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Language and Logics An
Introduction to the Logical Foundations of Language
Howard Gregory
Edinburgh Advanced
Textbooks in Linguistics Edinburgh: University
Press, 2015 Paperback. viii+315
p. ISBN 978-0748691630. £24.99
Reviewed
by Christian Bassac Université
de Lyon2
Language and
Logics is
probably the most comprehensive textbook in Logics and Linguistics to date. Up
to now, students and instructors alike could mainly rely on Gamut (1990) or
Moot and Rétoré (2012), but the former, albeit a comprehensive account of
Montague semantics, was published 25 years ago and consequently could not include
the rich developments in logics of the post 1990s, and the latter, similar in
scope, concentrates more on categorial and Lambek grammars and is designed for
graduate students. As for Morril (2011), it includes a section on Natural
Language Processing, but it does not cover all the topics developed in Language and Logics. This is to say that
there was a crying need for a textbook that takes undergraduates from the
beginning of logical reasoning to the applications of contemporary logics to
linguistic analysis. In Part one, the classical picture of
logic is developed in four sections, from basic Aristotelian propositional logic
to Lambda Calculus. In these sections the author goes through the usual tree
method for testing the validity of logical formulae, briefly presents predicate
calculus and its interpretation, and offers a brief overview of quantification.
A necessary but brief algebraic background on relations, sets and lattices is
also presented. Part two is devoted to modality with
an introductory section on modal logics and a study of worlds and individuals. Part three takes on the study of
presuppositions, of many-valued logic, and culminates with the presentation of
the important tool known as Curry-Howard isomorphism, central to all logical
approaches to Natural Languages. This correspondence between the domains of
reasoning and typing associates a typed lambda-term of a given type to a proof
of the formula corresponding to this type. In algebraic background two,
structures of monoid and group are presented, and algebraic background three is
a quick note on Heyting algebras as models for the intuitionistic logic of
Brouwer, Heyting and Kolmogorov. The final part is devoted to the study
of Substructural Logics and Categorial grammars, with two important sections on
Linear Logic (from now on LL) and Combinators. Substructural Logics are Logics which
exclude some rules of classical logic. For instance LL, as it considers the
premises in a demonstration as a consumable resource and not as a lemma used in
a proof, rejects the rules of left and right contraction, as well as left and
right weakening (somewhat misleadingly named “W” and “K” respectively in the
book p. 225). One of the many undoubted merits of
the book is that the range of logical topics is wide enough to cover all a
linguistic student needs to know about logic(s). This book takes the reader
well beyond elementary logic, and most logics on the market in 2015 are
presented: the only exception I can think of is Preller’s pregroup grammars,
for instance as expressed in Preller (2007). The book is also carefully written,
each chapter contains useful and interesting notes and the final bibliography
is as comprehensive as a bibliography on such a wide subject can be. This quick
view shows that the book is welcome. However, I found this book disappointing
in several ways. For one thing, some statements are too cryptic for a
linguistic student to feel comfortable with the notion alluded to. Here I think
of note 7 p. 32 which is an interesting opening to the problems of
cryptology and where the reader is expected to know a bit of modulo
arithmetics: all this is very well, but a quick introduction to modulo
arithmetics would have been necessary. What proves even more unfortunate is
first that the deep linguistic motivation for a logical analysis is never
convincingly given, and second that the connection between linguistic problems
and their solutions in the various logical frameworks presented is not given
enough emphasis. Let me go into some detail here. 1) The
motivation for a logical analysis For instance p. 74, where the
following example is given (in exercise 5.1.3.1) (1) Margaret
teaches maths After reduction of the
lambda-expression associated to (1), the semantics of (1) that the student is
required to find is (2): (2) ((teach)maths)(margaret).
