exhibit when the antecedent set of the reciprocal expression denotes larger groups. Even for a simple reciprocal sentence with two individuals in the antecedent set, they assume it requires that each of the individuals is in the relation with the other one, ignoring cases that exhibit a weaker interpretation, such as the following example, in which the relation must hold only in one direction: (9) The two beds are stacked on top of each other. The sort of puzzles that are addressed by HLM s theory are represented in the following two sentences and the accompanying possible paraphrases of each: 13

23 (10) John and Mary told each other that they should leave. a. John told Mary that he should leave and Mary told John that she should leave. b. John told Mary that she should leave and Mary told John that he should leave. c. John told Mary, and Mary told John, We should leave. (11) John and Mary think they like each other. a. John and Mary think they (that is, John and Mary) like each other. b. John thinks that he likes Mary and Mary thinks that she likes John. In sentence (10), the different readings are attributed by HLM to different types of anaphoric bindings, not allowed in classical LF. In sentence (11) the ambiguity is attributed to narrow versus wide scope of the reciprocal. HLM assume that the semantic properties of reciprocal expressions such as each other and one another arise from the compositional interactions of the meanings that their constituent parts have in isolation. Thus, they decompose each other into the two constituents each and other. The resulting LF representation is exemplified in the following analysis of a simple reciprocal sentence: (12) The men saw each other. (13) [ S [ NP [ NP the men 1 ] each 2 ][ S e 2 [ V P [ NP e 2 other ] 3 [ V P saw e 3 ]]]] The constituent each is analyzed as an adnominal universal quantifier and assumed to undergo movement in the mapping onto its LF representation, to adjoin to the antecedent plural NP. Quantifier Raising operates on the subject and object NPs to yield the embedded S clause in (13). Adnominal each distributes over the atomic individuals that the plural NP denotes. The expression other turns the matrix VP into a reciprocated predicate, one that only considers relations between non-identical individuals. Importantly, the structure of the plural NP with its distributor each reflects according to HLM a more general property of the representation of plurals at LF. A covert distribution operator with similar semantics as those of the overt floated each in (13) is assumed by HLM, as well as in other works. However, HLM revise the indexing mechanism of plural NPs when a distributor is applied, such that both the original matrix NP and the distributed NP bear indices. 14

24 This allows different indexing choices at LF for sentence (10), each choice corresponding to one of the different paraphrases listed in (10): (14) [John and Mary 1 each 2 ] told [e 2 other] 3 that they 2 should leave (15) [John and Mary 1 each 2 ] told [e 2 other] 3 that they 3 should leave (16) [John and Mary 1 each 2 ] told [e 2 other] 3 that they 1 should leave The second puzzle, exemplified in sentence (11), is explained as a scope phenomenon with the following two possible LF forms: (17) [John and Mary 1 D] think [[they 1 each 2 ] like [e 2 other] 3 ] (18) [John and Mary 1 each 2 ] think [they 2 like [e 2 other] 3 ] In HLM s formulation of the possible LF representations of reciprocal sentences, one can differenciate between bound-variable anaphora and coreference anaphora. For example, sentence (17) manifests coreference anaphora, since the antecedent of the anaphora does not c-command the pronoun. In contrast, sentence (18) employs a bound-variable anaphora. This allows the use of binding theory to predict the possible meanings of such sentences. 2.3 Dalrymple et al. (1998) Dalrymple et al. (1998) address the problem of variation in the interpretation of reciprocal sentences when the antecedent set may contain more than two individuals. They contend that the reciprocal expression is not ambiguous, but has a context-sensitive interpretation. DKKMP analyze the possible contributions of the reciprocal expression, viewing it as an indivisible syntactic unit. They propose a parametrization of the range of possible meanings of the reciprocal expression, and a principle that predicts which of these meanings is selected in a given context. DKKMP show that the context of the reciprocal expression affects its interpretation by allowing weaker interpretations of the reciprocal expression when stronger interpretations are precluded by world knowledge. This is exemplified in the interpretation of the following sentences: (19) These three people like each other. (20) The three planks are stacked on top of each other. (21) The 3rd grade students gave each other measles. 15

25 Sentence (19) entails that each person in the group likes every other person in the group, while sentences (20) and (21), adapted from DKKMP, do not entail an analogous claim: Sentence (20) does not entail that each plank is stacked on top of each of the other planks, a physically impossible arrangement. Similarly, sentence (21) does not entail that each student gave each of the other students measles, again an impossible scenario, since one can only get measles once. In both these sentences, the actual interpretation is weaker: Sentence (20) only requires that the planks be arranged in one stack, such that each plank except for the lower one is stacked on top of one of the other planks. Sentence (21) is evaluated as true if each of the students, except for the one who had measles first, got measles from another one of the students. This observation about contextually-licensed weakening of the interpretation of the reciprocal is formulated in a new principle proposed in DKKMP, termed the Strongest Meaning Hypothesis (SMH): Definition 1 (Strongest Meaning Hypothesis) A reciprocal sentence S can be used felicitously in a context c, which supplies non-linguistic information I relevant to the reciprocal s interpretation, provided the set L c below has a member that entails every other one: L c = {p p is consistent with I and p is an interpretation of S obtained by interpreting the reciprocal as one of the six available quantifiers described below} In that case, the use of S in c expresses the logically strongest proposition in L c. The available six quantifiers referred to in the SMH are the ones generated using the following two parameters: 1. How the scope relation R should cover the domain A that the antecedent set denotes. The following three options are available: (a) Each pair of non-identical individuals in A is required to participate in the relation R directly (FUL); or (b) Each pair of non-identical individuals in A is required to participate in the relation R either directly or indirectly (LIN); or (c) Each single individual in A is required to participate in the relation R with another one (TOT). 16

