Introduction
Lambek introduced his homonymous calculus (originally called the ‘Syntactic Calculus’) for proposed applications in Linguistics. However the calculus got much of its cult following and reputation by being a convenient, wellbehaved prototype of a Gentzen sequent calculus without any structural rules.
This note recalls a Dialectica model of the Lambek Calculus presented by the first author in the Amsterdam Colloquium in 1991 [10]
. Here, like then, we approach the Lambek Calculus from the perspective of Linear Logic, so we are interested in the basic sequent calculus with no structural rules, except associativity of tensors. In that earlier work we took for granted the syntax of the calculus and only discussed the exciting possibilities of categorical models of linearlogiclike systems. Many years later we find that the work on models is still interesting and novel, and that it might inform some of the most recent work on the relationship between categorial grammars and notions of distributional semantics
[8].Moreover, using the Agda proof assistant [6] we verify the correctness of our categorical model (Section 4.1), and we add the type theoretical (Section 5) notions that were left undiscussed in the Amsterdam Colloquium presentation. All of the syntax in this paper was checked using Ott [23]. The goal is to show that our work can shed new light on some of the issues that remained open. Mostly we wanted to check the correctness of the semantic proposals put forward since Szabo’s seminal book [24] and, for future work, on the applicability and fit of the original systems to their intended uses.
Overview. The Syntactic Calculus was first introduced by Joachim Lambek in 1958 [19]. Since then the rechristened Lambek Calculus has had as its main motivation providing an explanation of the mathematics of sentence structure, starting from the author’s algebraic intuitions. The Lambek Calculus is the core of logical Categorial Grammar. The first use of the term “categorial grammar” seems to be in the title of BarHillel, Gaifman and Shamir (1960), but categorial grammar began with Ajdukiewicz (1935) quite a few years earlier. After a period of ostracism, around 1980 the Lambek Calculus was taken up by logicians interested in Computational Linguistics, especially the ones interested in Categorial Grammars.
The work on Categorial Grammar was given a serious impulse by the advent of Girard’s Linear Logic at the end of the 1980s. Girard [12] showed that there is a full embedding, preserving proofs, of Intuitionistic Logic into Linear Logic with a modality “!”. This meant that Linear Logic, while paying attention to resources, could always code up faithfully Classical Logic and hence one could, as Girard put it, ‘have one’s cake and eat it’, paying attention to resources, if desired, but always forgetting this accounting, if preferred. This meant also that several new systems of resource logics were conceived and developed and these refined resource logics were applied to several areas of Computer Science.
In Computational Linguistics, the Lambek Calculus has seen a significant number of works written about it, apart from a number of monographs that deal with logical and linguistic aspects of the generalized typelogical approach. For a shorter introduction, see Moortgat’s entry on the Stanford Encyclopedia of Philosophy on Type Logical Grammar [20]. Type Logical Grammar situates the typelogical approach within the framework of Montague’s Universal Grammar and presents detailed linguistic analyses for a substantive fragment of syntactic and semantic phenomena in the grammar of English. Type Logical Semantics offers a general introduction to natural language semantics studied from a typelogical perspective.
This meant that a series of systems, implemented or not, were devised that used the Lambek Calculus or its variants as their basis. These systems can be as expressive as Intuitionistic Logic and the claim is that they are more precise i.e. they make finer distinctions. Some of the landscape of calculi can be depicted as follows:
From the beginning it was clear that the Lambek Calculus is the multiplicative fragment of noncommutative Intuitionistic Linear Logic. In the diagrams stands for the Lambek Calculus, as expounded in [19] but with the unit added for the tensor connective (there was a certain amount of dispute on that, as the original system did not introduce the constant corresponding to the nullary case of the tensor product, here written as ). The system is the Lambek Calculus with permutation, sometimes called the van Benthem calculus. We are calling the Lambek Calculus with additives, the more usual algebraic connectives corresponding to meet and join. Hence adding both permutation and additives to the Lambek Calculus we get to intuitonistic linear logic. On the other diagram to the Lambek Calculus with permutation we add either weakening () or contraction ) or both to get the implicational fragment of Intuitionsitic Propositional Logic.
