From English to Logic: 
Context-Free Computation of 
"Conventional" Logical Translation 1 
Lenhart K. Schubert 
Department of Computing Science 
University of Alberta 
Francis Jeffry Pelletier 
Department of Philosophy 
University of Alberta 
Edmonton, Canada T6G 2H1 
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 reformulat- 
ed 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 recur- 
sive descent and left-corner parsers to demonstrate the practicality of our approach. 
1. Introduction 
Our ultimate objective is the design of a natural 
language understanding system whose syntactic, se- 
mantic and pragmatic capabilities are encoded in an 
easily comprehensible and extensible form. In addi- 
tion, these encodings should be capable of supporting 
efficient algorithms for parsing and comprehension. 
In our view, the achievement of the former objec- 
tive calls for a careful structural separation of the sub- 
systems 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, unam- 
biguous 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, devel- 
oped, and understood to a certain extent in 
isolation... Scientifically, a decomposable sys- 
tem 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 decom- 
position by no means precludes interleaving or paral- 
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26 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
lelism of the processes that draw on the various kinds 
of linguistic and non-linguistic knowledge. 
Note that we are making a distinction between the 
logical form that corresponds directly to surface struc- 
ture 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 <thel (little2 girl3)>, ambi- 
guous 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' mean- 
ing; rather, its meaning is determined by its logical 
form along with the context of its utterance. 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. 
All the experience gained within AI and linguistics 
suggests that bridging this final gap will be very diffi- 
cult. 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 con- 
ception of the constituent structure and logical form of 
the natural language input, in terms of which the prag- 
matic operations can in turn be precisely formulated. 
In AI research, the objectives of clarity and exten- 
sibility have often been sacrificed to immediate per- 
formance 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. 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 transfor- 
mational parsers seemed poor, given that transforma- 
tional grammars can in principle generate all recursive- 
ly enumerable languages. But most importantly, gen- 
erative 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. Katz and Fodor's 
markerese (Katz & Fodor 1963) was patently inade- 
quate 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. 
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 corre- 
sponds a structurally analogous semantic rule of logical 
composition; this is the so-called rule-to-rule hypothe- 
sis \[Bach 1976\]. 2 Furthermore, the translations of all 
consituents of a particular syntactic category are as- 
signed 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 Mo- 
ntague grammar accords by and large with intuitions 
about entailment, synonymy, ambiguity and other se- 
mantic phenomena. 
2 Interestingly enough, this linguistic hypothesis was anticipat- 
ed by Knuth's work on the semantics of attribute grammars (Knuth 
1968). Schwind (1978) has applied Knuth's insights to the devel- 
opment of a formal basis for question answering systems, anticipat- 
ing some of the work by Gazdar and others on which our own 
efforts are founded. 
There is also some similarity between the rule-to-rule hypothe- 
sis 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). 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 Wino- 
grad (1972) and Woods (1977) constructed input translations prior 
to evaluation, using semantic rules associated with particular syn- 
tactic structures. However, these rules neither corresponded one- 
to-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. 
American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 27 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
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 com- 
ponent (Partee 1976b, Cooper & Parsons 1976). At 
the same time, however, linguists dissatisfied with 
transformational theory were beginning to develop 
non-transformational alternatives to traditional genera- 
tive 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 struc- 
ture theory advanced by Gazdar and others (Gazdar 
1980, 1981, Gazdar, Pullum & Sag 1980, Gazdar & 
Sag 1980, Sag 1980, Gazdar, Klein, Pullum & Sag, to 
appear). The theory covers a wide range of the syn- 
tactic phenomena that have exercised transformation- 
alists from Chomsky onward, including subcategoriza- 
tion, coordination, passivization, and unbounded de- 
pendencies 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 node- 
admissibility rather than generative interpretation. For 
example, the rule \[(S) (NP) (VP)\] states that a frag- 
ment 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 per- 
mit the use of efficient context-free parsing methods. 
Moreover, the grammar realizes the rule-to-rule hy- 
pothesis, pairing each syntactic rule with a Montague- 
like semantic rule that supplies the intensional logic 
translation of the constituent admitted by the syntactic 
rule. 
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 metagrammat- 
ical devices to achieve descriptive elegance. These 
devices include rule-schemata (e.g., coordination sche- 
mata 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 corre- 
sponding passive-VP rule as 'output' by deleting the 
3 We use traditional category symbols in our exposition, occa- 
sionally followed by supplementary features, e.g., (V TRAN) for 
transitive verb. Gazdar actually assumes a two-bar X system (e.g., 
see Bresnan 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 informa- 
tion. 
object NP from the input rule and appending an op- 
tional by-PP). Although metarules resemble trans- 
formational rules, they map rules into rules rather than 
trees into trees, leaving the grammar itself context- 
free. 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 lev- 
el). A simple metarule and a few rule schemata are 
used to introduce rules involving such derived categor- 
ies, elegantly capturing unbounded dependencies. 
