{ "paper_id": "J82-1003", "header": { "generated_with": "S2ORC 1.0.0", "date_generated": "2023-01-19T03:07:09.747425Z" }, "title": "From English to Logic: Context-Free Computation of \"Conventional\" Logical Translation 1", "authors": [ { "first": "Lenhart", "middle": [ "K" ], "last": "Schubert", "suffix": "", "affiliation": {}, "email": "" }, { "first": "Francis", "middle": [ "Jeffry" ], "last": "Pelletier", "suffix": "", "affiliation": {}, "email": "" } ], "year": "", "venue": null, "identifiers": {}, "abstract": "We describe an approach to parsing and logical translation that was inspired by Gazdar's work on context-free grammar for English. Each grammar rule consists of a syntactic part that specifies an acceptable fragment of a parse tree, and a semantic part that specifies how the logical formulas corresponding to the constituents of the fragment are to be combined to yield the formula for the fragment. However, we have sought to reformulate Gazdar's semantic rules so as to obtain more or less 'conventional' logical translations of English sentences, avoiding the interpretation of NPs as property sets and the use of intensional functors other than certain propositional operators. The reformulated semantic rules often turn out to be slightly simpler than Gazdar's. Moreover, by using a semantically ambiguous logical syntax for the preliminary translations, we can account for quantifier and coordinator scope ambiguities in syntactically unambiguous sentences without recourse to multiple semantic rules, and are able to separate the disambiguation process from the operation of the parser-translator. We have implemented simple recursive descent and left-corner parsers to demonstrate the practicality of our approach.", "pdf_parse": { "paper_id": "J82-1003", "_pdf_hash": "", "abstract": [ { "text": "We describe an approach to parsing and logical translation that was inspired by Gazdar's work on context-free grammar for English. Each grammar rule consists of a syntactic part that specifies an acceptable fragment of a parse tree, and a semantic part that specifies how the logical formulas corresponding to the constituents of the fragment are to be combined to yield the formula for the fragment. However, we have sought to reformulate Gazdar's semantic rules so as to obtain more or less 'conventional' logical translations of English sentences, avoiding the interpretation of NPs as property sets and the use of intensional functors other than certain propositional operators. The reformulated semantic rules often turn out to be slightly simpler than Gazdar's. Moreover, by using a semantically ambiguous logical syntax for the preliminary translations, we can account for quantifier and coordinator scope ambiguities in syntactically unambiguous sentences without recourse to multiple semantic rules, and are able to separate the disambiguation process from the operation of the parser-translator. We have implemented simple recursive descent and left-corner parsers to demonstrate the practicality of our approach.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Abstract", "sec_num": null } ], "body_text": [ { "text": "Our ultimate objective is the design of a natural language understanding system whose syntactic, semantic and pragmatic capabilities are encoded in an easily comprehensible and extensible form. In addition, these encodings should be capable of supporting efficient algorithms for parsing and comprehension.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "In our view, the achievement of the former objective calls for a careful structural separation of the subsystems that specify possible constituent structure (syntax), possible mappings from constituent structure to underlying logical form (part of semantics), and possible mappings from logical form to deeper, unambiguous representations as a function of discourse context and world knowledge (part of pragmatics and l Submitted August 1981; revised July 1982. inference). This sort of view is now widely held, as evidenced by a recent panel discussion on parsing issues (Robinson 1981) . In the words of one of the panelists, \"I take it to be uncontroversial that, other things being equal, a homogenized system is less preferable on both practical and scientific grounds than one that naturally decomposes. Practically, such a system is easier to build and maintain, since the parts can be designed, developed, and understood to a certain extent in isolation... Scientifically, a decomposable system is much more likely to provide insight into the process of natural language comprehension, whether by machines or people.\" (Kaplan 1981 ) The panelists also emphasized that structural decomposition by no means precludes interleaving or paral-lelism of the processes that draw on the various kinds of linguistic and non-linguistic knowledge.", "cite_spans": [ { "start": 420, "end": 442, "text": "Submitted August 1981;", "ref_id": null }, { "start": 443, "end": 461, "text": "revised July 1982.", "ref_id": null }, { "start": 572, "end": 587, "text": "(Robinson 1981)", "ref_id": "BIBREF43" }, { "start": 1126, "end": 1138, "text": "(Kaplan 1981", "ref_id": "BIBREF26" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "Note that we are making a distinction between the logical form that corresponds directly to surface structure on the one hand, and an unambiguous deeper representation on the other. Indeed, at the level of logical form our theory of logical translation admits ambiguities in all of the formal building blocks (terms, functions, predicates, connectives, and quantifiers), as well as in the scopes of quantifiers and coordinators. For example, logical-form translations may contain terms such as Mary2 and , ambiguous between various referents (e.g., MARY5 and MARY17), and quasi-predicates such as has3, good2, cold5, and recovers l, ambiguous between various proper predicates (e.g., has3: OWNS1, AFFLICTED-WITH .... ; good2: VIRTUOUS, GOOD-TASTING .... ; cold5: COLD1, EMOTIONLESS .... ; and recoversl: RE-COVERS1, REGAINS .... ). In other words, we do not regard the logical form of a sentence as fully determining its meaning -not even its 'literal' meaning; rather, its meaning is determined by its logical form along with the context of its utterance.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "Thus \"She is becoming cold\" might convey on one occasion that Lady Godiva is beginning to feel cold, on another that Queen Victoria is becoming emotionless, and on a third that Mount St. Helens is cooling off; but the logical form does no more than specify the feminine gender of the referent and its property of \"becoming cold (in some sense) at the time of utterance\". Our primary concern in this paper will be with the semantic rules that define immediate logical form, although we attempt to define this form in a way that minimizes the remaining gap to the deeper representation.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "All the experience gained within AI and linguistics suggests that bridging this final gap will be very difficult. Some would take as their lesson that research efforts should concentrate on the last, pragmatic phase of comprehension, where 'the real problems' lie. We believe on the contrary that the only way to make the pragmatic problems tractable is to have a precise conception of the constituent structure and logical form of the natural language input, in terms of which the pragmatic operations can in turn be precisely formulated.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "In AI research, the objectives of clarity and extensibility have often been sacrificed to immediate performance goals. One reason for this may have been the need to establish the credibility of a relatively young and controversial discipline. In any case, the state of linguistic theory until fairly recently left no real alternatives.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "The transformational grammars whose study dominated theoretical linguistics seemed a poor prospect even for the limited goal of describing natural language syntax, because of the subtlety of transformational rules and supplementary devices such as co-indexing procedures, filters and constraints on movement, and the complexity of their interactions. Moreover, the prospects for writing efficient transformational parsers seemed poor, given that transformational grammars can in principle generate all recursively enumerable languages. But most importantly, generative grammarians developed syntactic theories more or less independently of any semantic considerations, offering no guidance to AI researchers whose primary objective was to compute 'meaning representations' for natural language utterances.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "Katz and Fodor's markerese (Katz & Fodor 1963) was patently inadequate as a meaning representation language from an AI point of view, and Generative Semantics (Lakoff 1971) never did develop into a formal theory of the relation between surface form and meaning.", "cite_spans": [ { "start": 27, "end": 46, "text": "(Katz & Fodor 1963)", "ref_id": "BIBREF27" }, { "start": 159, "end": 172, "text": "(Lakoff 1971)", "ref_id": "BIBREF29" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "Theoretical linguistics took an important new turn with the work of Montague on the logic of English and later expansions and variants of his theory (e.g., see Thomason 1974a , Partee 1976a , and Cresswell 1973 . According to Montague grammar the correspondence between syntactic structure and logical form is much simpler than had generally been supposed: to each lexeme there corresponds a logical term or functor and to each rule of syntactic composition there corresponds a structurally analogous semantic rule of logical composition; this is the so-called rule-to-rule hypothesis [Bach 1976] . 