The Lambek Calculus also has the potential for many applications in other areas of computer science, such as, modeling processes. Linear Logic has been at the forefront of the study of process calculi for many years [14, 22, 1]. We can think of the commutative tensor product of linear logic as a parallel operator. For example, given a process and a process , then we can form the process which runs both processes in parallel. If we remove commutativity from the tensor product we obtain a sequential composition instead of parallel composition. That is, the process first runs process and then process in that order. Paraphrasing Vaughan Pratt, “The sequential composition operation has no evident counterpart in type theory” see page 11 of [22]. We believe that the Lambek Calculus will lead to filling this hole, and the results of this paper as a means of obtaining a theory with both a parallel operator and a sequential composition operator. This work thus has a potential to impact research in programming languages and computer security where both linear logic and sequential composition play important roles.
There are several interesting questions, considered for Linear Logic, that could also be asked of the Lambek Calculus or its extensions. One of them, posed by Morrill et al is whether we can extend the Lambek Calculus with a modality that does for the structural rule of (exchange) what the modality of course ‘!’ does for the rules of (weakening) and (contraction). A preliminary proposal, which answers this question affirmatively, is set forward in this paper. The answer was provided in semantical terms in the first version of this work. Here we provide also the more syntactic description of these modalities. Building up from work of Ciabattoni, Galatos and Terui in [7] and others that describe how to transform systems of axioms into cutfree sequent rules, we aim to refine the algebraization of proof theory.
1 Related Work
Lamarche and Retoré [18] give an overview of proof nets for the Lambek Calculus where they discuss the addition of various exchange rules to the calculus. For example, the following permutation and cycle rules:
Taken in isolation each rule does not imply general exchange, but taken together they do. Thus, it is possible to take a restricted notion of exchange to the Lambek Calculus without taking general exchange. However, applications where one needs a completely noncommutative tensor product and a commutative tensor product cannot be modeled in the Lambek Calculus with these rules.
Polakow and Pfenning [21] combine intuitionistic, commutative linear, and noncommutative linear logic into a system called Ordered Linear Logic (OLL). Polakow and Pfenning then extend OLL into two new systems: a term assignment of OLL called OLLi, and a logical framework in the tradition of LF called OLF.
OLL’s sequent is of the form where is a multiset of intuitionistic assumptions, is a multiset of commutative linear assumptions, and is a list of noncommutative linear assumptions. Furthermore, OLL contains logical connectives from each logic. For example, there are two different tensor products and three different implications. Thus, the systems developed here are a simplification of OLL showing how to get all of these systems using modalities.
Greco and Palmigiano [13] give a type of logic called a proper display logic for the Lambek Calculus with exponentials. However, they decompose the linear exponential into an adjunction in the spirit Benton’s LNL models. In this paper we concentrate on sequent calculi rather than display logic.
2 The Lambek Calculus
The Lambek Calculus, formerly the Syntactic Calculus , due to J. Lambek [19], was created to capture the logical structure of sentences. Lambek introduced what we think of as a substructural logic with an operator denoting concatenation, , and two implications relating the order of phrases, and . The first implication corresponds to a phrase of type when followed by a phrase of type , and the second is a phrase of type when preceded by a phrase of type .
The Lambek Calculus, defined in Figure 1, can be presented as a noncommutative intuitionistic multiplicative linear logic. Contexts are sequences of formulas, and we denote mapping the modalities over an arbitrary context by and .
Because the operator denotes the type of concatenations the types and are not equivalent, and hence, L is noncommutative which explains why implication must be broken up into two operators and . In the following subsections we give two extensions of L: one with the wellknown modality ofcourse of linear logic which adds weakening and contraction, and a second with a new modality adding exchange.
3 Extensions to the Lambek Calculus
The linear modality, , read “ofcourse ” was first proposed by Girard [12] as a means of encoding nonlinear logic in both classical and intuitionistic forms in linear logic. For example, nonlinear implication is usually encoded into linear logic by . Since we have based L on noncommutative intuitionistic linear logic it is straightforward to add the ofcourse modality to L. The rules for the ofcourse modality are defined by the following rules:
The rules C and W add contraction and weakening to L in a controlled way. Then the other two rules allow for linear formulas to be injected into the modality; and essentially correspond to the rules for necessitation found in S4 [5]. Thus, under the ofcourse modality the logic becomes nonlinear. We will see in Section 4.1 that these rules define a comonad. We call the extension of L with the ofcourse modality .
