Download CATEGORICAL MODELS OF FIRST

CATEGORICAL MODELS OF
FIRST-ORDER CLASSICAL PROOFS
Submitted by Richard Mc Kinley
for the degree of
Doctor of Philosophy
of the University of Bath
2006
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Abstract
This thesis introduces the notion of a classical doctrine: a semantics for proofs in firstorder classical logic derived from the classical categories of Führmann and Pym, using
Lawvere’s notion of hyperdoctrine. We introduce a hierarchy of classes of model,
increasing in the strength of cut-reduction theory they model; the weakest captures
cut reduction, and the strongest gives De Morgan duality between quantifiers as an
isomorphism. Whereas classical categories admit the elimination of logical cuts as
equalities, (and cuts against structural rules as inequalities), classical doctrines admit
certain logical cuts as inequalities only. This is a result of the additive character of the
quantifier introduction rules, as is illustrated by a concrete model based on families of
sets and relations, using an abstract Geometry of Interaction construction.
We establish that each class of models is sound and complete with respect to the relevant cut-reduction theory on proof nets based on those of Robinson for propositional
classical logic. We show also that classical categories and classical doctrines are not
only a class of models for the sequent calculus, but also for deep inference calculi
due to Brünnler for classical logic. Of particular interest are the local systems for
classical logic, which we show are modelled by categorical models with an additional
axiom forcing monoidality of certain functors; these categorical models correspond to
multiplicative presentations of the sequent calculus with additional additive features.
Acknowledgements
There are many without whom this work would languish unfinished.
I must begin by thanking David Pym for introducing to me the core ideas in this work,
and for the guidance he has provided. In matters both sacred and profane, both integral and peripheral, he has provided invaluable, fruitful and enjoyable discussion and
advice. To him I am deeply grateful.
I thank Carsten Führmann for too many helpful discussions to remember, and his for
his encouragement. I am very grateful to John Power and Daniel Richardson for agreeing to examine the thesis, and for their comments. I have also benefited from discussions with Kai Brünnler, Matthew Collinson, Alessio Guglielmi, Francois Lamarche,
Edmund Robinson and Lutz Strassburger.
Thanks to my friends and family, who have continued to cheer this thesis on, despite
resolutely refusing to read it.
The greatest thanks must go to my wife, Mary. Throughout she has been a constant
source of love and encouragement.
Diagrams were typeset using Paul Taylor’s diagrams and prooftree packages. The
author was supported by a University Studentship from the University of Bath.
1
Contents
1 Introduction
1.0.1
6
Structure of this thesis . . . . . . . . . . . . . . . . . . . . .
2 Pre-background: established theory
2.1
11
13
Sequent calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.1.1
Multiplicative and additive presentations . . . . . . . . . . .
16
2.1.2
One-sided calculi . . . . . . . . . . . . . . . . . . . . . . . .
17
2.1.3
Calculi for intuitionistic logic . . . . . . . . . . . . . . . . .
18
2.1.4
Cut-elimination . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.1.5
First-order LK . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.2
Proof nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.3
Categorical logic and categorical preliminaries . . . . . . . . . . . . .
34
2.3.1
Interpretation of proofs in a category . . . . . . . . . . . . . .
35
2.3.2
Indexed categories and hyperdoctrines . . . . . . . . . . . . .
36
2
2.3.3
Monoidal categories and functors . . . . . . . . . . . . . . .
39
2.3.4
Symmetric linearly distributive and ∗-autonomous categories .
45
3 Background: recent work
54
3.1
Proof nets for propositional classical logic . . . . . . . . . . . . . . .
54
3.2
Classical categories . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.3
The calculus of structures for classical logic . . . . . . . . . . . . . .
70
3.3.1
System SKSg . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.3.2
A local system . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.3.3
First-order logic in the calculus of structures . . . . . . . . .
76
3.3.4
Formalism A and Formalism B . . . . . . . . . . . . . . . . .
79
4 Proof nets for first-order classical logic
4.1
81
First-order proof nets and quantifier boxes . . . . . . . . . . . . . . .
82
4.1.1
Static properties . . . . . . . . . . . . . . . . . . . . . . . .
84
4.2
Cut-elimination in first-order proof nets . . . . . . . . . . . . . . . .
85
4.3
Other (in)equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
5 Classical doctrines
92
5.1
Categorical axioms for quantifiers . . . . . . . . . . . . . . . . . . .
93
5.2
Classical doctrines . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
3
5.3
Some properties of classical doctrines . . . . . . . . . . . . . . . . . 100
5.4
Sharp Classical Doctrines . . . . . . . . . . . . . . . . . . . . . . . . 110
5.5
Linear functors, duality and M IX . . . . . . . . . . . . . . . . . . . . 117
5.6
A concrete example . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6 Modelling classical first-order proofs
144
6.1
Interpretation of the sequent calculus in classical doctrines . . . . . . 144
6.2
Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3
The term model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4
Permutations with structural rules and sharp doctrines . . . . . . . . . 173
6.5
Permutations between quantifiers . . . . . . . . . . . . . . . . . . . . 183
7 Categorical semantics and the calculus of structures
189
7.1
Providing a semantics for deep inference systems . . . . . . . . . . . 189
7.2
Locality and categorical semantics . . . . . . . . . . . . . . . . . . . 193
7.2.1
7.3
The first-order medial rules . . . . . . . . . . . . . . . . . . . 200
Towards a native notion of proof equality for SKS . . . . . . . . . . . 200
8 Classical doctrines with equality and axiomatic theories
203
8.1
Nonlogical axioms in sequent calculi . . . . . . . . . . . . . . . . . . 204
8.2
Equality and algebraic theories . . . . . . . . . . . . . . . . . . . . . 205
4
8.3
Theories with relations . . . . . . . . . . . . . . . . . . . . . . . . . 210
9 Conclusions and further work
215
References
220
A Monoidal categories
226
A.1 Adjunctions in monoidal categories . . . . . . . . . . . . . . . . . . 226
A.1.1 Proof of Theorem 2.3.3 . . . . . . . . . . . . . . . . . . . . . 226
A.1.2 Proof of Lemma 2.3.15 . . . . . . . . . . . . . . . . . . . . . 233
B Generalized quantifiers
235
5
Chapter 1
Introduction
This thesis is concerned with the denotations of propositions and of proofs in classical
logic. This question seems, on the face of things, to be answered very quickly; every
first-year mathematics undergraduate is familiar with the notion of a truth value: to
every atomic proposition we may assign either the value ⊤ (true) or ⊥ (false). The
truth values of compound propositions are then determined by truth tables, as functions
of the truth values of their constituent atomic propositions. For propositional logic, this
is enough; a model of propositional classical logic in this sense is a Boolean algebra.
The simplest model of this type is given by the Boolean algebra {⊤, ⊥}. We use
the notation ⌊φ⌋ for the object of a Boolean algebra representing the meaning of a
proposition φ.
This mathematical structure may be seen as a lattice, with ⌊φ⌋ ≤ ⌊ψ⌋ if φ ⇒ ψ. Treating a logic as an ordering on propositions scales well to a number of non-classical
logics; for example, associated to intuitionistic logic is the notion of Heyting semilattices.
While this very simple notion of model has been very successful as a tool, allowing
the development of model theory, it is unsatisfying from the perspective of modelling
logic itself. Any two logically equivalent propositions are equal, and in particular
the denotation of any tautology is ⊤. Moreover, the model gives us no information
about the structure of proof. For this reason, we will refer to this paradigm for the
semantics of logic as provability semantics, contrasted with the subject of this thesis:
the semantics of proofs.
6
The notion that the meaning of a proposition should be tied to its set of proofs originates with Heyting. To Heyting, a proof of a formula φ ∧ ψ should consist of nothing
more or less than a proof of φ and a proof of ψ, a proof of φ ∨ ψ should be either a
proof of φ or a proof of ψ (plus an indication of which we have proved), and a proof
of φ ⇒ ψ should be a function mapping proofs of φ to proofs of ψ.
It seems intuitive that under this interpretation classes of logically equivalent propositions are no longer identified: a proof of ψ ∧ ψ is not a proof of ψ. What is not immediately obvious is how we might formalize this notion; in order to do so, we must
first know when two proofs are identical. The first steps toward this were made by
Prawitz, who suggested that, for natural deduction (for intuitionistic logic), two proofs
should be considered equal if they have the same normal form under normalization
(βη-reduction).
A natural setting in which to evaluate proofs is a category: a categorical semantics for
set of proofs in a category C consists of
1. Evaluating each proposition φ as an object ⌊φ⌋ of C, and
2. each proof Φ from φ to ψ as a morphism ⌊Φ⌋ : ⌊φ⌋ → ⌊ψ⌋.
The provability semantics mentioned above is retrieved by insisting C be a pre-order.
Soon after Prawitz suggested normalization as a source for proof equality, it was noticed by Lambek that this is precisely the equality on proofs given by interpreting
proofs as morphisms in a cartesian closed category. Thus, it is often said a categorical
model of intuitionistic logic is a cartesian closed category. This is somewhat misleading: there may be several proof theories for each logic. For example, the multipleconclusioned sequent calculus for intuitionistic logic [21] has “more” proofs than the
usual single conclusioned calculus, as we will see below, and at a sufficient level of
detail even the correspondence between natural deduction and the sequent calculus is
not precise, although it is very close [59]. Thus it is technically incorrect to say that,
for example, that cartesian closed categories are a class of models of propositional intuitionistic logic; they are a model of natural deduction for propositional intuitionistic
logic.
The structure of our calculus imposes constraints on proof-equality; for example, in
7
the sequent calculus, we require that the rules of the calculus preserve proof equality
(and in general, cut-reducibility).
Taking the reductions on a proof calculus as equalities is enough to give a semantics
for the single-conclusioned calculus of intuitionistic logic, and for linear logic, but in
general it fails; in particular for a multiple-conclusioned sequent calculus with weakening and contraction. The problem is illustrated by the following example, usually
attributed to Lafont [28].
Given two proofs Φ1 and Φ2 of the same sequent,consider the proof
Φ1
·
·
·
Γ⊢∆
Γ ⊢ φ, ∆
Φ2
·
·
·
Γ⊢∆
WR
Γ, φ ⊢ ∆
Γ, Γ ⊢ ∆, ∆
Γ⊢∆
WL
(1.1)
C UT
CL, CR.
Suppose we want our semantics to admit cut-reduction as equality. Then this proof,
under the usual set of cut-reductions, reduces to both Φ1 and Φ2 . Therefore, this naı̈ve
approach identifies every proof of a given sequent, so obtain, not a semantics of proofs
as desired, but nothing more than provability semantics.
It is important to note that, while this example was conceived as a demonstration of the
“inconsistencies” in the proof theory of classical logic, it is equally applicable to the
multiple conclusioned calculus for intuitionistic logic, since it makes no use, explicit
or implicit, of negation. The collapse arises from unrestricted use of weakening and
contraction.
One tactic for repairing this lack of a denotational semantics is allow a restricted set
of reductions; in the above example, we might allow the left-hand reduction but not
the right-hand. This is the approach taken in the λµ calculus [55] for classical logic.
The two choices correspond to a call-by-name or a call-by-value evaluation strategy.
This gives rise to models of classical proofs, either via fibrations of cartesian closed
categories [57] or control and co-control categories [61]. This is at the expense of
breaking symmetry.
8
Recently, several successful attempts to give a non-trivial semantics to the classical
logic have appeared which preserve the symmetry of cut-reduction. Classical categories [23, 22, 24] have morphisms that denote proofs in Gentzen’s LK, and an order
enrichment modelling the standard theory of cut-reduction in LK. These models are
sound and complete with respect to the ordering on proofs.
Other notable projects in the semantics of classical proofs include the work of Hyland
[38] on abstract interpretation of classical proof, which are similar to a classical category for which ∧ and ∨ are modelled by the same monoidal product. The difference
lies in the omission of one axiom on the structure of the ordering on proofs. The work
of Bellin, Hyland, Robinson and Urban [7] is ambitious in considering models of classical logic which reject the functoriality of conjunction and disjunction (corresponding
to the rejection of axiom expansion in the proof theory). This leads to a more general
notion of model, but one for which there are few natural examples. Dosen and Petric [20], by analyzing the coherence of distributive categories via the category of sets
and relations, derive a notion of Boolean category, which is somewhat degenerate in
requiring finite products and coproducts.
The work of Lamarche and Strassburger [44, 45], does not analyze the symmetric cut
reduction system of LK, nor the restricted, asymmetric reduction of the λµ-calculus,
but instead a reduction system which rejects both reductions in Lafont’s example, introducing (implicitly) a primitive mix law [26] , to which a degenerate cut reduces:
Φ1
·
·
·
Γ⊢∆
Γ ⊢ φ, ∆
WR
Φ2
·
·
·
′
Γ ⊢ ∆′
′
Γ ,φ ⊢ ∆
Γ, Γ′ ⊢ ∆, ∆′
′
4
WL
Φ1
·
·
·
Γ⊢∆
Φ2
·
·
·
′
Γ ⊢ ∆′
Γ, Γ′ ⊢ ∆, ∆′
C UT
Mix
In a classical category the mix rule is denotationally equivalent to the degenerate cut
with φ = ⊤, or alternatively with φ = ⊥. Mix also gives rise to a product on proof
spaces:
Φ∗Ψ=
Ψ
·
·
·
Γ⊢∆
Φ
·
·
·
Γ⊢∆
Γ, Γ ⊢ ∆, ∆
Γ⊢∆
9
M ix
CL, CR
In classical categories this operation is idempotent; Φ ∗ Φ = Φ. In the Boolean categories of [45], there are given examples where this is also the case (which are a special
case of classical categories) and also models where this equality does not hold; these
non-idempotent models are intruiguing but not well understood.
This thesis concentrates on the classical categories of Führmann and Pym, and extends
their results to first-order LK, in the spirit of Lawvere [49] and Seely [60]. In particular, we will see that the “Adjointness in Foundations” view of quantification (where
quantifiers are adjoints to substitution) can be unified with the naı̈ve view of quantifiers
as an infinitary conjunction or disjunction over the domain of quantification. While it
does not directly address the other notions of model above, it is worth noting that
techniques developed in this thesis should adapt successfully to them.
The issue of finding a notion of equality on proofs is separate from, but related to,
the issue of finding canonical objects representing those proofs; the “essence” of the
proof. For multiplicative linear logic (without units), proof nets [26] are such canonical
objects, but for classical logic (for any logic with weakening) the usual approach to
weakening [58, 12] relies on attaching a weakening to a particular node of the net.
Since we do not wish to distinguish between proofs which differ only by the attachment
of a weakening, we must quotient by a re-wiring relation on nets. The search for a
notion of proof net which gives canonical cut-free proofs has led to the development
by Hughes of abstract combinatorial proofs [36, 37].
Nevertheless, the non-canonical proof nets are technically useful objects in the study
of the sequent calculus; the proof of completeness for classical categories [23] relies
on the proof nets of Robinson [58], with a cut-reduction equivalent to that for LK,
to significantly simplify the calculations. Our first task will be to develop a similar
system for first-order logic.
Another area of research which has of late been very active is the search for new
deductive systems, with the hope being that the problem of identity of proofs is more
easily solved in those systems. These deep inference systems [33, 13, 63] have much
in common with categorical logic; in particular, the notions of “type A bureacracy” and
“type B bureacracy” correspond precisely to the categorical notions of bifunctoriality
and naturality [35, 52, 64]. For some of these systems, categorical coherence gives
a natural equational (or inequational) theory on proofs, but the most sucessful deep
inference systems make use of a notion previously uninvestigated in categorical logic
10
— the medial rule. Moreover, while coherence correpsonds precisley to cut-reduction
in sequent calculi, it is not clear how it relates to notions of reduction in deep inference
systems. We take in this thesis some small steps to addressing this problem; we give
conditions for the models of Führmann and Pym to capture the medial rule, and show
how coherence may connect to reduction in deep inference systems.
A large part of this thesis will be concerned with De Morgan duality: in classical logic
A ∨ B is logically equivalent to ¬(¬A ∧ ¬B). When using classical logic in a practical
setting, it is often a useful short-cut to consider only one half of any De Morgan dual
pair, and/or use a one-sided presentation of the logic; this will, for example, half the
length of any presentation of the system. However, when considering the system for
its own merits, one may appreciate the beauty of the symmetry (which is, of course,
hidden in a one-sided approach). Moreover, in categorical logic logical equivalence
does not imply isomorphism; for example φ is not ismorphic to φ ∧ φ We will refer
to the concept of isomorphic De Morgan duality as proof-theoretic duality. We will
continue to use a two sided system in this thesis.
By using a setting in which duality is not assumed (symmetric linearly distributive
categories) rather than one where it is (∗-autonomous categories), we can analyze the
properties needed of the theory on proofs to derive duality of, for example, existential
and universal quantification.
Note that this does not stop us from using categorical duality; the structure used to
model a connective (for example, a monoidal functor) will be categorically dual to
that of its De Morgan dual (in this case, a comonoidal functor), and so we will be
able to economize on paper by concluding a proof about the existential quantifier with
the words “and dually for the universal quantifier”, without committing to their being
proof-theoretically dual.
1.0.1 Structure of this thesis
Chapter 2: We recall some established results in proof-theory and categorical logic.
Chapter 3: We cover some more recent developments: proof nets for classical logic,
classical categories, and deep inference systems.
11
Chapter 4: We give details of a calculus of proof nets for first-order classical logic
with quantifiers, and demonstrate the key properties: sequentialization and cut-elimination.
Chapter 5: We give three notions of model of first order classical proofs, in order of increasing strength. The first, classical doctrines, models quantifiers as functors monoidal in one connective. The second, sharp classical doctrines, embodies
the well-behavedness of quantifiers with respect to structural rules and has quantifiers (co)monoidal in both connectives. The third, dual classical doctrines, introduces
enough extra structure to ensure that existential quantification is De Morgan dual to
universal quantification.
Chapter 6: We introduce, for each notion of model, a corresponding notion of theory
on proofs, giving soundness and completeness result for each class. For sharp doctrines/theories, we observe that ideas from the calculus of structures allow a smoother
axiomatization for monoidality in the proof nets, associated to the additive nature of
the sequent calculus quantifier rules.