2 Furthermore, the translations of all consituents of a particular syntactic category are assigned formal meanings of the same set-theoretic type; for example, all NPs, be they names or definite or indefinite descriptions, are taken to denote property sets. Crucially, the formal semantics of the logical translations produced by the semantic rules of Montague grammar accords by and large with intuitions about entailment, synonymy, ambiguity and other semantic phenomena.", "cite_spans": [ { "start": 160, "end": 174, "text": "Thomason 1974a", "ref_id": "BIBREF54" }, { "start": 175, "end": 189, "text": ", Partee 1976a", "ref_id": "BIBREF40" }, { "start": 190, "end": 210, "text": ", and Cresswell 1973", "ref_id": "BIBREF7" }, { "start": 585, "end": 596, "text": "[Bach 1976]", "ref_id": "BIBREF0" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "2 Interestingly enough, this linguistic hypothesis was anticipated by Knuth's work on the semantics of attribute grammars (Knuth 1968) . Schwind (1978) has applied Knuth's insights to the development of a formal basis for question answering systems, anticipating some of the work by Gazdar and others on which our own efforts are founded.", "cite_spans": [ { "start": 122, "end": 134, "text": "(Knuth 1968)", "ref_id": "BIBREF28" }, { "start": 137, "end": 151, "text": "Schwind (1978)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "There is also some similarity between the rule-to-rule hypothesis and the rule-based approach to the interpretation of syntactic structures that emerged within AI during the 1960's and early 70's. The idea of pairing semantic rules with phrase structure rules was at the heart of DEACON (Craig et al. 1966) , a system based on F. B. Thompson's proposal to formalize English by limiting its subject matter to well-defined computer memory structures (Thompson 1966) .", "cite_spans": [ { "start": 287, "end": 306, "text": "(Craig et al. 1966)", "ref_id": "BIBREF6" }, { "start": 448, "end": 463, "text": "(Thompson 1966)", "ref_id": "BIBREF57" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "However, DEACON's semantic rules performed direct semantic evaluation of sorts (via computations over a data base) rather than constructing logical translations. The systems of Winograd (1972) and Woods (1977) constructed input translations prior to evaluation, using semantic rules associated with particular syntactic structures. However, these rules neither corresponded oneto-one to syntactic rules nor limited interpretive operations to composition of logical expressions; for example, they incorporated tests for selectional restrictions and other forms of inference, with unrestricted use of the computational power of LISP.", "cite_spans": [ { "start": 177, "end": 192, "text": "Winograd (1972)", "ref_id": "BIBREF60" }, { "start": 197, "end": 209, "text": "Woods (1977)", "ref_id": "BIBREF61" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "The chief limitation of Montague's grammar was that it treated only very small, syntactically (though not semantically) simple fragments of English, and efforts were soon under way to extend the fragments, in some cases by addition of a transformational component (Partee 1976b , Cooper & Parsons 1976 . At the same time, however, linguists dissatisfied with transformational theory were beginning to develop non-transformational alternatives to traditional generative grammars (e.g., Peters & Ritchie 1969 , Bresnan 1978 , Lapointe 1977 , Brame 1978 , Langendoen 1979 . A particularly promising theory that emerged from this development, and explicitly incorporates Montague's approach to semantics, is the phrase structure theory advanced by Gazdar and others , Gazdar, Pullum & Sag 1980 . The theory covers a wide range of the syntactic phenomena that have exercised transformationalists from Chomsky onward, including subcategorization, coordination, passivization, and unbounded dependencies such as those occurring in topicalization, relative clause constructions and comparatives. Yet the grammar itself makes no use of transformations; it consists entirely of phrase structure rules, with a nodeadmissibility rather than generative interpretation. For example, the rule [(S) (NP) (VP)] states that a fragment with root S, left branch NP and right branch VP is an admissible fragment of a syntactic tree. 3 Such phrase structure rules are easy to understand and permit the use of efficient context-free parsing methods. Moreover, the grammar realizes the rule-to-rule hypothesis, pairing each syntactic rule with a Montaguelike semantic rule that supplies the intensional logic translation of the constituent admitted by the syntactic rule.", "cite_spans": [ { "start": 264, "end": 277, "text": "(Partee 1976b", "ref_id": "BIBREF40" }, { "start": 278, "end": 301, "text": ", Cooper & Parsons 1976", "ref_id": "BIBREF5" }, { "start": 485, "end": 506, "text": "Peters & Ritchie 1969", "ref_id": "BIBREF41" }, { "start": 507, "end": 521, "text": ", Bresnan 1978", "ref_id": "BIBREF4" }, { "start": 522, "end": 537, "text": ", Lapointe 1977", "ref_id": "BIBREF31" }, { "start": 538, "end": 550, "text": ", Brame 1978", "ref_id": null }, { "start": 551, "end": 568, "text": ", Langendoen 1979", "ref_id": "BIBREF30" }, { "start": 762, "end": 789, "text": ", Gazdar, Pullum & Sag 1980", "ref_id": "BIBREF17" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "It has long been assumed by transformationalists that linguistic generalizations cannot be adequately captured in a grammar devoid of transformations. Gazdar refutes the assumption by using metagrammatical devices to achieve descriptive elegance. These devices include rule-schemata (e.g., coordination schemata that yield the rules of coordinate structure for all coordinators and all syntactic categories), and metarules (e.g., a passive metarule that takes any transitive-VP rule as 'input' and generates a corresponding passive-VP rule as 'output' by deleting the object NP from the input rule and appending an optional by-PP). Although metarules resemble transformational rules, they map rules into rules rather than trees into trees, leaving the grammar itself contextfree. Another key innovation is the use of categories with 'gaps', such as NP/PP, denoting a NP from which a PP has been deleted (not necessarily at the top level). A simple metarule and a few rule schemata are used to introduce rules involving such derived categories, elegantly capturing unbounded dependencies.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "The character of the syntactic theory will become clearer in Section 4, where we supply a sampling of grammatical rules (with our variants of the semantic rules), along with the basic metarule for passives and the coordination schemata. First, however, we would like to motivate our attempt to reformulate Gazdar's semantic rules so as to yield 'conventional' logical translations (Section 2), and to explain the syntactic and semantic idiosyncrasies of our target logic (Section 3).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "By 'conventional' logics we mean first-order (and perhaps second-order) predicate logics, augmented with a lambda operator, necessity operator, propositional attitude operators and perhaps other nonextensional propositional operators, and with a Kripkestyle possible-worlds semantics (Hughes & Cresswell 1968) . 4 The logic employed by Montague in his first formal fragment of English comes rather close to what we have in mind (Montague 1970a) , while the intensional logics of the later fragments introduce the unconventional features we hope to avoid (1970b,c) . It is the treatment in these later fragments that is usually referred to by the term \"Montague grammar\". (For a detailed discussion of the distinction between conventional logics in the above sense and intensional logics, see Guenthner 1978) .", "cite_spans": [ { "start": 284, "end": 309, "text": "(Hughes & Cresswell 1968)", "ref_id": "BIBREF24" }, { "start": 428, "end": 444, "text": "(Montague 1970a)", "ref_id": "BIBREF35" }, { "start": 554, "end": 563, "text": "(1970b,c)", "ref_id": null }, { "start": 792, "end": 807, "text": "Guenthner 1978)", "ref_id": "BIBREF22" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "We should stress that it is semantics, not syntax, which is the crux of the distinction. We shall take certain liberties with conventional logical syntax, aligning it more nearly with the surface structure; but this will not lead to major departures from conventional semantics. ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1." }, { "text": "We should emphasize at the outset that our objective is not to impugn Montague grammar, but merely to make the point that the choice between intensional and conventional translations is as yet unclear. Given that the conventional approach appears to have certain advantages, it is worth finding out where it leads; but we are not irrevocably committed to this approach. Fortunately, the translation component of a parser for a Gazdar-style grammar is easily replaced.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "Montague grammarians assume that natural languages closely resemble formal logical systems; more specifically, they postulate a strict homomorphism from the syntactic categories and rules of a natural language to the semantic categories and rules required for its formal interpretation. This postulate has led them to an analysis of the logical content of natural language sentences which differs in important respects from the sorts of analyses traditionally employed by philosophers of language (as well as linguists and AI researchers, when they have explicitly concerned themselves with logical content).