As we remarked above, one leading question of the Lambek Calculus is: can exchange be added in a similar way to weakening and contraction? That is, can we add a new modality that adds the exchange rule to L in a controlled way? The answer to this question is positive, and the rules for this new modality are as follows:
The first two rules are similar to ofcourse, but the last two add exchange to formulas under the modality. We call L with the exchange modality . Thus, unlike intuitionistic linear logic where any two formulas can be exchanged, restricts exchange to only formulas under the exchange modality. Just like ofcourse the exchange modality is modeled categorically as a comonad; see Section 4.1.
4 Categorical Models
We now turn to the categorical models, ones where one considers different proofs of the same theorem. Since the Lambek Calculus itself came from its categorical models, biclosed monoidal categories, there is no shortage of these models. However, Girard’s insight of relating logical systems via modalities should also be considered in this context.
Lambek’s work on monoidal biclosed categories happened almost three decades before Girard introduced Linear Logic, hence there were no modalities or exponentials in Lambek’s setting. The categorical modelling of the modalities (ofcourse! and whynot?) was the difficult issue with Linear Logic. This is where there are design decisions to be made.
4.1 Dialectica Lambek Spaces
A sound and complete categorical model of the Lambek Calculi can be given using a modification of de Paiva’s dialectica categories [9]. Dialectica categories arose from de Paiva’s thesis on a categorical model of Gödel’s Dialectica interpretation, hence the name. Dialectica categories were one of the first sound categorical models of intuitionistic linear logic, with linear modalities. We show in this section that they can be adapted to become a sound and complete model for the Lambek Calculus, with both the exchange and ofcourse modalities. We call this model dialectica Lambek spaces.
Due to the complexities of working with dialectica categories we have formally verified^{1}^{1}1The complete formalization can be found online at https://github.com/heades/dialecticaspaces/blob/Lambek/NCDialSets.agda. this section in the proof assistant Agda [6]. Dialectica categories arise as constructions over a given monoidal category. Suppose is such a category. Then in complete generality the objects of the dialectica category over are triples where and are objects of , and is a subobject of the tensor product in of and . Thus, we can think of as a relation over . If we specialize the category to the category of sets and functions, , then we obtain what is called a dialectica space. Dialectica spaces are a useful model of full intuitionistic linear logic [15].
Morphisms between objects and are pairs where and are morphisms of such that the pullback condition holds. In dialectica spaces this condition becomes . The latter reveals that we can think of the condition on morphisms as a weak form of an adjoint condition. Finally, through some nontrivial reasoning on this structure we can show that this is indeed a category; for the details see the formal development. Dialectica categories are related to the Chu construction [11] and to Lafont and Streicher’s category of games [17].
To some extent the underlying category controls the kind of structure we can expect in the dialectica category over . However, de Paiva showed [11] that by changing the relations used in the objects and the order used in the ‘adjoint condition’ (which also controls the type of structure) we can obtain a nonsymmetric tensor in the dialectica category, if the structure of the underlying category and the structure of the underlying relations are compatible. She also showed that one can abstract the notion of relation out as a parameter in the dialectica construction, so long as this has enough structure, i.e. so long as you have an algebra (that she called a lineale) to evaluate the relations at. We denote this construction by where is the lineale controlling the relations coming from the monoidal category . For example, is the category of usual dialectica spaces of sets over the Heyting (or Boolean) algebra .
This way we can see dialectica categories as a framework of categorical models of various logics, varying the underlying category as well as the underlying lineale or algebra of relations . Depending on which category we start with and which structure we use for the relations in the construction we will obtain different models for different logics.
The underlying category we will choose here is the category , but the structure we will define our relations over will be a biclosed poset, defined in the next definition.
Definition 1
Suppose is an ordered noncommutative monoid. If there exists a largest such that for any , then we denote by and called it the leftpseudocomplement of w.r.t . Additionally, if there exists a largest such that for any , then we denote by and called it the rightpseudocomplement of w.r.t .
A biclosed poset, , is an ordered noncommutative monoid, , such that and exist for any .
Now using the previous definition we define dialectica Lambek spaces.
Definition 2
Suppose is a biclosed poset. Then we define the category of dialectica Lambek spaces, , as follows:

objects, or dialectica Lambek spaces, are triples where and are sets, and is a generalized relation over , and

maps that are pairs where , and are functions such that the weak adjointness condition holds.
Notice that the biclosed poset is used here as the target of the relations in objects, but also as providing the order relation in the weak adjoint condition on morphisms. This will allow the structure of the biclosed poset to lift up into .
We will show that is a model of the Lambek Calculus with modalities. First, we show that it is a model of the Lambek Calculus without modalities. Thus, we must show that is monoidal biclosed.
Definition 3
Suppose and are two objects of . Then their tensor product is defined as follows:
where is the function space from , and .
The identity of the tensor product just defined is , where is the terminal object in , and is the unit of the biclosed poset. It is straightforward to show that the tensor product is functorial, one can define the left and right unitors, and the associator for tensor; see the formalization for the definitions. In addition, all of the usual monoidal diagrams hold [9]. Take note of the fact that this tensor product is indeed noncommutative, because the noncommutative multiplication of the biclosed poset is used to define the relation of the tensor product.
The tensor product has two right adjoints making biclosed.
Definition 4
Suppose and are two objects of . Then two internalhoms can be defined as follows:
These two definitions are functorial, where the first is contravariant in the first argument and covariant in the second, but the second internalhom is covariant in the first argument and contravariant in the second. The relations in the previous two definitions prevent these two from collapsing into the same object, because of the use of the left and right pseudocomplement. It is straightforward to show that the following bijections hold:
Therefore, is biclosed, and we obtain the following result.
Theorem 4.1
is a sound and complete model for the Lambek Calculus without modalities.
We now extend with two modalities: the usual modality, ofcourse, denoted , and the exchange modality denoted . However, we must first extended biclosed posets to include an exchange operation.
Definition 5
A biclosed poset with exchange is a biclosed poset equipped with an unary operation satisfying the following:
Compatibility results in being a functor in the biclosed poset, and the remainder of the axioms imply that is a comonad extending the biclosed poset with left and right exchange.
We can now define the two modalities in where is a biclosed poset with exchange; clearly we know is also a model of the Lambek Calculus without modalities by Theorem 4.1 because is a biclosed poset.
Definition 6
Suppose is an object of where is a biclosed poset with exchange. Then the ofcourse and exchange modalities can be defined as and where is the free commutative monoid on , for , and .
This definition highlights a fundamental difference between the two modalities. The definition of the exchange modality relies on an extension of biclosed posets with essentially the exchange modality in the category of posets. However, the ofcourse modality is defined by the structure already present in , specifically, the structure of .
Both of the modalities have the structure of a comonad. That is, there are monoidal natural transformations , , , and which satisfy the appropriate diagrams; see the formalization for the full proofs. Furthermore, these comonads come equipped with arrows , , , and . Thus, we arrive at the following result.
Theorem 4.2
Suppose is a biclosed poset with exchange. Then is a sound and complete model for the Lambek Calculi , , and .
5 Type Theory for Lambek Systems
In this section we introduce typed calculi for each of the logics discussed so far. Each type system is based on the term assignment for Intuitionistic Linear Logic introduced in [2]. We show that they are all strongly normalizing and confluent, but we do not give full detailed proofs of each of these properties, because they are straightforward consequences of the proofs of strong normalization and confluence for intuitionistic linear logic. In fact, we will reference Bierman’s thesis often within this section. The reader may wish to review Section 3.5 on page 88 of [4].
5.1 The Typed Lambek Calculus:
The first system we cover is the Lambek Calculus without modalities. This system can be seen as the initial core of each of the other systems we introduce below, and thus, we will simply extend the results here to three other systems.
The syntax for patterns, terms, and contexts are described by the following grammar:
Contexts are sequences of pairs of free variables and types. Patterns are only used in the letexpression which is itself used to eliminate logical connectives within the left rules of L. All variables in the pattern of a letexpression are bound. The remainder of the terms are straightforward.
The typing rules can be found in the in Figure 2 and the reduction rules in Figure 3. The typing rules are as one might expect. The reduction rules were extracted from the cutelimination procedure for L.
We denote the reflexive and transitive closure of the by . We call a term with no redexes a normal form, and we denote normal forms by . In the interest of space we omit the congruence rules from the definition of the reduction relation; we will do this for each calculi introduced throughout this section. The other typed calculi we introduce below will be extensions of , thus, we do not reintroduce these rules each time for readability.
Strong normalization. It is well known that intuitionistic linear logic (ILL) is strongly normalizing, for example, see Bierman’s thesis [4] or Benton’s beautiful embedding of ILL into system F [3].
It is fairly straightforward to define a reduction preserving embedding of into ILL. Intuitionistic linear logic can be obtained from by replacing the rules T_IRl, T_ILl, T_IRr, and T_ILr with the following two rules:
In addition, contexts are considered multisets, and hence, exchange is handled implicitly. Then we can reuse the idea of Benton’s embeddings to show type preservation and type reduction.
At this point we define the following embeddings.
Definition 7
We embed types and terms of into ILL as follows:
The previous embeddings can be extended to contexts in the straightforward way, and to sequents as follows:
We can now prove strong normalization using the embedding preserves.
Theorem 5.1 (Strong Normalization)

