Moortgat, M., Categorial type logics, in: J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, Elsevier, 1997 .  Home/Search   Document Details and Download   Summary   Related Articles   Check

....calculus are also valid in pregroups under the above interpretation of residuals anb and a=b. The converse is not true even for product free types [5] Grammars based on the Lambek calculus are a kind of categorial grammars, extensively studied nowadays as a framework for deductive parsing (see [14, 15, 3]) Lambek [12] proposes alternative parsing strategies, based on the calculus of free pregroups (also see [2, 7] We discuss these matters in section 3. Using (1) one easily proves the following equalities hold in every pregroup: 1 = 1; a lr rl = a; ab) ab) 3) ....

M. Moortgat, Categorial Type Logics, in [17], 93-177.

....we show that for Nonassociative Lambek Calculus the situation is di erent: all systems of Nonassociative Lambek Calculus with nitely many nonlogical axioms are decidable in polynomial time and generate context free languages. The same holds for systems with unary modalities, advocated by Moortgat [20, 21], n ary operations (i.e. for the Generalized Lambek Calculus, introduced in [5] and studied in e.g. 15, 13, 10] and the rule of permutation [12, 10] These results are new; they do not rely on cut elimination which is not available for systems with nonlogical axioms. Further, our results from ....

....description. For example, Lambek [18, 19, 7] uses axioms of the form i to express the inclusion of the class of personal pronouns in i th Person in the class of personal pronouns; di erent kinds of subcategorization can be found in Keenan and Faltz [14] More in the style of Moortgat [21], one might take Nonassociative Lambek Calculus as the basic logic and, besides modalities, use axioms of the form (A B) C A (B C) or A B B A, for some concrete types A; B; C, to admit associativity or permutation in some special cases. A limited usage of contraction can also be helpful: ....

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M. Moortgat, Categorial Type Logics, in: J. van Benthem and A. ter Meulen (eds.), Handbook of Logic and Language, Elsevier, Amsterdam, The MIT Press, Cambridge Mass., 1997, 93-177.

....shown below conform to the logical style of Gentzen s [4] sequent calculus. Definition 2. 2 (Axiom) A #[A] #[#] Cut) #[ A B #) L) #[ # B A) L) # B) A B ( R) B #) B A ( R) Extensions of NL have been developed in the literature [11,13] which employ unary operators that are analogous to those found in modal logics. Here are the sequent rules of inference governing the unary modal operators, using their corresponding structural operator. #[#A#] #[#A] #L) #[A] #[## # A#] # # L) #A (#R) # # A (# # R) ....

....A general algorithm has been developed [3] for learning a syntactic category system for a natural language together with a lexical assignment that determines a completely descriptive grammar. The syntax can in principle be handled using any cut free decidable multimodal type logical grammar [11]. The learning data consists of term labeled strings, i.e. sentences annotated by typed lambda calculus meaning recipes with either variable semantic types on the subterms, or with no types on the subterms. The lambda calculus is used in a standard fashion to model the compositional meaning ....

Moortgat, M., Categorial type logics, in: J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, Elsevier, 1997 .

....carlos science.uva.nl Raffaella Bernardi UiL OTS, University of Utrecht, Raffaella.Bernardi let.uu. nl ABSTRACT: Even though residuation is at the core of Categorial Grammar [14] it is not always immediate to realize how standard logical systems like Multi modal Categorial Type Logics (MCTL) [20] actually embody this property. In this paper, we focus on the basic system NL [15] and its extension with unary modalities NL(3) 19] and we spell things out by means of Display Calculi (DC) 5, 13] The use of structural operators in DC permits a sharp distinction between the core properties we ....

....from the types assigned to its lexical components. In addition to this pure proof theoretical use of categorial systems, there is also an important semantic byproduct: the interpretation of (formally, the lambda term associated to) a linguistic expression can be obtained while inferring its type [20]. This connection offers a rich framework where linguistic issues can be investigated from all three different points of view: purely syntactic checking of type composition, the compositional meaning of the linguistic expression, and their interface. Actually, the CTL approach can be understood as ....