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "The most obvious difference is that intensional logic translations of natural language sentences conform closely with the surface structure of those sentences, except for some re-ordering of phrases, the introduction of brackets, variables and certain logical operators, and (perhaps) the reduction of idioms. For example, since the constituent structure of \"John loves Mary\" is [John [loves Mary]], the intensional logic translation likewise isolates a component translating the VP \"loves Mary\", composing this VP-translation with the translation of \"John\" to give the sentence formula. By contrast, a conventional translation will have the structure [John loves Mary], in which \"John\" and \"Mary\" combine with the verb at the same level of constituent structure.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "In itself, this difference is not important. It only becomes important when syntactic composition is assumed to correspond to function application in the semantic domain. This is done in Montague grammar 5 We consistently use infix form (with the predicate following its first argument) and square brackets for complete sentential formulas.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "by resort to the Schoenfinkel-Church treatment of many-place functions as one-place functions (Schoenfinkel 1924 , Church 1941 . For example, the predicate \"loves\" in the above sentence is interpreted as a one-place function that yields a one-place function when applied to its argument (in this instance, when applied to the semantic value of \"Mary\", it yields the function that is the semantic value of \"loves Mary\"). The resultant function in turn yields a sentence value when applied to its argument (in this instance, when applied to the semantic value of \"John\", it yields the proposition expressed by \"John loves Mary\"). Thus, a dyadic predicator like \"loves\" is no longer interpreted as a set of pairs of individuals (at each possible world or index), but rather as a function into functions. Similarly a triadic predicator like \"gives\" is interpreted as a function into functions into functions.", "cite_spans": [ { "start": 94, "end": 112, "text": "(Schoenfinkel 1924", "ref_id": "BIBREF48" }, { "start": 113, "end": 126, "text": ", Church 1941", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "Moreover, the arguments of these functions are not individuals, because NPs in general and names in particular are assumed to denote property sets (or truth functions over properties) rather than individuals. It is easy to see how the postulate of syntactic-semantic homomorphism leads to this further retreat from traditional semantics. Consider Gazdar's top-level rule of declarative sentence structure and meaning:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "<10, [(S) (NP) (VP)], (VP' NP\")>.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "The first element of this triple supplies the rule number (which we have set to 10 for consistency with the sample grammar of Section 4), the second the syntactic rule and the third the semantic rule. The semantic rule states that the intensional logic translation of the S-constituent is compounded of the VP-translation (as functor) and the NP-translation (as operand), where the latter is first to be prefixed with the intension operator A. In general, a primed syntactic symbol denotes the logical translation of the corresponding constituent, and a double-primed symbol the logical translation prefixed with the intension operator (thus, NP\" stands for ANP').", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "For example, if the NP dominates \"John\" and the VP dominates \"loves Mary\", then S' (the translation of S) is ((loves' AMary') A John').", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "Similarly the translation of \"Every boy loves Mary\" comes out as ((loves' AMary') A(every' boy')),", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "given suitable rules of NP and VP formation. 6 Note the uniform treatment of NPs in the logical formulas, i.e., (every' boy') is treated as being of the same semantic category as John', namely the (unique) seman-tic category corresponding to the syntactic category NP. What is the set-theoretic type of that category? Since (every v boy t) cannot be interpreted as denoting an individual (at least not without making the rules of semantic valuation for formulas depend on the structure of the terms they contain), neither can John w. The solution is to regard NPs as denoting sets of properties, where a property determines a set of individuals at each index, and VPs as sets of such property sets (or in functional terms, as truth functions over truth functions over properties). Thus John v does not denote an individual, but rather a set of properties, namely those which John has; (every w boy t) denotes the set of properties shared by all boys, (a v boy w) the set of all properties possessed by at least one boy, and so on. It is not hard to see that the interpretation of VPs as sets of property sets then leads to the appropriate truth conditions for sentences. 7", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "With respect to our objective of building a comprehensible, expandable natural language understanding system, the simplicity of Gazdar's semantic rules and their one-to-one correspondence to phrase structure rules is extremely attractive; however, the semantics of the intensional logic translations, as sketched above, seems to us quite unnatural.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "Admittedly naturalness is partly a matter of familiarity, and we are not about to fault Montague grammar for having novel features (as some writers do, e.g., . But Montague's semantics is at variance with pretheoretical intuitions as well as philosophical tradition, as Montague himself acknowledged (1970c:268) .", "cite_spans": [ { "start": 270, "end": 311, "text": "Montague himself acknowledged (1970c:268)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "Intuitively, names denote individuals (when they denote anything real), not sets of properties of individuals; extensional transitive verbs express relations between pairs of individuals, not between pairs of property sets, and so on; and intuitively, quantified terms such as \"everyone\" and \"no-one\" simply don't bear the same sort of relationship to objects in the world as names, even though the evidence for placing them in the same syntactic category is overwhelming. Such objections would carry no weight if the sole purpose of formal semantics were to provide an explication of intuitions about truth and logical consequence, for in that area intensional logic is remarkably successful. But formal semantics should also do justice to our intuitions about the relationship between word and object, where those intuitions are clear -and intensional logic seems at odds with some of the clearest of those intuitions. 8", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "There is also a computational objection to intensional logic translations. As indicated in our introductory remarks, a natural language understanding system must be able to make inferences that relate the natural language input to the system's stored knowledge and discourse model. A great deal of work in AI has focused on inference during language understanding and on the organization of the base of stored knowledge on which the comprehension process draws. Almost all of this work has employed more or less conventional logics for expressing the stored knowledge. (Even such idiosyncratic formalisms as Schank's conceptual depen~lency theory (Schank 1973) are much more akin to, say, first order modal logic than to any form of intensional logic -see Schubert 1976 ). How are intensional logic formulas to be connected up with stored knowledge of this conventional type?", "cite_spans": [ { "start": 647, "end": 660, "text": "(Schank 1973)", "ref_id": "BIBREF46" }, { "start": 756, "end": 769, "text": "Schubert 1976", "ref_id": "BIBREF49" } ], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "One possible answer is that the stored knowledge should not be of the conventional type at all, but should itself be expressed in intensional logic. However, the history of automatic deduction suggests that higher-order logics are significantly harder to mechanize than lower-order logics.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "Developing efficient inference rules and strategies for intensional logics, with their arbitrarily complex types and their intension, extension and lambda abstraction operators in addition to the usual modal operators, promises to be very difficult indeed.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "Another possible answer is that the intensional logic translations of input sentences should be postprocessed to yield translations expressed in the lowerorder, more conventional logic of the system's knowledge base. A difficulty with this answer is that discourse inferences need to be computed 'on the fly' to guide syntactic choices. For example, in the sentences \"John saw the bird without binoculars\" and \"John saw the bird without tail feathers\" the syntactic roles of the prepositional phrases (i.e., whether they modify \"saw\" or \"the bird\") can only be determined by inference. One could uncouple inference from parsing by computing all possible parses and choosing among the resultant translations, but this would be cumbersome and psychologically implausible at best.