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). 
By 'conventional' logics we mean first-order (and 
perhaps second-order) predicate logics, augmented 
with a lambda operator, necessity operator, proposi- 
tional attitude operators and perhaps other non- 
extensional propositional operators, and with a Kripke- 
style 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 inten- 
sional logics of the later fragments introduce the un- 
conventional 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 conven- 
tional logics in the above sense and intensional logics, 
see Guenthner 1978). 
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, align- 
ing it more nearly with the surface structure; but this 
will not lead to major departures from conventional 
semantics. For example, our syntax of terms allows 
syntactically unfamiliar formulas such as 
\[<alll man2> mortal3\]. 
4 We admit predicate modifiers and some second-order predi- 
cate constants into our logical vocabulary, and may ultimately want 
to employ a full-fledged second-order logic, in view of such sen- 
tences 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 proper- 
ties, propositions and other intensional entities as first-order indi- 
viduals. 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. 
28 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
But the formula derives its interpretation from its sti- 
pulated logical equivalence to 
(alll x:\[x man2\])\[x mortal3\], 
which may in turn become 
Vx\[\[x HUMAN\] => \[x MORTAL\]\], 
after disambiguation. 5 
2. Intensional and 'Conventional' Translations 
We should emphasize at the outset that our objec- 
tive 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 cer- 
tain 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. 
Montague grammarians assume that natural lan- 
guages 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). 
The most obvious difference is that intensional 
logic translations of natural language sentences con- 
form closely with the surface structure of those sen- 
tences, 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", compos- 
ing this VP-translation with the translation of "John" 
to give the sentence formula. By contrast, a conven- 
tional translation will have the structure 
\[John loves Mary\], 
in which "John" and "Mary" combine with the verb at 
the same level of constituent structure. 
In itself, this difference is not important. It only 
becomes important when syntactic composition is as- 
sumed 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. 
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. 
Moreover, the arguments of these functions are not 
individuals, because NPs in general and names in par- 
ticular 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 tradi- 
tional semantics. Consider Gazdar's top-level rule of 
declarative sentence structure and meaning: 
<10, \[(S) (NP) (VP)\], (VP' NP")>. 
The first element of this triple supplies the rule num- 
ber (which we have set to 10 for consistency with the 
sample grammar of Section 4), the second the syntac- 
tic 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 op- 
erator A. In general, a primed syntactic symbol de- 
notes the logical translation of the corresponding con- 
stituent, and a double-primed symbol the logical trans- 
lation prefixed with the intension operator (thus, NP" 
stands for ANP'). 
For example, if the NP dominates "John" and the 
VP dominates "loves Mary", then S' (the translation 
of S) is 
((loves' AMary') A John'). 
Similarly the translation of "Every boy loves Mary" 
comes out as 
((loves' AMary') A(every' boy')), 
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 se- 
mantic category as John', namely the (unique) seman- 
6 The exact function of the intension operator need not con- 
cern 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. 
American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 29 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
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 struc- 
ture of the terms they contain), neither can John w. 
The solution is to regard NPs as denoting sets of prop- 
erties, 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 de- 
note 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 appro- 
priate truth conditions for sentences. 7 
With respect to our objective of building a compre- 
hensible, 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. 
Admittedly naturalness is partly a matter of famili- 
arity, and we are not about to fault Montague gram- 
mar for having novel features (as some writers do, 
e.g., Harman 1975). But Montague's semantics is at 
variance with pretheoretical intuitions as well as philo- 
sophical tradition, as Montague himself acknowledged 
(1970c:268). Intuitively, names denote individuals 
(when they denote anything real), not sets of proper- 
ties 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 ob- 
jects 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 pro- 
vide an explication of intuitions about truth and logical 
consequence, for in that area intensional logic is re- 
markably successful. But formal semantics should also 
do justice to our intuitions about the relationship be- 
tween word and object, where those intuitions are 
clear - and intensional logic seems at odds with some 
of the clearest of those intuitions. 8 
There is also a computational objection to inten- 
sional logic translations. As indicated in our introduc- 
tory 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 fo- 
cused 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 convention- 
al 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 in- 
tensional logic formulas to be connected up with 
stored knowledge of this conventional type? 
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. Howev- 
er, the history of automatic deduction suggests that 
higher-order logics are significantly harder to mecha- 
nize than lower-order logics. Developing efficient 
inference rules and strategies for intensional logics, 
with their arbitrarily complex types and their inten- 
sion, extension and lambda abstraction operators in 
addition to the usual modal operators, promises to be 
very difficult indeed. 