Chapter 7: We comment on the interplay between deep inference systems and our
categorical models, and observe that the models give a natural notion of proof equality
on calculus of structures proofs.
Chapter 8: We extend our models to include equality predicates, algebraic and nonalgebraic signatures, and non-logical axioms. We provide models of modal and (multiple conclusioned) intuitionistic propositional logics by viewing them in terms of their
first-order Kripke semantics.
Chapter 9: We give conclusions and ideas for future work.
12
Chapter 2
Pre-background: established theory
2.1 Sequent calculi
The sequent calculus was introduced in Gentzen’s seminal paper [25] as a setting for
studying the properties of formal proof. He gave systems LJ (for intuitionistic logic)
and LK (for classical logic), which have since formed the template for numerous calculi for other logics. A key feature of sequent systems is the cut rule, by which two
proofs in the system may be composed, and a key property of sequent systems is that
judgments that can be derived with cut may also be derived without cut; the procedure for deriving the cut-free proof being known as cut-elimination. The lack of a
cut-elimination procedure is considered, in most cases, a serious defect of a sequent
calculus.
A sequent calculus is a system for deriving judgments of the form
Γ ⊢ ∆,
where Γ and ∆ are sequences1 of formulae, with formulae given by (a sub-grammar
of) the grammar
A := a | A ∧ A | A ∨ A | ¬A | ⊤ | ⊥ | ∃yA | ∀yA,
1
In certain contexts Γ and ∆ are taken to be multisets or sets, making weakening and contraction
implicit; in the context of semantics of proofs this is fatal, since we must keep track of where weakenings
are made.
13
AX
ϕ⊢ϕ
Γ, ϕ, ψ ⊢ ∆
Γ′ ⊢ ψ, ∆′
Γ ⊢ ϕ, ∆
∧L
Γ, ϕ ∧ ψ ⊢ ∆
Γ ⊢ ϕ, ψ, ∆
Γ, Γ′ ⊢ ϕ ∧ ψ, ∆, ∆′
Γ′ , ψ ⊢ ∆′
Γ, ϕ ⊢ ∆
∨R
Γ ⊢ ϕ ∨ ψ, ∆
∧R
Γ, Γ′ , ϕ ∨ ψ ⊢ ∆, ∆′
∨L
⊤R
⊢⊤
⊥L
⊥⊢
Γ, ϕ, ψ, Γ′ ⊢ ∆
Γ, ψ, ϕ, Γ′ ⊢ ∆
Γ ⊢ ∆, ϕ, ψ, ∆′
EL
Γ ⊢ ϕ, ∆
Γ ⊢ ∆, ψ, ϕ, ∆′
ER
Γ′ , ϕ ⊢ ∆′
Γ, Γ′ ⊢ ∆, ∆′
C UT
Table 2.1: LB: a sequent calculus for the “core system”
and the intended reading is that the conjunction of the formulae in Γ entails the disjunction of the formulae in ∆.
The simplest logic we will consider in this thesis has the sequent calculus given in
Table 2.1. We will call this the “core system”, and the calculus LB. The rules of the
core system will be a subset every other system we consider in this thesis. Using the
rule ∨L rule as an example, we give some terminology:
Γ, ϕ ⊢ ∆
Γ′ , ψ ⊢ ∆′
Γ, Γ′ , ϕ ∨ ψ ⊢ ∆, ∆′
∨L.
We call the formula introduced by the rule (in this case ϕ ∨ ψ) the principal formula.
14
Γ, ϕ ⊢ ∆
Γ ⊢ ¬ϕ, ∆
Γ ⊢ ϕ, ∆
¬L
Γ, ¬ϕ ⊢ ∆
¬R
Table 2.2: Optional rules for negation
The formulae above the line which make up the principal formula are the active formulae, and the other formulae are the context.
An alternative presentation for LB (the efficacy of which was demonstrated in [18] and
then [23]) is to replace the rules ∧R and ∨L by constants:
φ, ψ ⊢ φ ∧ ψ
K∧ ,
φ ∨ ψ ⊢ φ, ψ
K∨ ,
and simulate ∧R and ∨L by cutting against these constants.
We can extend LB by adding more constants, or alternatively more rules. For example,
Table 2.1 gives rules for negation, equivalent to adding the constants
⊢ ¬φ, φ
K¬R ,
¬φ, φ ⊢
K¬L ,
giving multiplicative linear logic (MLL).
Table 3.4 gives structural rules, equivalent to adding the constants
φ ⊢ φ, φ
⊥⊢φ
KCL ,
φ, φ ⊢ φ
KW R ,
φ⊢⊤
KCR ,
KW L .
We may use these constants to define a calculus for classical logic, which we will call
LK:
15
Γ, ϕ, ϕ ⊢ ∆
Γ, ϕ ⊢ ∆
Γ⊢∆
Γ, ϕ ⊢ ∆
Γ ⊢ ∆, ϕ, ϕ
CL
Γ ⊢ ∆, ϕ
Γ⊢∆
WL
Γ ⊢ ∆, ϕ
CR
WR
Table 2.3: Optional rules for weakening, contraction
Definition 2.1.1. The system LK is given by LB, plus the constants (alternatively, rules)
for contraction, weakening, and negation.
2.1.1 Multiplicative and additive presentations
The definition given by Gentzen in his original paper for LK differs from that given
above; not only in the (somewhat technical) use of constants, but more substantially.
Gentzen’s original formulation for the rule introducing conjunction on the left, for
instance, is the pair of rules
Γ, ψ ⊢ ∆
Γ, ϕ ∧ ψ ⊢ ∆
Γ, ϕ ⊢ ∆
∧La ,
Γ, ϕ ∧ ψ ⊢ ∆
∧La ,
while the formulation of the rule introducing conjunction on the right is
Γ ⊢ ϕ, ∆
Γ ⊢ ψ, ∆
Γ ⊢ ϕ ∧ ψ, ∆
∧Ra
(These formulations are known as “additive” rules, and the previous presentation “multiplicative” rules.) In a system with with weakening and contraction, ∧L and ∧La are
inter-derivable, as are ∧R and ∧Ra . In the absence of these structural rules, however,
the connectives that these two families of rules define are very different. These logics
without structural rules are known as linear logics [26], which might contain both multiplicative and additive conjunction and disjunction. Of principal importance to us will
be the logic MLL, given by adding negation to LB. It has a very well behaved proof
16
⊢ ϕ, ¬ϕ
⊢ ∆, ϕ
AxGS1
⊢ ∆′ , ¬ϕ
⊢ ∆, ∆′
CU TGS1p
Table 2.4: System GS1p: Axiom and C UT
theory and model theory, which we will discuss below.
2.1.2 One-sided calculi
Given the duality of the classical sequent calculus, it is common to consider Gentzen–
Schütte style systems, where sequents ⊢ ∆ have no antecedant, only a conclusion. For
example, the system GS1p is given by removing the axiom, cut and left rules of LK,
and adding the rules in table 2.4
It will be useful for later chapters to see how a proof in GS1p may be translated into
LK:
Lemma 2.1.2. Any proof Φ in GS1p can be transformed into a proof in LK.
Proof. Suppose we have a GS1p proof Φ. Since GS1p and LK differ only on cut and
axiom, we transform only instance of these rules.
We replace each occurrence of an axiom
⊢ ¬ϕ, ϕ
AxGS1p
with
ϕ⊢ϕ
AxLK .
⊢ ¬ϕ, ϕ
¬R
(2.1)
The case of cut is little more complicated. There are three obvious ways of defining
the GS1p cut rule in LK:
17
ϕ⊢ϕ
⊢ ¬ϕ, ∆′
ϕ ⊢ ∆′
⊢ ϕ, ∆
⊢ ∆, ∆′
⊢ ϕ, ∆
⊢ ¬ϕ, ∆′ , ¬ϕ ⊢ ∆
⊢ ∆, ∆′
AxLK
¬ϕ, ϕ ⊢
¬L
C UTLK
(2.2)
C UTLK ,
¬L
(2.3)
C UTLK ,
and
ϕ⊢ϕ
⊢ ¬ ϕ, ∆′
⊢ ϕ, ∆
⊢ (¬ϕ ∧ ϕ), ∆, ∆
′
∧R
⊢ ∆, ∆′
AxLK
¬ϕ, ϕ ⊢
¬L
¬ϕ ∧ ϕ ⊢
∧L
(2.4)
C UTLK .
These definitions are coherent, in the sense that they are identified in a semantics that
validates logical cut as equality (in particular the semantics of Führmann and Pym
[23]). This is clear from eliminating the logical cut in (2.4), and representing negation
as cut against a constant.
2.1.3 Calculi for intuitionistic logic
In addition to LK for classical logic, Gentzen [25] introduced a sequent calculus LJ
for intuitionistic logic. The calculus is a restriction of the additive calculus, minus
negation, in which the right hand of the sequent consists of at most one formula. We
give the calculus in Table 2.5.
It is not, however, necessary to restrict all the rules of the sequent calculus in this way.
Dummett [21] gives a calculus for intuitionistic logic for which, in the propositional
fragment, only negation and implication are restricted. The calculus is obtained by
adding the rules in Table 2.6 to LB, plus structural rules.
18
AX
ϕ⊢ϕ
Γ, ψi ⊢ ∆
Γ⊢ϕ
Γ⊢ψ
∧L
∧R
Γ, ψ1 ∧ ψ2 ⊢ ∆
Γ ⊢ ϕ∧ψ
Γ, ϕ ⊢ ∆
Γ ⊢ ψi
Γ, ψ ⊢ ∆
∨R
∨L
Γ ⊢ ψ1 ∨ ψ2
Γ, ϕ ∨ ψ ⊢ ∆
Γ, φ ⊢ ψ
Γ⊢ϕ
Γ, ψ ⊢ ∆
⇒L
⇒R
Γ⊢φ⇒ψ
Γ, ϕ ⇒ ψ ⊢ ∆
Γ, ϕ, ϕ ⊢ ∆
CL
Γ, ϕ ⊢ ∆
Γ⊢∆
Γ⊢
WL
WR
Γ, ϕ ⊢ ∆
Γ⊢ϕ
Γ ⊢ ϕ, ∆
Γ′ , ϕ ⊢ ∆
Γ, Γ′ ⊢ ∆
C UT
(Where ∆ is either empty, or contains at most one formula.)
Table 2.5: LJ: a sequent calculus for intuitionistic logic
19
Γ, φ ⊢ ψ
Γ′ , ψ ⊢ ∆′
Γ ⊢ ϕ, ∆
⇒R
⇒L
Γ, Γ′ ϕ ⇒ ψ ⊢ ∆, ∆′
Γ⊢φ⇒ψ
Γ ⊢ ϕ, ∆
Γ, φ ⊢
¬R
¬L
Γ ⊢ ¬φ
Γ, ¬ϕ ⊢ ∆
Table 2.6: Negation and implication in Dummett’s calculus for intuitionistic logic.
This calculus is sound and complete for intuitionistic logic (with respect to, say, the
usual Kripke semantics) but its proof theory is very different to that of the singleconclusioned system, as we noted in the introduction.
2.1.4 Cut-elimination
Conceptually, the proof of cut-elimination for a sequent system is very simple; the underlying notion is the logical cut. A logical cut applies in the following circumstance,
where the cut formula is introduced, in both premises of the cut, by the introduction
rule for its main connective:
Γ, φ ⊢ ∆
L
Γ′ ⊢ φ, ∆′
′
′
Γ, Γ ⊢ ∆, ∆
R
C UT
.
The reduct of a logical cut has a cut or cuts between the principal formulae of the cut
formula. For example, the logical cut for ∨ reduces
Γ, φ ⊢ ∆
Γ′ , ψ ⊢ ∆′
Γ, Γ′ φ ∨ ψ ⊢ ∆, ∆′
∨L
Γ′′ ⊢ φ, ψ, ∆′′
Γ′′ ⊢ φ ∨ ψ, ∆′′
Γ, Γ′ , Γ′′ ⊢ ∆, ∆′ , ∆′′
20
∨R
C UT
(2.5)
to
Γ′′ ⊢ φ, ψ, ∆′′
Γ, φ ⊢ ∆
Γ′ , ψ ⊢ ∆′
Γ, Γ′′ ⊢ ψ, ∆, ∆′′
Γ, Γ′ , Γ′′ ⊢ ∆, ∆′ , ∆′′
C UT
C UT .
(2.6)
Assuming the top cut is also a logical cut, the procedure may continue by induction on
the structure of the cut formula.
Of course, we may not assume that all cuts are logical. In the lower cut of 2.6 the cut
formula is a result of another application of cut. The proof of cut-elimination therefore
requires a number of permutation laws or commuting conversions. It is for this reason
that a calculus of proof nets (see Section 2.2) is essential in giving short proofs of
meta-theorems such as completeness; proof nets allow cuts to appear “in parallel”.
The final obstacle to a simple proof of cut-elimination lies in the structural rules. Because of these, the cut-formula might not be the consequence of a logical rule, even
after the application of commuting conversions. Consider a proof of the following
shape:
Γ′ ⊢ ∆′
Γ, φ ⊢ ∆
Γ′ ⊢ φ, ∆′
Γ, Γ′ ⊢ ∆, ∆′
WR
.
C UT
(Here φ does not result from any logical rule). In eliminating the cut, we must keep
the context Γ and ∆. This is achieved by successive weakenings:
Γ′ ⊢ ∆′
′
′
Γ, Γ ⊢ ∆, ∆
WL∗ , WR∗ .
We have deleted a proof of arbitrary size. Now consider a proof of the shape
Γ′ ⊢ φ, φ, ∆′
Γ, φ ⊢ ∆
Γ′ ⊢ φ, ∆′
Γ, Γ′ ⊢ ∆, ∆′
CR
.
C UT
If we push the cut above the contraction, we copy the contexts Γ and ∆, since we
must make two copies of the proof of Γ, φ ⊢ ∆. These copies must be removed by
21
contraction:
Γ′ ⊢ φ, φ, ∆′
Γ, φ ⊢ ∆
Γ, Γ′ ⊢ φ, ∆, ∆′
Γ, φ ⊢ ∆
Γ, Γ, Γ′ ⊢ ∆, ∆, ∆′
Γ, Γ′ ⊢, ∆, ∆′
C UT
C UT
CL, CR.
By contrast with the logical cuts, the elimination of these cuts against structural rules
removes or copies a subproof of arbitrary size. An elimination of this kind is referred
to as non-local, while a reduction which removes only a bounded subproof is called
local.
The multiplicative system with the cut rule we have seen so far behaves much better
with respect to the required permutation rules than the additive system with the same
rule, in the sense that if a permutation of rule order is possible, it is possible without
any extra weakenings and contractions. By contrast, to permute the cut above the
logical rule in the proof
Γ′ , φ ⊢ θ, ∆′
Γ′ , ψ ⊢ θ, ∆′
Γ′ , φ ∨ ψ ⊢ θ, ∆′
Γ, θ ⊢ ∆
Γ, Γ′ , φ ∨ ψ ⊢ ∆, ∆′
∨La
C UT ,
.
we must both duplicate one proof:
Γ′ , φ ⊢ θ, ∆′
Γ, θ ⊢ ∆
Γ, Γ′ , φ ⊢ ∆, ∆′
Γ′ , ψ ⊢ θ, ∆′
C UT
Γ, θ ⊢ ∆
Γ, Γ′ , φ ⊢ ∆, ∆′
Γ, Γ′ , φ ∨ ψ ⊢ ∆, ∆′
C UT
∨Ra ,
so the permutation is no longer local. Indeed, if one translates the ∨Ra rule into the
multiplicative calclulus, the connection to the non-local rule for cut against contraction
becomes clear.
The purely multiplicative system is conceptually clean, in the sense that we can simulate the rules ∨L and ∨R by cut against a constant, without additional weakening or
contraction. With this observation, we can derive the well-behavedness of the other
connectives from the fact that cuts can permute past each other.
By contrast, to simulate the additive rule ∧Ra with a multiplicative cut, we need con-
22
traction:
Γ ⊢ φ, ∆
φ, ψ ⊢ φ ∧ ψ
Γ, ψ ⊢ φ ∧ ψ, ∆
Γ ⊢ ψ, ∆
Γ, Γ ⊢ φ ∧ ψ, ∆, ∆
Γ ⊢ φ ∧ ψ, ∆
C UT
C UT
.
CL, CR
Thus, there is no easy way of relating the properties of ∧Ra to cut in the same way we
can with ∧R.
The additive cut rule,
Γ, φ ⊢ ∆
Γ ⊢ φ, ∆
Γ⊢∆
C UT a
fits more naturally with the additive rules. We can simulate ∧Ra with weakening (and
without contraction). Permuting an additive cut above another additive cut is a nonlocal process.
This presents us with a question. The additive formulation of classical logic, with an
additive cut, is a widely used system with a perfectly good theory of cut-reduction, but
one with different properties to those of the multiplicative system. How compatible
are the two systems; i.e., does the system of proof inequality given by Führmann and
Pym collapse in the presence of additive cut-reduction? In this thesis we will begin
to answer that question, by showing that the additition of reductions from the additive
theory yields models of local proof systems in the calculus of structures for classical
logic [15, 13] which do not collapse.
We do not give a detailed account of the admissibility of cut in a sequent system; for accounts of the proof for various different formulations of classical logic, see Gentzen’s
original paper [25], and also [66] and [54].
2.1.5 First-order LK
Given a first-order language, we now give the formulation of first-order classical proofs
over that language. To make the transition from the syntax to the semantics easier, we
23
introduce a slight modification to the language of formulae; we add notation to make
the dummying of variables in a formula explicit, and we restrict the propositional connectives to act only on formulae over the same variables (including dummied variables). Note that this change is not intended to give any extra proofs, but merely to
make some information implicit in the usual grammar explicit, and to make the structure of formulae more suited to the treatment we give in Chapter 6.
A first-order language L = (X , (VX ), A, F ) consists of a nonempty set X of variable
sorts, an infinite set VX of variable symbols for each sort X in X , a set A of sorted
atomic formulae, and a set F of sorted function symbols.
As is standard (see [60]), we introduce the notion of “type” as a finite sequence of sorts.
(This will allow us to think of formulae and terms as being unary, which will unify the
standard quantifiers we give now with the generalised quantifiers given in chapter 8).