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "As a final remark on the disadvantages of intensional translations, we note that Montague grammar relies heavily on meaning postulates to deliver simple consequences, such as A boy smiles -There is a boy;", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "(in this instance an extensionalizing postulate is required for \"smiles\" -see Montague 1970c:263) .", "cite_spans": [ { "start": 78, "end": 97, "text": "Montague 1970c:263)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "A conventional approach dispensing with postulates of this type would be preferable.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "Having stated our misgivings about Montague grammar, we need to confront the evidence in its favour. Are there compelling reasons for regarding sentential constituents as more or less directly and uniformly interpretable? In support of the affirmative, one can point out the simplicity and elegance of this strategy from a logical point of view. More tellingly, one can cite its success record: it has made possible for the first time the formal characterization of nontrivial fragments of natural languages, with precisely defined syntactic-semantic mappings; and as one would hope, the formal semantics accounts for many cases of entailment, ambiguity, contradictoriness, and other semantic phenomena, including some of the subtlest arising from intensional locutions.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "Concerning the simplicity of the strategy, we note that the connection between language and the world could be just as simple as Montague grammar would have it, without being quite so direct. Suppose, for a moment, that people communicated in first-order logic. Then, to express that A1, Bill and Clyde were born and raised in New York, we would have to say, in effect, \"Al was born in New York. A1 was raised in New York. Bill was born in New York ....", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "Clyde was raised in New York.\" The pressure to condense such redundant verbalizations would be great, and might well lead to 'overlay' verbalizations in which lists enumerating the non-repeated constituents were fitted into a common sentential matrix. In other words, it might lead to something like constituent coordination. But unlike simple constituents, coordinated constituents would not be meaningful in isolation; they would realize their meaning only upon expansion of the embedding overlay verbalization into a set of firstorder formulas. Yet the connection between language and the world would remain simple, assuming that the syntactic relation between overlay verbalizations and their first-order translations were simple. It would be quite pointless to reeonstrue the semantics of the enhanced language so as to align the denotations of names with the denotations of coordinated names, for example, as is done in Montague grammar. While formally simplifying the semantic mapping function, such a move would lead to complex and counterintuitive semantic types.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "The success of Montague grammar in characterizing fragments of natural languages, with a proper account of logical relations such as entailment, is indeed strong evidence in its favour. The only way of challenging this success is to offer an equally simple, equally viable alternative. In part, this paper is intended as a move in that direction.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "While we do not explicitly discuss logical relations between the translations of sentences, the kinds of translations produced by the sample grammar in Section 4 should at least provide some basis for discussion.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "To the extent that the translations are of a conventional type (or easily converted to conventional form), the entailment relations should be more or less self-evident.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Intensional and 'Conventional' Translations", "sec_num": "2." }, { "text": "is one linguistic phenomenon, however, which deserves preliminary comment since it might be thought to provide conclusive evidence in favour of Montague grammar, or at least in favour of the intensional treatment of NPs. This concerns intensional verbs such as those in sentences (1) and 2, and perhaps (3):", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "(1) John looks for a unicorn, (2) John imagines a unicorn, (3) John worships a unicorn.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "These sentences admit non-referential readings with respect to the NP \"a unicorn\", i.e., readings that do not entail the existence of a unicorn which is the referent of the NP. In intensional logic the nonreferential reading of the first sentence would simply be ((looks-for t ^(a v unicornt)) ^Johnt).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "The formal semantic analysis of this formula turns out just as required; that is, its value can be \"true\" or \"false\" (in a given possible world) irrespective of whether or not there are unicorns (in that world). The referential reading is a little more complicated, but presents no difficulties.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "It is the non-referential reading which is troublesome for conventional logics. For the first sentence, there seems to be only one conventional translation,", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "viz., 3x[[John looks-for x] & [x unicorn]],", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "and of course, this is the referential reading. There is no direct way of representing the non-referential reading, since the scope of a quantifier in conventional logics is always a sentence, never a term.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "The only possible escape from the difficulty lies in translating intensional verbs as complex(non-atomic) logical expressions involving opaque sentential operators. 9 The extant literature on this subject supports the view that a satisfactory decomposition cannot be supplied in all cases (Montague 1970c , Bennett 1974 , Partee 1974 , Dowty 1978 , Dowty, Wall & Peters 1981 . A review of this literature would be out of place here; but we would like to indicate that the case against decomposition (and hence against conventional translations) is not closed, by offering the fol-lowing paraphrases of the three sample sentences. (Paraphrase (1)w is well-known, except perhaps for the particular form of adverbial (Quine 1960 , Bennett 1974 , Partee 1974 , while (2)I-(3) '' are original). These could be formalized within a conventional logical framework allowing for non-truth-functional sentential operators:", "cite_spans": [ { "start": 289, "end": 304, "text": "(Montague 1970c", "ref_id": "BIBREF37" }, { "start": 305, "end": 319, "text": ", Bennett 1974", "ref_id": "BIBREF1" }, { "start": 320, "end": 333, "text": ", Partee 1974", "ref_id": "BIBREF38" }, { "start": 334, "end": 346, "text": ", Dowty 1978", "ref_id": "BIBREF7" }, { "start": 347, "end": 374, "text": ", Dowty, Wall & Peters 1981", "ref_id": null }, { "start": 714, "end": 725, "text": "(Quine 1960", "ref_id": "BIBREF42" }, { "start": 726, "end": 740, "text": ", Bennett 1974", "ref_id": "BIBREF1" }, { "start": 741, "end": 754, "text": ", Partee 1974", "ref_id": "BIBREF38" } ], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "(1)' John tries to find a unicorn (by looking around), (2) I John forms a mental description which could apply to a unicorn, (3) 1 John acts, thinks and feels as if he worshipped a unicorn.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "(3)\" John worships an entity which he believes to be a unicorn.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "In each case the operator that is the key to the translation is italicized. Note that the original ambiguity of (1) and (2) has been preserved, but can now be construed as a quantifier scope ambiguity in the conventional fashion.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "In (3) 1 and (3)\" the embedded \"worships\" is to be taken in a veridical sense that entails the existence of the worshippee. It is important to understand that the translations corresponding to (3) 1 and (3)\" would not be obtained directly by applying the rules of the grammar to the original sentence; rather, they would be obtained by amending the direct translation, which is patently false for a hearer who interprets \"worships\" veridically and does not believe in unicorns. Thus we are presupposing a mechanism similar to that required to interpret metaphor on a Gricean account (Grice 1975) . The notion of \"acting, thinking and feeling as if...