Another possible answer is that the intensional 
logic translations of input sentences should be post- 
processed to yield translations expressed in the lower- 
order, more conventional logic of the system's knowl- 
edge base. A difficulty with this answer is that dis- 
course 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 infer- 
ence. 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. 
As a final remark on the disadvantages of inten- 
sional translations, we note that Montague grammar 
relies heavily on meaning postulates to deliver simple 
consequences, such as 
A boy smiles - There is a boy; 
7 This was the approach in Montague (1970b) and is adopted 
in Gazdar (1981a). In another, less commonly adopted approach 
NPs are still interpreted as sets of properties but VPs are interpret- 
ed 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). 
30 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
(in this instance an extensionalizing postulate is re- 
quired for "smiles" - see Montague 1970c:263). A 
conventional approach dispensing with postulates of 
this type would be preferable. 
Having stated our misgivings about Montague 
grammar, we need to confront the evidence in its fav- 
our. Are there compelling reasons for regarding sen- 
tential constituents as more or less directly and uni- 
formly 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 non- 
trivial 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 sub- 
tlest arising from intensional locutions. 
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 .... 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 fit- 
ted into a common sentential matrix. In other words, 
it might lead to something like constituent coordina- 
tion. But unlike simple constituents, coordinated con- 
stituents would not be meaningful in isolation; they 
would realize their meaning only upon expansion of 
the embedding overlay verbalization into a set of first- 
order 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 en- 
hanced 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 counterintui- 
tive semantic types. 
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 via- 
ble alternative. In part, this paper is intended as a 
move in that direction. 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. To the extent that the 
translations are of a conventional type (or easily con- 
verted to conventional form), the entailment relations 
should be more or less self-evident. 
There 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 inten- 
sional treatment of NPs. This concerns intensional 
verbs such as those in sentences (1) and (2), and per- 
haps (3): 
(1) John looks for a unicorn, 
(2) John imagines a unicorn, 
(3) John worships a unicorn. 
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 refer- 
ent of the NP. In intensional logic the nonreferential 
reading of the first sentence would simply be 
((looks-for t ^(a v unicornt)) ^Johnt). 
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. 
It is the non-referential reading which is trouble- 
some for conventional logics. For the first sentence, 
there seems to be only one conventional translation, 
viz., 
3x\[\[John looks-for x\] & \[x unicorn\]\], 
and of course, this is the referential reading. There is 
no direct way of representing the non-referential read- 
ing, since the scope of a quantifier in conventional 
logics is always a sentence, never a term. 
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 sup- 
ports the view that a satisfactory decomposition can- 
not be supplied in all cases (Montague 1970c, Bennett 
1974, Partee 1974, Dowty 1978, 1979, 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 conven- 
tional translations) is not closed, by offering the fol- 
9 With regard to our system-building objectives, such resort to 
lexical decomposition is no liability: the need for some use of lexi- 
cal decomposition to obtain "canonical" representations that facili- 
tate inference is widely acknowledged by AI researchers, and car- 
ried to extremes by some (e.g., Wilks 1974, Schank 1975). 
American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 31 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
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 logi- 
cal framework allowing for non-truth-functional sen- 
tential operators: 
(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. 
(3)" John worships an entity which he believes to be 
a unicorn. 
In each case the operator that is the key to the trans- 
lation is italicized. Note that the original ambiguity of 
(1) and (2) has been preserved, but can now be con- 
strued as a quantifier scope ambiguity in the conven- 
tional fashion. 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 apply- 
ing 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 ac- 
tion 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 ac- 
ceptable 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, hypotheti- 
cal, 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 untrans- 
latable 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). 
In the light of these considerations, the convention- 
al approach to logical translation seems well worth 
pursuing. The simplicity of the semantic rules to 
which we are led encourages us in this pursuit. 
3. Syntactic and Semantic Preliminaries 
The logical-form syntax provides for the formation 
of simple terms such as 
John1, x, 
quantified terms such as 
<somel man2>, <thel (little2 boy3)>, 
simple predicate formulas such as 
man2, loves3, P4, 
compound predicate formulas such as 
(loves2 Mary3), ((loves2 Mary3) Johnl), 
\[Johnl loves2 Mary3\], 
modified predicate formulas such as 
(bright3 red4), (passionately2 (loves3 Mary4)), 
and lambda abstracts such as 
~x\[x shaves2 x\], Xy\[y expects2 \[y wins4\]\]. 
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 formu- 
las with all arguments in place) as operands of senten- 
tial operators, as in 
\[\[John5 loves6 Mary7\] possible3\], 
\[Suel believes2 \[John5 loves5 Mary6\]\], 
\[\[Johnl feverish3\] because4 
\[Johnl has5 malaria6\]\]. 
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 
<or2 manyl few3>, <and2 Johnl Bill3>, 
<and4 (hugs2 Mary3) (kisses5 Sue6)>. 