We may now replace the notion of a sorted formula/function symbol with that of a
typed formula/function symbol; to each atomic formula a and function symbol f we
associate a type for the sorts of the arguments, and in addition for a function symbol we
associate a sort for the “result” of the function symbol. We introduce the type 1 (which
one may think of as the empty sequence) as the type of nullary formulae (sentences)
and nullary terms (constants). Each instance of an atomic formula and function symbol
has a tuple/tuples of variable symbols associated to it, so that from an atomic a with
type X, X, we may form instances a(x, y), a(y, x) etc., where x, y are variables from
VX .
We will denote by T the set of terms, by which we mean the set formed by closing F
under substitution. We will assume that each instance of a term has a tuple of properly
typed variables associated to its domain and codomain. It is clear, then, how typing
extends to instances of terms, but it is not so clear how it extends to the usual grammar
of formula instances. A substitution instance of an atom is given by the grammar
A := a(x̄) | [t/y]A,
where a has type X1 , X2 , ..., Xn and x̄ is an n-tuple of variable names such that xi has
type Xi , and where y is a variable name of type Y and t is a term of type Y .
Formula instances are now given by the grammar
B := b| (B ∧ B) | (B ∨ B) | ¬B | ⊤ | ⊥ | ∃y.B | ∀y.B,
24
where b is a substitution instance of an atom. Note that substitution is defined here
on atoms. It is trivial to extend the definition of substitution to formulae, only being
careful to note that some alpha conversion implicitly occurs when a term with free
variable x is applied to a formula which has x bound. For example, if we wish to
substitute a term t with domain x and codomain y into the formula ∃x.a(x, y), then we
must first rename the bound x to, say, x′ , to avoid capturing the free x in t.
If φ and ψ are formulae of type X, what is the type of (φ ∧ φ)? If we are able to give
an answer, it is only for a particular instance of (φ ∧ φ) ( the formula (φ(x) ∧ φ(x))
has type X, and the formula (φ(x) ∧ φ(y)) has type X, X). Yet, when considered on
their own, clearly φ(x) and φ(y) should have the same interpretation. That is, if I have
one proof containing φ(x) and no other variables than x, and another containing φ(y)
and y, If I can interpret φ(x) as a particular function from objects to truth values (in
the standard Tarskian semantics) then I can interpret φ(y) as the same function. In
each other’s presence, one must interpret each by a different function, and by default
we must dummy the argument of each onto the other (so both φ(x) and ψ(y) are
interpreted as binary functions, but each actually depends on only one of its inputs.
To ensure that the meaning of a formula is independent of context, we must annotate
a formula with information about the variables present in context on which it does not
depend. Following [60], we assume that the function symbols of our language include
binary projections πX,Y : X, Y → X for each pair X and Y of sorts, and nullary
projections πX : X → 1 for each sort X. Given a formula φ over a type X, W we
will use the abbreviation [y]φ for the substitution [πX,Y /x]φ (a formula over X, Y, W ),
where the variable associated to the sort Y does not appear associated to the type
X, W . (If φ is over 1, we will use [y]φ as shorthand for πY φ.)
Similarly, given Γ = φ1 , φ2 , . . . φn , where each φi does not depend on x, we define
[x]Γ = [x]φ1 , [x]φ2 , . . . [x]φn . By contrast, a formula φ that genuinely does depend on
x we will refer to as being properly over x.
Remark 2.1.3. The requirement that φ not be over y before we may form Y would
be redundant if terms have types rather than sorts as codomains. Then we way form
projections πX,Y → X where X is a type, and define the abbreviation [y]φ to be
[πX,Y /x̄]φ, where X is the type of φ, and the variable associated to X is different to
those appearing in Y . Here Y cannot possible have associated to it the same variable
as any of the sorts in X.
25
Alternatively, one may define [x]φ to be the result of making, for each sort Z in the
type X of φ, the substitution [πX,Z /z], where z is the variable associated to Z. Again,
we must specify that the variable associated to X is not the same as that associated to
Z (so, in this case, the projection really is a projection from X, X to X, and not just
the identity term on X. (Clearly the order of these substitutions is irrelevant, so we
may think of the occuring simultaneously). Once again, if φ contains the variable x
then we will be unable to carry out the substitution necessary on x, and so if φ is over
a type containing x, the expression [x]φ is not well-formed.
With this notation in place, we may restrict conjunction and disjunction to acting
“fibrewise”; that is, we may form the conjunction and disjunction only of formulae
over the same type. We recover the conjunction x = 1 ∧ y = 1, for example, as
[y]x = 1 ∧ [x]y = 1. the operators ∧ and ∨ to act only on formulae of the same type.
A typed substitution instance of an atom is given by the grammars below:
A1 := a | [s/y]Ay:Y ,
where a is an atom of type 1 and s is a term with codomain Y and domain 1, with the
y the variable associated to the domain of s;
Ax̄:X := a(x̄) | [t/y]Az̄,y,v̄:Z,Y,V ,
where x̄ is a tuple of variables of type X, t is a term with domain variables w̄ of type
W and codomain variable y of sort Y , z̄ is a tuple of variables of type Z, the type X
equals Z, W, V., and x̄ = z̄, w̄, v̄.
That is, a typed substitution instance over x̄ : X of an atom is either an atom of type
X with free variables x̄, or the result of substituting a term into a typed substitution
instance of a different type which has type X and variables x̄.
Grammars for typed formula instances over 1 (sentences) and over X (formulae with
free variables x̄ of type X), are given below:
B1 := b1 | (B1 ∧ B1 ) | (B1 ∨ B1 ) | ¬B1 | ∃z.Bz:Z | ∀z.Bz:Z | ⊤ | ⊥,
where b1 is a typed substitution instance over 1 of an atom and z is a variable of type
26
Z;
Bx̄:X := bx̄:X | (Bx̄:X ∧Bx̄:X ) | (Bx̄:X ∨Bx̄:X ) | ¬Bx̄:X | ∃z.Bw̄,z ȳ:W,Z,Y | ∀z.Bw̄,z ȳ:W,Z,Y .
where bx̄:X is a typed substitution instance over x̄ : X of an atom, and z is a variable
name of sort Z, w̄ is a sequence of varibles of type W , ȳ is a sequence of variables of
type Y , and x̄ = w̄, ȳ.
As above, we extend the definition of substitution from typed instances of atoms to
typed instances of formulae by induction on the structure of the formula (once again
taking care to apply any necessary alpha conversion where necessary). Since the notation [x] is shorthand for a particular substitution, it is defined on typed formula instances.
The only change this notation makes to the propositional fragment of LK is that we
may consider each rule therein as acting fibrewise; each formula in such a rule is over
the same type. The usual formulation of LK for first-order logic is given in Table
2.7. The Table 2.8 shows how these rules may be adapted to our modified grammar
of formulae. The notation allows us to state the side conditions for ∃L and ∀R within
the body of the rules; recall from abaove that the definition of [x]φ requires that x not
appear as the variable associated to any sort in the type of φ. This is clearly no great
innovation, but it does make the adjunction present more obvious. These rules take a
proof over one type (perhaps X, Y ) to another (in this case, Y ).
The rules ∃R and ∀L, like the propositional rules, act fibrewise. To maintain the
fibrewise nature of the rules we must decorate the active formula with copies of some
of the [yi], since it no longer depends on the variable quantified over.
To ensure the well-behavedness of substitution in our proofs, we require an eigenvariable condition. To each instance of ∃L and ∀R we associate a unique variable (
eigenvaraiable) and we require that this variable only appear above the relevant quantifier in the proof. Clearly we may alpha-rename any proof which does not satisfy this
property so that the renamed proof does. Given this condition, the following result is
standard:
Lemma 2.1.4. The sequent Γ ⊢ ∆ is provable in LK iff the sequent [t/x]Γ ⊢ [t/x]∆ is
provable.
27
Γ, φ ⊢ ∆
Γ, ∃x.φ ⊢ ∆
Γ, [t/x]φ ⊢ ∆
Γ, ∀x.φ ⊢ ∆
Γ ⊢ [t/x]φ, ∆
∃xL
Γ ⊢ ∃x.φ, ∆
Γ ⊢ φ, ∆
∀xL
Γ ⊢ ∀x.φ, ∆
∃xR
∀xR
Where for ∃L and ∀R, x is not free in Γ and in ∆.
Table 2.7: System LK: quantifiers
[x]Γ, φ ⊢ [x]∆
Γ, ∃x.φ ⊢ ∆
Γ ⊢ [t/x]φ, ∆
∃xL
Γ, [t/x]φ ⊢ ∆
Γ, [yi1 ][yi2 ]...[yim ]∀x′ .φ ⊢ ∆
Γ ⊢ [yi1 ][yi2 ]...[yim ]∃x′ .[x′ /x]φ, ∆
[x]Γ ⊢ φ, [x]∆
∀xL
Γ ⊢ ∀x.φ, ∆
∃xR
∀xR
Where φ is over a type containing x of sort X , t is term with domain variables
y1 , y2 , ..., yn of type Y = Y1 , Y2 , ...Yn and codomain variable x, x′ is a variable name
of type X, and yi1 , yi2 ...yim is the subsequence of (yi) not free in ∃x′ .φ,.
Table 2.8: System LK: quantifiers with typing
28
Proof. If x is not free in Γ ⊢ ∆, the two sequents are clearly the same. Otherwise, x is
not an eigenvariable of any quantifier, and so any proof of Γ ⊢ ∆ will remain a proof
if we replace x with t.
2.2 Proof nets
Proof nets were introduced as a syntax for linear logic by Girard [27], initially for the
fragment MLL− of linear logic; that is, MLL without units. In many ways a proof
net is a sequent proof without the sequentiality, as applications of rules which in the
sequent calculus may be permuted exists in parallel in a proof net. A proof net is ,
informally, a bipartite directed graph, with one sort of nodes labeled with propositions
and the other with names of sequent calculus rules, such that locally the graph has
structure as given in Table 2.9. However, not all such graphs will represent proofs in
the sequent calculus. We will refer to the class of all these graphs (whether or not they
represent proofs) proof structures.
Remark 2.2.1. It is customary to distinguish between the conjunction and disjunction
of MLL and those of classical/intuitionistic logic; here we follow the custom of [18]
and use ⊗ for the multiplicative linear conjunction (“tensor”) and ⊕ for the multiplicative linear disjunction (“par”).
For example, the following is a proof net demonstrating an entailment that is very
important in the categorical semantics of classical proofs: A⊗(B ⊕C) ⊢ (A⊗B)⊕C.
Ax
B :L
Ax
A:R
C :L
⊕L
A:L
Ax
B:R
⊗R
B⊕C :L
A⊗B :R
⊗L
C :R
⊕R
A ⊗ (B ⊕ C) : L
(A ⊗ B) ⊕ C
We can, however, construct more structures than there are proofs; specifically, we can
form graphs such as
29
Ax
φ:L
φ:L
φ:R
ψ:L
φ:R
∧R
∧L
φ∧ψ :R
φ∧ψ :L
φ:L
ψ:R
ψ:L
φ:R
∨L
ψ:R
∨R
φ∨ψ :L
φ∨ψ :R
φ:L
φ:R
¬L
¬R
¬φ : L
¬φ : R
φ:L
φ:R
Cut
Table 2.9: Proof nets for MLL−
30
Ax
A:L
Ax
B :L
A:R
⊕L
⊗R
A⊕B
A⊗B
B:R
which conform to the given specification, but represent unsound proofs. To eliminate
such structures, we need to introduce a correctness criterion. Such a criterion for MLLnets is given in [26], and refined by Danos and Regnier [19].
Definition 2.2.2. A (Danos-Regnier) switching σ is the choice of one of the hypotheses for each node of the following forms: ∧L, ∨R, CL, CR. We shall say that the
remaining nodes are unswitched.
Definition 2.2.3. Let S be a proof structure and σ a switching on it. Then the (DanosRegnier) graph of σ, DR(σ, S), is the following undirected graph:
• Its vertices are the propositional vertices of S;
• Its edges join conclusions of rule nodes to hypotheses as follows. If the rule node
is unswitched, then each conclusion is joined to each hypothesis. If the rule node
is switched, then the conclusion is joined only to the hypothesis chosen by σ. The
exceptions are axioms and cut, where the two formul are joined.
Definition 2.2.4. A proof structure S is a proof-net if, for every switching σ, DR(σ, S)
is directed and acyclic.
Clearly, from any sequent proof we can inductively generate a proof net: The converse
is a non-trivial result, known as sequentialization:
Theorem 2.2.5. Each correct proof net S is the image of some sequent proof.
For example, the net 2.2 sequentializes to
A⊢A
B⊢B
A, B ⊢ A ⊗ B
⊗R
C⊢C
A, B ⊕ C ⊢ A ⊗ B, C
A, B ⊕ C ⊢ (A ⊗ B) ⊕ C
⊕R
A ⊗ (B ⊕ C) ⊢ (A ⊗ B) ⊕ C
31
⊕L
⊗L
(but also to three other sequent proofs which differ on the ordering of ⊗L and ⊕R and
of ⊗R and ⊕L.)
Note that the representation for proof nets we have given is somewhat wasteful; we
could drop the propositional nodes, and arrive at nets as follows
⊕L
⊗R
⊗L
⊕R
as used in [16]. This makes for nets with which calculations may more easily be made.
Alternatively, we can drop the rule nodes and arrive at nets of the form
B:L
A:L
C :L
A:R
B⊕C :L
B:R
A⊗B :R
A ⊗ (B ⊕ C) : L
C :R
(A ⊗ B) ⊕ C
These nets are less cluttered, but retain the character of the sequent calculus, and it is
this style we will use.
Cut-reduction for MLL proof nets is particularly simple. Begin by expanding all logical cuts:
→
φ:L
ψ:L
φ⊗ψ :L
φ:R
ψ:R
φ:L
φ⊗ψ :R
and then all that remains are cuts of the form
32
ψ:L
φ:R
ψ:R
φ:L
φ:R
φ:L
φ:R
...
φ:R
φ:L
φ:R
which can be reduced to axiom links. This procedure is deterministic and terminating,
and we acquire a unique normal form for each net.
If we limit axioms to occurring only on atoms, the normal form will be a linking of
atoms on the left to those on the right, and a series of introductions of conjunctions
and disjunctions.
Extending proof nets for MLL− to logics with weakening and contraction has proven
difficult. While MLL− proof nets are simple and act as canonical objects for proofs,
proofs involving contraction or weakening involve a certain amount of bureaucracy,
associated with the order in which contractions occur and the the location to which a
weakening is “attached”. For example, if we add units to MLL− , we need a representation of the rule
Γ⊢∆
Γ ⊢ ∆, ⊥
In general, weakenings (of which this is a special case) must be anchored to some other
node of the net
M
φ:X
φ:X
ψ :L
since unanchored nets implicitly admit mix (and mix is not sound in MLL). These
links need to be rewired, as otherwise they obstruct the flow of cut reduction; the net
φ:R
φ:L
φ:R
33
ψ:L
cannot be reduced without moving the weakening. These rewirings are necessary, so
we have lost uniqueness of normal forms. (However, see [36] for a solution to this
problem.)
As proof nets diverge further from those for MLL− , the approaches separate into two
distinct families:
1. Syntactic proof nets, where the goal is to provide a representation of proofs as
close to the sequentialized proofs as possible, while still eliminating the bureaucracy associated with ordering of rules; and
2. Semantic proof nets, where the aim is to give objects as close to canonical objects
as possible, at the expense (perhaps) of giving intelligible proofs.
Our intention is to use proof-nets to eliminate commuting conversions in our calculations, rather than as canonical objects of a category. In fact, we will want to model
some commuting conversions non-trivially in our categories. The nets we use will be
unapologetically those of the first type.
2.3 Categorical logic and categorical preliminaries
We give some preliminaries specific to the semantics of proofs; for more a more general introduction to category theory, and further details, see [50]. The field of categorical proof theory, in which one considers the propositions of a logic as objects and
the proofs as morphisms in a category began with Lambek [46]. It is most richly developed for intuitionistic natural deduction, where a model is given by a bicartesian
closed category; conjunction is modeled by categorical product, disjunction by coproduct, and implication by exponentiation (for details see [47]). We give details now
of some extensions to this basic idea.
34
2.3.1 Interpretation of proofs in a category
Given a set P of proofs derived in a given calculus, let a theory T on those terms be a
relation on proofs of the same entailment which is reflexive and transitive. The relation
given by Prawitz on natural deduction proofs (two proofs are related if they have the
same normal form) is an example of such a theory. We recall in this section what it
means to model P or T , for a model to be sound, for a class of models to be complete,
and for a model to be initial.
Remark 2.3.1. We assume, for this discussion, that the calculus in question is propositional, leaving the discussion of first-order soundness etc. for the specific case of
classical doctrines.
Definition 2.3.2. Given a category C and a set P of proofs, an interpretation of P in
C is given by:
1. A function C ⌊−⌋ : φ → ⌊φ⌋, mapping propositions to objects of C; and
2. A function C ⌊−⌋ : (Φ : φ → ψ) → ⌊Φ⌋ : ⌊φ⌋ → ⌊ψ⌋, mapping proofs from P
to morphisms of C
- that is, it is a functor from P to C.
For the interpretation C ⌊−⌋ to be a model of T , all the relations holding in T should
hold of their images under C ⌊−⌋. Of course, if C is an ordinary category, the only
relationship that can hold between morphisms is equality, and so we may only model
equivalence relations.
Definition 2.3.3. An interpretation C ⌊−⌋ is sound with respect to an equivalence relation T if, for every equivalence Φ ≡ Ψ in T , C ⌊Φ⌋ = C ⌊Ψ⌋.
We use the term completeness, in this context, as a property of a class of models.
Definition 2.3.4. A class M of models is complete with respect to a theory T if we
have the following: if C ⌊Φ⌋ = C ⌊Ψ⌋ holds of every model in M, then Φ ≡ Ψ is in T .