\" may seem rather ad hoc, but appears to be applicable in a wide variety of cases where (arguably) non-intensional verbs of human action and attitude are used non-referentially, as perhaps in \"John is communing with a spirit\", \"John is afraid of the boogie-man in the attic\", or \"John is tracking down a sasquatch\". Formulation (3)\" represents a more radical alternative, since it supplies an acceptable interpretation of (3) only if the entity actually worshipped by John may be an 'imaginary unicorn'. But we may need to add imaginary entities to our 'ontological stable' in any event, since entities may be explicitly described as imaginary (fictitious, hypothetical, supposed) and yet be freely referred to in ordinary discourse. Also, sentences such as \"John frequently dreams about a certain unicorn\" (based on an example in Dowty, Wall and Peters 1981) seem to be untranslatable into any logic without recourse to imaginary entities. Our paraphrases of (3) have the important advantage of entailing that John has a specific unicorn in mind, as intuitively required (in contrast with (1) and (2)). This is not the case for the intensional logic translation of (3) analogous to that of (1), a fact that led Bennett to regard \"worships\" -correctly, we think -as extensional (Bennett 1974 ).", "cite_spans": [ { "start": 583, "end": 595, "text": "(Grice 1975)", "ref_id": "BIBREF21" }, { "start": 1928, "end": 1941, "text": "(Bennett 1974", "ref_id": "BIBREF1" } ], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "In the light of these considerations, the conventional approach to logical translation seems well worth pursuing.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "The simplicity of the semantic rules to which we are led encourages us in this pursuit.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "There", "sec_num": null }, { "text": "The logical-form syntax provides for the formation of simple terms such as Note the use of sharp angle brackets for quantified terms, square brackets or blunt angle brackets for compound predicate formulas, and round brackets for modified predicate formulas. (We explain the use of square brackets and blunt angle brackets below.) We also permit sentences (i.e., compound predicate formulas with all arguments in place) as operands of sentential operators, as in", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "[[John5 loves6 Mary7] possible3], [Suel believes2 [John5 loves5 Mary6]], [[Johnl feverish3] because4 [Johnl has5 malaria6]].", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "For coordination of expressions of all types (quantifiers, terms, predicate formulas, modifiers, and sentential operators) we use sharp angle brackets and prefix form, as in , , .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "The resemblance of coordinated expressions to quantified terms is intentional: in both cases the sharp angle brackets signal the presence of an unscoped operator (viz., the first element in brackets) to be scoped later on.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "Finally, we may want to admit second-order predicates with first-order predicate arguments, as in", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "[Fidol little-for3 dog5], [bluel colour4], [Xx[x kissesl Mary2] is-fun3],", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "though it remains to be seen whether such secondorder predications adequately capture the meaning of English sentences involving implicit comparatives and nominalization.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "Fuller explanations of several of the above features follow. In outline, we first delve a little further into the syntax and semantics of predicate formulas; then we discuss the sources and significance of ambiguities in the formulas.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "Atomic sentences are of the form [t n P t 1 ... tn_l], (equivalently, (P t 1 ... tn)), where t I .... , t n are terms and P is a predicate constant, and the square brackets and blunt angle brackets distinguish infix and prefix syntax respectively. We regard this sentential form as equivalent to As in Montague grammar, this predicate application syntax helps to keep the rules of translation simple: in most cases the translation of a phrase is just the composition of the translations of its top-level constituents. However, we saw earlier that a functional interpretation of predicate application leads to the interpretation of predicates as telescoped function-valued functions, whereas we wish to interpret predicates as n-ary relations (in each possible world) in the conventional way. where I is an interpretation, I(x:d) is an interpretation identical to I except that x denotes d, and X denotes Cartesian product (and a particular possible world is 10 We provide the double syntax for purely cosmetic reasons. In our use of the notation, expressions delimited by square brackets will generally be complete open or closed sentences, while expressions delimited by blunt angle brackets will be 'incomplete sentences', i.e., predicates with one or more arguments missing (and denoting a relation with adicity = number of missing arguments). understood). It can be verified that the usual lambdaconversion identities hold, i.e., ()~x(P...x...) t) = (P...t...), and P = )tx(P x) = ~,xhy(P x y) .....", "cite_spans": [ { "start": 958, "end": 960, "text": "10", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "where P is a predicate of any adicity (including null, if we use{<>}XA= A for any set A).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "As far as modified predicate formulas such as (bright3 red4) are concerned, we can interpret the modifiers as functions from n-ary relations to n-ary relations (perhaps with n restricted to 1).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "We now turn to a consideration of the potential sources of ambiguity in the formulas. One source of ambiguity noted in the Introduction lies in the primitive logical symbols themselves, which may correspond ambiguously to various proper logical symbols. The ambiguous symbols are obtained by the translator via the first stage of a two-stage lexicon (and with the aid of morphological analysis, not discussed here). This first stage merely distinguishes the formal logical roles of a lexeme, supplying a distinct (but in general still ambiguous) symbol or compound expression for each role, along with syntactic information. For example, the entry for \"recover\" might distinguish (i) a predicate role with preliminary translation \"recovers-from\" and the syntactic information that this is a V admissible in the rule that expands a VP as a V optionally followed by a (PP from); (this information is supplied via the appropriate rule number); and (ii) a predicate role with preliminary translation \"recovers\" and the syntactic information that this is a V admissible in the rule that expands a VP as a V followed by an NP.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "Having obtained a preliminary translation of a lexeme in keeping with its apparent syntactic role, the translator affixes an index to it which has not yet been used in the current sentence (or if the translation is a compound expression, it affixes the same index to all of its primitive symbols). In this way indexed preliminary translations such as Maryl, good2, and recovers3 are obtained. For example, the verb translation selected for \"recovers\" in the sentence context \"John recovers the sofa\" would be recovers2, recovers-from2 being ruled out by the presence of the NP complement. The second stage of the lexicon supplies alternative final translations of the first-stage symbols, which in the case of \"recovers\" might be RE-COVERS, REGAINS, and so on. Naturally, the processors that choose among these final symbols would have to draw on knowledge stored in the propositional data base and in the representation of the discourse context.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "A second source of ambiguity lies in quantified terms. The sentence Someone loves every man illustrates a quantifier scope ambiguity arising from a syntactically unambiguous construction. Its logicalform translation is [ loves3 ], wherein the relative scopes of the quantifiers somel and every4 are ambiguous. Quantified terms are intended to be 'extracted' in the postprocessing phase to positions left-adjacent to sentential formulas (which may already be prefixed with other quantifiers). A new variable is introduced into each extracted quantifier expression, the angle brackets are changed to round brackets, and the new variable is substituted for all occurrences of the extracted term. (Thus the level of extraction must be 'high' enough to encompass all of these occurrences.) In the above formula, quantifier extraction reveals the implicit ambiguity, yielding either Assuming that somel and every4 correspond to the standard existential and universal quantifiers, these translations could be further processed to yield ~x", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "[[x one2] & Vy[[y man5] => [x loves3 y]]] and Vy[[y man5] => 3x[[x one2] & [x loves3 y]]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "]. However, we may not implement this last conversion step, since it cannot be carried out for all quantifiers. For example, as Cresswell remarks, \"most A's are B's\" cannot be rendered as \"for most x, either x is not an A or x is a B\" (Cresswell 1973: 137) . (Consider, for instance, A = dog and B = beagle; then the last statement is true merely because most things are not dogsirrespective of whether or not most dogs are in fact beagles.) It appears from recent work by Goebel (to appear) that standard mechanical inference methods can readily be extended to deal with formulas with restricted quantifiers.", "cite_spans": [ { "start": 235, "end": 256, "text": "(Cresswell 1973: 137)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "A third source of ambiguity lies in coordinated expressions. For example, the logical form of the sentence \"Every man loves Peggy or Sue\" is The postprocessing steps required to scope coordinators are similar to those for quantifiers and are illustrated in Section 4.1 l An important constraint on the disambiguation of the basic symbols as well as quantified terms and coordinated expressions is that identical expressions (i.e., expressions with identical constituent structure, including indices) must be identically disambiguated. For example, \"John shaves himself\" and \"John shaves John\" translate respectively into The stated constraint ensures that both occurrences of Johnl in the first formula will ultimately be replaced by the same unambiguous constant. Similarly \"Someone shaves himself\" and \"Someone shaves someone\" translate initially into [ shaves3 ] and [ shaves3 ] respectively, and these translations become respectively after quantifier extraction. Note that the two occurrences of in the first formula are extracted in unison and replaced by a common variable. Indexing will be seen to play a similar role in the distribution of coordinators that coordinate nonsentential constituents.