The resemblance of coordinated expressions to quanti- 
fied 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. 
Finally, we may want to admit second-order predi- 
cates with first-order predicate arguments, as in 
\[Fidol little-for3 dog5\], \[bluel colour4\], 
\[Xx\[x kissesl Mary2\] is-fun3\], 
though it remains to be seen whether such second- 
order predications adequately capture the meaning of 
English sentences involving implicit comparatives and 
nominalization. 
32 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
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. 
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 con- 
stant, and the square brackets and blunt angle brack- 
ets distinguish infix and prefix syntax respectively. 
We regard this sentential form as equivalent to 
\[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 
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 com- 
position of the translations of its top-level constitu- 
ents. However, we saw earlier that a functional inter- 
pretation of predicate application leads to the interpre- 
tation of predicates as telescoped function-valued 
functions, whereas we wish to interpret predicates as 
n-ary relations (in each possible world) in the conven- 
tional way. 
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 particu- 
lar interpretation and possible world understood) and 
let 
V(P) = {<a,b,c>, <a,b,d>, <e,f,g>}, 
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)) = {<b,c>, <b,d>}, 
V((P x y)) = V(((P x) y) = {<c>, <d>}, 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 • D}, 
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 expres- 
sions 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 lambda- 
conversion identities hold, i.e., 
()~x(P...x...) t) = (P...t...), and 
P = )tx(P x) = ~,xhy(P x y) ..... 
where P is a predicate of any adicity (including null, if 
we use{<>}XA= A for any set A). 
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). 
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 primi- 
tive 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 predi- 
cate role with preliminary translation "recovers-from" 
and the syntactic information that this is a V admissi- 
ble 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. 
Having obtained a preliminary translation of a lex- 
eme 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 pre- 
liminary translations such as Maryl, good2, and 
recovers3 are obtained. For example, the verb trans- 
lation 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 sup- 
plies 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 propo- 
sitional data base and in the representation of the 
discourse context. 
A second source of ambiguity lies in quantified 
terms. The sentence 
Someone loves every man 
American Journal of Computational Linguistics, Volume 8, Number i, January-March 1982 33 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
illustrates a quantifier scope ambiguity arising from a 
syntactically unambiguous construction. Its logical- 
form translation is 
\[<somel one2> loves3 <every4 man5>\], 
wherein the relative scopes of the quantifiers somel 
and every4 are ambiguous. Quantified terms are in- 
tended 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 quanti- 
fier 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 ei- 
ther 
(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. 
Assuming that somel and every4 correspond to 
the standard existential and universal quantifiers, these 
translations could be further processed to yield 
~x\[\[x one2\] & Vy\[\[y man5\] => \[x loves3 y\]\]\] and 
Vy\[\[y man5\] => 3x\[\[x one2\] & \[x loves3 y\]\]\]. 
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 state- 
ment is true merely because most things are not dogs - 
irrespective 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. 
A third source of ambiguity lies in coordinated 
expressions. For example, the logical form of the 
sentence "Every man loves Peggy or Sue" is 
\[<everyl man2> loves3 <or5 Peggy4 Sue6>\], 
which is open to the readings 
(everyl x:\[x man2\])\[ \[x loves3 Peggy4\] or5 
\[x loves3 Sue6\]\] 
and 
\[(everyl x:\[x man2\])\[x loves3 Peggy4\] 
or5 (everyl x:\[x man2\])\[x loves3 Sue6\]\]. 
The postprocessing steps required to scope coordina- 
tors are similar to those for quantifiers and are illus- 
trated in Section 4.1 l 
An important constraint on the disambiguation of 
the basic symbols as well as quantified terms and coor- 
dinated expressions is that identical expressions (i.e., 
expressions with identical constituent structure, includ- 
ing indices) must be identically disambiguated. For 
example, "John shaves himself" and "John shaves 
John" translate respectively into 
\[Johnl hx\[x shaves2 x\]\] = \[Johnl shaves2 Johnl\], 
and 
\[Johnl shaves2 John3\]. 
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 
\[<somel one2> shaves3 <somel one2>\] and 
\[<somel one2> shaves3 <some4 one5>\] 
respectively, and these translations become 
(somel x:\[x one2\])\[x shaves3 x\] and 
(somel x:\[x one2\])(some4 y:\[y one5\])\[x shaves3 y\] 
respectively after quantifier extraction. Note that the 
two occurrences of <somel one2> 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 non- 
sentential constituents. 
