The document discusses modeling the Lambek Calculus, a categorial grammar, using Dialectica Categories. It introduces the Lambek Calculus and categorical proof theory. Dialectica Categories are then introduced as a way to model linear logic and its non-commutative version. The document describes how the categories Dial2(Sets) and DialM(Sets) can model exponential-free multiplicative linear logic and non-commutative linear logic respectively, providing the necessary categorical structures. This allows the modeling of the Lambek Calculus in a category with the needed implications and non-commutativity.

1. 1/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Dialectica Categories for the Lambek Calculus
Valeria de Paiva
Topos Institute, CA
(joint work with Harley Eades III, Augusta, GA)
June, 2020
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

2. 2/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Thanks Shawn for the invite!
thanks Shay for all the work on the Logic SuperGroup!
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

3. 3/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Thanks Harley for the collaboration!
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

4. 4/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Introduction
I want to talk to you about modeling the Lambek Calculus,
using Dialectica Categories.
(This work is dedicated to Jim Lambek, 1922–2014)
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

5. 5/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Introduction
Lambek Calculus (1958, 1988, 1993, 2012)
Categorical Proof Theory
Dialectica Categories
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

6. 5/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Introduction
Lambek Calculus (1958, 1988, 1993, 2012)
Categorical Proof Theory
Dialectica Categories
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

7. 5/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Introduction
Lambek Calculus (1958, 1988, 1993, 2012)
Categorical Proof Theory
Dialectica Categories
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

8. 6/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Introduction
[..] However, use of the Lambek calculus for linguistic
work has generally been rather limited. There appear to
be two main reasons for this: the notations most
commonly used can sometimes obscure the structure of
proofs and fail to clearly convey linguistic structure, and
the calculus as it stands is apparently not powerful
enough to describe many phenomena encountered in
natural language.
“Categorial deductions and structural operations”, Morrill, Leslie,
Hepple, Barry, 1990
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

9. 7/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Introduction
[..] With grammar regarded as analogous to logic,
derivations are proofs; what we are advocating is
proof-reduction, and normal form proof; the invocation of
these logical techniques adds a further paragraph to the
story of parsing-as-deduction.
“Parsing and derivational equivalence”, Mark Hepple, Glyn Morrill
1989
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

10. 8/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Introduction
Lambek Calculus
Dialectica Categories
putting things together...
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

11. 9/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
What is the Lambek Calculus?
One of the several “type grammars”in use in Linguistics.
A long history: Ajdukiewicz [1935], Bar-Hillel [1953], Lambek
[1958, 1961], Ades-Steedman [1982], etc.
It provides a syntactic account of sentencehood.
Classes of type grammars:
1. Combinatory Categorial Grammar: Szabolcsi [1992],
Steedman-Baldridge [2011], etc..
2. Categorial Type Logics: van Benthem, Morrill [1994], Moortgat
[1994], etc..
Combinators/Lambda-calculus distiction.
Both classes worked on nowadays
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

12. 10/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
What is the Lambek Calculus?
Here a purely logical system, like usual propositional logic, but
with no structural rules at all.
Recall the basic logic ‘equation’:
A → (B → C) ⇐⇒ A ∧ B → C ⇐⇒ B → (A → C)
Now make your conjunction non-commutative, so that
A ⊗ B = B ⊗ A
Then you end up with two kinds of ‘implication’ ( , ):
A → (B C) ⇐⇒ A ⊗ B → C ⇐⇒ B → (A C)
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

13. 11/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Algebraic Proof Theory
[..] cut-elimination and completion, in the setting of
substructural logics and residuated lattices. We introduce
the substructural hierarchy – a new classification of
logical axioms (algebraic equations) over full Lambek
calculus FL, and show that a stronger form of
cut-elimination for extensions of FL and the MacNeille
completion for subvarieties of pointed residuated lattices
coincide up to the level N2 in the hierarchy.
“Algebraic proof theory for substructural logics: cut-elimination
and completions”, Ciabattoni, Galatos and Terui,
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

14. 12/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
What is the Lambek Calculus? sequent calculus
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

15. 13/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Why Dialectica?
For G¨odel (1958): a way to prove consistency of higher order
arithmetic
For Girard (1987): a way to show that Linear Logic had
serious pedigree
For Hyland (1987):
An intrinsic way modelling G¨odel’s Dialectica,
Proof theory in the abstract (Hyland, 2002)
Should produce a CCC, it wouldn’t for me
For me: a Swiss army knife...
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

16. 14/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Categorical Proof Theory
Types are formulae/objects in appropriate category,
Terms/programs are proofs/morphisms in the category,
Logical constructors are appropriate categorical constructions.
Most important: Reduction is proof normalization (Tait)
Outcome: Transfer results/tools from Logic to Categories to
Computing
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

17. 15/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Curry-Howard for Implication
Natural deduction rules for implication (without λ-terms)
A → B A
B
[A]
·
·
·
·
π
B
A → B
Natural deduction rules for implication (with λ-terms)
M : A → B N : A
M(N): B
[x : A]
·
·
·
·
π
M : B
λx.M : A → B
function application abstraction
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

18. 16/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

19. 17/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

20. 18/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

21. 19/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
The challenges of modeling Linear Logic
Traditional categorical modeling of intuitionistic logic:
formula A object A of appropriate category
A ∧ B A × B (real product)
A → B BA (set of functions from A to B)
But these are real products, so we have projections (A × B → A)
and diagonals (A → A × A) which correspond to deletion and
duplication of resources.
Not linear!!!
Need to use tensor products and internal homs in Category Theory.
Hard to define the “make-everything-as-usual”operator ”!”.
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