26 Note that in all of the options, it does not matter whether any pairs of identical individuals exist in R. Only pairs of non-identical individuals affect the evaluation of the quantifier. 2. How the reciprocal s scope determines the argument R of the quantifier chosen by the first parameter. There are two possibilities for this parameter: (a) The argument of the quantifier is the relation denoted by the predicate in the reciprocal expression s scope; or (b) The argument of the quantifier is obtained from the relation denoted by the predicate in the reciprocal expression s scope by ignoring the direction of the relation. Thus, if R is the relation denoted by the predicate in the reciprocal expression s scope, the argument of the quantifier is R, the symmetric closure of R. According to the SMH, the meaning of a reciprocal expression in a given context is the strongest available meaning that is consistent with the relevant contextual information. The six quantifiers generated by this system are the readings DKKMP assume for reciprocal expressions. For example, choosing FUL to operate on R, we get Strong Reciprocity (SR) (Fiengo & Lasnik, 1973), which requires that each pair of non-identical individuals in A be in R. Choosing TOT to operate on R we get Inclusive Alternative Ordering (IAO) (Kański, 1987), which requires that each individual in A be in the relation with at least one other individual in A as either the first or the second argument. Using the SMH and the list of available readings, and given the relevant contextual information, DKKMP s system generates a prediction of the interpretation of a specific reciprocal sentence. For example, sentence (20) is inconsistent with SR. It is, however, consistent with a weaker reading generated in this system by choosing LIN to operate on R. This reading is termed by DKKMP Intermediate Alternative Reciprocity (IAR). IAR requires that there be a path between each pair of non-identical individuals in A, in the undirected graph induced by R on A. Out of the six available quantifiers, IAR is the strongest reading that is consistent with (20). The system therefore predicts that the truth conditions of (20) match IAR. This is in fact consistent with the requirement for a single stack of planks. IAR is also the chosen quantifier for sentence (21). In some cases, the prediction produced by DKKMP s system is incorrect. Consider the following sentence from J.M. Barrie s Peter Pan, given in DKKMP. 17

27 (22) The captain! said the pirates, staring at each other in surprise. DKKMP s system expects this sentence to be consistent with Intermediate Reciprocity (IR), which results from LIN operating on R. IR requires that all the individuals be connected via the transitive closure of the relation R. This is possible in the case of (22), since it could be realized if the pirates stood in a circle and each pirate would stare at the pirate to his right. However, as DKKMP observe, the actual truth conditions of this sentence match the weaker One-way Weak Reciprocity (OWR), which only requires that each pirate stares at some other pirate. OWR is created in DKKMP s system by choosing TOT to operate on R. The possible choices for quantifier parameters that DKKMP propose, also generate a quantifier that was not found in any actual sentences by DKKMP or by others: This is the quantifier they term Strong Alternative Reciprocity (SAR). It is derived in DKKMP s system by choosing FUL to operate on R. SAR requires that each pair of non-identical individuals in the antecedent set participate in the relation in at least one of the two possible directions of the relation. DKKMP speculate that evidence might still be found to support the truth conditions generated by SAR. DKKMP s idea of a mechanism of choice that is guided by contextual factors is an idea that the present work aims to develop and expand. 2.4 Beck (2001) Beck (2001) develops the ideas of the compositional analysis proposed by HLM to show how the variety of meanings of reciprocal sentences, which are untreated by HLM, can be derived using known properties of plurals. Beck combines HLM s work with that of Sternefeld (1998). Sternefeld attempts to fill in a formal gap in the theory of Langendoen (1978), by suggesting a formal theory to derive the truth conditions assumed by Langendoen (1978) for Elementary Plural Reciprocal Sentences from those of Elementary Relational Sentences (see section 2.1). This derivation is done at LF using the known plurality operators and, which are defined below. Sternefeld does not address the variation in the interpretations of reciprocal sentences. As in HLM and Sternefeld (1998), Beck assumes that the semantics of the reciprocal expression interacts compositionally at LF with known operators on plurals and with contextual factors. Almost all the variation in reciprocal meaning is claimed to be governed by these factors. Unlike HLM, Beck does not associate either the antecedent set or the reciprocal expression with quantificational force. Instead the reciprocal is taken to be a group-denoting 18

28 individual: given a specific individual from the antecedent set, the denotation of the reciprocal expression is assumed to be the antecedent set minus the given individual. Beck uses previously suggested mechanisms that operate on sentences that predicate over plurals to derive the required readings. The first is the distribution operator suggested in Link (1983). This operator performs a function similar to the distributor assumed by HLM (see section 2.2), however it operates on the predicate, and not on the argument of the predicate. This operator is applied to a predicate P to generate a new predicate P such that the following holds: P(x) P(x) u, v[(x = u v) F(u) F(v)] The second mechanism is a pluralization operator that is meant to capture cumulative readings. This operator was suggested by Sternefeld (1998) following Krifka (1986), to explain the truth conditions in (8), assumed by Langendoen (1978) for elementary plural relational sentences (see section 2.1). This operator is applied to a binary relation R to generate a new relation R that is defined as follows: R(x, y) R(x, y) x 1, x 2, y 1, y 2 [(x = x 1 x 2 ) (y = y 1 y 2 ) R(x 1, y 1 ) R(x 2, y 2 )] The third mechanism is the cover mechanism proposed by Schwarzschild (1996). The cover mechanism allows groups denoted by plural NPs to distribute into contextually salient subgroups. For instance, this mechanism allows the following sentence to be understood to mean that the cows were separated from the pigs: (23) The cows and the pigs were separated. This is obtained by assuming there is a free variable Cov that represents the cover: its value is a set of subsets of the domain of entities, that cover the entire domain. The subsets in Cov are contextually salient sets. In the case of sentence (23), the cover is assumed to include a set that denotes the cows and a set that denotes the pigs. Strong Reciprocity is derived in Beck s work by simply applying the distribution operator to the assumed denotation of the reciprocal. Beck contends that the only other independent readings that should be derived are Weak Reciprocity (see the definition in 2.1), and Partitioned Strong Reciprocity (PSR), which requires that Strong Reciprocity hold within subsets that partition the antecedent set. PSR is needed in Beck s system to explain sentences such as the following from DKKMP: 19