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 pars- 
ing 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 whimsical- 
ly), 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 decomposi- 
tion, such as the decomposition of "seeks" into 
hyhx\[x tries \[x finds y\]\], 
as a result of which a sentence like 
John seeks a unicorn 
admits the alternative translations 
3x\[\[x unicorn\] & \[John tries \[John finds x\]\]\], and 
\[John tries 3x\[\[x unicorn\] & \[John finds x\]\]\], 
neglecting indices. It is simpler to produce a single 
output which can then be subjected to pragmatic post- 
11 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: \[P...<Q R>...\] -~ ~kx(Q 
y:\[y Rl)\[x P...y...\], <C P R> ~ ~kx\[\[x PI C \[x R\]\]. 
34 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
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. 
We also note that a grammar designed to generate 
all possible unambiguous translations of English phras- 
es and sentences would have to supply multiple seman- 
tic rules for certain syntactic rules. For example, no 
one semantic rule can translate a quantifier-noun com- 
bination (rule 3 in Section 4) so as to deliver both 
readings of "Someone loves every man" upon combi- 
nation of the verb translation with the translations of 
the NPs. Our use of an ambiguous logical form pre- 
serves the rule-to-rule hypothesis. 
4. Sample Grammar 
Our syntactic rules do not depart significantly from 
Gazdar's. The semantic rules formally resemble 
Gazdar's as well, but of course produce conventionally 
interpretable translations of the type described in the 
preceding section. As in Gazdar's semantic rules, 
constituent translations are denoted by primed catego- 
ry 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. The transla- 
tions of individual lexemes are obtained as described 
above. 
In operation, the translator generates the minimum 
number of brackets consistent with the notational 
equivalences stated earlier. For example, in assem- 
bling \[NP' VP'\], with NP' = Johnl and VP t = 
\[loves2 Mary3\], the result is 
\[Johnl loves2 Mary3\], 
rather than 
\[Johnl (loves2 Mary3)\]. 
Also, in binding a variable with lambda, the translator 
replaces all occurrences of the variable with a previ- 
ously unused variable, thus minimizing the need for 
later renaming. Finally, it performs lambda conver- 
sions on the fly. For example, the result of assembling 
\[NP' VP'\] with NP' = Johnl and 
VPV= ?~x\[x shaves2 x\], 
is 
\[Johnl shaves2 Johnl\]. 
The rules that follow have been adapted from Gaz- 
dar (1981a). Note that each rule that involves a lexi- 
cal category such as PN, N or V is accompanied by a 
specification of the subset of lexical items of that cate- 
gory admissible in the rule. This feature is particularly 
important for verb subcategorization. In addition, 
each rule is followed by (a) a sample phrase accepted 
by the rule, (b) an indication of how the logical trans- 
lation of the phrase is obtained, and possibly (c) some 
words of further explanation. 
<I, \[(NP) (PN)\], PN'>, PN(1) = \[John, Mary, New York .... \] 
(a) Mary 
(b) with PN' : Mary6, NP' becomes Mary6. 
<2, \[(AN) (ADJP) (N)\], (ADJP' N')>, N(2) = \[boy, game, noise, 
(a) little boy 
(b) with ADJP' : little2, N' = boy3, 
AN' becomes (little2 boy3); 
(c) "little" is taken as a predicate modifier. \]2 
<3, \[(NP) (Q) (AN)\], <Q' AN'>>, Q(3) : \[a, the, all, many, 
(a) the little boy 
(b) with Q' = thel, AN' = (little2 boy3), 
NP' -> <thel (little2 boy3)>. 
<4, \[(PP to) (to) (NP)\], NP'> 
(a) to Mary 
(b) with NP' = Mary6, PP' -> Mary6; 
(c) PP verb complements have the same meaning as their NP, 
<5, \[(VP) (V)\], V'>, V(5) = {run, smile, disappear .... \] 
(a) smiles 
(b) with V' = smiles4, VP' -> smiles4. 
• . .\] 
...\] 
as per Gazdar (1981a). 
12 Siegel (1979) argues rather persuasively that measure adjec- 
tives, 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 predi- 
cate. 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 35 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
<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' -> (gives4 Mary6 Fido5>. 
<7, \[(VP INF) (to) (VP BASE)\], VP'> 
(a) to give Fido to Mary 
(b) with VP' : (gives4 Mary6 Fido5>, the resultant infinitive has the 
same meaning. 
<8, \[(VP) (V) (VP INF)\], Ax\[x V' \[x VP'\]\]>, 
V(8) : {want, expect, try .... \] 
(a) wants to give Fido to Mary 
(b) with V' : wants2, VP' = {gives4 Mary6 Fido5\], 
VP' -> Ax3\[x3 wants2 \[x3 gives4 Mary6 Fido5\]\]; 
(c) The formal lambda variable x given in the semantic rule has been replaced by 
the new variable x3. Two pairs of square brackets have been deleted, in 
accordance with the simplification rules stated earlier. 