35
The actual proof of completeness is usually achieved by what might reasonably be
considered sleight of hand; one constructs a term model CT from P and T , in which
CT ⌊Φ⌋ = CT ⌊Ψ⌋ if and only if Φ ≡ Ψ. If this model lies in a class M, then we
immediately derive completeness of M. The real content in the existence of a term
model is initiality:
Definition 2.3.5. A model D ⌊ ⌋ is initial if, given any other model C ⌊ ⌋, that model
factors through D ⌊ ⌋
F
D
-
6
-
C
⌊−
⌋
D ⌊−⌋
C
T
such that the functor F preserves all the structure of D.
Clearly, exactly what is being preserved depends on what is being modelled, and how.
2.3.2 Indexed categories and hyperdoctrines
Indexed categories are an attempt to capture, abstractly, the notion of a set-indexed
family. Formally, an indexed category is a (pseudo-)functor C : Bop → Cat, where the
category B is often taken to have finite products. Figure 2.1 shows intuition behind
this picture. For every object X in B there is a category C(X) — the “fibre” over X.
For every morphism a : X → Y , we define a functor a∗ : C(Y ) → C(X), referred to
as variously “substitution” or “re-indexing”.
The notion of a hyperdoctrine [49, 48], introduced by Lawvere, captures the categorical structure of quantification with an indexed category. The observation which led to
Lawvere’s advances in categorical logic is that many examples of logical structure can
be captured by a categorical adjunction. In particular, the form the rule ∃xL (without
contexts)
φ ⊢ a∗ ψ
∃xL
∃xφ ⊢ ψ
36
is that of a categorical adjunction:
A → GB
FA → B
.
It seems natural, therefore, to treat the ∃xL rule as a natural bijection between proofs
of φ ⊢ a∗ ψ and proofs of ∃xφ ⊢ ψ.
In addition to the structure of an indexed category a hyperdoctrine has, for certain
morphisms a in B, a functor Σa left adjoint to a∗ , and (possibly) a functor Πa right
adjoint to a∗ , as shown in Figure 2.1 . These functors are intended to model, respectively, generalized existential and universal quantification “along a”. We recover the
usual quantifiers from the adjoints to π ∗ , where π is a projection. The connection
between hyperdoctrines and intuitionistic first-order logic was made precise by Seely
[60], where the following definition is given of a hyperdoctrine for natural deduction:
Definition 2.3.6. A hyperdoctrine for natural deduction is an indexed category in
which
1. each fibre is a cartesian closed category (and thereby a model of intuitionistic
propositional natural deduction);
2. there exists a left and right adjoint to each substitution, as described above;
3. Frobenius reciprocity: the canonical morphism from Σa (A × a∗ B) to ΣA × B
is an isomorphism; and
4. The Beck-Chevalley condition: if
A
t
r
-
B
s
?
C
u
37
-
?
D
is a pullback in B, then
t∗
CA CB
Σs
Σr
?
?
u∗
CC CD
commutes.
This axiomatization is remarkably compact, in light of the results we can derive from
it:
1. Since it is a left adjoint, the functor Σ preserves coproducts, and so (the interpretation of) existential quantification commutes with disjunction (and dually
for universal quantification and conjunction);
2. Frobenius reciprocity for a functor implies that it is cartesian closed, so (the
interpretation of) existential quantification preserves implication; and
3. The Beck condition ensures that the interaction between quantifiers and term
substitution is well behaved, insofar as the order of quantifying over non-interfering
variables is irrelevant, as given by the pullback
Xs × X × Y
πY-
Xs × X
′
πX
πX
?
Xs × Y
πY - ?
Xs,
the sentence “x = x” is equivalent to ⊤, given by the pullback
X
id X
∆
id
?
X
∆-
?
X × X,
and that the functor Σt may be interpreted as ∃x(tx = y ∧ φ(x)) given by the
38
C(X)
C(Y )
Σa
a∗
Πa
B
a
X
Y
Figure 2.1: Indexed categories and hyperdoctrines: the set-up
pullback
X
hid, ti
-
X ×Y
t × id
t
?
Y
∆-
?
Y × Y.
In [48], Lawvere demonstrates that this is enough structure to ensure that substitution
is a cartesian closed functor. Seely additionally shows that each of these structures
generates a theory of first-order intuitionistic logic, and that each instance of a firstorder intuitionistic theory can, given a suitable system of proof reductions, form a
hyperdoctrine: the semantics is sound and complete.
Remark 2.3.7. In fact, Seely proves in [60] that insisting on the Beck condition for the
above pullbacks, and that satisfying the condition is preserved by cartesian product
and composition of pullbacks, is enough for soundness and completeness of the pure
logic (i.e., the logic without non-logical axioms).
2.3.3 Monoidal categories and functors
In order to model more general logics, a more general notion of multiplication on a
category is needed — the categorical product ties us to intuitionistic logic. In particular, to model logics without structural rules, we need a notion of multiplication in
which projection is not a basic concept.
39
The notion of a monoidal product is the one required in this context. Categories
with monoidal products are in fact intimately linked to sequent calculi by coherence
[50, 41, 40] . A coherence theorem for a particular class of monoidal categories tells
when a diagram in the free category of that type commutes; for monoidal, symmetric monoidal, symmetric monoidal closed and (symmetric) linearly distributive categories, coherence can be proved by cut reduction in a corresponding calculus of proof
nets [43, 11, 10].
Definition 2.3.8. A monoidal product (⊙, I, α, λ, ρ) on a category C is a functor ⊙ :
C ⊗ C → C, together with natural isomorphisms
α : (A ⊙ B) ⊙ C → A ⊙ (B ⊙ C),
λ:I ⊙A∼
= A,
ρ:A⊙I ∼
=A
such that the following coherence conditions hold:
α
α
((A ⊙ B) ⊙ C) ⊙ D - (A ⊙ B) ⊙ (C ⊙ D) - A ⊙ (B ⊙ (C ⊙ D))
α ⊙ id
id ⊙ α
?
α
(A ⊙ (B ⊙ C)) ⊙ D
(A ⊙ I) ⊙ B
ρ⊙
6
- A ⊙ ((B ⊙ C) ⊙ D)
α A ⊙ (I ⊙ B)
λ
⊙
id
d
i
A⊙B
A symmetric monoidal product on C is a monoidal product with an additional natural
transformation σ : A ⊙ B ∼
= B ⊙ A such that
σ
A⊙B - B⊙A
id
σ
-
?
A⊙B
(C ⊙ A) ⊙ B
α
?
C ⊙ (A ⊙ B)
σ ⊙ id
-
α
(A ⊙ C) ⊙ B - A ⊙ (C ⊙ B)
id ⊙ σ
?
σα
(A ⊙ B) ⊙ C - A ⊙ (B ⊙ C)
40
We use the notation (C, ⊙, I, α, λ, ρ, σ) to denote a symmetric monoidal category.
(However, where possible we will suppress the isomorphisms and refer to a symmetric monoidal category (C, ⊙, I), or even just C where the other structure is clear from
context.)
Definition 2.3.9. A monoidal functor
(F, µA,B , µI ) : (C, ⊙, I, α, λ, ρ) → (C ′ , ⊙′ , I ′, α′ , λ′ , ρ′ )
is a functor F : C → C ′ , together with the following natural transformations
µA,B : F A ⊙′ F B → F (A ⊙ B) µI : I ′ → F I
such that the following diagrams commute:
F A ⊙′ (F B ⊙′ F C)
id ⊙′ µ
′
α-
(F A ⊙′ F B) ⊙′ F C
µ ⊙′ id
?
F A ⊙′ F (B ⊙ C)
µ
?
F (A ⊙ (B ⊙ C))
F A ⊙′ I ′
id ⊙′ µI
?
F (A ⊙ B) ⊙′ C
µ
F α-
ρ FA
?
F ((A ⊙ B) ⊙ C)
I ′ ⊙′ F B
6
µI ⊙′ id
F (ρ)
?
µ
F A ⊙′ F I - F (A ⊙ I)
λ′ FA
6
F (λ)
?
µ
F I ⊙′ F B - F (I ⊙ B)
A symmetric monoidal functor between two symmetric monoidal categories is a monoidal
functor that, in addition, satisfies
41
F A ⊙′ F B
µ
′
σ-
F B ⊙′ F A
µ
?
F (A ⊙ B)
Fσ
?
F (B ⊙ A).
A monoidal natural transformation
θ : (F, µA,B , µI ) - (G, νA,B , νI )
between two monoidal functors is a natural transformation between the underlying
functors satisfying
µ
F A ⊙′ F B - F (A ⊙ B)
θ A ⊙′ θ B
?
θA⊙B
I′
id
?
ν
GA ⊙′ GB - G(A ⊙ B)
?
I′
µI-
FI
θI
νI-
?
GI
Theorem 2.3.10 (Kelly). Symmetric monoidal categories, symmetric monoidal functors and monoidal natural transformations form a 2-category.
Definition 2.3.11. We call a symmetric monoidal functor strong if µI and µ are natural
isomorphisms.
Lemma 2.3.12 ([42]). Let (C, ⊙, I, α, λ, ρ, σ) and (D, ⊙′ , I ′ , α′ , λ′ , ρ′ , σ ′ ) be symmetric monoidal categories, and let F ⊣ G be an adjunction between the underlying
categories:
F
C
⊥
G
-
D
Suppose further that F is a strong symmetric monoidal functor, with invertible natural
transformations βA,B and βI . Then G is a symmetric monoidal functor on D, and the
42
unit and co-unit of the adjunction are monoidal natural transformations (that is, the
adjunction is symmetric monoidal).
Proof. The strong monoidal structure of F and the adjunction induce the following
morphisms in C:
η
Gβ −1
G(ε ⊙ ε)
- G(F GA ⊙ F GB)
- G(A ⊙ B)
GA ⊙ GB - GF (GA ⊙ GB)
GβI−1 η GF I
GI
I
That this structure gives a symmetric monoidal adjunction is shown in the appendix,
section A.1.1
Definition 2.3.13 (Commutative monoids on a monoidal category). The notion of
multiplication given by a monoidal category is very general; in particular, a given
object may have no diagonal or unit maps (as required for multiplicative linear logic)
but it may also have several, with no universal property to pick out a specific (logically
significant) family. Additionally, there is no guarantee that the diagonal and unit maps
interact together in a sensible way.
A commutative monoid on a symmetric monoidal category (C, ⊙, I, α, λ, ρ, σ) consists
of an object A, and morphisms ∇A : A ⊙ A → A and []A : I → A, such that the
following diagrams commute:
(A ⊙ A) ⊙ A
∇A ⊙ id
-
A⊙A
∇A
-
α
∇
A
-
A
?
A ⊙ (A ⊙ A)
id ⊙ A-
43
A⊙A
A⊙I
id ⊙ []
-
ρ −1
[] ⊙ id
A⊙A I ⊙A
−1
∇
- ?
λ
A
A⊙A
∇
-
A
∇
-
σ
?
A⊙A
Definition 2.3.14. A symmetric monoidal category C as above has monoids if each
object A of C has a specified monoid, such that the following diagrams commute:
A⊙B⊙A⊙B
∇
A⊙
id ⊙ σ ⊙ id
?
∇A
⊙
I
[]A
-
B
A⊙B
∇ B-
⊙B
id
?
A⊙A⊙B⊙B
[] A
⊙
-
A⊙B
[] B -
I
id = []I : I → I
Lemma 2.3.15. Let (F, κA,B , κI ) : C → D be a comonoidal functor, and let (A, ∆, hi)
be a commutative comonoid in C. The object FA with morphisms dA and eA below
define a commutative comonoid in D.
dA = F A
F∆
-
eA = F A
κ
F (A ⊙ A) - F A ⊙ F A
F hi FI
44
κI
-
I
Proof. See appendix section A.1.2
2.3.4 Symmetric linearly distributive and ∗-autonomous categories
Introduced by Barr [5], ∗-autonomous categories are the commonest categorical formulation of a model of MLL.
Definition 2.3.16. A ∗-autonomous category (C, ⊗, I, (−)⊥ ) is a monoidal category
(C, ⊗, I), together with:
1. A functor (−)⊥ : C op → C;
2. A natural isomorphism (A⊥ )⊥ → A; and
3. A natural bijection
A⊗B →C
A→B⊸C
where B ⊸ C is shorthand for (B ⊥ ⊗ C)⊥ .
This is a model of MLL in the spirit of the one-sided sequent calculus; disjunction is
defined in terms of conjunction and negation as B ⊕ C := (B ⊥ ⊗ C ⊥ )⊥ .
Linearly distributive categories (previously weakly distributive categories) were introduced by Cockett and Seely [18]. They capture precisely the behavior of the two sided
multiplicative sequent calculus, in the sense that a linearly distributive category is a
model of LB.
How the cut rule is modeled is of particular interest. Given a proof Φ of Γ ⊢ ∆, φ and
a proof Ψ of φ, Γ′ ⊢ ∆′ , with denotations C⌊Φ⌋, C⌊Ψ⌋ in a classical category C, we
denote the cutting together of these two proofs by:
′
⌊Γ⌋ ⊗ ⌊Γ ⌋
⌊Φ⌋ ⊗ id
-
′
(⌊∆⌋ ⊕ ⌊φ⌋) ⊗ ⌊Γ ⌋
′
⌊∆⌋ ⊕ (⌊φ⌋ ⊗ ⌊Γ ⌋)
C⌊Ψ⌋
-
δ′
′
(2.7)
⌊∆⌋ ⊗ ⌊∆ ⌋,
where δ ′ is the evident morphism obtained from δ and symmetric monoidal isomorphisms, and id is identity. In this setting, cut is a generalized composition.
45
Unlike in a ∗-autonomous category, conjunction and disjunction are both given as
primitives. Duality is, nonetheless, built in, in the sense that if we add dualizing negation object-wise, that negation forms a functor with the required categorical properties.
In fact the category of linearly distributive categories with negation is equivalent to the
category of *-autonomous categories. An advantage of this approach is that we can
observe structural rules in the absence of dualizing negation (as with the Dummett
categories of Führmann and Pym, discussed in the following chapter.)
A category C is symmetric linearly distributive if it admits two monoidal structures
(C, ⊕, 0, α⊕ , λ⊕ , ρ⊕ , σ⊕ ) and (C, ⊗, 1, α⊗ , λ⊗ , ρ⊗ , σ⊗ ) linked by a natural transformation
δ : A ⊗ (B ⊕ C) → (A ⊗ B) ⊕ C
called a linear distribution. plus some coherence conditions: Before stating them, we
define natural transformations δLL , δRL , δRR , and δLR as follows:
A ⊗ (B ⊕ C)
id ⊗ σ⊕
δ = δLL
=
??
(A ⊗ B) ⊕ C
δRL
-
σ⊕
A ⊗ (C ⊕ B)
σ⊗ σ⊕ ⊗ id
- (B ⊕ C) ⊗ A
(C ⊕ B) ⊗ A
δRR
=
?
C ⊕ (A ⊗ B)
-
id ⊕ σ⊗
?
C ⊕ (B ⊗ A)
δLR
=
-
σ⊕
?
(B ⊗ A) ⊕ C
In our statement of the coherence conditions, we shall use the following three symmetries (taken from [18]):
op′ Reverse the arrows and swap ⊗ and ⊕, as well as 1 and 0. This gives the following
assignment of maps:
−1
−1
δLL ↔ δRR α⊗ →
7
α⊕
α⊕ →
7
α⊗
δRL 7→ δRL ρ⊗ →
7
ρ−1
ρ⊕ →
7
ρ−1
⊕
⊗
−1
R
R
δL 7→ δL λ⊗ 7→ λ⊕ λ⊕ 7→ λ−1
⊗
−1
−1
σ⊗ 7→ σ⊕
σ⊕ 7→ σ⊗
⊗′ Reverse the tensor ⊗; this assigns
−1
δLL ↔ idR
α⊕ 7→ α⊕
L α⊗ 7→ α⊗
δRL ↔
δRR
ρ⊗ ↔
λ⊗
−1
σ⊗ 7→ σ⊗
46
ρ⊕ 7→ ρ⊕
λ⊕ 7→ λ⊕
σ⊕ 7→ σ⊕
⊕′ Reverse the tensor ⊕; this assigns
−1
δLL ↔ δRL α⊗ 7→ α⊗ α⊕ →
7
α⊕
δLR ↔ δRR ρ⊗ 7→ ρ⊗ ρ⊕ ↔ λ⊕
λ⊗ 7→ λ⊗
−1
σ⊗ 7→ σ⊗ σ⊕ 7→ σ⊕
The coherence laws are as follows, where for each law we also require all versions
generated by the symmetries op′ , ⊗′ , and ⊕′ :
1 ⊗ (A ⊕ B)
δLL
λ⊗
(1 ⊗ A) ⊕ B
(2.8)
- A⊕B
?
λ⊗ ⊕ id
α⊗ A ⊗ (B ⊗ (C ⊕ D))
(A ⊗ B) ⊗ (C ⊕ D)
id ⊗ δLL
?
δLL
A ⊗ ((B ⊗ C) ⊕ D)
(2.9)
δLL
?
-
((A ⊗ B) ⊗ C) ⊕ D
α⊗ ⊕ id
?
(A ⊗ (B ⊗ C)) ⊕ D
(A ⊕ B) ⊗ (C ⊕ D)
δRR
L
δL
-
((A ⊕ B) ⊗ C) ⊕ D
δRR ⊕ id
A ⊕ (B ⊗ (C ⊕ D)) (2.10)
id ⊕ δLL
?
-
(A ⊕ (B ⊗ C)) ⊕ D
α⊕
47
?
A ⊕ ((B ⊗ C) ⊕ D)
A ⊗ ((B ⊕ C) ⊕ D)
δLL
id ⊗ α⊕
δRL
?
?
(A ⊗ (B ⊕ C)) ⊕ D
δRL ⊕ id
A ⊗ (B ⊕ (C ⊕ D))
B ⊕ (A ⊗ (C ⊕ D))
(2.11)
id ⊕ δLL
?
-
(B ⊕ (A ⊗ C)) ⊕ D
α⊕
?