If in , then in ILL.

If in , then in ILL.

If , then is strongly normalizing.
Proof
The first two cases hold by straightforward induction on the form of the assumed typing or reduction derivation. They then imply the third.
Confluence. The ChurchRosser property is well known to hold for ILL modulo commuting conversions, for example, see Theorem 19 of [4] on page 96. Since is essentially a subsystem of ILL, it is straightforward, albeit lengthly, to simply redefine Bierman’s candidates and carry out a similar proof as Bierman’s (Theorem 19 on page 96 of ibid.).
Theorem 5.2 (Confluence)
The reduction relation, , modulo the commuting conversions is confluent.
5.2 The Typed Lambek Calculus:
The calculus we introduce in this section is an extension of with the ofcourse modality . This extension follows from ILL exactly. The syntax of types and terms of are extended as follows:
The new type and terms are what one might expect, and are the traditional syntax used for the ofcourse modality. We add the following typing rules to : Γ1, x : ! A , Γ2, y : ! A , Γ3⊢t : BΓ1, z : ! A , Γ2, Γ3⊢copy z as x , y in t : B T_C Γ1, Γ2⊢t : BΓ1, x : ! A , Γ2⊢discard x in t : B T_W : ! Γ⊢t : B : ! Γ⊢promote! for in t : ! B T_Br Γ1, x : A , Γ2⊢t : BΓ1, y : ! A , Γ2⊢[ derelict!y / x ] t : B T_Bl
Finally, the reduction rules can be found in Figure 4. The equality used in the R_BetaC rule is definitional, meaning, that the rule simply gives the terms on the right side of the equation the name on the left side, and nothing more. This makes the rule easier to read.
Strong normalization. Showing strong normalization for easily follows by a straightforward extension of the embedding we gave for .
Definition 8
The following is an extension of the embedding of into ILL resulting in an embedding of types and terms of into ILL. First, we define , then the following defines the embedding of terms:
Just as before this embedding is type preserving and reduction preserving.
Theorem 5.3 (Type and Reduction Preserving Embedding)

If in , then in ILL.

If in , then in ILL.