<9, \[(VP) (V) (NP) (VP INF)\], (V' \[NP' VP'\])>, 
V(9) = {want, expect, imagine .... \] 
(a) wants Bill to give Fido to Mary 
(b) with V' : wants2, NP' : Bill3, VP' = (gives~ Mary6 Fido5), 
VP' -> (wants2 {Bill3 gives4 Mary6 Fido5\]). 
<10, \[(S DECL) (NP) (VP)\], \[NP' VP'\]> 
(a) the little boy smiles 
(b) with NP' = <the\] (little2 boy3)> and VP' = smiles4, the result is 
S' -> \[<thel (little2 boy3)> smiles4\]. After pragmatic postprocessing 
to extract quantifiers, the result might be 
S' = (thel x5:\[x5 (little2 boy3)\]) \[x5 smiles4\]. 
Further postprocessing to determine referents and disambiguate operators 
and predicates might then yield 
S' = \[INDIVI7 SMILESl\], 
where INDIV17 is a (possibly new) logical constant unambiguously denoting 
the referent of (the\] x5:\[x5 (little2 boy3)\]) and SMILESl is an unambiguous 
logical predicate. 13 If constant INDIV17 is new, i.e., if the context provided 
no referent for the definite description, a supplementary assertion like 
\[INDIV17 (LITTLE2 BOYI)\] 
would be added to the context representation. 
(a)' John wants to give Fido to Mary 
(b)' with NP' = Johnl, 
VP' : lx3\[x3 wants2 \[x3 gives4 Mary6 FidoS\]\], 
S' -> \[Johnl wants2 \[Johnl gives4 Mary6 FidoS\]\]; 
(c) ' Note that Johnl becomes the subject of both the main clause and the 
embedded (subordinate) clause. 
The reader will observe that we have more or less 
fully traced the derivation and translation of the sen- 
tences "The little boy smiles" and "John wants to give 
Fido to Mary" in the course of the above examples. 
The resultant phrase structure trees, with rule numbers 
and translations indicated at each node, are shown in 
Figs. 1 and 2. 
13 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. 
36 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
rule 1: NP' 
=PN'=dohnl 
PN'=dohnl I 
dohn 
© 
rule 10: S' = \[NP' VP'\] 
= \[dohnl wants2 \[dohnl gives4 Mary6 Fido5\]\] 
rule 8: VP' = Xx\[x V' \[x VP'\]I 
= ~x3\[x3 wants2 \[x3 gives4 Mary6 Fido5\]\] \ 
(vp IN© U 
V' =wants2 I 
wants 
rule 7" (VP INF)'= (VP BASE)' 
= <gives4 Mary6 Fido5> / 
to (_VP BASE) 
rule 6" VP' = ( V' PP' NP' ) 
/~(gives4 Mary6 Fido5> 
(V BASE) 
V' -gives4 
I 
give 
rule 1" NP' =PN' 
= Fido5 
PN' =Fido5 
Fido 
Cpp to) 
rule 4 PP' =NP' 
= Mar't6 / 
rule 1: NP'=PN' 
= Mary6 
I Mary 
Figure 1. Phrase structure and translation of the sentence 
"John wants to give Fido to Mary". 
American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 37 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
rule 
= <the1 
Q'=thel I 
the 
Q 
rule 10: S' = \[NP' VP' \] 
= \[<the1 (little2 boy3)> smiles4\] 
3: NP'= <O' AN' > rule 5: VP' = 
(little2 boy3)> = smiles4 
srniles 
rule 2: AN'= (ADJP' N' ) 
= (little2 boy3) 
rule n: ADdP' = ADd' N' =boy3 
= little2 I 
boy 
ADd'= ittle2 
V, 
little 
Figure 2. Phrase structure and translation of the sentence 
"The little boy smiles." 
38 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
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 exam- 
ple, our semantic rule 8, i.e., Ax\[x V' \[x VP'\]\], may be 
contrasted with the corresponding rule suggested by 
Gazdar: 
XP{P Xx\[(V' A(Vp' XP(P x))) XP(P x)\]}. 
Here the lambda variable x, as in our formula, is used 
to feed a common logical subject to V' (the transla- 
tion 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 dispens- 
es 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 ex- 
traction 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 propa- 
gating 'gaps' from level to level and linking rules for 
creating and filling them. The linking rules are ob- 
tained from the correspondingly numbered basic rules 
by means of the metarule 
\[AXCY\] ==> \[A/BXC/BY\], 
where 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 bas- 
ic categories. The linking rules for topicalization are 
obtained from the rule schemata 
<I I, \[B/B t\] , h>, and 
<12, \[(S) B (S)/B\], <AhS' B')>, 
where B ranges over all basic phrasal categories, and t 
is a dummy element (trace). The first of these sche- 
mata 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 varia- 
ble, 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 particu- 
lar derived rules required to propagate the PP-gap in 
our example and 11 and 12 the particular linking rules 
that create and fill it: 
<11, \[(PP to)/(PP to) 
(a) t 
(b) PP' -> h 
t\], h> 
<6, \[(VP)/(PP to) (V) (NP) (PP to)/(PP to)\], (V' PP' NP')> 
(a) give Fido 
(b) with V' : gives5, NP' : Fido6, PP' = h, 
VP' -> (gives5 h Fido6) 
(c) Note that the semantic rule is unchanged. 