B ⊕ ((A ⊗ C) ⊕ D)
A linearly distributive category with negation is a linearly distributive category together with, for every object A, an object Ā, and maps
γ L : Ā ⊗ A
-
0
τR : 1
-
A ⊕ Ā
0
τL : 1
-
Ā ⊕ A
Together with the induced maps
γ R : A ⊗ Ā
-
the following coherence conditions are required:
A⊗1
L
id ⊗ τ-
δLL
γ R ⊕ id
- 0⊕A
A ⊗ (Ā ⊕ A) (A ⊗ Ā) ⊕ A
ρ⊗
λ⊕
-
A
Ā ⊗ 1
R
id ⊗ τ-
(2.12)
δLL
γ L ⊕ id
- 0 ⊕ Ā
Ā ⊗ (A ⊕ Ā) (Ā ⊗ A) ⊕ Ā
ρ⊗
λ⊕
-
Ā
(2.13)
Theorem 2.3.17. A symmetric linearly distributive category with negation is equivalent to a *-autonomous category.
Coherence for SLDCs corresponds precisely to a system of cut-reduction on the proof
nets of [12] (which in turn is equivalent to cut-reduction on a two sided sequent system
for MLL), this result states that given that theory of cut-reduction, the two sided and
48
one-sided systems are equivalent.
The equivalence between ∗-autonomous categories and SLDCs with negation extends
to an equivalence of 2-categories with the introduction of linear functors and linear
transformations [16]. This notion of morphisms between symmetric linearly distributive categories reduces, in the presence of negation, to that of a monoidal functor. A
linear functor F : C → D consists of a functors F⊕ , F⊗ : C → D such that F⊕ is
comonoidal w.r.t ⊕ and F⊗ is monoidal w.r.t. ⊗, , with natural transformations
mL : F⊕ A ⊗ F⊗ B → F⊕ (A ⊗ B)
nR : F⊗ (A ⊕ B) → F⊕ A ⊕ F⊗ B
(which we call linear strengths such that certain coherence conditions hold. Once
again, before we give the conditions we define, via the symmetries of monoidal categories, morphisms
mR : F⊗ A ⊗ F⊕ B → F⊕ (A ⊗ B)
nL : F⊗ (A ⊕ B) → F⊗ A ⊕ F⊕ B
and add to the permutations op′ ,⊗′ and ⊕′ :
op′ Additionally, swap F⊗ A and F⊕ B, making the following additional changes to
arrows
mL ↔ nL
µ⊕ ↔ µ⊗
µ⊥ ↔ µ⊤
⊗′ Swap the following linear strengths
mL ↔ mR
⊕′ Swap the following linear strengths
nL ↔ nR
The following, plus their images under the permutations above, are the axioms of a
linear functor:
49
n
G(⊥ ⊕ A) - F ⊥ ⊕ GA
µ⊥ ⊕ id
G(λ)
?
GA G((A ⊕ B) ⊕ C)
(2.14)
?
λ
⊥ ⊕ GA
Gα-
G(A ⊕ (B ⊕ C))
n
n
?
?
F A ⊕ G(B ⊕ C)
F (A ⊕ B) ⊕ GC
(2.15)
id ⊕ n
ν ⊕ id
?
?
(F A ⊕ F B) ⊕ GC
α-
G((A ⊕ B) ⊕ C)
Gα-
F A ⊕ (F B ⊕ GC)
G(A ⊕ (B ⊕ C))
n
n
?
?
F A ⊕ G(B ⊕ C)
G(A ⊕ B) ⊕ F C
n ⊕ id
id ⊕ n
?
(F A ⊕ GB) ⊕ F C
Gα
-
50
?
F A ⊕ (GB ⊕ F C)
(2.16)
id ⊗ n
GA ⊗ G(B ⊕ C)
GA ⊗ (GB ⊕ F C)
ν
Gδ
?
?
G(A ⊗ (B ⊕ C))
(GA ⊗ GB) ⊕ F C
(2.17)
ν ⊕ id
Gδ
?
n G(A ⊗ B) ⊕ F C
?
G((A ⊗ B) ⊕ C)
id ⊗ n
GA ⊗ G(B ⊕ C)
GA ⊗ (F B ⊕ GC)
δ
ν
?
?
(GA ⊗ F B) ⊕ GC
G(A ⊗ (B ⊕ C))
(2.18)
n ⊕ id
Gδ
?
?
n F (A ⊗ B) ⊕ GC
G((A ⊗ B) ⊕ C)
This notion of functor is sound and complete with respect to a class of proof nets with
linear functor boxes. A linear functor box takes a net M to a net with applications of
the functors F⊗ and F⊕ as doors, with the following formation rule (and its dual)
F⊗ A
F⊗ B
F⊗ C
B
C
A
D
E
F⊕ D F⊗ E
H
J
F⊗ H F⊗ J
51
(notice that the wire passing though the semicircle is labeled with F⊗ , and each other
wire from the bottom of the proof is labeled with F⊕ .) The rewrites corresponding
to the axioms of a linear functor include two rewrites where two boxes rewrite to one
box:
F⊗
F⊗
F⊗
F⊗
F⊗
F⊕
(For full details see [16].)
MIX in linearly distributive categories
One of the striking features of MLL is that the tensor ⊗ is not stronger than ⊕ in the
same way that ∧ is stronger than ∨; we do not have a proof of A ⊗ B ⊢ A ⊕ B. In fact,
52
we do not even have a proof of ⊥ ⊢ ⊤. The two are equivalent, however, and adding
the latter we may derive the former, in the following two ways:
-
(id ⊕ m) ⊗ id
δL
L
(A ⊕ ⊤) ⊗ B
A ⊕ (⊤ ⊗ B)
λ
1
⊥
−
ρ⊤
⊗
⊕
id
id
-
(A ⊕ ⊥) ⊗ B
A⊕B
(2.19)
ρ⊥
λ−
⊤ 1
⊗
⊕
id
id
-
A⊗B
A ⊗ (⊥ ⊕ B)
-
id ⊗ (m ⊗ id)
δR
R
A ⊗ (⊤ ⊕ B)
(A ⊗ ⊤) ⊕ B
A symmetric linearly distributive category is “mix” [17] if diagram (2.19) commutes,
giving a canonical morphism mixA,B : A ⊗ B → A ⊕ B. This is in reference to
Girard’s mix rule [26]:
Φ1
·
·
·
Γ⊢∆
Φ2
·
·
·
′
Γ ⊢ ∆′
Γ, Γ′ ⊢ ∆, ∆′
Mix
which is also derivable from the morphism m.
The following surprising result is initially due to Hasegawa and refined by Führmann
and Pym:
Proposition 2.3.18. A symmetric linearly distributive category with a comonoid on ⊥
(dually, a monoid on ⊤) is mix.
Proof. See [24].
53
Chapter 3
Background: recent work
3.1 Proof nets for propositional classical logic
The work of Robinson [58] extends the notion of proof net from linear logic to classical
logic, preserving the structure of proofs from the sequent calculus in the sense that each
rule in the sequent calclulus corresponds to a node in the [roof nets.
A net is a bipartite graphs labelled with propositions and rules, whose local structure
is made from the parts given in Tables 3.1 and 3.2, which satisfies a certain global
correctness criterion. We call the graphs made from the figures in Tables 3.1 and 3.2
proof structures.
The definition of a switching is extended to certain structural rules.
Definition 3.1.1. A (Danos-Regnier) switching σ is the choice of one of the hypotheses for each node of the following forms: ∧L, ∨R, CL, CR. We shall say that the
remaining nodes are unswitched.
Definition 3.1.2. Let S be a proof structure and σ a switching on it. Then the (DanosRegnier) graph of σ, DR(σ, S), is the following undirected graph:
• Its vertices are the propositional vertices of S;
• Its edges join conclusions of rule nodes to hypotheses as follows. If the rule node
is unswitched, then each conclusion is joined to each hypothesis. If the rule node
54
Ax
φ:L
φ:L
φ:R
ψ:L
φ:R
∧R
∧L
φ∧ψ :R
φ∧ψ :L
φ:L
ψ:R
ψ:L
φ:R
∨L
ψ:R
∨R
φ∨ψ :L
φ∨ψ :R
⊤R
⊤:R
⊥L
⊥:L
φ:L
φ:R
¬L
¬R
¬φ : L
¬φ : R
φ:L
φ:R
Cut
Table 3.1: Proof nets for classical logic: linear rules
55
φ:X
φ:X
WL
WLR
φ:X
ψ:L
φ:X
ψ :R
φ:L
φ:L
φ:R
φ:R
CL
CR
φ:L
φ:R
Table 3.2: Proof nets for classical logic: structural rules
is switched, then the conclusion is joined only to the hypothesis chosen by σ. The
exceptions are axioms and cut, where the two formul are joined.
This allows the definition of correctness for classical nets.
Definition 3.1.3. A proof structure S is a proof-net if, for every switching σ, DR(σ, S)
is directed and acyclic.
Theorem 3.1.4. (Robinson) Each correct proof net S is the image of some sequent
proof.
Sequentialization allows us to present proof nets as an (ambiguous) grammar, given
in Tables 3.3 and 3.4, corresponding to the rules of the sequent calculus. It is this
representation that we choose to present proofs. The presentation also removes the
rule nodes (which can be deduced from context).
Cut-elimination for these proof nets was established in [23]. The reductions involved
in the cut-elimination procedure are given in in Tables 3.1 and 3.7. For each connective
and unit we have a local cut-reduction step, e.g. Cut∨, in which the same structure is
introduced on both sides of the cut, and so can be eliminated on both sides of the cut.
Cuts against axioms are trivial and are eliminated by CutAx. By contrast, cuts against
structural rules are “non-local”, in the sense that one application can either delete or
copy a sub-net of arbitrary size.
56
φ:L
φ:R
M
N
M
φ:R
φ:L
ψ:R
ψ :L
φ∧ψ : R
φ∧ψ :L
M
N
M
φ:L
ψ :L
φ:R
ψ:R
φ∨ψ :R
φ∨ψ :L
M
φ:X
φ:X
⊤:R
⊤:L
M
φ:X
⊥:L
φ:X
M
⊥:R
M
φ:R
φ:L
¬φ : L
¬φ : R
N
M
φ:L
φ:R
Table 3.3: Proof net grammar: linear rules
M
M
φ:X
φ:X
φ:X
ψ:L
φ:X
M
φ:L
ψ:R
M
φ:L
φ:R
φ:L
φ:R
φ:R
Table 3.4: Proof net grammar: structural rules
57
In addition, the lemma requires some coherence laws to manage certain aspects of
structural rules. An important property derived from these conditions is empire rewiring:
Lemma 3.1.5. A weakening may be moved (rewired) so it is connected to any node in
the largest subnet containing the conclusion φ of the weakening as a door(the empire
of φ).
With these in place, the following principal lemma is proved:
Lemma 3.1.6. The net
M
φ:L
...
N
φ:L
φ:R
φ:L
...
φ:R
φ:R
where M and N are cut-free, reduces to a cut-free net.
The proof is by induction on (degree, rank) of the cut, lexicographically ordered, where
the degree is the number of logical symbols in the cut formula, and the rank is, informally, the maximum distance (in the sequent calculus) from the cut formula to the
logical rule that introduces that formula.
The formal definition of rank depends on marking some formulae of a net, and then
propagating the marking through a sequence of decompositions (given in Table 3.5);
the left (resp. right) rank of the formula on the left (right) is the maximum length n of
a decomposition M = M1 ⇒ M2 ⇒ · · · ⇒ Mn for which each Mi has at least one
marked door. Given a cut such as in the principal lemma, the rank is the left rank of
M with all the φ : L marked plus the right rank of N with all the φ : R marked.
Proof. (Principal lemma)
(We summarize the details: a full proof is given in [23].) As mentioned above, it is an
induction on (degree, rank), ordered lexicographically.
58
M
φ:L
ψ:L
M
⇒
φ ∧ ψ : L/x
φ : L/0
ψ : L/0
M
φ:R
ψ:R
M
⇒
φ ∨ ψ : R/x
φ : R/0
ψ : R/0
M
M
φ:X
⇒
¬φ : X̄/x
φ : X/0
M
φ:X
φ : X/x
M
ψ : Y /y
⇒
φ : X/x
M
φ:X
φ:X
M
⇒
φ : X/x
φ : X/x
M
φ : R/0
N
φ:R
M
⇐
ψ :R
φ : L/0
ψ : R/0
N
φ:L
⇐
N
⇒
φ ∧ ψ : R/x
M
M
φ : X/x
ψ :L
φ ∨ ψ : L/x
N
⇒
ψ : L/0
Table 3.5: Decomposition relation for determining left and right rank of marked nets
59
In the case where rank is minimal cut reduction is either against a weakening or axiom (in which case the cut may immediately be eliminated) or logical (so the relevant
reduction may be applied, reducing degree).
In the case of non-minimal rank, if the last rule is a weakening, we use empire rewiring
to either move the weakening to an unmarked door (in the case where the door introduced is unmarked) or to a marked door (in the case where the formula introduced is
marked — an application of WC then removes the weakening).
Otherwise the last rule is the introduction of some logical symbol. All cases follow the
same general pattern; we give the case for negation:
Given the net
M
¬φ : L
...
¬φ : L
N
φ:R
¬φ : R
¬φ : R
¬φ : L
¬φ : L
¬φ : R
¬φ : L
we apply the reduction C UT C, and obtain
N
N
M
¬φ : R
¬φ : L
...
¬φ : L
¬φ : L
¬φ : R
¬φ : R
¬φ : R
φ:R
¬φ : L
¬φ : R
60
¬φ : R
The empire of φ satisfies the induction hypothesis, and so we can eliminate the cut.
This results in another net that satisfies the induction hypothesis, and so the final cut
can be eliminated.
We carry over our notion of replacing an inference rule with a constant, so that for
example ∧R can be seen as a cut against the constant node K∧ :
K∧
M
φ:L
φ:L
N
ψ:L
φ:L
φ∧ψ :R
3.2 Classical categories
Classical categories [23, 22, 24] are a sound and complete semantics of LK. The soundness and completeness is with respect to cut-reduction, via an ordering on proofs.
A classical category is a poset-enriched symmetric linearly distributive category with
negation, with extra structure necessary to model weakening and contraction - it has
symmetric monoids and symmetric comonoids, and some additional axioms such that
cut-reduction is sound as an inequality on the hom-sets of the category.
Definition 3.2.1. A classical category is an order-enriched symmetric linearly distributive category with negation, such that:
1. The symmetric monoidal category (C, ⊕, 0) has symmetric monoids;
2. The symmetric monoidal category (C, ⊗, 1) has symmetric comonoids;
3. The object indexed families of maps ∆A , ∇A , hiA and [ ]A are lax natural transformations, in the sense that for every morphism f we have
∆ ◦ f ≤ (f ⊗ f ) ◦ ∆ f ◦ ∇ ≤ ∇ ◦ (f ⊕ f )
hi ◦ f ≤ hi
f ◦ [] ≤ []
4. The inequalities in Table 3.8 hold, where δ̂ and δ̌ are the evident morphisms
obtained from δ and symmetric monoidal isomorphisms
61
≡
φ:R ψ:R
φ:L
φ∨ψ :R
ψ:L
φ:R ψ:R
φ:L
ψ:L
φ∨ψ : L
C UT ∨
≡
φ:L
ψ:L
φ:R
φ∧ψ :L
ψ:R
φ:L
ψ:L
φ:R
ψ:R
φ∧ψ :R
C UT ∧
≡
φ:L
φ:R
¬φ : R
¬φ : L
φ:L
φ:R
C UT ¬
≡
ψ:L
ψ :R
ψ:L
ψ:L
C UTA X
≡
φ:X
φ:X
φ:X
⊤:L
⊤:R
C UT ⊤
≡
φ:X
φ:X
φ:X
⊥:R
⊥:L
C UT ⊥
≡
φ:X
φ:X
φ:X
φ:X
WC
φ:X
≡
ψ:X φ:Y :
ψ :X
ψ:X φ:Y :
φ:Y : θ:Z
θ:Z
Table 3.6: Rules for cut-elimination: local rules
62
W MOVE
4
ψ:X
ψ:X
θ : Z ′ φ1 : Y1
ψ:X
φn : Yn
ψ:X
θ:Z
φ1 : Y1
...
ψ:X
ψ1 : Y1ψn : Yn φ : X ′
φ:X φ:X
C UT W
φn : Yn
4
φ:X
φ:X φ:X
ψ1 : Y1 ψn : Yn φ : X ′
ψ1 : Y1 ψn : Yn φ : X ′
ψ1 : Y1
ψn : Yn
C UT C
Table 3.7: Rules for cut elimination: structural rules
5. Composition of morphisms, and the functors ⊕, ⊗ are monotonic in all arguments.
Examples
An immediate example of a classical category is the trivial, collapsed case:
Example 3.2.2. A boolean lattice B is a classical category, with meet as ⊗ and join
as ⊕.
We can extract a non-trivial model from sets and relations in the following manner:
Example 3.2.3. Rel⊗ is a classical category with objects sets and morphisms binary
relations, in which both ⊗ and ⊕ are given by the set theoretic product. Both 0 and 1
are given by the singleton set {∗}. Negation is identity on objects, and the excluded
middle on a set A is the relation {(∗, (x, x)) : x ∈ A} from {∗} to A × A. The map
∇A is {((x, x), x) : x ∈ A} and [ ]A is {(∗, x) : x ∈ A}. The order on hom-sets is
set-theoretic inclusion of relations.
63
∆(A ⊕ C) ⊗ (A ⊕ C)
A⊕C
∆∇
id ⊕ ∆
?
A ⊕ (C ⊗ C)
A⊗C ≤
∇ ⊕ id
∇
id ⊗ ∇
A ⊗ (C ⊕ C)
≤
∆ ⊗ id
-
hi
-
1
(A ⊕ A) ⊕ (C ⊗ C)
∼
≤
=
?
[
]
⊕
id
A⊕1 0⊕1
(A ⊗ C) ⊕ (A ⊗ C)
[]
A⊗C δ̂
6
∇∆
A⊕C
δ̌
?
6
(A ⊗ A) ⊗ (C ⊕ C)
id ⊕ hi
?
6
0
6
∼
≤
=
hi ⊗ id
- 1⊗0
A⊗0
id ⊗ [ ]
Table 3.8: Inequalities of a classical category
Example 3.2.4. Given any two classical categories C and D, their product C × D is
a classical category. Thus, given any boolean lattice B, the product Rel⊗ × B is a
non-compact, nontrivial classical category.