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M. Moortgat. Categorial Type Logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, pages 93--178. The MIT Press, Cambridge, Massachusetts, Cambridge, 1997.

.... Lambek in [Lam58] shows that CG besides the functional operators n and = contains the composition connective . Making this explicit gives a logic system, the Lambek calculus (L) van Benthem in [vB88] proves that L and the lambda calculus are CurryHoward isomorphic. Moortgat in [Moo96, Moo97] shows that L contains unary operators as well 3; 2 . Making this explicit we obtain NL(3) Similarly, Areces and Bernardi in [AB01] show that the same principle which characterizes n, can give their unary counterpart, The full system is referred to as NL(3, and studied ....

M. Moortgat. Categorial Type Logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, pages 93-178. The MIT Press, Cambridge, Massachusetts, Cambridge, 1997.

....and semantics in logical systems. By working out the details of the distinction where the connection is close, we help avoiding the confusion between these levels in other cases as well. The link between logic and linguistics is at the heart of Categorial Grammar frameworks as shown by Lambek [5] [7]. We propose to teach the Lambek calculus to students attending logic courses. The Lambek calculus has interesting logical properties with di erent levels of diculty. It o ers quite a broad range of logical topics, from the more elementary one, such as modus ponens and hypothetical reasoning, to ....

M. Moortgat. Categorial Type Logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language. Elsevier, 1997.

....and one which extends I am grateful to Richard Oehrle for his helpful suggestions on this work. the power of the structural component allowing to permute and change the order of the composed structures. Following the rst solution we obtain Multimodal (non associative) Lambek Calculus (MMNL) [Moortgat,1997], whereas adding permutation and associativity we reach Multiplicative Linear Logic (MLL) Danos and Regnier,1989] The aim of this paper is to show the advantages of having unary operators at our disposal when reasoning with linguistic structures. In particular, we will look at expressions, ....

....(mary works) s, since mary has type np and works has type npns. In a logical approach to natural language an important goal is to assign types which can be logically related to each other. In particular, when reasoning with PIs phenomena, the following logical implications, discussed in [Moortgat,1997], will be used: 1a. Application: A (AnB) B (A=B) B ) A 2a. Co application: A ) Bn(B A) A ) A B) B 3a. Monotonicity n; if A ) B and C ) D, then (DnA) CnB) if A ) B and C ) D, then (A=D) B=C) where, for example, the rst theorem means that if a linguistic structure has ....

M. Moortgat. Categorial Type Logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, pages 93-178. The MIT Press, Cambridge, Massachusetts, Cambridge, 1997.

.... of simplicity we abbreviate the logical types using their corresponding sentential question level: q 1 q 2 q 3 where the relation between types of different levels is the logical derivability relation discussed above; and the one between sentences and questions is the lifting theorem [19]. Hence, q 1 stands for s 1 (s 1 1 ) which in turn abbreviates ## # s (## # s ## # s) and q 2 stands for s 2 (s 2 2 ) viz. s (s s) Reading out this logical types, a yes no question is seen as a function which takes a sentential modifier and yields a sentence. As might be clear, the two ....

M. Moortgat. Categorial Type Logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, pages 93--178. The MIT Press, Cambridge, Massachusetts, Cambridge, 1997.

....to belong to the domain of formal semantics. Most of the literature in natural reasoning assumes a model theoretic perspective and uses a formal language as an intermediate step into which natural language expressions are translated. However, within the generalized type logical framework [Moo97] the challenge is to assume a direct proof theoretic perspective; this entails that instead of employing logical forms as vehicles of inference, natural language expressions are used directly, and instead of taking models into account, the validity of an inference is read o the derivation. A ....

....base logic, with its Introduction, Elimination rules for the logical constants, and their Curry Howard interpretation captures invariants of grammatical composition. The structural packages make it possible for the form meaning correspondence to be realized in di erent ways across languages. See [Moo97] for more details. We will use the 3, 2 , and the structural rules for these operators to propagate the monotonicity markers and to calculate the polarity of the single nodes step by step in the process of dynamically building a proof, so that the output sentence will have its polarity already ....