<8, \[(VP)/(PP to) (V) (VP INF)/(PP to)\], Ax\[x V' 
(a) wants to give Fido 
(b) with V' = wants3, VP' = (gives5 h Fido6), 
VP' -> Ax4\[x4 wants3 \[x4 gives5 h Fido6\]\] 
\[x VP'\]\]> 
<10, \[(S)/(PP to) (NP) (VP)/(PP to)\], \[NP' VP'\]> 
(a) John wants to give Fido 
(b) with NP' : John2, 
VP' = Ax4\[x4 wants3 Ix4 gives5 h Fido6\]\], 
S' -> \[John2 wants3 \[John2 gives5 h Fido6\]\] 
<12, 
(a) 
(b) 
(c) 
\[(S) (PP to) (S)/(PP to)\], (AhS' PP')> 
To Mary John wants to give Fido 
With S' as in 10 (b) above and PP' : Maryl, 
S' -> \[John2 wants3 \[John2 gives5 Maryl Fido6\] \] . 
This translation is logically indistinguishable from the 
translation of the untopicalized sentence. However, the 
fronting of "to Mary" has left a pragmatic trace: the 
American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 39 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
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). In subsequent pragmatic 
processing, this feature could be used to detect the special 
salience of Maryl, without re-examination of the superficial 
sentence form. 
Another example of a sentence that can be ana- 
lyzed by such methods, using relative clause rules simi- 
lar to those for topicalization, is 
Every dog Mary wants to buy is small. 
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 
¥y\[\[\[y dog\] & \[Mary wants \[Mary buys y\]\]\] 
=> \[y (small P)\]\], 
where P is an undetermined predicate (= dog, in the 
absence of contrary contextual information). 
As a further illustration of Gazdar's approach and 
how easily it is adapted to our purposes, we consider 
his metarule for passives: 
<\[(VP)(V TRAN) (NP) X\], (St NP")> ==> 
<\[(VP PASS) (V) X {(PP by)}\], ~,p((~r p) pp")>; 
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, fol- 
lowed 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 cor- 
responding to our rule 6 which will account for sen- 
tences such as "Fido was given to Mary by John". 
Moreover, if we introduce a ditransitive rule 
<14, \[ (VP) (V TRAN) (NP) (NP) \] , 
(V' NP' NP')>I4 
to allow for sentences such as "John gave Mary Fido", 
the metarule will generate a passive VP rule that ac- 
counts for "Mary was given Fido by John", in which 
the indirect rather than direct object has been turned 
into the sentence subject. 
The only change needed for our purposes is the 
replacement of the property variable P introduced by 
the metarule by an individual variable x: 
...(~r NP' ) ..... > ...hx((~ r x) PP' )... 
Once the subject NP of the sentence is supplied via 
rule 10, x is replaced by the translation of that NP 
upon lambda conversion. 
Finally in this section, we shall briefly consider 
coordination. Gazdar has supplied general coordina- 
tion rule schemata along with a cross-categorical se- 
mantics 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 
<and John Bill>, <or many few>, 
<and (hugs Mary) (kisses Sue)>, 
echoing the surface forms. However, it should be 
clear from our discussion in Section 2 that direct inter- 
pretation of expressions translating, say, coordinated 
NPs or VPs is not compatible with our conventional 
conception of formal semantics. For example, no for- 
mal semantic value is assigned directly to the coordi- 
nated term in the formula 
\[<and John Bill> loves Mary\]. 
Rather, interpretation is deferred until the pragmatic 
processor has extracted the coordinator from the em- 
bedding sentence (much as in the case of quantified 
14 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. 
40 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
terms) and distributed the coordinated terms over 
duplicate copies of that sentence, yielding 
\[\[John loves Mary\] and \[Bill loves Mary\]\]. 
We adopt the following coordination schemata 
without change. The superscript denotes sequences of 
length > 1 of the superscripted element. The schema- 
ta are accompanied by examples of phrases they admit, 
along with (unindexed) translations. The bracketing 
in (a) and (a)' indicates syntactic structure. 