An informative class of models is given in [22] and extended in [24], based on the
abstract Geometry of Interaction (GoI) construction of [2, 1]. The construction begins with the notion of a Dummett category, which is a model of the negation and
implication free fragment of Dummett’s multiple conclusioned calculus (which, in our
notation, is LB + structural rules) for intuitionistic logic. Recall from the introduction
that this setting is already enough to allow the collapse caused by Lafont’s example.
Example 3.2.5. A model of LB+structural rules is given by Rel⊕ , in which both conjunction and disjunction are modelled by disjoint union.
The general definition of a model is given by removing negation from classical categories.
Definition 3.2.6. A Dummett category is an order-enriched symmetric linearly distributive category, such that:
1. The symmetric monoidal category (C, ⊕, 0) has symmetric monoids;
2. The symmetric monoidal category (C, ⊗, 1) has symmetric comonoids;
64
hi[ ]
[ ]hi
3. The object indexed families of maps ∆A , ∇A , hiA and [ ]A are lax natural transformations, in the sense that for every morphism f we have
∆ ◦ f ≤ (f ⊗ f ) ◦ ∆ f ◦ ∇ ≤ ∇ ◦ (f ⊕ f )
hi ◦ f ≤ hi
f ◦ [] ≤ []
4. The inequalities in Table 3.8 hold.
5. Composition of morphisms, and the functors ⊕, ⊗ are monotonic in all arguments.
The construction used to form classical categories concerns a Dummett category D
which is compact and traced; full details of the construction G yielding a classical
category are given in [24]. We present the case where the monoidal product on D is
a categorical biproduct; this simplifies the presentation of Dummett categories significantly:
Proposition 3.2.7 (Hasegawa). A category with biproducts forms a Dummett category
if and only if the equation ∆ ◦ ∇ = id holds.
Proof. See [24].
Categories with biproducts are simple to deal with, since a morphism f from A1 ⊕
A2 ⊕ · · · ⊕ An to B1 ⊕ B2 ⊕ · · · ⊕ Bm consists of an m by n matrix of relations

f11 f12 . . . f1m

 ..
..
 .
.

fn1 fn2 . . . fnm





In this case we also have an intuitive notion of trace; D has a trace if its hom-spaces
are complete semi-lattices. In particular, disjoint union of sets is a biproduct in Rel,
and Rel⊕ has arbitrary unions of relations , so this construction will yield a concrete
example.
Given a Dummett category with biproducts and arbitrary unions of morphisms, the
category G(D) is as follows:
65
• as objects, pairs (A+ , A− ) of objects of D, where the first element is intended to
model the positive literals in a formula, and the second the negative;
• a morphism (A+ , A− ) → (B + , B − ) is a morphism in D from A+ ⊗ B − to
B + ⊗ A− , which we may represent (thanks to the biproduct) as
fAB-
A+
B+
6
fAA
fBB
?
fBA −
A− B ,
where each f is a morphism of D ; and
• composition (given in the general case by the trace) is given by the sum of the
paths in the following diagram:
A+
fAA
?
−
A
fAB - +
B
gBC-
6
fBB
f
BA
6
fCC
gBB
B
C+
?
−
gCB
C −.
For example, the component (g ◦ f )AA is given by the relation
fAA ⊔ fBA ◦ (gBB ◦ fBB )∗ ◦ gBB ◦ (fBB ◦ gBB )∗ ◦ fAB .
where f ∗ denotes the limit ⊔(id, f, f 2 , f 3 . . . ).
This model (in the case D = Rel⊕ ) gives a nice characterization of those morphisms
that behave well with respect to cut against structural rules. For example, the morphism
representing ∆ in G(Rel) is the square
A+
∅
?
−
A
∆-
A+ ⊕ A+
∅
∆◦
6
A− ⊕ A− .
Given a general morphism g from (A+ , A− ) to (B + , B − ), if we compare the morphism
(g ⊕ g) ◦ ∆, given by
66
A+
∅
gAB ⊕ gAB-
∆- +
A ⊕ A+
?
∅
◦
∆
A− B+ ⊕ B+
6
6
gBB ⊕ gBB
gAA ⊕ gAA
?
gBA ⊕ gBA
A− ⊕ A− B− ⊕ B−,
(3.1)
with the morphism ∆ ◦ g, given by
A+
gAA
gAB -
∆-
B+
6
gBB
?
gBA
A− B+ ⊕ B+
∅
∅
?
◦
− ∆
B
6
(3.2)
B− ⊕ B−,
we notice that the two are equal as morphisms
a-
A+
b
B+ ⊕ B+
6
c
?
d
A− B − ⊕ B −
iff gBB is empty. To see this, notice that 3.1 gives the relation c as


gBB
∅
∅
gBB

,
while 3.2 gives it to be


gBB gBB
gBB gBB

.
This is an example of the data flow interpretation of the proof theory of classical logic
[22]. Put simply, this says that a proof Φ is larger than a proof Ψ if, when we connect
premises to conclusions in a sequent proof, the connections in Φ are a superset of those
in Ψ.
67
Structural properties of Dummett categories
The following are a series of properties of Dummett categories (and so, clearly, also
on classical categories.)
We already know, from Proposition 2.3.18 that Dummett categories are mix; this result
from [24] shows that they are canonically so:
Lemma 3.2.8. Every Dummett category has a greatest morphism ⊥ → ⊤, given by
[]⊤ = hi⊥ .
Proof. In a Dummett category C, hi⊥ = id⊤ , by the definition of having comonoids.
Now, by the lax naturality of hi, we have for any f : ⊥ → ⊤,
f = id ◦ f = hi⊤ ◦ f ≤ hi⊥ .
Dually, we get f ≤ []⊤ .
The mix structure allows the definition of a convolution on morphisms: given two
morphisms f and g : A → B, let f ∗ g : A → B be given by:
-
A⊗A
B⊕B
∇
-
B
⊗
m
ix
B
f
,B
-
∆-
g
A
⊕
m
ix
A
f
,A
-
A⊕A
g
-
B⊗B
.
The convolution corresponds precisely to Lafont’s example in the case where the cut
is against ⊥ or ⊤, and so the following inequalities hold:
f ≥ f ∗ g ≤ g.
68
The following is proved in [24]:
Lemma 3.2.9. The convolution f ∗ f of a morphism with itself is equal to f .
Moreover, we may induce from ∗ a partial order by setting f ⊏ g iff f = f ∗ g.
Theorem 3.2.10. In every Dummett category, the ordering ⊏ induced by ∗ agrees with
the ordering ≤.
This result allows for definitions of Dummett categories and classical categories which
are fully equational (by translating the inequalities into facts about the convolution.)
Soundness and Completeness
The following is the notion of theory with respect to which classical categories are
sound and complete:
Definition 3.2.11. A sequent theory over a collection of atoms A is a set of inequalities
Φ 4 Ψ, where both Φ and Ψ are proofs of the same sequent Γ ⊢ ∆ over A, such that:
1. The relation 4 is reflexive, transitive, and compatible (i.e. all inference rules are
“monotonic w.r.t. 4);
2. The relation holds for both directions of the usual cut-reduction rules for eliminating logical cuts (Table 3.1) .
3. The relation holds in both directions for a number of coherence rules: axiom
expansions and coherences for structural rules (Tables 3.9, 3.10 and3.11 );
4. The usual rules for eliminating cut against weakening and contraction hold in
only one direction: from redex to reduct.
Theorem 3.2.12 (Soundness). Let P be a set of proofs over a set of atoms A. Then
for every interpretation C⌊−⌋ in a classical category C, the judgements Φ 4 Ψ such
that ⌊Φ⌋ ≤ ⌊Ψ⌋ form a sequent theory.
Proof. By verifying that each reduction in the definition of a net theory holds in each
classical category. See [23]
69
φ:L
φ∧ψ :L
φ∨ψ :L
φ∧ψ :R
φ∨ψ :R
ψ:L
ψ:R
φ∧ψ :L
φ∧ψ :R
φ:L
φ:R
ψ:L
ψ:R
≡
φ∨ψ :L
≡
¬φ : L
φ:R
≡
¬φ : R
φ:L
¬φ : L
φ∨ψ :R
φ:R
¬φ : R
Table 3.9: Axiom expansions for sequent theories
Theorem 3.2.13 (Completeness). Let T be a sequent theory, and suppose that ⌊Φ⌋ ≤
⌊Ψ⌋ holds for every interpretation in a classical category C. Then the judgement Φ 4
Ψ is in T .
Proof. By constructing a term model from T in which morphisms are equivalence
classes of proof nets under T .
3.3 The calculus of structures for classical logic
3.3.1 System SKSg
Deep inference is a new paradigm for designing formalisms in proof theory; it is easiest
to see how it differs from traditional formalisms by seeing that it does not apply in
those formalisms. The sequent calculus LK does not exhibit deep inference. Consider
the sequent
⊢ C ∧ (A ∨ A), B, B.
70
M
M
≡
⊤:L
⊤:L
⊤:L
⊤:L
⊤:L
⊤:L
M
twist⊤
M
≡
⊥:R
⊥:R
⊥:R
⊥:R
⊥:R
⊥:R
twist⊥
≡
φ:X
φ:X
φ:X
ψ:L
ψ:L
φ:X
ψ∧θ :L
θ:L
W∧
ψ∧θ :L
≡
φ:X
φ:X
φ:X
ψ:R
ψ :R
φ:X
ψ∧θ :R
θ:R
W∨
ψ∨θ:R
Table 3.10: Coherence for weakening
71
≡
φ : L ψ : Lφ : L
φ∧ψ : L
φ : L ψ : Lφ : L
ψ : L
ψ : L
φ∧ψ : L
φ :L
ψ : L
C∧
φ∧ψ : L
φ∧ψ : L
≡
φ:R
ψ:R
φ∧ψ :R
φ:R ψ:R
φ : R ψ : Rφ : R
ψ:R
φ∧ψ :R
φ:R
φ∧ψ :R
ψ:R
φ∧ψ :R
C∨
Table 3.11: Coherence for contraction
We can apply contraction across the comma, but not across the disjunction. Semantically, comma on the right is the same as disjunction: the syntactic distinction is required for completeness of the sequent calculus. The calculus of structures [33, 30] is
a formalism that employs deep inference. It removes the distinction between comma
and ∨, and any calculus of structures inference rule operates arbitrarily deeply in a
formula: this is deep inference. Proofs in the calculus of structures are not trees as in
the sequent calculus, but are linear.
We now present the syntax for classical logic in a deep inference setting:
Definition 3.3.1. Given a set V of propositional variables, we consider formulæ given
by the grammar
F ::= f | t | v | [F, . . . , F ] | (F, . . . , F ) | v̄,
where v is a variable, t and f are true and false, [. . . ] and (. . . ) are disjunction and
conjunction, and v̄ is the negation of v (negation on general formulae being inductively
defined). We use {} to denote a hole in a formula: a context S{} is a formula with one
72
S {t}
S (R, R̄ )
i↓
i↑
S [R, R̄ ]
S {f}
S ([R, U ], T )
s
S [(R, T ), U ]
S {f}
S (R)
w↓
w↑
S {t}
S [R]
S {R}
S [R, R ]
c↓
c↑
S {R}
S (R, R )
Table 3.12: System SKSg
occurrence of the hole, and F {R} is that context with the hole filled with a formula R.
We take associativity and commutativity of connectives, De Morgan duality, and the
usual behaviour of units, to hold at the level of syntactic equivalence. The set S of
structures is the quotient of the set F by the smallest relation containing these syntactic
equivalences and closed under formation of formulæ from contexts.
The deep symmetric system SKSg for classical propositional logic is given in Table
3.12. Each rule is unary, and is either self dual or comes as one of a dual pair. A
derivation from A to B in SKSg is a sequence of applications of rules starting with A
and ending with B. A proof of A in SKSg is a derivation from t to A.
The rules correspond closely to those of the one-sided sequent system GS1p. For
example:
⊢ Γ, ϕ
⊢ Γ′ , ψ
⊢ Γ, Γ′ , ϕ ∧ ψ
([Γ, ϕ], [Γ, ψ])
s
∧R
→
[Γ, (ϕ, [Γ′ , ψ])]
s
[Γ, Γ′ , (ϕ, ψ)]
and
73
([Γ, ϕ], [Γ, ϕ̄])
s
⊢ Γ, ϕ
⊢ Γ′ , ¬ϕ
⊢ Γ, Γ′
[Γ, (ϕ, [Γ′ , ϕ̄])]
s
CU T
→
[Γ, Γ′ , (ϕ, ϕ̄)]
i↑
[Γ, Γ′ , f]
[Γ, Γ′ ]
=
The cases for axiom, weakening and contraction are similar. We now have the essential
tools for the proof of the following theorem from [15]:
Theorem 3.3.2 (SKSg and GS1p I). For every derivation Φ of ⊢ ∆ in GS1p there
exists a proof
t
··
· SKSg \ {c↑ , w↑ }.
∆
That derivation has the same number of occurrences of i↑ as there are occurrences of
cut in Φ.
Remark 3.3.3. Only the translated cut rule requires an application of i↓ : hence the
preservation result. Since the i↑ rule is so closely related to the cut rule in GS1p, it is
often referred to as the cut rule for the calculus of structures.
The following theorem (from [15]) gives us a partial converse — partial, since the
number of cuts is not preserved:
t
·
Theorem 3.3.4 (SKSg and GS1p II). For every proof ·· in SKSg there exists a proof
S
of ⊢ S in GS1p.
Given a proof in SKSg, we can translate it into a GS1p proof. Since cut is admissible
in GS1p, we can obtain a cut-free proof. Translating that proof back into SKSg, the
proof we obtain contains no instance of i↑ . This proves the following,
Theorem 3.3.5 (Cut-elimination). Any structure provable in SKSg is provable without the use of i↑ (and without w↑ or c↑ ).
We call the system SKSg \{i↑ , w↑ , c↑ } KSg; the preceding theorem demonstrates
that KSg is complete for provability.
Remark 3.3.6. Notice that this notion of cut-elimination applies only to proofs, and
not to derivations.
74
S {t}
S (a, ā )
ai↓
ai↑
S {f}
S [a, ā ]
S ([R, U ], T )
s
S [(R, T ), U ]
S [(A, B), (C, D)]
S ([A, C], [B, D])
S {f}
m.
S (a)
aw↓
aw↑
S {t}
S [a]
S {a}
S [a, a ]
ac↓
c↑
S {a}
S (a, a )
Table 3.13: System SKS
3.3.2 A local system
The major technical achievement of the calculus of structures (at least for classical
logic) is the reduction of contraction to atomic form. To see what this means, recall
that in the sequent calculus we may replace the axiom φ ⊢ φ in LK by the atomic axiom
a ⊢ a, and recover the general axiom from the atomic by induction on the structure of
φ. However, in general we cannot, in a sequent calculus, reduce contraction to such an
atomic form [14].
In SKSg (and the sequent calculus) , we may make some progress:
S[[P, Q], [P, Q]]
S[[P, P ], [Q, Q]
S[[P, P ], Q]
S[P, Q]
=
↓c
≡
S[[P, Q], [P, Q]]
↓c
S[P, Q]
↓c
but the sticking point is the reduction of a contraction on a conjunction:
S [(A, B), (A, B)]
S (A, B)
75
c↓
The system SKS [15] is derived from SKSg by making two changes. We restrict i↓ ,
c↓ and w↓ and their duals to atomic formulæ (these atomic forms are known as ai↓ ,
ac↓ and aw↓ respectively), and we introduce a new rule called medial:
S [(A, B), (C, D)]
S ([A, C], [B, D])
m.
(3.3)
which is admissible in SKSg In fact, it is admissible in two ways:
[(A, B), (C, D)]
([(A, B), (C, D)], [(A, B), (C, D)])
([A, C], [B, D])
[(A, B), (C, D)]
c↑
[([A, C], [B, D]), ([A, C], [B, D])]
w↑ ×4
([A, C], [B, D])
With this rule we may reduce contraction to its atomic form:
S[(P, Q), (P, Q)]
S([P, P ], [Q, Q])
S([P, P ], Q)
S(P, Q)
m
↓c
(3.4)
↓c
Theorem 3.3.7 (Locality). A consequence is derivable in SKSg if and only if it is
derivable in SKS
For the local system, there is a procedure for eliminating i↑ from a proof without reference to the sequent calculus, called splitting [13, 15]), which works by transforming
a proof
·
·
· Φ ∈ KS
· Ψ ∈ KS
·
·
into a proof
[R, (a, ā)]
[R, R]
i↑
c↓
R
R
(Having shown that each application of i↑ can be replaced by a shallow application as
above, this is enough to eliminate all the cuts in a proof.)
3.3.3 First-order logic in the calculus of structures
The systems SKSgq (Table 3.14) and SKSq (Table 3.15) are the nonlocal and local
systems in the calculus of structures for first-order classical logic [15]. We spend a
little time discussing the rules involved.
76
w↓ ×4
c↓
S {t}
S (R, R̄ )
i↓
i↑
S [R, R̄ ]
S {f}
S ([R, U ], T )
s
S [(R, T ), U ]
S ∀x[R, T ]
S (∃xR, ∀xT )
u↓
S [∀xR, ∃xT ]
u↑
S ∃x(R, T )
S {f}
S (R)
w↓
w↑
S {t}
S [R]
S {R}
S [R, R ]
c↓
c↑
S {R}
S {R[x/t]
S (R, R )
S {∀xT }
n↓
n↑
S {∃xR}
S {T [x/t]}
Table 3.14: System SKSgq
The sequent calculus rule ∃R is represented by the n ↓ rule. The more difficult rule
in the sequent calculus is the ∀R rule, which has a side condition. The corresponding
rule in the calculus of structures is u↓ . Adding an equality
∀xT = ∃xT = T where T does not depend on x
we may recover the behaviour of ∀R as follows:
S ∀x[R, T ]
S [∀xR, ∃xT ]
u↓
S [∀xR, T ]
where T does not depend on x.