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "By allowing the above types of ambiguities in the logical form translations, we are able to separate the problem of disambiguation from the problems of parsing and translation. This is an important advantage, since disambiguation depends upon pragmatic factors. For example, \"John admires John\" may refer to two distinct individuals or just to one (perhaps whimsically), depending on such factors as whether more than one individual named John has been mentioned in the current context. Examples involving ambiguities in nouns, verbs, determiners, etc., are easily supplied. Similarly, the determination of relative quantifier scopes involves pragmatic considerations in addition to level of syntactic embedding and surface order. This is true both for explicit quantifier scope ambiguities such as in the sentence \"Someone loves every man\", and for scope ambiguities introduced by decomposition, such as the decomposition of \"seeks\" into neglecting indices. It is simpler to produce a single output which can then be subjected to pragmatic post-processing to determine likely quantifier scopes, than to generate all possible orderings and then to make a pragmatic choice among them. Much the same can be said about scoping of coordinators.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "We also note that a grammar designed to generate all possible unambiguous translations of English phrases and sentences would have to supply multiple semantic rules for certain syntactic rules. For example, no one semantic rule can translate a quantifier-noun combination (rule 3 in Section 4) so as to deliver both readings of \"Someone loves every man\" upon combination of the verb translation with the translations of the NPs. Our use of an ambiguous logical form preserves the rule-to-rule hypothesis.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntactic and Semantic Preliminaries", "sec_num": "3." }, { "text": "Our syntactic rules do not depart significantly from Gazdar's.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "The semantic rules formally resemble Gazdar's as well, but of course produce conventionally interpretable translations of the type described in the preceding section.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "As in Gazdar's semantic rules, constituent translations are denoted by primed category symbols such as NP' and V'. The semantic rules show how to assemble such translations (along with the occasional variable and lambda operator) to form the translations of larger constituents.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "The translations of individual lexemes are obtained as described above.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "In operation, the translator generates the minimum number of brackets consistent with the notational equivalences stated earlier. The rules that follow have been adapted from Gazdar (1981a) . Note that each rule that involves a lexical category such as PN, N or V is accompanied by a specification of the subset of lexical items of that category admissible in the rule. This feature is particularly important for verb subcategorization.", "cite_spans": [ { "start": 175, "end": 189, "text": "Gazdar (1981a)", "ref_id": "BIBREF14" } ], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "In addition, each rule is followed by (a) a sample phrase accepted by the rule, (b) an indication of how the logical translation of the phrase is obtained, and possibly (c) some words of further explanation. .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "..]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "as per Gazdar (1981a) . corresponding argument Maryl has the lowest index, lower than that of the subject translation John2 (assuming that symbols are indexed in the order of occurrence of the lexical items they translate).", "cite_spans": [ { "start": 7, "end": 21, "text": "Gazdar (1981a)", "ref_id": "BIBREF14" } ], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "In subsequent pragmatic processing, this feature could be used to detect the special salience of Maryl, without re-examination of the superficial sentence form.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "Another example of a sentence that can be analyzed by such methods, using relative clause rules similar to those for topicalization, is where P is an undetermined predicate (= dog, in the absence of contrary contextual information).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "As a further illustration of Gazdar's approach and how easily it is adapted to our purposes, we consider his metarule for passives:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "<[(VP)(V TRAN) (NP) X], (St NP\")> ==> <[(VP PASS) (V) X {(PP by)}],", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "~,p((~r p) pp\")>;", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "i.e., \"for every active VP rule that expands VP as a transitive verb followed by NP, there is to be a passive VP rule that expands VP as V followed by what, if anything, followed the NP in the active VP rule, followed optionally by a by-PP\" (Gazdar 1981a ). In the original and resultant semantic rules, (~\" ...) represents the original rule matrix in which NP\" is embedded; thus (~r p) is the result of substituting the lambda variable P (which varies over NP intensions) for NP\" in the original rule. Intuitively, the lambda variable 'reserves' the NP\" argument position for later binding by the subject of the passive sentence. It can be seen that the metarule will generate a passive VP rule corresponding to our rule 6 which will account for sentences such as \"Fido was given to Mary by John\". Moreover, if we introduce a ditransitive rule", "cite_spans": [ { "start": 241, "end": 254, "text": "(Gazdar 1981a", "ref_id": "BIBREF14" } ], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "<14, [ (VP) (V TRAN) (NP) (NP) ] , (V' NP' NP')>I4", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "to allow for sentences such as \"John gave Mary Fido\", the metarule will generate a passive VP rule that accounts for \"Mary was given Fido by John\", in which the indirect rather than direct object has been turned into the sentence subject.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "The only change needed for our purposes is the replacement of the property variable P introduced by the metarule by an individual variable x:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "...(~r NP' ) ..... > ...hx((~ r x) PP' )...", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "Once the subject NP of the sentence is supplied via rule 10, x is replaced by the translation of that NP upon lambda conversion.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "Finally in this section, we shall briefly consider coordination. Gazdar has supplied general coordination rule schemata along with a cross-categorical semantics that assigns appropriate formal meanings to coordinate structures of any category (Gazdar 1980b) . Like Gazdar's rules, our rules generate logical-form translations of coordinated constituents such as , , , echoing the surface forms. However, it should be clear from our discussion in Section 2 that direct interpretation of expressions translating, say, coordinated NPs or VPs is not compatible with our conventional conception of formal semantics. For example, no formal semantic value is assigned directly to the coordinated term in the formula Rather, interpretation is deferred until the pragmatic processor has extracted the coordinator from the embedding sentence (much as in the case of quantified The reader can verify that the first and second readings, but not the third, could have been obtained by extracting the coordinator first and the quantifier second.", "cite_spans": [ { "start": 243, "end": 257, "text": "(Gazdar 1980b)", "ref_id": "BIBREF13" } ], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "Finally, we should remark that the distributive rules are not appropriate for the group reading of coordinate structures in sentences such as John and Mary carried the sofa (together).