If , then is strongly normalizing.
Proof
The first two cases hold by straightforward induction on the form of the assumed typing or reduction derivation. They then imply the third.
Confluence. The ChurchRosser property also holds for , and can be shown by straightforwardly applying a slightly modified version of Bierman’s proof [4] just as we did for . Thus, we have the following:
Theorem 5.4 (Confluence)
The reduction relation, , modulo the commuting conversions is confluent.
5.3 The Typed Lambek Calculus:
The next calculus we introduce is also an extension of with a modality that adds exchange to denoted . It is perhaps the most novel of the calculi we have introduced.
The syntax of types and terms of are extended as follows:
The syntax for types has been extended to include the exchange modality, and the syntax of terms follow suit. The terms and are used to explicitly track uses of exchange throughout proofs.
We add the following typing rules to : Γ1, x1: κA , y1: B , Γ2⊢t : CΓ1, y2: B , x2: κA , Γ2⊢exchangely2, x2with x1, y1in t : C T_E1 Γ1, x1: A , y1: κB , Γ2⊢t : CΓ1, y2: κB , x2: A , Γ2⊢exchangery2, x2with x1, y1in t : C T_E2 : κΓ⊢t : B : κΓ⊢promoteκ for in t : κB T_Er Γ1, x : A , Γ2⊢t : BΓ1, y : κA , Γ2⊢[ derelictκy / x ] t : B T_El
The reduction rules are in Figure 5, and are vary similar to the rules from .
Strong normalization. Similarly, we show that we can embed into ILL, but the embedding is a bit more interesting.
Definition 9
The following is an extension of the embedding of into ILL resulting in an embedding of types and terms of into ILL. First, we define , and then the following defines the embedding of terms:
The embedding translates the exchange modality into the ofcourse modality of ILL. We do this so as to preserve the comonadic structure of the exchange modality. One might think that we could simply translate the exchange modality to the identity, but as Benton showed [3], this would result in an embedding that does not preserve reductions. Furthermore, the left and right exchange terms are translated away completely, but this works because ILL contains exchange in general, and hence, does not need to be tracked explicitly. We now have strong normalization and confluence.
Theorem 5.5 (Strong Normalization)

If in , then in ILL.

If in , then in ILL.

If , then is strongly normalizing.

The reduction relation, , modulo the commuting conversions is confluent.
Proof
The first two cases hold by straightforward induction on the form of the assumed typing or reductions derivation. They then imply the third case.
5.4 The Typed Lambek Calculus:
If we combine all three of the previous typed Lambek Calculi, then we obtain the typed Lambek Calculus . The main characteristics of this system are that it provides the benefits of the nonsymmetric adjoint structure of the Lambek Calculus with the ability of having exchange, and the ofcourse modality, but both are carefully tracked within the proofs.
Strong normalization for this calculus can be proved similarly to the previous calculi by simply merging the embeddings together. Thus, both modalities of would merge into the ofcourse modality of ILL. The ChurchRosser property also holds for by extending the proof of confluence for ILL by Bierman [4] just as we did for the other systems. Thus, we have the following results.
Theorem 5.6 (Strong Normalization)
If , then is strongly normalizing.
Theorem 5.7 (Confluence)
The reduction relation, , modulo the commuting conversions is confluent.
6 Acknowledgments
The authors would like to thank the anonymous reviewers for their feedback which did make this a better paper. The second author was partially supported by the NSF grant #1565557.
7 Conclusions
We have recalled how to use biclosed posets and sets to construct dialecticalike models of the Lambek Calculus. This construction is admittedly not the easiest one, which is the reason why we use automated tools to verify our definitions, but it has one striking advantage. It shows how to introduce modalities to recover the expressive power of intuitionistic (and a posteriori classical) propositional logic to the system. We know of no other construction of models of Lambek Calculus that does model modalities, not using their syntactic properties. (The traditional view in algebraic semantics is to consider idempotent operators for modalities like !). The categorical semantics here has been described before [10], but the syntactic treatment of the lambdacalculi, in the style of [2] had not been done and there were doubts about its validity, given the results of Jay [16]. We are glad to put this on a firm footing, using another one of Benton’s ideas: his embedding of intuitionistic linear logic into system F. Finally, we envisage more work, along the lines of algebraic proof theory, for modalities and nonsymmetric type systems.
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