29 (24) To muddy the ballot waters further, at least four sets of propositions compete with each other. According to DKKMP, from the context of this sentence it is clear that each of the members in a specific set of propositions competes with each of the other members in that set. PSR is derived in Beck s system by combining the derivation of Strong Reciprocity with distribution of the antecedent set into subgroups using the cover mechanism. Weak Reciprocity is derived using the operator on the predicate in the scope of the reciprocal expression. Let us exemplify the derivation on the following sentence: (25) The children like each other. Assume this sentence has the truth conditions implied by WR, so that it entails that each child likes at least one other child and that each child is liked by at least one other child. These truth conditions are derived by assuming the following LF form of the sentence: (26) [Pro 1 [[ the children ] 3 [ [1 [ Cov[2 [t 2 [ Cov[ like [max[ [[ other x 1 ] (of) t 2 ]]]]]]]]]]]] This form is reached by applying Quantifier Raising to both the subject and to the pronoun other, and applying the cover and the distribution and cumulation operators and. Assuming the cover includes only individuals, the following logical formula is the translation of this LF form: (27) A, A λyλx[r(x, x)] In this formula A denotes the antecedent set The children and R denotes the predicate like. The argument of operator is a presupposition, rather than an assertion. The importance of this distinction is a delicate matter we shall not elaborate upon here. This logical formula means that the set of children is in the cumulated relation R with itself. By definition of the operator, this is equivalent to the truth conditions of WR. Beck argues that other readings suggested by DKKMP are unnecessary as independent readings. Instead they are explained as special cases of the readings that are derived in Beck s system. For example, DKKMP assume that the sentence below is an example for Intermediate Reciprocity (IR), one of the quantifiers available in their system (see section 2.3). (28) The phone poles are spaced 500 feet from each other. 20

30 In contrast, Beck analyzes this sentence as a special case of Strong Reciprocity in which contextually salient distribution causes reciprocity to be required only between adjacent poles: Applying the cover operator to the subject the phone poles distributes the set of poles into pairs of adjacent poles, such that SR holds only within each pair. Beck claims that in general IR is not an independent reading and that all sentences that appear to exhibit it can be analyzed in a similar manner. She also explains away OWR, assumed by DKKMP to be an independent reading, as a special case of WR with exceptions. For example, sentence (22) from DKKMP is analyzed as exhibiting WR, with exceptions that allow some of the pirates not to be stared at by other pirates. For the reading Inclusive Alternative Ordering that is also part of DKKMP s system (see section 2.3), Beck claims that this interpretation only arises with specific predicates (usually spatial or temporal). Beck speculates that it is the result of a different lexical process than ordinary reciprocity. Beck s system generates a set of readings that is different from the set assumed by DKKMP. Each of these readings may be derived compositionally in any context. Beck assumes that DKKMP s SMH is then applied as an orthogonal principle that chooses among these readings in each specific context. 2.5 Assessment The analyses of reciprocals proposed in Langendoen (1978), Sternefeld (1998) and Heim et al. (1991) advance our understanding as to the nature of reciprocal expressions and their interaction with the semantics of plurals in general and with scope and anaphora effects, but these works do not account for the wide variation in the interpretation of the reciprocals in different contexts. Beck (2001) does attempt to account for this variation, but the analyses proposed in her work for different cases are ad-hoc and rely on complicated combinations of many operators, arranged independently for each case. We aim for a simpler system that can directly predict the truth conditions of a reciprocal sentence, based on as little external information as possible. The SMH principle proposed by DKKMP is promising and shows good predictions in several cases. In our work we develop and expand the work done in DKKMP. The system proposed by DKKMP is composed of two components: One is the SMH, and the other is the set of readings the SMH chooses from. We propose a new system for predicting the interpretation of a reciprocal expression in a given setting. We base our system on the critical observation made in DKKMP about the weakening of the interpretation of a 21

31 reciprocal sentence by contextual factors. However, unlike DKKMP, we implement the SMH as a mapping from semantic restrictions on the predicate s denotation into the interpretation of the reciprocal, with no independent assumptions about available reciprocal meanings. In addition to producing a simpler system that employs fewer assumptions, this new implementation defines the loose notion of relevant contextual information used by DKKMP, and identifies this information solely with a formal concept of semantic restrictions of predicates. We note that along the lines of Beck s remark, the SMH is found most easily with spatial predicates such as sit alongside and stand on top, and with temporal predicates such as temporal follow. Very often the SMH is not manifested in kinship relations as well as some other types of relations. The following contrast demonstrates this: (29) The two chairs are stacked on top of each other. (30) During the past year, historic events followed each other non-stop. (31) #Ruth and Beth are each other s mother. A weakening effect allows sentence (29) and sentence (30) to be felicitous, but a similar effect does not occur in sentence (31), although world knowledge precludes both two-way stacking and two-way mothering. Our system allows describing this phenomenon formally within the system, using the notion of semantic restrictions. 22