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M. Moortgat. Categorial Type Logics. In Handbook of Logic and Language, pages 93-178. J. van Benthem and A. ter Meulen, Cambridge, 1997.

....Natural Logic which derives (monotonicity) inferences using as vehicle the parsed output. The monotonicity markers assigned in the lexicon are propagated through the proofs via a combination of the structural and the logical rules for the unary operators of Multimodal Categorial Grammar (MMCG) [Moo97]. We have chosen to work with an expressive grammar logic , in order to avoid the use of extra logical marking devices and extra logical structural reasoning. Having MMCG as parser, our system is able to make the derivations simply within the logic. This new approach makes the implementation of ....

....position. As we will better explain in the last section a controlled form of [P1] will solve the problems. 2.2 Multimodal Categorial Grammar In the last decade new versions of the formalism we have described, has been developed. In this paper we work with Multimodal Categorial Grammar (MMCG) [Moo97] which overcomes some expressive limitations of the Lambek calculus. Following the insight that languages are different from each other basically in the way they structurally realize the form meaning correspondence, MMCG consists of two independent parts: i) a base logic, in which no structural ....

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M. Moortgat. Categorial Type Logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, pages 93--178. The MIT Press, Cambridge, Massachusetts, Cambridge, 1997.

....grammars from function argument structures in Gold s model. Section 4 concludes. 2 Background 2. 1 Categorial Grammars The reader not familiar with Lambek Calculus and its non associative version will find nice presentation in the first articles written by Lambek [14,15] or more recently in [13,1,5,16,7,8]. We use in the paper non associative Lambek calculus without empty sequence and without product. Types. The types Tp, or formulas, are generated from a set of primitive types Pr, or atomic formulas, by two binary connectives (over) and (under) Tp : Pr Tp Tp Tp Tp Rigid and ....

Moortgat, M., Categorial type logic, in: van Benthem and ter Meulen [21] pp. 93--177.

....carlos science.uva.nl Ra aella Bernardi UiL OTS, University of Utrecht Raffaella.Bernardi let.uu. nl Abstract Even though residuation is at the core of Categorial Grammar [11] it is not always immediate to realize how standard logic systems like Multi modal Categorial Type Logics (MCTL) [17] actually embody this property. In this paper we focus on the basic system NL [12] and its extension with unary modalities NL(3) 16] and we spell things out by means of Display Calculi (DC) 3, 10] The use of structural operators in DC permits a sharp distinction between the core ....

....for this task, being able to handle di erent CTL s. In addition to this pure proof theoretic use of categorial systems, there is also an important semantic byproduct: the interpretation of (formally, the lambda term associated to) a linguistic expression can be obtained while inferring its type [17]. This connection o ers a rich framework where linguistic issues can be investigated from all three di erent points of view: purely syntactic checking of type composition, the compositional meaning of the linguistic expression, and their interface. Actually, the CTL approach can be understood as ....

[Article contains additional citation context not shown here]

M. Moortgat. Categorial Type Logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, pages 93-178. The MIT Press, Cambridge, Massachusetts, Cambridge, 1997.

....above, we consider objects being associated with tuples PHON, UNINT, SEM . The most natural solution for including semantical features is to treat them as another kind of labels on which we shall return later on. Borrowing from works inside the categorial or the type theoretical framework ([11], 14] 13] we simply 11 assume that semantical representations are given by proofs 7 . We can now see that the object that we build step by step in such a deductive system and was assumed to be of no use on the syntactic side (except for keeping track of the order of hypotheses) is in fact ....

....As a matter of fact, because we know a head movement can cross a phrasal one, that can be too strong a constraint. It is therefore necessary to introduce a kind of structural rule which allows to perform associativity w.r.t. and , a rule that we name mixed associativity, along the lines of [11], 9] etc. Mixed Associativity rule: Theta[ Gamma; Delta; Delta 0 ) A [MA] Theta[ Gamma; Delta) Delta 0 ) A We can then get the following deduction: p : k Omega d y 2 : k y 2 : k likes : knt) vp) ffl ( kn(dnvp) d) subproof [fflE] x 2 : d likes x ....