The rules m1 ↑ ,m2 ↑ , m1 ↓ and m2 ↓ , which we will refer to as the first-order medial
rules, allow for reduction of contraction to atoms, as in the propositional case. For
example,
77
S {t}
S (a, ā )
ai↓
ai↑
S {f}
S [a, ā ]
S ([R, U ], T )
s
S [(R, T ), U ]
S ∀x[R, T ]
S (∃xR, ∀xT )
u↓
u↑
S [∀xR, ∃xT ]
S [∃xR, ∃xT ]
S {∃x[R, T ]}
S [∀xR, ∀xT ]
S {∀x[R, T ]}
S ∃x(R, T )
S {∀x(R, T )}
m1↓
S [(A, B), (C, D)]
S ([A, C], [B, D])
S (∀xR, ∀xT )
m
S {∃x(R, T )}
m2↓
S (∃xR, ∃xT )
S {f}
m2↑
S (a)
aw↓
aw↑
S {t}
S [a]
S {a}
S [a, a ]
ac↓
ac↑
S {a}
S {R[x/t]
m1↑
S (a, a )
S {∀xT }
n↓
n↑
S {∃xR}
S {T [x/t]}
Table 3.15: System SKSq
78
S{[∃xR, ∃xR]
S{∃x[R, R]}
S{∃xR}
m1↓
c↓
As one might expect, the first-order medial rules are derivable for contraction and
weakening in SKSgq:
S{[∃xR, ∃xT ]}
S{[∃x[R, T ], ∃x[R, T ]]}
S{∃x[R, T ]}
w↓ ×2
c↓
S{[∀xR, ∀xT ]}
S{[∀x[R, T ], ∀x[R, T ]]}
S{∀x[R, T ]}
w↓ ×2
c↓
Curiously, however, m2 ↓ is derivable in a system without structural rules, in the up
fragment of SKSq:
S{[∀xR, ∀xT ]}
=
S{∀x[∀xR, , ∀xT ]}
n↑ ×2
S{∀x[R, T ]}
where the equality holds since [∀xR, ∀xT ] does not depend on x. This observation
will have consequences for the semantics both of the sequent calculus and SKSq.
3.3.4 Formalism A and Formalism B
The calculus of structures allows more freedom in the application of inference rules.
As a result, lays bare more bureaucracy than many other systems. Formalisms A [31]
and B [32] are suggestions for ways to design new systems which lack this bureaucracy.
Here we will describe the ideas behind these systems and the types of bureaucracy
they avoid. Later, we will ask what light a categorical semantics sheds on the issues
involved.
When dealing with the sequent calculus, one has to deal with an enormous number
of commuting conversions; this is why proof nets, which validate these conversions as
identities, are so useful. In the calculus of structures we may see even more bureacracy.
79
For example, in the sequent calculus derivation,
·
·Ψ
·
⊢ Γ′ , ψ
∧R
⊢ Γ, Γ′ , ϕ ∧ ψ
·
·Φ
·
⊢ Γ, ϕ
(3.5)
the two sub-proofs act in parallel. In the calculus of structures, the same derivation can
be written by applying the sub-proofs sequentially, as either
t
(t, t)
t
=
([Γ, ϕ], t)
(t, t)
Φ
([Γ, ϕ], [Γ, ψ])
[Γ, (ϕ, [Γ′ , ψ])]
[Γ, Γ′ , (ϕ, ψ)]
Ψ or
s
=
(t, [Γ, ψ])
Ψ
([Γ, ϕ], [Γ, ψ])
[Γ, (ϕ, [Γ′ , ψ])]
s
[Γ, Γ′ , (ϕ, ψ)]
Φ
(3.6)
s
s,
or by interleaving the inference rules from each proof, in a number of possible ways.
All of these should, morally, represent the same proof. Formalism A would be a system
in which applications of inference rules can be made in parallel, as they are in the
sequent calculus or proof nets.
Formalism B caters to another, more subtle, form of bureaucracy. Suppose we have a
derivation Φ of A′ from A in the calculus of structures. The syntactic objects
(A, [B, C])
(A, [B, C])
s
[(A, B), C]
[(A′ , B), C]
and
Φ
should denote the same proof.
80
(A′ , [B, C])
Φ
(3.7)
s
[(A′ , B), C]
Chapter 4
Proof nets for first-order classical logic
We discuss proof nets for first-order classical logic, and in particular quantifiers in
proof nets as boxes. Boxes were the first attempt to add additive content to proof
nets, as they appeared in [26], at first to add units and exponentials, and then additive
connectives and quantifiers. A box is the rendering of a sequent calculus rule in proof
net syntax: the proof net (or nets) contained in the box represent the premises of the
rule, and the resulting box and its doors the conclusion. The box is a point at which the
parallelism of the proof net system breaks down, and it has been an ongoing project
to remove them from nets altogether. An alternative presentation is eigenweights [27],
or in the context of quantifiers eigenvariables. This approach (refined for first-order
MLL− by Bellin and van de Wiele [8]), has unary rule nodes for the quantifier rules ∃L
and ∀R. The correctness criterion is in the style of Danos and Regnier: the switching
condition for the quantifiers connects the conclusion of the quantifier node to any node
in which the eigenvariable of the quantifier is free. This approach treats quantifiers and
propositional connectives together, with a single correctness criterion, and the many
commuting conversions and rule permutations of the sequent calculus are identified. In
fact, too much structure is identified; our models will distinguish between cuts inside
and outside the scope of a particular quantifier, whereas these proof nets admit that
permutation as an equality.
Our treatment of quantifiers using boxes is much closer in spirit to both the sequent calculus and the categorical structure of classical doctrines. The presentation is based on
that for functor boxes of Cockett and Seely [16] — functor boxes are used to represent
linear functors on a linearly distributive category. (In the case where we have proof
81
theoretic duality, the quantifiers on a classical doctrine from a linear functor: see chapter 6.5) A quantifier box associated with a particular variable x takes a proof net over
the variables (X, x), with at most one door properly over x, to a proof net over (X).
This condition is the precise analogue of the side conditions of the sequent calculus.
The only correctness condition on the quantifiers is the verification of this condition.
Applications of quantifiers have a definite order, and permutations of propositional
rules with quantifier boxes, or of quantifier boxes with other quantifier boxes (even if
valid), are not given as equalities: if they are needed they must be added separately as
an equational theory.
4.1 First-order proof nets and quantifier boxes
Definition 4.1.1. Given a set A = {φ, (Y )} of atomic predicates φ over a sequence of
variables Y (the arity of φ) , the set AY of (pseudo)atomic predicates over a particular
sequence (X) of variables is given by the following grammar:
A(X) := φ(X)|[x]A(X \ x)|∃yA(X, y))|∀yA(X, y)
As the name suggests, an instance of a box rule is written as a box, decorated with a
quantifier (∃ or ∀) and an eigenvariable x, surrounding a proof net, which has at most
one door not of the form [x]ψ . Each door inside the box is connected to a new door
outside of the box, in the following way:
• One door, labeled with the formula φ, (on the left for ∃, on the right for ∀) is
connected to a door outside the box labeled with Qx.φ, where Q is the relevant
quantifier;
• Each other node inside the box is of the form [x]ψ, and is connected to a node
decorated with ψ outside the box.
For example, the net
82
∀x
φ(x) : R
φ(x) : R
ψ(x) : R
φ ∨ ψ(x) : R
[x]∃y.φ : R
ψ(x) : R
[x]∀yφ ∨ ψ : R
∀x.(φ ∨ ψ) : L
∃y.φ : R
∀x.ψ : R
∃y.φ ∨ ∀x.ψ : R
is equivalent to a proof of the sequent
∀x(φ ∨ ψ) ⊢ ∃xφ ∨ ∀xψ.
In addition, we add two new families of logical constants,Kφ∃x and Kφ∀x , to model the
sequent calculus laws ∃R and ∀L.
Definition 4.1.2. The proof nets over a particular sequence of variables (X) and a
set of atomic predicates A = {(A, Y )} are defined inductively over the following
formation rules:
1. A propositional net over the set AX is a proof net over (X)
2. For each x in (X), the constants Kφ∀x
∀x
Kφ
[x]∀yφ : L
φ(x) : R
and Kφ∃x
∃x
Kφ
φ(x) : L
are nets over (X).
83
[x]∃yφ : R
3. Given a net M over the sequence of variables (X, x), with at most one left door
properly over x, the result of placing M in an ∃x-box is a proof net over (X)
4. Given a net M over the sequence of variables (X, x), with at most one right
door properly over x, the result of placing M in an ∀x-box is a proof net over
(X)
4.1.1 Static properties
The static properties of proof nets are all those unrelated to proof equality and cut
elimination. In particular, we are interested in the sequentialization of proof nets:
given a proof net M, can we construct (non-deterministically) a sequent calculus proof
of the same entailment that, when the sequential information is stripped away, renders
the original proof net M?
Definition 4.1.3 (Switching). Given a first-order proof net M, a switching σ is as for
a propositional proof net: a choice of parent node for each switched rule. The graph
DR(σ, M) is defined in the same way as for propositional nets, with the additional
condition that nodes are connected across the boundary of a quantifier box (and so in
effect boxes are ignored for the purposes of switching conditions).
Definition 4.1.4. Given a first-order proof net M, a switching σ, and a node A of M,
define σ(M, A) to be the subgraph of DR(σ, M) defined as follows. If A is the premise
of a rule node, remove all the links in DR(σ, M) which pass from A to a corresponding
conclusion. If A is a cut formula, remove the link to the other cut formula. If A is a
premise of a quantifier box, remove the box and all links to its conclusions. Take the
component containing A (If A is not the premise of a rule node, take DR(σ, M))
Theorem 4.1.5 (Sequentialization). Given a first-order proof net N, there is an LK
derivation whose when translation into a proof net is N.
Proof. By induction on the greatest number of nested box rules, and the number of
rule nodes outside any proof boxes. If there are no box rules, the propositional sequentialization algorithm is used. Now assume that a proof net with n nested boxes
can be sequentialized. We sequentialize a proof net M with n + 1 nested boxes as
follows: if there are no propositional rule nodes outside any quantifier boxes, then M
is the result of applying a quantifier box to a proof net N with n nested boxes, and so
84
is clearly can be sequentialized by applying the relevant sequent calculus rule to the
sequentialization of N. Now suppose we can sequentialize if there are m propositional
nodes outside n nested boxes: the case for m + 1 propositional nodes is dealt with as
for the propositional case. If any of the final rule nodes is switched it can be removed
to leave a valid proof net, to which we apply the induction hypothesis, and if not we
can remove the greatest unswitched rule (w.r.t to ordering given by kingdoms) to give
two sub-nets each of which can be sequentialized.
4.2 Cut-elimination in first-order proof nets
In this section we give a system of inequational laws on proof nets sufficient to prove
cut-elimination.
Before we begin, we require a few technical definitions:
Definition 4.2.1. Given a proof net M over variables (X), the net [y]M, where y does
not appear in any node of M, is obtained by replacing each node φ of M with [x]φ. The
net [t/z]M, where z : Z appears in (X) and t : Z is a term, is given by substituting t
for z.
There are two cut-reduction steps involving each quantifier. This arises since we take
greater care over the handling of substitution. We give the examples for the existential
85
quantifier. In the case of ∃, the first is called ∃G ROW :
∃x
M
N
[x]φ(x) : X
ψ:L
∃xψ : L
φ:R
φ:L
∃x
M
[x]N
4
ψ:L
[x]φ(x) : X
[x]φ : L
∃xψ : L
and the second is called C UT ∃
∃y : L
M
[t/x][x]N
φ(t/x) : R
[t/x][x]φ(y) : L
[t/x][x]∃yφ : R
[t/x][x]∃yφ : L
M
4
[t/y]N
φ(t/x) : R
φ(t/y) : L
In addition we have the rule ∃W–Y
∃xL
∃xL
M
M
≡
φ : L
[x]γ : R
φ : L
[x]ψ : R [x]γ : R
∃xφ : L
ψ :R
[x]γ : R
γ :R
∃xφ : L
γ : R
ψ : R γ : R
allowing us to move a weakening into and out of a quantifier box. This law is analogous
to W-move in the propositional equalities.
86
Remark 4.2.2. Why are the cut-reductions given as inequalities? We will see a semantic argument in later sections, but for a syntactic reason consider the following
net:
Q1 x
Q2 y
M
N
[x]φ : L
[y]φ : R
φ:L
phi : R
This has two possible reducts:
Q1 x
Q2 y
M
[x]N
[x]φ : L
[x, y]φ : R
[x]φ : R
and
Q2 y
Q1 x
[y]M
N
[x, y]φ : L
[y]φ : R
[y]φ : L
.
There is no particular reason why we should identify these proofs (or, rather, we might
want them to be different).
To guarantee that we can use the propositional cut-reduction theorem, we must extend
the empire rewiring theorem:
Lemma 4.2.3 (Empire rewiring). A weakening link in a proof net M can be moved
anywhere within the empire of the node A introduced (with the moved weakening introducing an appropriate substitution instance of the label of A (φ), where necessary)
Proof. By induction on the maximum number of nested boxes in M. If M contains no
boxes then the propositional result holds. Now suppose the result holds for n nested
boxes, and let M have n + 1 nested boxes. A weakening node outside the boxes
87
may, by the propositional rewiring result, be moved to the conclusion of any of the
outermost quantifier boxes. Using the rule ∃W X, we move the weakening inside the
box (and here the substitution instance of the introduced formula changes), and from
there (by the induction hypothesis) the weakening may be moved to any point within
the quantifier box. All the operations involved are invertible.
Cut-elimination relies on the following principal lemma:
Lemma 4.2.4. The net
M
φ:L
...
N
φ:L
φ:R
φ:L
...
φ:R
φ:R
where M and N are cut-free, reduces to a cut-free net.
Proof. The proof is an induction over (δ, q, r), ordered lexicographically, where δ, the
degree of the cut, is the number of propositional logical symbols in the cut formula, r
is the sum of the number of quantifier boxes in the kingdoms of the two cut formulae,
and r is the sum of the right rank of M and the left rank of N.
If rankR (M) = rankL (N) = 1, and q = 0, cut-elimination is propositional. Given
a proof with minimal rank but involving some quantifier boxes, we show that we can
reduce either δ or q (and then apply the induction hypothesis).
If M ends with a weakening or propositional rule, we can reduce the degree of the cut
in precisely the manner above. If M ends with an application of a quantifier box, then
there are two sub-cases. If the cut formula is not the principal conclusion of that box,
we can apply the relevant permutation to bring the cut inside the box. This reduces
value of q by one, so we can apply the induction hypothesis.
If the cut formula of M is the principal conclusion of the box rule, then either the other
cut formula is the non-principal conclusion of another box rule (in which case the dual
of the previous case applies) or of a non-box quantifier rule, in which case the cut is
88
logical and can be eliminated by ∃cut or ∀cut, reducing the value of q and enabling
the induction hypothesis.
Now assume that, without loss of generality, that rankR (M) > 1. The inductive steps
for the propositional connectives follow from the proof of the propositional principal
lemma. Since M has non-minimal rank, the final rule cannot be an ∃L or a ∀R box.
What remains are the cases where M ends with ∃R. The net in the principal lemma is
of the form
M
N
[x]∃yφ : R... [x]∃yφ : R φ : R
[x]∃yφ : R
[x]∃yφ : L
...
[x]∃yφ : L
[x]∃yφ : R
[x]∃yφ : L
[x]∃yφ : R
An application of C UT Cyields
M
N
[x]∃yφ : R... [x]∃yφ : R φ : R
[x]∃yφ : R
[x]∃yφ : L
...
N
[x]∃yφ : L
[x]∃yφ : L
...
[x]∃yφ : L
[x]∃yφ : R
[x]∃yφ : L
[x]∃yφ : L
The empire of φ satisfies the induction hypothesis, and so we can eliminate the cut.
This results in another net that satisfies the induction hypothesis, and so the final cut
can be eliminated.
We now have the following:
Theorem 4.2.5 (Cut Elimination). Given a first-order proof net, the cut-reduction
steps yield a cut free form.
89
∃∨R:
∃xL
∃xL
M
M
≡
φ:L
[x]ψ : L
∃xφ : L
ψ:L
[x]γ : L
[x]ψ : L
φ:L
γ:L
[x]γ : L
[x]ψ ∨ γ : L
ψ∧γ :L
ψ∧γ :L
∃xφ : L
∃∧R:
∃xL
∃xL
M
[x]N
γ : L [x]φ : R
[x]ψ : R
M
4
γ : L
N
[x]φ : R
φ : R
[x]φ ∧ ψ : R
ψ : R
φ∧ψ : R
∃xγ : L φ ∧ ψ : L
∃xγ : L
φ∧ψ : L
∃C–Y
∃xL
∃xL
M
M
≡
φ :L
[x]ψ : Y[x]ψ : Y
[x]ψ : Y
φ : L
[x]ψ : Y
[x]ψ : Y
∃xφ : L
ψ :Y ψ :Y
ψ :Y
∃xφ : L
ψ : Y
Table 4.1: Multiplicative box-rule permutations
4.3 Other (in)equations
Tables 4.3 and 4.3give a number of putative equalities/reductions that we would expect to hold between first-order proof nets, where the action of a propositional rule is
permuted with that of a box. We will see in later sections that our choice of theory on
nets dictates which of these will hold.
90
W∃
∃xL
M
[x]M
≡
φ : X
[a]φ : X
ψ : L
∃xψ : L
[x]φ : X
∃xψ : L
φ : X
φ : X
C∃
∃x
∃y
∃x
N
M
M
(x/y)N
4
′
φ(x) : L [x]ψ : Y [y]ψ : Y φ(y)
: L
∃xφ : L
ψ : Y
ψ : Y′
∃xφ : L
′
φ(x) : L[x]ψ : Y[x]ψ : Y φ(x)
: L
φ(x) : L
∃xφ : L
∃xφ : L
Table 4.2: Additive box-rule permutations
91
Chapter 5
Classical doctrines
In this chapter we outline three notions of classical doctrine, of increasing strength,
corresponding to three notions of proof equality on first-order classical proofs (given
in the next chapter). We begin by considering how the core concepts of doctrinal semantics translate into the setting of monoidal categories, and show how the properties
we might desire of a model of classical logic arise from structures analogous to those
for intuitionistic logic.
We then examine what additional structure we might need to add for the quantifiers
to interact well with the structural rules, and arrive the stronger Sharpened classical
doctrines.
Finally, we compare the structures we have presented with linear functors, and arrive
at a definition of a dual classical doctrine.