32 Chapter 3 Semantic Restrictions and Reciprocal Meanings The quantifiers chosen by the SMH are picked out of a list of given quantifiers that are consistent with the context. We propose a new system for predicting the truth conditions of a simple reciprocal sentence. In the proposed system, no specific quantifier is assumed a priori for reciprocals. To achieve this purpose, we first propose formal definitions for concepts that pertain to the interpretation of reciprocal sentences, and relate them to concepts in previous works. DKKMP propose that the SMH uses relevant information that guides the choice of the quantifier that matches the reciprocal sentence in a specific context. In this chapter we define the notion of semantic restriction and show its relevance in delimiting the range of interpretations available for a reciprocal in a given sentence, instead of the relevant information of DKKMP. Then we define the notion of reciprocal meaning as a set of quantifiers of the appropriate type, with natural restrictions from generalized quantifier theory. We subsequently show that for every reciprocal interpretation there is exactly one minimal reciprocal meaning that extends it, thereby proposing a method for attesting reciprocal meanings using natural language sentences. The implications of this method for the study of meaning variation with reciprocals are studied in the next chapters. We will denote the general form of simple reciprocal sentences by NP P each other, where NP is the subject noun phrase that denotes the antecedent set of the reciprocal expression, and P is the binary predicate in the scope of the reciprocal. The set of entities denoted by the NP in a specific model will be marked by A, and the binary relation over entities denoted by the predicate P in a specific model will be marked by R. 23

33 3.1 Semantic Restrictions We first take a closer look at the informal concept of relevant information which is used by DKKMP in their formulation of the SMH. Clearly, not all kinds of contextual information allow weakening of the reciprocal meaning. Otherwise, according to the SMH as formulated by DKKMP, the two sentences in (32) below would not be contradictory, since the information given in the first sentence would cause the reciprocal in the second sentence to require weaker truth conditions. (32) # John and Bill don t know each other. John, Bill and Dan know each other. As a minimal notion of relevant information, we propose to only consider semantic restrictions of the binary predicate in the scope of the reciprocal, along the lines of Winter (2001). A semantic restriction of a binary predicate P over the domain of entities E is a set Θ P of binary relations over E: Θ P (E 2 ). This is the set of relations that are possible as denotations of the predicate. For example, the denotation of the predicate stare at is limited to relations that are also (possibly partial) functions, since one cannot stare at more than one person at a time. Therefore Θ stare at is the set of binary relations over E that are (possibly partial) functions. We know of very few works that attempt to formulate constraints on the available semantic restrictions on natural language predicates. One such work is Rubinstein (1996), which ventures to explain why some restrictions such as symmetry and transitivity are common in natural language predicates while others are not, based on optimality principles. In the present work we only touch this topic as far as it relates to our work on reciprocals. In a simple reciprocal sentence of the form NP P each other, we assume that NP denotes a set of entities and P denotes a binary relation over entities. The denotation of the reciprocal expression each other is accordingly assumed to be a relation between sets of entities and binary relations. Note that its denotation in a given sentence cannot be determined for binary relations outside the semantic restriction of P: Suppose R is such a relation, R / Θ P. Then P cannot denote R in any real world model, therefore we cannot empirically tell what the truth value of the sentence would be if P denoted R. Therefore, we define the interpretation of the reciprocal expression only for relations in the semantic restriction of the predicate. Given a semantic restriction Θ, the interpretation of a reciprocal expression relative to Θ is a binary relation I Θ (E) Θ. The reciprocal interpretation domain of Θ, 24

34 denoted recip Θ, is the set of all possible reciprocal interpretations relative def to Θ: recip Θ = ( (E) Θ). As we exemplified in section 2.5, a weakening of the truth conditions does not occur for predicates such as kinship relations. We therefore conjecture that semantic restrictions are not always an exact representation of world knowledge, and are more refined for some classes of predicates than for others. Further research is required to understand the reasons for this differentiation between classes of predicates. 3.2 Reciprocal Meanings The interpretation of a reciprocal expression relative to a semantic restriction, as defined above, is a novel notion and central to our analysis of reciprocals in general. This notion allows us to investigate and to formulate the effect of a semantic restriction on the truth conditions of a sentence with a reciprocal expression. Unlike reciprocal meanings, the reciprocal interpretation of a given sentence is absolutely determined by the truth conditions of that sentence. The discussion of the semantics of reciprocals in the literature mostly concerns their meanings, which, in contrast with reciprocal interpretations, are defined for all binary relations and not only for relations in a given semantic restriction. As a preliminary to our analysis of the meanings available for reciprocals, we propose a formal definition of the notion of reciprocal meaning. The definition suggests semantic properties that a reciprocal meaning must have, partly adapted from known constrains in generalized quantifier theory (see (Keenan & Westerståhl, 1996)). Unlike DKKMP, our notion of reciprocal meaning is only meant to delimit the range of a priori possible meanings, with no assumptions about which of these meanings are actually manifested in natural language. A reciprocal meaning is a relation Π (E) (E 2 ). Thus, reciprocal meanings are all in the domain recip Θ with Θ = (E 2 ). We assume that a reciprocal meaning must be conservative on its first argument, as expected of any natural language determiner (Keenan & Westerståhl, 1996). Formally: A E, R E 2 [ Π(A, R) Π(A, R A 2 ) ] The conservativity condition is illustrated in the validity of the following entailment: (33) If every blik plips every blik, then The bliks birr each other The bliks birr and plip each other. 25