<15, \[(A ~) (~) (A)\], A'>, 
where A is any syntactic category and ~ E {and, or\] 
(a) and admires 
(b) admires 
(a)' or Mary 
(b)' Mary 
<16, \[(A) (A)+ (A ~)\], <~' A'A'...A'>> 
(a) loves \[and admires\] 
(b) <and loves admires> 
(a)' \[Fido Kim\] \[or Mary\] 
(b)' <or Fido Kim Mary> 
<17, \[(A) (A) (A ~)+\], <~' A'A'...A'>> 
(a) Fido \[\[or Kim\] \[or Mary\]\] 
(b) <or Fido Kim Mary> 
The order in which coordinators are extracted and 
distributed is a matter of pragmatic choice. However, 
a crucial constraint is that multiple occurrences of a 
particular coordinated expression (with particular ind- 
ices) must be extracted and distributed in a single 
operation, at the level of a sentential formula whose 
scope encompasses all of those occurrences (much as 
in the case of quantifier extraction). The following 
examples illustrate this process. 
(a) John loves and admires Fido or Kim 
(b) \[Johnl <and3 loves2 admires4> <or6 Fido5 Kim7>\] -> 
\[\[Johnl loves2 <or6 Fido5 Kim7>\] and3 
\[Johnl admires4 <or6 Fido5 Kim7>\]\] -> 
\[\[\[Johnl loves2 Fido5\] and3 
\[Johnl admires4 Fido5\]\] or6 
\[\[Johnl loves2 Kim7\] and3 
\[Johnl admires4 Kim7\]\]\]. 
(c) 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). 
(a) ' All men want to marry Peggy or Sue 
(b)' \[<alll man2> wants3 \[<alll man2> marries4 <or6 Peggy5 Sue7>\]\]-> 
(alll x: \[x man2\]) Ix wants3 \[x marries4 <or6 Peggy5 Sue7>\]\] -> 
(alll x: \[x man2\]) \[x wants3 
\[\[x marries4 Peggy5\] or6 \[x marries4 Sue7\]\]\]. 
(c) ' In the second step above, the coordinator or6 might instead 
have been raised to the second highest sentential level, yielding 
(alll x: \[x man2\]) \[Ix wants3 Ix marries4 Peggy5\]\] or6 
\[x wants3 \[x marries4 Sue7\]\]\], 
or to the highest sentential level, yielding 
American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 41 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
\[(allq x:\[x man2\]) \[x wants3 \[x marries4 Peggy5\]\] or6 
(alll x:\[x man2\]) \[x wants3 \[x marries4 Sue7\]\]\]. 
The three readings are logically distinct and all are quite 
plausible (in the absence of additional context). 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. 
Finally, we should remark that the distributive rules 
are not appropriate for the group reading of coordi- 
nate structures in sentences such as 
John and Mary carried the sofa (together). 
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). 
5. Parsing 
Phrase structure grammars are relatively easy to 
parse. The most advanced parser for Gazdar-style 
grammars that we are aware of is Thompson's chart- 
parser (Thompson 1981), which provides for slash 
categories and coordination, but does not (as of this 
writing) generate logical translations. We have imple- 
mented two small parser-translators for preliminary 
experimentation, one written in SNOBOL and the 
other in MACLISP. The former uses a recursive de- 
scent algorithm and generates intensional logic transla- 
tions. The latter is a 'left corner' parser that uses our 
reformulated semantic rules to generate conventional 
translations. It begins by finding a sequence of left- 
most 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 han- 
dles slash categories and coordination (although they 
could be handled inefficiently by resort to closure of 
the grammar under metarules and rule schemata). Ex- 
traction of quantifiers from the logical-form transla- 
tions is at present based on the level of syntactic em- 
bedding and left-to-right order alone, and no other 
form of postprocessing is attempted.l 5 
15 Since submission of this paper for publication, we have 
become aware of several additional papers on parser-translators 
similar to ours. One is by Rosenschein & Shieber (1982), another 
by Gawron 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 by McCord (1981, 1982). 
It has been gratifyingly easy to write these parser- 
translators, 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 re- 
suits, 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 imple- 
mented; this work, as well as some ideas on the con- 
ventional translation of negative adjective prefixes, 
plurals, and tense/aspect structure, is reported in 
Schubert (1982). 
6. Concluding Remarks 
From the point of view of theoretical and com- 
putational linguistics, Gazdar's approach to grammar 
offers profound advantages over traditional ap- 
proaches: 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. 
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 moti- 
vate 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 fav- 
our 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 metal- 
inguistic constructs). 
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 compre- 
hensible 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 
42 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 
Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic 
bring a measure of precision and clarity to the rather 
murky area of natural language interpretation by ma- 
chine. 
Acknowledgements 
The authors are indebted to Ivan Sag for a series of 
very stimulating seminars held by him at the Universi- 
ty 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 au- 
thors (LKS) under an Alexander von Humboldt fel- 
lowship in 1978-79. 

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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. 
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. 