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "We envisage a mereological interpretation in which John and Mary together comprise a two-component entity. However, we refrain from introducing a logical syntax for such entities here (but see the treatment of plurals in Schubert, 1982) .", "cite_spans": [ { "start": 221, "end": 236, "text": "Schubert, 1982)", "ref_id": "BIBREF50" } ], "ref_spans": [], "eq_spans": [], "section": "Sample Grammar", "sec_num": "4." }, { "text": "Phrase structure grammars are relatively easy to parse. The most advanced parser for Gazdar-style grammars that we are aware of is Thompson's chartparser (Thompson 1981) , which provides for slash categories and coordination, but does not (as of this writing) generate logical translations. We have implemented two small parser-translators for preliminary experimentation, one written in SNOBOL and the other in MACLISP. The former uses a recursive descent algorithm and generates intensional logic translations. The latter is a 'left corner' parser that uses our reformulated semantic rules to generate conventional translations. It begins by finding a sequence of leftmost phrase-structure-rule branches that lead from the first word upward to the sentence node. (e.g., Mary -~ PN -~ NP -~ S). The remaining branches of the phrase structure rules thus selected form a \"frontier\" of expectations. Next a similar initial-unit sequence is found to connect the second word of the sentence to the lowest-level (most immediate) expectation, and so on. There is provision for the definition and use of systems of features, although we find that the parser needs to do very little feature checking to stay on the right syntactic track. Neither parser at present handles slash categories and coordination (although they could be handled inefficiently by resort to closure of the grammar under metarules and rule schemata). Extraction of quantifiers from the logical-form translations is at present based on the level of syntactic embedding and left-to-right order alone, and no other form of postprocessing is attempted.l 5 It has been gratifyingly easy to write these parsertranslators, confirming us in the conviction that Gazdar-style grammars hold great promise for the design of natural language understanding systems. It is particularly noteworthy that we found the design of the translator component an almost trivial task; no modification of this component will be required even when the parser is expanded to handle slash categories and coordination directly. Encouraged by these resuits, we have begun to build a full-scale left-corner parser. A morphological analyzer that can work with arbitrary sets of formal affix rules is partially implemented; this work, as well as some ideas on the conventional translation of negative adjective prefixes, plurals, and tense/aspect structure, is reported in Schubert (1982) .", "cite_spans": [ { "start": 154, "end": 169, "text": "(Thompson 1981)", "ref_id": "BIBREF58" }, { "start": 2403, "end": 2418, "text": "Schubert (1982)", "ref_id": "BIBREF50" } ], "ref_spans": [], "eq_spans": [], "section": "Parsing", "sec_num": "5." }, { "text": "From the point of view of theoretical and computational linguistics, Gazdar's approach to grammar offers profound advantages over traditional approaches: it dispenses with transformations without loss of insight, offers large linguistic coverage, and couples simple, semantically well-motivated rules of translation to the syntactic rules.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Concluding Remarks", "sec_num": "6." }, { "text": "We have attempted to show that the advantages of Gazdar's approach to grammar can be secured without commitment to an intensional target logic for the translations of natural language sentences. To motivate this endeavour, we have argued that there are philosophical and practical reasons for preferring a conventional target logic, and that there are as yet no compelling reasons for abandoning such logics in favour of intensional ones. More concretely, we have shown how to reformulate Gazdar's semantic rules to yield conventional translations, and have briefly described some extant PSG parsers, including one that is capable of parsing and translating in accordance with the reformulated Gazdar grammar (minus metalinguistic constructs).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Concluding Remarks", "sec_num": "6." }, { "text": "We believe that a parser-interpreter of this type will prove very useful as the first stage of a natural language understanding system. Since the grammar rules are expressed in a concise, individually comprehensible form, such a system will be easy to expand indefinitely. The assignment of a well-defined logical form to input sentences, compatible with favoured knowledge representation formalisms, should help to ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Concluding Remarks", "sec_num": "6." }, { "text": "American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "We use traditional category symbols in our exposition, occasionally followed by supplementary features, e.g., (V TRAN) for transitive verb. Gazdar actually assumes a two-bar X system (e.g., seeBresnan 1976, Jackendoff 1977) that distinguishes between X, ,~ and X categories (e.g., ~, V, and V, equivalent to the traditional S, VP and V respectively) and employs complex symbols whose first component specifies the 'number of bars' and whose second component supplies a feature bundle encoding syntactic category, subcategorization, and morphosyntactic and morphological information.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "We admit predicate modifiers and some second-order predicate constants into our logical vocabulary, and may ultimately want to employ a full-fledged second-order logic, in view of such sentences as \"Every good general has at least some of Napoleon's qualities\". On the other hand, we may pare down rather than expand the logical apparatus, opting for a logic that treats properties, propositions and other intensional entities as first-order individuals. This type of treatment, which avoids the unwanted identity of logically equivalent propositions, appears to be gaining currency (e.g.,Fodor 1978, McCarthy 1979, Thomason 1980, Chierchia 1981. Some minor adjustments would be required in our rules of logical translation.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "The exact function of the intension operator need not concern us here. Roughly speaking, it is used to bring meanings within the domain of discourse; e.g,, while an NP t denotes a property set at each index, the corresponding ANp~ denotes the entire NP intension (mapping from indices to property sets) at each index.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "This was the approach inMontague (1970b) and is adopted inGazdar (1981a). In another, less commonly adopted approach NPs are still interpreted as sets of properties but VPs are interpreted simply as properties, the truth condition for a sentence being that the property denoted by the VP be in the set of properties denoted by the NP(Montague 1970c, Cresswell 1973.In other words, the NP is thought of as predicating something about the VP, rather than the other way around.8 Thomason reminds us that \"...we should not forget the firmest and most irrefragable kind of data with which a semantic theory must cope. The theory must harmonize with the actual denotations taken by the expressions of natural languages,...\", but confines his further remarks to sentence denotations, i.e., truth values(Thomason, 1974b:54).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "With regard to our system-building objectives, such resort to lexical decomposition is no liability: the need for some use of lexical decomposition to obtain \"canonical\" representations that facilitate inference is widely acknowledged by AI researchers, and carried to extremes by some (e.g.,Wilks 1974, Schank 1975.American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "American Journal of Computational Linguistics, Volume 8, Number i, January-March 1982", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "If first-order predicates are to be allowed as arguments of second-order predicates, then quantifier and coordinator scoping of the following types must also be allowed:[P......] -~ ~kx(Q y:[y Rl)[x P...y...], ~ ~kx[[x PI C [x R]].", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "Siegel (1979) argues rather persuasively that measure adjectives, unlike genuine predicate modifiers such as \"consummate\", actually combine with terms. For such adjectives we might employ the semantic rule ~.x[[x ADJP'] & [x N']]; in the case of \"little\", we would use ADJP' = (little-for P), where P is an indeterminate predicate to be replaced pragmatically by a comparison-class predicate. Thus the translation of \"little boy\" (neglecting indices) would be ~kx[[x little-for PI & [x boyl].American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "Definite singular terms often serve as descriptions to be used for referent determination, and in such cases it is the name of the referent, rather than the description itself, which is ultimately wanted in the formula.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "In the computational version of the semantic rules, primed symbols are actually represented as numbers giving the positions of the corresponding constituents, e.g., (1 2 3) in rule 14. Thus no ambiguity can arise.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "Since submission of this paper for publication, we have become aware of several additional papers on parser-translators similar to ours. One is byRosenschein & Shieber (1982), another byGawron et al. (1982); in conception these are based quite directly on the generalized phrase structure grammar of Gazdar and his collaborators, and use reeursive descent parsers. A related Prolog-based approach is described byMcCord (1981McCord ( , 1982.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null } ], "back_matter": [ { "text": "The authors are indebted to Ivan Sag for a series of very stimulating seminars held by him at the University of Alberta on his linguistic research, and valuable follow-up discussions.The helpful comments of the referees and of Lotfi Zadeh are also appreciated.The research was supported in part by NSERC Operating Grants A8818 and A2252; preliminary work on the left-corner parser was carried out by one of the authors (LKS) under an Alexander von Humboldt fellowship in 1978-79.