35 Furthermore, reciprocal meanings are never sensitive to relations between identical pairs. Formally: A E, R E 2 [Π(A, R) Π(A, R \ I)] Where I is the identity relation: I def = {(x, x) x E}. Insensitivity to relations between identical pairs is illustrated in the validity of the following entailment: (34) If every blik plips only itself, then The bliks birr each other The bliks birr and don t plip each other. In addition, all reciprocal meanings suggested so far in the literature are upward monotonic in the second argument, and we expect this to be true in general. Formally: R, R E 2 [(Π(A, R) R R ) Π(A, R )] Upward monotonicity in the second argument is illustrated in the validity of the following entailment: (35) The bliks birr each other. Birring is a type of pliping. = The bliks plip each other. These three properties are all subsumed by the following single property of argument monotonicity: Definition 2 A binary relation D (E) β, where β (E 2 ), is argumentmonotonic if and only if the following holds: A E [ R, R β [(D(A, R) R A R ) D(A, R )]] Where R A R is defined as R A \ I R A \ I. Argument monotonicity is assumed to be the underlying property of reciprocal meanings, with β = (E 2 ). Thus, we henceforth refer to a relation Π (E) (E 2 ) as a reciprocal meaning if Π is argument-monotonic. We furthermore assume that reciprocal interpretations are also argument monotonic. In the case of reciprocal interpretations, β = Θ where Θ is a semantic restriction of a natural language predicate. To illustrate the above properties, let us give some examples for artificial binary relations that are not argument-monotonic, and thereby not considered reciprocal meanings. 26

36 Example 1 Let D 1 (E) (E 2 ) be defined as follows: A E [ R E 2 [(D 1 (A, R) R \ I 5] ] In words, D 1 holds for A and R if there are at least 5 non-identical pairs in R. D 1 is consistent with two of the properties we listed above: It is not sensitive to identical pairs in R, and it is upward monotonic in the second argument. However, it is not conservative, since it considers all pairs in R and not only pairs in A. In fact, there is no dependence on A at all in the condition that defines D 1. The conservativity property precludes relations such as D 1 from being considered as reciprocal meanings. Example 2 Let D 2 (E) (E 2 ) be defined as follows: A E [ R E 2 [ (D 2 (A, R) R = A 2]] In words, D 2 holds for A and R only if all the pairs of individuals in A are in R. D 2 is conservative and upward monotonic, but it is sensitive to identical pairs in R. As has been observed before (see for example Langendoen (1978)), a sentence with reciprocal expression never entails any claim about the relation between an individual and itself. For example, the following simple sentence does not entail anything about whether Dan or Ruth like themselves: (36) Dan and Ruth like each other. Therefore the property of insensitivity to identical pairs disqualifies D 2 from being considered a reciprocal meaning. Example 3 Let D 3 (E) (E 2 ) be defined as follows: A E [ R E 2 [(D 3 (A, R) R A \ I 3] ] In words, D 3 holds for A and R if there are no more than 3 non-identical pairs in R over A. D 3 is conservative, and it is not sensitive to identical pairs in R. However, it is not upward monotonic in the second argument: D 3 may hold for some A and R but not for A with some R R. Reciprocal expressions are always upward monotonic in the second argument, as the following entailment, that rests on the fact that the denotation of hit is a subset of the denotation of touch, exemplifies: (37) The snooker balls hit each other. The snooker balls touched each other. We therefore preclude relations such as D 3 from being considered as reciprocal meanings. 27

37 3.3 When is a Reciprocal Meaning Attested? When presented with a potential reciprocal meaning, we would like to find out in which settings we can test whether this meaning is indeed available. In other words: what semantic restrictions of binary predicates would allow us to attest a given reciprocal meaning? Due to the semantic restrictions on the denotation of two-place predicates in natural language, we cannot always directly extract a meaning for a reciprocal expression using the truth-conditions of reciprocal sentences. Consider for instance the following sentence: (38) Proposals 1 through n are similar to each other. Given that the predicate be similar is symmetric, the interpretation of the reciprocal in (38) is in recip SY M where SY M is defined by: SY M = {R E 2 x, y E[R(x, y) R(y, x)]} Since (38) is true only if every proposal is similar to every other proposal, the interpretation of each other in (38) is the relation ISY 0 M recip SY M defined by: ISY 0 def M = {(A,R) (E) SY M x,y A[x y R(x,y)]}. This interpretation is consistent with at least two reciprocal meanings proposed in the literature: Both Strong Reciprocity (SR) and Strong Alternative Reciprocity (SAR) match. However, the discussion of reciprocals in the literature often strives to decide on available reciprocal meanings using such examples, where more than one meaning is consistent with the interpretation of the reciprocal. It is reasonable to assume that in such situations, the reciprocal interpretation realizes the strongest meaning that is consistent with the sentence. For instance in (38) there is little reason to assume that the meaning of the reciprocal is anything but Strong Reciprocity. We therefore suggest that the meaning of the reciprocal in a given sentence is the strongest possible reciprocal meaning that coincides with the reciprocal interpretation of this sentence. To be able to formally define this rule, we define a notion of congruence between a reciprocal meaning and a reciprocal interpretation I Θ recip Θ, for a semantic restriction Θ: 28