22. 20/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
The ‘simplest’ Dialectica Category
The Dialectica category Dial2(Sets) objects are triples, an object is
A = (U, X, R), where U and X are sets and R ⊆ U × X is a
set-theoretic relation (think of it as U × X → 2). A morphism
from A to B = (V , Y , S) is a pair of functions f : U → V and
F : Y → X such that a ‘semi-adjunction condition’ is satisfied. For
u ∈ U, y ∈ Y , uαFy implies fuβy.
Theorem 1: You just have to find the right structure. . .
(de Paiva 1989) The category Dial2(Sets) has a symmetric mo-
noidal closed structure, and involution which makes it a model of
(exponential-free) multiplicative linear logic.
Theorem 2 (Hard part): You still want usual logic. . .
There is a comonad ! which models exponentials/modalities, hence
recovers Intuitionistic and Classical Logic.
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

23. 21/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Can we give some intuition for these categories?
Blass makes the case for thinking of problems in computational
complexity. Intuitively an object of Dial2(Sets)
(U, X, R)
can be seen as representing a problem.
The elements of U are instances of the problem, while the
elements of X are possible answers to the problem instances.
The relation R says whether the answer is correct for that instance
of the problem or not.
The morphisms between these problems have two components:
while f maps instances of a problem to instances of another, F
maps solutions ‘backwards’.
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

24. 22/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Examples of objects in Dial2(Sets)
1. The object (N, N, =) where n is related to m iff n = m.
2. The object (NN, N, α) where f is α-related to n iff f (n) = n.
3. The object (R, R, ≤) where r1 and r2 are related iff r1 ≤ r2
4. The objects (2, 2, =) and (2, 2, =) with usual equality inequality.
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

25. 23/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
The ‘simplest’ is not very simple...
What do we need to show to prove
Theorem 1 (de Paiva 1989)
The category Dial2(Sets) has a symmetric monoidal closed struc-
ture, and an involution which makes it a model of (exponential-free)
multiplicative linear logic.
Need to show:
Dial2(Sets) is a category (easy),
Dial2(Sets) has an internal-hom (not so much)
Dial2(Sets) has a tensor product, (ok)
the usual adjunction (pretty)
(A ⊗ B, C) ∼= (A, [B C])
involution and par
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

26. 24/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Internal-hom in Dial2(Sets)
To “internalize”the notion of map between problems, we need to
consider the collection of all maps from U to V , V U, the collection
of all maps from Y to X, XY and we need to make sure that a
pair f : U → V and F : Y → X in that set, satisfies the dialectica
condition:
∀u ∈ U, y ∈ Y , uαFy → fuβy
This give us an object in DC (V U × XY , U × Y , βα)
The relation βα : V U × XY × (U × Y ) → 2 evaluates a pair (f , F)
of maps on the pair of elements (u, y) and checks the dialectica
implication between the relations.
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

27. 25/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
The Right Structure
Because it’s fun, let us calculate the “reverse engineering”
necessary for this as a model of Linear Logic
A ⊗ B → C if and only if A → [B −◦ C]
U × V (α ⊗ β)XV
× Y U
U α X
⇓ ⇓
W
f
?
γ Z
6
(g1, g2)
W V
× Y Z
?
(β −◦ γ)V × Z
6
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

28. 26/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
What’s different for the Lambek calculus?
Need to have a non-commutative tensor ⊗.
Need to have two (left and right) implications.
Can we have these disturbing minimally the (admitedly)
complicated structures?
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

29. 27/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
The non-commutative Dialectica Category
(de Paiva 1992, Amsterdam Colloquium) Category
DialM(Sets), objects are A = (U, X, R), where U and X are sets
and U × X → M is a M-valued relation. A morphism from A to
B = (V , Y , S) is a pair of functions f : U → V and F : Y → X
such that R(u, Fy) ≤M S(fu, y).
Theorem 3: have the right strux. . .
The category DialM(Sets) has a non-symmetric monoidal closed
structure, hence it is a model of (exponential-free) non-commutative
multiplicative linear logic.
Theorem 4 (Hard part): You still want usual logic. . .
There is a comonad ! which models exponentials/modalities, and a
comonad κ (Yetter) that brings back commutativity. Putting the
two together we recover Intuitionistic and Classical Logic.
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

30. 28/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Conclusions
Introduced you to the Lambek calculus, as a relative of Linear
Logic
Introduced you to Dialectica categories
(there’s much more to say...)
Described one example of Dialectica categories DialM(Sets),
a non-commutative case. Should’ve shown you the modalities
that make it work.
Advantages over previous work:
1. Proved syntax works as expected.
2. Working on Agda implementation of syntactical results
use this system for PLs? Jiaming Jiang and Harley’s work
To do: comparison with pregroups; why I don’t use it
in my language work, etc
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

31. 29/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Thank you!
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS

32. 30/30
Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Some References
J. Lambek, The Mathematics of Sentence Structure. American
Mathematical Monthly, pages 154–170, 1958.
de Paiva, The Dialectica Categories, Cambridge University DPMMS PhD
thesis, Technical Report 213, 1991.
de Paiva, A Dialectica Model of the Lambek Calculus, In Proc Eighth
Amsterdam Colloquium, December 17–20, 1991. eds Martin Stokhof and
Paul Dekker, ILLC, University of Amsterdam, 1992, pp. 445-462.
Hyland, J. Martin E. Proof theory in the abstract, Annals of pure and
applied logic 114.1-3, 2002, pp. 43-78.
de Paiva, Eades III, Dialectica Categories for the Lambek Calculus. Proc
LFCS 2018, 01 February 2018.
Jiang, Eades III, de Paiva On the Lambek Calculus with an Exchange
Modality. Proc Linearity 2018, Oxford.
Valeria de Paiva Logic SuperGroup2020 – Melbourne, AUS