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Acknowledgements", "sec_num": null }, { "text": "<6, [(VP) (V) (NP) (PP to)], (V' PP' NP')> V(6) = {give, hand, tell .... ] (a) gives Fido to Mary (b) with V' = gives4, NP' : Fido5, PP' : Mary6, VP [(VP INF) The reader will observe that we have more or less fully traced the derivation and translation of the sentences \"The little boy smiles\" and \"John wants to give Fido to Mary\" in the course of the above examples. All of the above rules, as well as our versions of the remaining rules in Gazdar (1981a) , are as simple as the intensional logic versions or simpler. For example, our semantic rule 8, i.e., Ax[x V' [x VP'] ], may be contrasted with the corresponding rule suggested by Gazdar: XP{P", "cite_spans": [ { "start": 75, "end": 78, "text": "(a)", "ref_id": null }, { "start": 85, "end": 101, "text": "Fido to Mary (b)", "ref_id": null }, { "start": 149, "end": 158, "text": "[(VP INF)", "ref_id": null }, { "start": 443, "end": 457, "text": "Gazdar (1981a)", "ref_id": "BIBREF14" }, { "start": 560, "end": 575, "text": "Ax[x V' [x VP']", "ref_id": null }, { "start": 638, "end": 645, "text": "Gazdar:", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "annex", "sec_num": null }, { "text": "Here the lambda variable x, as in our formula, is used to feed a common logical subject to V' (the translation of the main verb) and to VP' (the translation of the embedded infinitive); the variables P and P, on the other hand, serve to ensure that the arguments of the V' and VP' functions will be of the correct type. Our 'conventional' rule is simpler because it makes no such use of lambda abstraction for type-raising and dispenses with the intension operator.Gazdar's approach to unbounded dependencies carries over virtually unchanged and can be illustrated with the sentence To Mary John wants to give Fido. Here the PP \"to Mary\" has been topicalized by extraction from \"John wants to give Fido to Mary\", leaving a PP 'gap' at the extraction site. This 'gap' is syntactically embedded within the infinitive VP \"to give Fido\", within the main VP \"wants to give Fido\", and at the highest level, within the sentence \"John wants to give Fido\". In general, the analysis of un-bounded dependencies requires derived rules for propagating 'gaps' from level to level and linking rules for creating and filling them. The linking rules are obtained from the correspondingly numbered basic rules by means of the metarulewhere A, B and C may be any basic (i.e., non-slash) syntactic categories such that C can dominate B, and X, Y may be any sequences (possibly empty) of basic categories. The linking rules for topicalization are obtained from the rule schemata, h>, andwhere B ranges over all basic phrasal categories, and t is a dummy element (trace). The first of these schemata introduces the free variable h as the translation of the gap, while the second lambda-abstracts on h and then supplies B' as the value of the lambda variable, thus 'filling the gap' at the sentence level. At syntactic nodes intermediate between those admitted by schemata 11 and 12, the B-gap is transmitted by derived rules and h is still free.Of the following rules, 6, 8, and 10 are the particular derived rules required to propagate the PP-gap in our example and 11 and 12 the particular linking rules that create and fill it: Note that once the and3-conjunction has been chosen for initial extraction and distribution, the simultaneous extraction and distribution of both occurrences of the or6-disjunction at the highest sentential level is compulsory. The resultant formula expresses the sense of \"John loves and admires Fido or loves and admires Kim\".Initial extraction of the or6-disjunction would have led to the (implausible) reading \"John loves Fido or Kim and admires Fido or Kim\" (which is true even if John loves only Fido and admires only Kim). 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He received the Ph.D. degree in computer science from the University of Toronto", "authors": [ { "first": "K", "middle": [], "last": "Lenhart", "suffix": "" } ], "year": null, "venue": "", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Lenhart K. Schubert is an associate professor of computer science at the University of Alberta, Edmonton. He received the Ph.D. degree in computer science from the University of Toronto.", "links": null }, "BIBREF63": { "ref_id": "b63", "title": "Francis Jeffry Pelletier is a professor of philosophy at the University of Alberta. He received the Ph.D. degree from the University of California at Los Angeles", "authors": [], "year": null, "venue": "", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Francis Jeffry Pelletier is a professor of philosophy at the University of Alberta. He received the Ph.D. degree from the University of California at Los Angeles.", "links": null } }, "ref_entries": { "FIGREF0": { "type_str": "figure", "num": null, "text": "), (passionately2 (loves3 Mary4)), and lambda abstracts such as ~x[x shaves2 x], Xy[y expects2 [y wins4]].", "uris": null }, "FIGREF1": { "type_str": "figure", "num": null, "text": "[t n (...((e t 1) t 2) ... tn_ 1 )1, i.e., as obtained by applying an n-ary predicate successively to n terms. For example,[John loves Mary] = Cloves Mary John) =[John Cloves Mary)] = ((loves Mary) John). 10", "uris": null }, "FIGREF2": { "type_str": "figure", "num": null, "text": "We can satisfy this requirement by interpreting predicate application not as function application, but rather as leftmost section of the associated relation at the value of the given argument. For example, let V denote the semantic valuation function (with a particular interpretation and possible world understood) and letV(P) = {, , }, V(x) = a, V(y) = b, and V(z) = d, where P is a triadic predicate symbol, x, y, and z are individual constants or variables, and a, b ..... g are elements of the individual domain D. Then V((P x)) = {, }, V((P x y)) = V(((P x) y) = {, }, and V([z V x y]) = V((((P x) y) z)) = {<>}. We use the convention {<>} = true, {} = false. Lambda abstraction can be defined compatibly by Vl(~,x~b) = {{d} X Vi(x:d) (~b) I d \u2022 D},", "uris": null }, "FIGREF3": { "type_str": "figure", "num": null, "text": "somel x:[x one2])(every4 y:[y man5])[x loves3 y] or (every4 y:[y man5])(somel x:[x one2])[x loves3 y], depending on the order of extraction.", "uris": null }, "FIGREF4": { "type_str": "figure", "num": null, "text": "Johnl hx[x shaves2 x]] = [Johnl shaves2 Johnl], and [Johnl shaves2 John3].", "uris": null }, "FIGREF5": { "type_str": "figure", "num": null, "text": "somel x:[x one2])[x shaves3 x] and (somel x:[x one2])(some4 y:[y one5])[x shaves3 y]", "uris": null }, "FIGREF6": { "type_str": "figure", "num": null, "text": "For example, in assembling [NP' VP'], with NP' = Johnl and VP t binding a variable with lambda, the translator replaces all occurrences of the variable with a previously unused variable, thus minimizing the need for later renaming. Finally, it performs lambda conversions on the fly. For example, the result of assembling[NP' VP'] with NP' = Johnl and VPV= ?~x[x shaves2 x], is [Johnl shaves2 Johnl].", "uris": null }, "FIGREF7": { "type_str": "figure", "num": null, "text": ", PN(1) = [John, Mary, New York .... ] is taken as a predicate modifier. ]2 <3, [(NP) (Q) (AN)], >, Q(3) : [a, the, all, many,", "uris": null }, "FIGREF8": { "type_str": "figure", "num": null, "text": "wants2 [dohnl gives4 Mary6 Fido5]] rule 8: VP' = Xx[x V' [x VP']I = ~x3[x3 wants2 [x3 gives4 Mary6 Fido5]] Phrase structure and translation of the sentence \"John wants to give Fido to Mary\". Phrase structure and translation of the sentence \"The little boy smiles.\"", "uris": null }, "FIGREF9": { "type_str": "figure", "num": null, "text": " loves Mary].", "uris": null }, "FIGREF10": { "type_str": "figure", "num": null, "text": "of precision and clarity to the rather murky area of natural language interpretation by machine.", "uris": null }, "TABREF3": { "type_str": "table", "content": "
The rules analyze \"Mary wants to buy\" as an S/NP
with translation
[Mary wants [Mary buys h]],
neglecting indices. A further rule reduces the S/NP to
an R (relative clause), and its semantic part abstracts
on h to yield the predicate
R' = Xh[Mary wants [Mary buys h]]
as the translation of the relative clause. The rules for
NPs can be formulated in such a way that \"every dog\"
will be translated as
<every kx[[x dog] & [x R]]>
where R is a free predicate variable that is replaced by
the translation of the relative clause when the NP-R
rule
<13, [(NP) (NP) (R)], <XRNP' R'>>
is applied (cf., Gazdar 1981b; we have ignored multi-
ple relative clauses). The resulting NP translation is
<every hx[[x dog] & [Mary wants
[Mary buys x]]]>.
The translation of the complete sentence, after extrac-
tion of the quantifier and conversion of the constraint
on the universally quantified variable to an implicative
antecedent, would be
\u00a5y[[[y dog] & [Mary wants [Mary buys y]]]
=> [y (small P)]],
", "html": null, "num": null, "text": "Every dog Mary wants to buy is small." }, "TABREF4": { "type_str": "table", "content": "
[(allq x:[x man2]) [x wants3 [x marries4 Peggy5]] or6
(alll x:[x man2]) [x wants3 [x marries4 Sue7]]].
", "html": null, "num": null, "text": "The three readings are logically distinct and all are quite plausible (in the absence of additional context)." } } } }