38 Definition 3 Let Θ be a semantic restriction over E. A reciprocal meaning Π over E is congruent with a reciprocal interpretation I Θ recip Θ if Π is a minimal reciprocal meaning that extends I Θ. Formally, Π satisfies: 1. A E, R Θ [I Θ (A, R) Π(A, R)], and 2. Any reciprocal meaning Π that satisfies 1, also satisfies Π Π. According to the following two propositions, if I Θ is argument-monotonic, it is congruent with exactly one reciprocal meaning. Our rule for matching a reciprocal meaning to a given reciprocal interpretation is therefore to choose the unique meaning that is congruent with the interpretation of the reciprocal expression. Proposition 1 For every semantic restriction Θ over E and a reciprocal interpretation I Θ recip Θ, there is at most one reciprocal meaning Π over E that is congruent with I Θ. Proposition 2 For every semantic restriction Θ over E and an argumentmonotonic reciprocal interpretation I Θ recip Θ, there exists a reciprocal meaning Π over E that is congruent with I Θ. The proofs of these propositions can be found in section 3.4 Returning to our example above of possible reciprocal meanings for sentence (38), by our definition of congruence, SR is congruent with I 0 SY M while SAR is not. Therefore (38) cannot be considered evidence for the existence of SAR. More generally, we assume that any meaning associated with reciprocals should be congruent with the interpretation of the reciprocal in at least one natural language sentence. In this case we can say that a sentence attests the meaning in question. By Propositions 1 and 2, for any semantic restriction Θ over E and an argument-monotonic reciprocal interpretation I Θ recip Θ, there is a unique reciprocal meaning that is congruent with I Θ. On the empirical side, this result means that when given a sentence with a reciprocal expression, when Θ is the semantic restriction of the predicate in the sentence, the important semantic decision concerns the interpretation of the reciprocal chosen from the domain recip Θ. In other words, we are looking for a mapping from the semantic restriction of a predicate to the truth conditions of a reciprocal sentence with that predicate. In the next chapter we propose a new realization of the SMH that functions as exactly this needed mapping. 29

39 3.4 Proofs of Propositions in this Chapter In this section we supply the proofs for Propositions 1 and 2 above. Proposition 1 For every semantic restriction Θ over E and a reciprocal interpretation I Θ recip Θ, there is at most one reciprocal meaning Π over E that is congruent with I Θ. Proof: Assume for contradiction that there are two different reciprocal meanings Π 1 and Π 2 such that Π 1 and Π 2 are both congruent with I Θ. Then the relation Π 3, defined by Π 3 def = Π 1 Π 2 is also a reciprocal meaning. It extends I Θ, and it is stronger than at least one of Π 1 and Π 2. Therefore at least one of Π 1 and Π 2 is not congruent with I Θ, a contradiction. Proposition 2 For every semantic restriction Θ over E and an argumentmonotonic reciprocal interpretation I Θ recip Θ, there exists a reciprocal meaning Π over E that is congruent with I Θ. Proof: Let Ω be the set of reciprocal meanings that extend I Θ. First, we show that Ω : Let Π (E) (E 2 ) be the relation such that A E, R E 2 [Π(A, R) S I Θ (A) [S A R]] Π is a clearly argument monotonic, and is therefore a reciprocal meaning. Π also extends I Θ : A E, R Θ [I Θ (A, R) Π(A, R)]. The left-to-right implication trivially follows from the definition of Π, and the right-to-left implication follows from the definition of Π and the argument-monotonicity of I Θ. Hence Ω. Let Π (E) (E 2 ) be the relation defined by: A E, R E 2 [Π (A, R) Π Ω[Π(A, R)]] Π is argument monotonic, therefore it is a reciprocal meaning. By the definition of Π, there is no reciprocal meaning stronger than Π that extends I Θ. Therefore Π is congruent with I Θ. 30

40 Chapter 4 The SMH-Based Interpretation of Reciprocals In the previous chapter we introduced a formal framework for analyzing the interpretation of reciprocal sentences. In this chapter we proceed to propose a new realization of the SMH within this framework. This new realization disposes with the need for an independent set of reciprocal meanings to choose from as in DKKMP s system. Instead the interpretation of the reciprocal sentence is directly derived from the semantic restriction of the predicate in the scope of the reciprocal. 4.1 A New Realization of the SMH We propose that the SMH is realized as a local maximality principle: a reciprocal sentence is consistent with models in which no pairs of non-identical individuals in the antecedent set can be added to the denotation of the predicate within its semantic restriction. Formally: Definition 4 Let Θ be a semantic restriction over E. The SMH-based interpretation of the reciprocal is the relation R Θ recip Θ, defined as follows: A E, R Θ [R Θ (A, R) R Θ [(R A R ) (R = A R )]] This definition of the SMH leads to correct interpretations in many cases, as the following sections show. Other reformulations of the SMH as a maximality principle have been suggested in Gardent & Konrad (2000) and in Winter (2001). Unlike the current proposal, these works assume a principle of global maximality, in which the SMH maximizes the number of pairs in the 31

41 (a) Brick Tower (b) Brick Pyramid Figure 4.1: Two settings of bricks Ñ Ô Ð ÐÝ Ñ Ü Õ ÔÝ relation. We contend that local maximality is more likely both because of cognitive computation considerations and because of empirical reasons. From the cognitive computation aspect, computing a global maximum is harder than computing a local maximum: To compute whether a specific relation is a global maximum, all other possible configurations must be considered, while to compute whether a relation is a local maximum it is enough to consider configurations that result from adding pairs to the examined relation. From the empirical aspect, consider the following example: (39) The bricks are laid on top of each other. This sentence is evaluated as true both in figure 4.1(a) and in figure 4.1(b). In the former figure there are five pairs in the relation denoted by laid on top, while in the latter figure there are six pairs in the relation. Each of this figures depicts a locally maximal relation, but figure 4.1(a) is not globally maximal. This example supports a preference to the local maximality principle over the global maximality principle. The example also illustrates the larger cognitive effort involved in identifying global maximality: To identify that figure 4.1(a) is not globally maximal, one has to realize that the configuration in figure 4.1(b) is also possible, while finding out whether it is locally maximal or not is much easier: it consists of realizing that no additional pair can be added to the laid on top relation in this specific configuration. From Definition 4, it is clear that R Θ is argument-monotonic for any semantic restriction Θ. Therefore, by Propositions 1 and 2, for each semantic restriction Θ there is exactly one reciprocal meaning congruent with R Θ. 32