M. Moortgat. Categorial type logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, chapter 2, pages 93--178. Elsevier, 1997.

....of linguistic phenomena when interested either in its syntactic or semantic properties. These two aspects of natural language are strictly connected in the logical system we employ as parser, viz. Multimodal Categorial Logic (MMCL) a type logic suitable for reasoning with linguistic signs [Moo97]. A well known property of this logic, and of the type logic family in general, is the Curry Howard correspondence between proofs and lambda terms [How80, Ben88] Thanks to this relation, proofs of the grammaticality of a string correspond to lambda terms. We will use this property to give an ....

....about natural language, scope ambiguity phenomena must be taken into consideration. We can satisfy this requirement by using a logical theorem prover as a parser which produces the di erent possible meanings of a sentence. Type logics are systems suitable to reasoning with linguistic signs [Moo97]. We have chosen Multimodal Categorial Logic (MMCL) as the system to describe ambiguity phenomena because we believe it has the right kind of expressiveness for the topics we wish to cover. In this section we rst brie y present the system, and then we show how we can infer the linguistic data ....

[Article contains additional citation context not shown here]

M. Moortgat. Categorial Type Logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, pages 93-178. The MIT Press, Cambridge, Massachusetts, 1997.

....These make it possible to dynamically relate structures during on line computation, or to establish o# line lexical generalizations. We report on the initial experiments in [15] to apply this method in the context of the Spoken Dutch Corpus. For the general type logical background, we refer to [12]; 1hasabrief recap of some key features. 1 Constants and variation One can think of type logical grammar as a functional programming language with some special purpose features to customize it for natural language processing tasks. Basic constructs are demonstrations of the form # A, ....

Moortgat, M., `Categorial type logics'. Chapter 2 in Van Benthem and ter Meulen (eds.) Handbook of Logic and Language. Elsevier, 1997, pp. 93--177.

....and describe a typology of polarity items. In this way we improve on the analysis of [4] which was given in terms of the residuated operators 3; 2 . 2 Formal preliminaries 2. 1 Axiomatic presentation, completeness There are two ways to extend the standard axiomatic presentation of NL(3) see [19]) with Galois operators. The system NL(3, can be obtained by extending NL(3) with the axioms (A1) A2) and the rules (R1) R2) below. It is easy to show that (GC) is a derived rule in this setting. Alternatively, one adds (GC) to NL(3) It can be shown then that (A1) A2) and the rules ....

Moortgat, M., Categorial type logics, in: J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, The MIT Press, Cambridge, Massachusetts, Cambridge, 1997 pp. 93-178.

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Moortgat, M., Categorial type logics, in: J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, Elsevier, 1997 .

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M. Moortgat. Categorial type logics. In Handbook of Logic and Language, pages 93-178. J. van Benthem and A. ter Meulen, Cambridge, 1997.

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Moortgat, M. (1997), `Categorial Type Logics', in J. van Benthem and A. ter Meulen (eds.), Handbook of Logic and Language, Elsevier, Amsterdam, 93-177.

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M. Moortgat. Categorial Type Logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, pages 93--178. The MIT Press, Cambridge, Massachusetts, Cambridge, 1997.

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M. Moortgat. Categorial Type Logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, chapter 2. Elsevier, 1996.

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M. Moortgat. Categorial Type Logics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, chapter 2. Elsevier, 1996.

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M. Moortgat, Categorial type logic, in: van Benthem and ter Meulen [30], Ch. 2, pp. 93--177.

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M. Moortgat, Categorial Type Logics, in: J. van Benthem and A. ter Meulen (eds.), Handbook of Logic and Language, Elsevier, Amsterdam, 1997, 93-177.

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Moortgat, Categorial type logics. In Handbook of Logic and Language, Elsevier /MIT Press.

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