It would have been interesting to
compare this with the semantics of the complement constituent of (3): (3) I
wonder what Margaret teaches The classical syntactic analysis of (3) is (4), and its Logical Form is (5): (4) [I
wonder [whati [Margaret [teaches
ti]]]] (5) [I
wonder [whatx [Margaret [teaches
x]]]] In the lambda-reduction process that
yields (2), the lambda-operator binds a variable exactly like the operator-like
what in (5). But never is the lambda-operator moved from a position on its
right, contrary to the operator-like what in (5). Consequently, the semantics
for the sentence in (2) is more straightforward and economical than its
counterpart in (5). This exercise then could have been the ideal place to offer
a strong motivation for all logical analyses presented in the book and to
strongly state the fact that contrary to what goes on in syntactic theories
where the semantics of a sentence is obtained via sequences of structural
operations on strings of words or trees, the syntax and the semantics of a
sentence (or any constituent of a sentence) can be considered as proofs in a
logical demonstration. A few pages should also have been devoted
to a study of the link between mainstream
generative syntax and logical
analyses, in order to show how the
structure building operations Merge and
Move in the Minimalist Program can be
cast in a categorical-like framework. Here, it seems to me that papers by
Amblard et al. (2010), Lecomte & Rétoré
(2001), or Stabler (1997) which all explore a possible way of encoding operations
of minimalist syntax in a Lambek Grammar formalism should have been quoted and
analyzed. 2)
Applications of logic(s) to Natural Languages. Here I have in mind the examples of application
of LL to the syntax and semantics of Natural Languages. Precise linguistic
phenomena should be given an important place in a textbook like this one. For
instance p. 257, the reader is only asked to manipulate logical formulae,
but no linguistic phenomenon is studied. The impression then is that the
presentation remains somewhat cut off from precise syntactic phenomena. Furthermore, in the chapter devoted to
LL, the author states [248] that “implication and fusion (tensor) are the only
connectives normally (emphasis mine) used
in linguistic applications”. The question of course is: what does normally mean? An interesting reason why implication
is widely used is that applications of multiplicative connective “par” are
subsumed by linear implication (which is not a primitive in LL), provided
linear negation is used, as an important theorem of LL states that: A linearly
implies B is equivalent to linear negation of A par B. This should have been clearly stated. Also, the additive disjunctive
connective plus of LL has been used to encode type coercion (cf. Pustejovsky 1995 : 111), and
exponentials (and not additives or multiplicatives only) do have linguistic
applications (provided of course that you are not working with a fragment of LL
without exponentials). For instance exponential “!” can be used to encode the
blocking of extraction from a subject NP, as in (7): (6) [The
book that Jane read [ ]] is long (7) *[The
book that [the story in [ ]]] is long This analysis goes back to Hodas
(1997 : 170) (not in the bibliography), who suggests that in order to
block extraction from the subject NP in (7), the type assignment of NP islands
should be: !np There was room for a deeper study of
this syntactic phenomenon, all the more so as the issue is evoked (lightly) later
on [275]. It cannot be said that lexical ambiguity or extraction from an NP are
marginal linguistic phenomena. What the author means by “normally” remains
mysterious to me then. Probably the scope and the ambitious
purpose of the book prevented the author from devoting the necessary space to
other important topics. For instance,
what I found missing in the linear logic section is the link between a formula
with linear implication and classical/intuitionistic logic implication. The translation
formula is given [251] but what the fundamental opposition between the two
means is not provided. A word here would have been in order to show that
classical/intuitionistic logic is perfect for mathematical theorems (they
express stable truths), but inadequate when it comes to causal implication, as
causal implication cannot be iterated. This is implicit (for instance p. 248),
when the author states that “linear implication can be read as a process which
consumes (one instance of) p to produce (one instance of) q”. But as this is a
crucial motivation for the emergence of LL, this again should have been made
more explicit. What is missing too is a few pages on
models for LL. Whereas models for other logical frameworks are presented
(monoids, groups, Heyting algebra), no model is given for the interpretation of
LL and I think a word on coherent spaces would have been in order here. I must add that I find the development
in (3.16) p. 51 a bit confusing and incomplete. It is confusing, as the
formal definition of a homomorphism is not given, and the only formal
definition that appears is that of an isomorphism (modulo the condition that h in the example given is an onto). It
is incomplete, as the next step should have been to give the definition of an
automorphism here. The notion of
automorphism is presented only p. 84, with no formal definition and no
formal connection with the definition of a homomorphism. It would have been
more coherent and convenient to show that it is a particular case of an
isomorphism, right from 3.16. Aside from the criticism expressed
above I must say that I consider that there is much to recommend this book,
which presents a wealth of logical material in a clear way. References Gamut, L.T.F (1990),
Language, Logic and Meaning.
University of Chicago Press. Hodas, Joshua
(1997), “A Linear Logic treatment of Phrase Structure Grammars for unbounded
dependencies”. In Lecomte, A.,
Lamarche, F., Perrier G., (Eds) Logical
Aspects of Computational Linguistics, Second conference, Nancy, France, September 1997.
Berlin : Springer : 160-179. Moot Richard & Rétoré, Christian (2012), The Logic
of Categorial Grammars : A Deductive Account of Natural Language Syntax
and Semantics, Springer. Lecomte, Alain
& Rétoré, Christian (2001), “Extending Lambek grammars : A logical account
of minimalist grammars”. Proceedings of the
39th Annual Meeting of the Association for Computaional Linguistics : 354-361. Morril, Glynn
(2011), Categorial Grammar : Logical
Syntax, Semantics, and Processing. Oxford: University Press. Preller,
Anne (2007), “Linear Processing with Pregroup Grammars”. In W. Buszkowski et al., Eds, Studia
Logica 87 2/3 : 171-197. Pustejovsky,
James (1995), The Generative Lexicon. Cambridge: The MIT Press. Stabler,
Edward (1997), “Derivational Minimalism”. Logical
Aspects of Computational Linguistics, Vol. 1328 : 68-95.
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