42 In the following sections we present some predictions that arise from our system using this definition of the SMH-based interpretation of reciprocal expressions. 4.2 Applying the SMH-based Interpretation Our definition of the SMH-based interpretation of reciprocal expressions allows a correct prediction of the meaning of sentences presented in DKKMP and analyzed there using their system. Let us review examples (19)-(21) from section 2.3, repeated here as (40)-(42): (40) These three people like each other. (41) The three planks are stacked on top of each other. (42) The 3rd grade students gave each other measles. According to our system, the interpretation of the reciprocal in each sentence is determined by the semantic restrictions of the predicate. In sentence (40), the predicate like may denote any binary relation: Θ like = (E 2 ). Hence, R Θlike (A, R) R A 2 \ I, i.e. the sentence is deemed true only if each person in the antecedent set likes each of the others. This corresponds to Strong Reciprocity, the strongest meaning suggested for reciprocal expressions in the literature. The same interpretation is given for sentences with other predicates which also have no restrictions on the relations they may denote, such as know and see: (43) These children know each other. (44) The four contestants see each other. As we saw above, Strong Reciprocity also matches the interpretation of reciprocal sentences with other predicates that do have some restrictions on the relations they denote. This is exemplified in the following sentences: (45) The kittens born yesterday are similar to each other. (46) The citizens of a democratic state are equal to each other. The predicate be similar to is restricted to symmetric relations, and the predicate be equal to is furthermore restricted to transitive relations. For both predicates, the SMH-based interpretation of the reciprocal is consistent with Strong Reciprocity. 33

43 In sentence (41), we assume that the semantic restriction of the predicate stack on top is the set Θ stack on top that includes all the relations R E 2 such that R and R 1 are (possibly partial) functions, and R is acyclic. Consequently, R Θstack on top (A, R) holds if and only if the elements of A are arranged into one sequential stack, as expected. In sentence (42), the predicate give measles may only denote acyclic relations which are the inverse of a function: one cannot get measles twice, and measles can t be passed circularly among individuals. Using this semantic restriction, we find that the sentence is predicted to be true if and only if each 3rd grade student is connected to each other 3rd grade student by the transitive and symmetric closure of the denotation of give measles. These are in fact the expected truth conditions of this sentence. They match the reciprocal meaning Intermediate Alternative Reciprocity (IAR) suggested by DKKMP (see section 2.3). A different example can be found in the following sentence from Langendoen (1978), with the predicate follow: (47) The guests followed each other into the room. The semantic restrictions of follow can be assumed to be the same as those of stack on top above. Using this assumption, the sentence is predicted to be true if the guests walk single file, analogously to sentence (41). Another example from Langendoen (1978) is the following sentence: (48) The prisoners released each other. In both Langendoen (1978) and Beck (2001) it is claimed that for this sentence to be true, ignoring the possibility of group releasings, each prisoner must have released one other prisoner, and must have been released by anther prisoner. These are the requirements of Weak Reciprocity (see section 2.1). However, in any reasonable situation of a jail rebellion, at least one of the prisoners would have had to be the first to escape from his cell on his own, and at least one of the prisoners would have been the last to be released. In this situation the sentence would still be evaluated as true. Therefore we contend that the truth conditions of sentence (48) do not actually match WR. Instead, we assume that the semantic restriction of release is the same as the one for give measles, for analogous reasons. Under this assumption the SMH-based interpretation of the sentence again matches Intermediate Alternative Reciprocity as in sentence (42), which is consistent with the actual truth conditions of this sentence. Unlike DKKMP, this proposal also gives a correct prediction of the truth conditions for the sentence (22) from DKKMP, that is repeated here as (49). (49) The captain! said the pirates, staring at each other in surprise. 34

44 This sentence is predicted in DKKMP s system to have the truth conditions of IR, instead of the correct truth conditions, that match OWR (see section 2.3). In the present proposal, we derive the correct OWR interpretation of the reciprocal expression, assuming that Θ stare at is the set of (possibly partial) functions over E. This corresponds to assuming that each individual can start at no more than one individual at a time. The OWR interpretation then matches the local maximality principle, since if each pirate is starting at a single other pirate, no more pairs can be added to the stare at relation without contradicting the semantic restriction we assumed. 4.3 Strong Alternative Reciprocity In this section we use our system to analyze the reciprocal meaning Strong Alternative Ordering (SAR), suggested in DKKMP (see section 2.3). This meaning is formally generated by DKKMP s system, but no evidence for its existence is given in their work. DKKMP conjecture that evidence may be found in the future to empirically support SAR. In our system it is easy to see that SAR is congruent with R ΘASYM, where Θ ASYM is the set of all asymmetric relations. We have not found a natural language predicate that allows all and only asymmetric relations: many predicates can denote only asymmetric relations, but usually the relations are also restricted to be acyclic, as in the case of the predicate stack on top and similar spatial predicates. Why strictly asymmetric predicates are so hard to come by in natural language remains unexplained. Consider, however, predicates that denote only acyclic and transitive relations. An example for such a predicate is contain: (50) The boxes are contained in each other Sentence (50) is true if the boxes are arranged as in figure 4.2, but false if they are arranged as in figure 4.3. In general, the sentence requires that the boxes be sorted sequentially. We formally characterize this requirement with a new reciprocal meaning we term SEQ: Definition 5 For a domain E, a finite subset A E and a binary relation R over E, SEQ(A, R) holds iff the individuals in A can be indexed such that A = {x 1, x 2,...x n }, where A = n, such that for every x i, x j, if i < j then R(x i, x j ). The truth conditions of (50) are correctly predicted both by DKKMP s system and by the SMH-based interpretation we propose. In DKKMP, SAR 35

45 Figure 4.2: Boxes Þ ÕØ Û Figure 4.3: Other boxes Þ Ü Þ ÕØ Û is the strongest reading possible for this predicate, since Strong Reciprocity is impossible. SEQ is stronger than SAR, but given that be contained in is acyclic and transitive, the truth conditions described by SAR are equivalent to those described by SEQ: If R is a transitive and acyclic relation over A, it can be viewed as a partial order relation over A 1. If SAR(A, R) holds, then every two elements in A are comparable using R, hence R is a total order. This order can be used to index the elements of A, given that A is finite as we assume in the definition of SEQ. Therefore SEQ(A, R) holds. Consequently, DKKMP s system predicts the correct interpretation of sentence (50). In the system we propose SEQ is directly derived as the interpretation of (50) by assuming that the semantic restriction of be contained in is: Θ be contained in def = {R E 2 R is acyclic and transitive} 1 The definition of a partial order also requires reflexivity, but this requirement is immaterial in the case of reciprocal meanings, since all reciprocal meanings are insensitive to identical pairs. 36

46 Proposition 3 For a finite domain E, SEQ is congruent with R Θbe contained in. The proof of this proposition relies upon Lemma 4, that is presented in chapter 5. Proof: By Lemma 4 it suffices to prove that A E [R Θbe contained in (A) = min(seq(a))] We prove the set equality in two stages. 1. Suppose R Θbe contained in (A, R) holds for some R. Then R is acyclic and transitive, hence R I is a partial order. Suppose there exists a pair (x, y) A 2 \I such that (x, y) / R and (y, x) / R. Then the transitive closure of R I {(x, y)} is also a partial order. Hence the transitive closure of R {(x, y)} is acyclic and intransitive. Consequently, it is in Θ be contained in. This contradicts the maximality of R in Θ be contained in, therefore there can be no such pair. Hence R I is a total order. A is finite, hence this order can be used to index the elements in A, therefore SEQ(A, R) holds. In addition, R min(seq(a)), since any relation in SEQ(A) \ min(seq(a)) has at least one cycle, and therefore it is not a total order. 2. Suppose R min(seq(a)). Then R I is a total order, hence R is a maximal acyclic and transitive relation in Θ be contained in. Therefore R Θ be contained in. 4.4 Some More Complicated Cases We find that for some predicates the actual interpretation matches a different semantic restriction than the one we might expect. Sentence (51) is evaluated as true if the children form a straight line. (51) The children are sitting alongside each other. According to world knowledge, the semantic restriction of sit alongside should be the set of symmetric relations in which each individual is in the relation with at most two other individuals, since one cannot sit alongside more than two people. Combining our SMH-based interpretation of reciprocals with this semantic restriction would predict that the sentence cannot be true if the children form a straight line, since a straight line is not locally 37

47 maximal with respect to this semantic restriction: one pair may be added to a relation that matches a straight line, to form a circle. A possible remedy to the interpretation our formulation predicts would be to assume that despite world knowledge, the semantic restriction for predicates like sit alongside does require that the relations should depict an arrangement of the individuals in a straight line. Formally, the relations are then required to be acyclic 2. A similar phenomenon can be found in the following sentence: (52) The children are holding each other s hands. The semantic restrictions we would assume for holding each other s hands are the same as those for sit alongside, and here as well sentence (52) will be satisfied if the children stand in a line and hold hands. The explanation in this case could, however, be a different one, given the different structure of (52) with the plural object NP hands. Cases such as sentence (31) in section 3.1 where no weakening of the truth conditions seems to occur also suggest that the relation between world knowledge and semantic restrictions is not entirely straight-forward. Further research into this relation would benefit from having an independent method for finding the semantic restriction of a predicate in natural language. However, we currently do not know of such a method. Some predicates produce an interpretation that is hard to explain by maximality principles over any semantic restriction. For these sentences, the actual interpretation is weaker than the interpretation predicted by the local maximality principle. The following sentences exemplify this: (53) The cubes are glued to each other. (54) The chains are connected to each other. These sentences are interpreted as true if the cubes/chains all form one glued/connected construction, as required by Intermediate Reciprocity, even though not all of the possible configurations that satisfy this requirement are maximal. For example, sentence (53) is true of figure 4.4, even though more glued pairs could feasibly be added to this configuration. The exact set of relations that these predicates may denote depends on the type of individuals in the antecedent set cubes allow different configurations than pyramids, for instance and in any case this set is not simple to calculate. 2 To be precise, since the sit alongside predicate is symmetric, the relation cannot be acyclic. Instead the underlying undirected graph induced by the relation is required to be acyclic. 38

48 We may conjecture that be glued to and be connected to are transitive at least to an extent, such that for instance in figure 4.4 it could be judged as true that the front cube is glued to the back cube. Assuming transitivity of the denoted relations, the SMH-based interpretation of the above sentences would be Strong Reciprocity, which would then induce Intermediate Reciprocity on the non-transitive version of the predicates. Another possible route to the analysis of these cases would be to assume that a collective interpretation of the predicate operates in these cases, an option that is not currently handled within our theory. The interplay between collective predication and reciprocal expressions has not been deeply investigated in any of the theories of the semantics of reciprocals, and it seems to be an almost uncharted territory. Figure 4.4: The cubes are glued to each other Ð Þ Û Ò Þ Û 4.5 Predictions of the System - A Summary of Examples Table 4.1 below summarizes the analysis of the examples presented above and the predictions made by our system for each example. In the next chapter we use the proposed framework and the definition of R Θ to examine the possibility of attesting two controversial meanings that have been suggested for reciprocals. 39