92
5.1 Categorical axioms for quantifiers
Recall the general setup for a hyperdoctrine model of first-order logic:
C(X)
C(Y )
Σa
a∗
Πa
B
X
a
Y
For now we will not consider functors Σa and Πa in cases where a is a general morphism, but only the case where a is a projection. Where possible we will suppress the
subscripts on Σ and Π.
We begin by presenting a naı̈ve axiomatization for classical doctrines; a first guess at
the required structure.
Definition 5.1.1 (Morphisms in a classical doctrine). We expect the following morphisms to appear (with the following names) in the fibres of a classical doctrine:
β : a∗ (A ⊗ B) →
˜ a∗ A ⊗ a∗ B
β⊤ : ⊤ →
˜ a∗ ⊤
γ : a∗ (A ⊕ B) →
˜ a∗ A ⊕ a∗ B
γ⊥ : ⊥ →
˜ a∗ ⊥
νΣ : Σ(A ⊕ B) → ΣA ⊕ ΣB
ν⊥ : Σ⊥ → ⊥
νΠ : ΠA ⊗ ΠB → Π(A ⊗ B)
ν⊤ : ⊤ → Π⊤
µΣ : Σ(A ⊗ B) → ΣA ⊗ ΣB
µΠ : ΠA ⊕ ΠB → Π(A ⊕ B)
µ⊤ : Σ⊤ → ⊤
µ⊥ : ⊥ → Π⊥
frobΣ : ΣA ⊗ B → Σ(A ⊗ π ∗ B)
frobΠ : Π(A ⊕ π ∗ B) → ΠA ⊕ B
where π is a projection in the base of the indexed category.
Remark 5.1.2. Note that via the structural rules of classical logic there are morphisms
in the opposite directions to νΣ , νΠ , ν⊤ and ν⊥ . For example, in the category of proof
93
nets, νΣ is given by
∃x
ψ:L
φL
φ∨ψ :L
φ:R
ψ:R
[x]∃yφ : R
[x]∃yψ : R
[x]∃y.φ ∨ ∃yψ : R
∃x.(φ ∨ ψ) : L
∃x.φ ∨ ∃xψ : R
(5.1)
with the reverse morphism given by
∃y
∃x
ψ:R
φ:R
φ:R
ψ:L
φ:L
ψ:R
φ∨ψ :R
∃yψ : L
[y]∃z.(φ ∨ ψ) : R
∃z.(φ ∨ ψ) : R
∃x.φ ∨ ∃yψ : L
ψ:R
φ∨ψ :R
[x]∃z.(φ ∨ ψ) : R
∃x.φ : L
φ:L
∃z.(φ ∨ ψ) : R
∃z.(φ ∨ ψ) : R
(5.2)
Note that these two are not, a priori, inverse, and that 5.2 is the interpretation in proof
nets of the calculus of structures rule m1↓ .
Remark 5.1.3. By commutativity, there is also a morphism
A ⊗ ΣB → Σ(π ∗ A ⊗ B)
L
which we will denote by frobR
Σ , and we will write frobΣ for frobΣ when we need to
make a distinction.
Following the example of hyperdoctrines for intuitionistic logic, and for algebraic rea94
,
sons, we might expect a classical doctrine to be an indexed category for which:
1. The base category B has finite products and a terminal object;
2. The fibres are classical categories;
3. The lifting a∗ of a morphism a in the base category preserves the structure of the
fibres;
4. There exist left (Σ) and right (Π) adjoints to π ∗ (where π is a projection in B)
which are (co)monoidal with respect to both connectives;
5. There are morphisms ΣA ⊗ B → Σ(A ⊗ π ∗ B) and Π(A ⊕ π ∗ B) → ΠA ⊕ B;
6. The Beck-Chevalley condition holds.
However, the situation is somewhat more subtle. The first point of departure from
hyperdoctrines for natural deduction is the move from categorical (co)products to
monoidal products. This necessitates points 3 and 4 of the definition. The axioms for
an existential quantifier in intuitionistic logic derive from the properties of substitution
in a hyperdoctrine. Given a substitution a∗ , a functor Σa is an existential quantifier for
a if:
1. Σa ⊣ a∗
This ensures that substitution preserves coproducts: this gives us that (the denotation of) φ(t/x) ∨ ψ(t/x) is isomorphic to (φ ∨ ψ)(t/x); and
2. The canonical morphism Σa (A ∧ a∗ B) → ΣA ∧ B is invertible. This property is
known as Frobenius reciprocity, and it ensures that a∗ preserves closures; since
closure denotes implication, this gives that (the denotation of) φ(t/x) → ψ(t/x)
is isomorphic to (φ → ψ)(t/x).
For classical categories, the case is rather more complicated: the presence of a left
adjoint to a functor does not imply preservation of monoidal products. We do, however,
have the following theorem:
Lemma 5.1.4. Let C and D be symmetric monoidal categories, and let F ⊣ G be an
adjunction between the underlying categories:
95
F
C
-
⊥
G
D
Let F be strong monoidal, with invertible natural transformations βA,B and β1 . Then
G is a monoidal functor on D, and the unit and co-unit of the adjunction are monoidal
natural transformations.
However, even this is not enough to, for example, ensure that the existential quantifier
is comonoidal. Another key difference between the intuitionistic and classical setup is
in the equalities between propositions/predicates to which we are prepared to commit.
In a hyperdoctrine, we would expect the denotation of ∃xφ and φ to be equal, in the
case where φ does not depend on x. As mentioned in the introduction, our models
should encompass the notion of a quantifier as an infinitary connective. We cannot,
L
L
in general, identify A with i A, and so the morphism ε : i A → A cannot be an
isomorphism. Indeed, in that case it should not even be natural, since it is the infinitary
generalization of contraction, but should instead be lax natural. In general, we cannot
derive, for instance, comonoidality of µΣ as we can for a similar morphism arising
from a (non lax) adjunction; the relevant diagram is an inequality:
Σ(A ⊗ B)
Σ(η ⊗ η)-
Σ(a∗ ΣA ⊗ a∗ ΣB)
≥ Σ(a∗ Σf ⊗ a∗ Σg)
Σ(f ⊗ g)
?
Σ(C ⊗ D)
Σ(η ⊗ η)-
Σβ
-
Σa∗ (ΣA ⊗ ΣB)
Σa∗ (Σf ⊗ Σg)
?
Σ(a∗ ΣC ⊗ a∗ ΣD)
Σβ
-
ε-
(ΣA ⊗ ΣB)
≥ Σf ⊗ Σg
?
?
ε
Σa∗ (ΣC ⊗ ΣD) - (ΣC ⊗ ΣD)
If we want (co)monoidality, we must either assume it or add some additional structure.
There is a clear reason to admit a morphism of type ΣA ⊗ B → Σ(A ⊗ a∗ B) (which
we will call a Frobenius morphism to classical doctrines (point 5). Given a morphism
f : A ⊗ a∗ B → a∗ C
we should be able to derive one of form
⌊∃L⌋f : ΣA ⊗ B → C.
96
denoting the application of the ∃L rule of the sequent calculus. This is given by
ΣA ⊗ B
frob
-
Σ(A ⊗ a∗ B)
Σf -
Σa∗ C
ε
-
C.
Unlike for Seely’s hyperdoctrines, we will also require a dual condition (which is
intuitionistically invalid) for the universal quantifier. Additionally, in general we will
L
not expect these morphisms to be invertible; if it were, we would have that i A is
isomorphic to A.
5.2 Classical doctrines
Before we define classical doctrines, we require some auxiliary definitions.
Definition 5.2.1. Given a morphism f in an order-enriched category C, the morphism
g is a left adjoint to f if
g ◦ f ≤ id
and
id ≤ f ◦ g.
Definition 5.2.2. We call a functor F on a symmetric linearly distributive category that
is strong monoidal in both tensors, and in addition preserves the linear distribution, in
the sense that the diagram
F (A ⊗ (B ⊕ C))
µ
F δ-
ν
?
F A ⊗ F (B ⊕ C)
id ⊗ ν
F ((A ⊗ B) ⊕ C)
?
F (A ⊗ B) ⊕ F C
µ ⊕ id
?
?
δ
F A ⊗ (F B ⊕ F C) - (F A ⊗ F B) ⊕ F C
commutes, a strongly self dual linear functor.
Definition 5.2.3. In the diagram below, let C and D be order enriched.
C
F
G
-
D
F is a lax left adjoint to G (F ⊣≤ G) if there are lax natural transformations ε :
97
F G → id and η : id → GF ; that is
f ◦ ε ≤ ε ◦ F Gf
and
GF g ◦ η ≤ η ◦ g.
We say that F is an oplax left adjoint to G ( F ⊣≥ G) if we have the same configuration
of functors and transformations, but with reversed inequalities.
Definition 5.2.4. If f : F → G is a lax natural transformation, we say that the
morphism g is f -strong if
fA GA
FA
Fg
Gg
?
fB - ?
GB
FB
commutes.
We now have enough terminology to give a first definition of a classical doctrine.
Definition 5.2.5. (Classical doctrine I)
1. A classical doctrine is an indexed category C : Bop → CAT in which the base
has finite products and each fibre is a classical category, such that the functor a∗
(defined for each morphism a in the base category) is strongly self dual linear
(see definition 5.2.2), and such that for every projection π : X × Y → X
2. there exists a lax left adjoint Σπ , and a right oplax-adjoint Ππ , to π ∗ ;
3. Σπ is comonoidal w.r.t. ⊕ and Ππ is monoidal w.r.t⊗ and the above lax adjunctions is symmetric monoidal in both tensors;
4. The linear distributivity δ in C(X) is εΣπ - and ηΣπ -strong;
5. Frobenius strengths The morphism
Σπ
∗
π
(id ⊗ εΣ
B ) ◦ µA,π ∗ B : Σπ (A ⊗ π B) → ΣA ⊗ B
π
has a left adjoint frobΣ
A,B such that
Σπ
Σπ
π
(id ⊗ εΣ
B ) ◦ µA,π ∗ B ◦ frobA,B = id
98
and
Σπ
Σπ
π
frobΣ
A,B ◦ (id ⊗ εB ) ◦ µA,π ∗ B ≤ id
and the morphism
Ππ
∗
π
µΠ
A,π ∗ B ◦ (id ⊗ ηB ) : Ππ A ⊕ B → Ππ (A ⊕ π B)
π
has a right adjoint frobΠ
A,B , such that
Ππ
Ππ
π
µΠ
A,π ∗ B ◦ (id ⊗ ηB ) ◦ frobA,B ≤ id
and
Ππ
Ππ
π
frobΠ
A,B ◦ µA,π ∗ B ◦ (id ⊗ ηB ) = id
And additionally,
6. Beck-Chevalley condition
if
t
A
-
B
s
r
?
u
C
-
?
D
is a pullback in B, and Σs , Σr are lax left adjoints to s∗ and r ∗ respectively, the
diagram
C
A
t∗
CB
Σs
Σr
?
u∗
CC ?
CD
commutes;
An equivalent set of axioms is given by replacing item [4] above:
Definition 5.2.6 (Classical Doctrines II).
Σπ (A ⊕ B)
Σπ (η ⊕ η)
-
Σπ (π ∗ Σπ A ⊕ π ∗ Σπ B)
99
4 (a) The family of morphisms
Σπβ
ε
Σπ π ∗ (Σπ A ⊕ Σπ B) - Σπ A ⊕ Σπ B
which we denote by νΣ , is natural and εΣ strong. Dually for νΠ .
(b) Each morphism
Σπ (A ⊗ B)
Σπ (η ⊗ η)
-
Σπ (π ∗ Σπ A ⊗ π ∗ Σπ B)
Σπγ
ε
Σπ π ∗ (Σπ A ⊗ Σπ B) - Σπ A ⊗ Σπ B
which we denote by µΣ , is εΣ strong. Dually for µΠ
We show below that this is a sufficient condition for monoidality of νA,B .
Definition 5.2.7. A classical doctrine is a µ-doctrine if the families µσ and µΠ are
natural for each projection π.
5.3 Some properties of classical doctrines
To show the equivalence of the two above definitions, we need to adapt the proof
of Theorem 2.3.12 to an order-enriched setting. This will be possible because the
structure forces certain morphisms to be strong:
Lemma 5.3.1. The morphism f is εΣ -strong (that is, the diagram
Σπ ∗ A
ε
Σπ ∗ f
-
A
f
?
Σπ ∗ B
ε
-
?
B
commutes) if f is a right adjoint. Similarly, f is εΠ , ∇- and []-strong if it is a right
adjoint, and is ηΠ -, ∆- and hi- and ηΣ -strong if it is a left adjoint.
Proof. We know that the clockwise direction is smaller than the anti-clockwise; if we
can prove the opposite ordering we are done. Let g be left adjoint to f , and consider
100
the following diagram:
ε
-A
-
-
Σπ ∗ A
id
ε
-
g
Σπ ∗ A
-
≤
Σπ ∗ f
Σπ ∗
g
≥
f
-
≤
id
ε
B
?
- B
-
?
ε
Σπ ∗ B
The left- and right-hand triangles are given by the two conditions for adjointness:
g ◦ f ≤ id and f ◦ g ≥ id. The central square, is the lax naturality of ε.
One consequence of the lax-adjunctions we use is that there is no longer an bijection
Σπ A → B
A → π∗B
.
(5.3)
so we require the following theorem:
Theorem 5.3.2. Let F ⊣≥ G. Given two η-strong morphisms f, g : A → GB, f = g
iff ε ◦ F f = ε ◦ F g.
Proof. This is a simple adaptation of the usual establishment of a bijection from a unit
and counit of an adjunction; the η-strength of f and g ensures that the usual proof
works.
By the triangle identities, f = Gε ◦ η ◦ f , and by the η-strength of f , this morphism is
equal to Gε ◦ GF f ◦ η, which by functoriality of G is equal to G(ε ◦ F f ) ◦ η. This is,
by our assumption, equal to G(ε ◦ F g) ◦ η, and by reversing our steps this is equal to
g.
We now have enough to adapt Theorem 2.3.12 to the order-enriched setting:
101
Lemma 5.3.3. In a classical doctrine, given a projection π the functor Σπ is comonoidal
with respect to ⊕ (and lax-comonoidal w.r.t. ⊗), and Ππ is monoidal with respect to ⊗
(and lax-comonoidal w.r.t. ⊕). In addition, Σπ and Ππ are compatible with δ; that is,
the diagram
Σδ-
Σ(A ⊗ (B ⊕ C))
µ
ν
?
ΣA ⊗ Σ(B ⊕ C)
id ⊗ ν
Σ((A ⊗ B) ⊕ C)
?
Σ(A ⊗ B) ⊕ ΣC
µ ⊕ id
?
δ-
ΣA ⊗ (ΣB ⊕ ΣC)
?
(ΣA ⊗ ΣB) ⊕ ΣC
commutes, as does the corresponding diagram for Ππ .
Proof. By adding (co)monoidality as an additional condition, the rest of the proof
of Lemma 2.3.12 goes through unaltered, since each time we require a commuting
diagram
Σπ ∗ A
ε
Σπ ∗ f
-
A
f
?
Σπ ∗ B
ε
-
?
B
the morphism f is an isomorphism, so is both a left and a right adjoint. By Lemma
5.3.1 they are strong with respect to the unit and co-unit of the lax adjunction, so the
square commutes. We may then apply Theorem 5.3.2.
The compatibility of this monoidal structure with the linear distribution morphism δ is
102
given by the diagram below:
103
(1) π ∗ is SSDL. (2) β −1 is natural. (3) ε is comonoidal w.r.t ⊕. (4) δ is natural. (5) ε
id
?
∗
π (ΠA ⊗ (ΠB ⊕ ΠC))
π ∗ (id ⊗ ν)
?
∗
π (ΠA ⊗ Π(B ⊕ C))
π∗ µ
?
∗
π Π(A ⊗ (B ⊕ C))
id
∗
?
π Π(A ⊗ (B ⊕ C))
-
π∗ δ
∗
π (ΠA ⊗ (ΠB ⊕ ΠC)
∗
π ((ΠA ⊗ ΠB) ⊕ ΠC)
(1)
-
β −1
∗
id ⊗ π ∗ ν
(2)
-
β −1
-
id ⊕ γ −1 ∗
∗
∗
π ΠA ⊗ π (ΠB ⊕ ΠC)
π ΠA ⊗ (π ΠB ⊕ π ΠC)
∗
∗
?
∗
π ΠA ⊗ π Π(B ⊕ C)
(5)
εΠ
(3)
id ⊗ ε
(6)
ε⊗
ε
id ⊗ (ε ⊕ ε)
-
?
δ
(4)
∗
ε ⊗ id
-
?
A ⊗ (B ⊕ C)
∗
(ε ⊗ ε) ⊕ id
∗
(A ⊗ B) ⊕ π ΠC
id ⊕ εΠ
δ
-
γ
−1
?
(A ⊗ B) ⊕ C
π ((ΠA ⊗ ΠB) ⊕ C)
π (Π(A ⊗ B) ⊕ ΠC)
ε
?
∗
π Π((A ⊗ B) ⊕ C)
(7)
π ∗ Π(δ)
?
∗
(8)
π∗ ν
?
∗
π ∗ (µ ⊕ id)
?
∗
π ΠA ⊗ (C ⊕ C)
-
id
β −1 ⊕ id ∗
∗
(π ΠA ⊗ π ΠB) ⊕ π ΠC
π (ΠA ⊗ ΠB) ⊕ π ΠC
∗
id
-
∗
6
π Π((A ⊗ B) ⊕ C)
is comonoidal w.r.t ⊗. (6) Clear. (7) δ is ε-strong. (8) Commutes by symmetry/duality
with the left-hand side of the diagram.
The image of this diagram under the bijection given by the adjunction is the required
diagram.
Corollary 5.3.4. The diagrams
Σπ ∗ A ⊙ Σπ ∗ B
ε ⊙ε
A⊙B
id -
I
I
6
?
Σ(π ∗ A ⊙ π ∗ B)
?
ε-
Σπ ∗ (A ⊙ B)
and
A⊙B
η ⊙η
ΣI
id
id
?
?
A ⊙ B Σπ ∗ I
π ∗ ΣA ⊙ π ∗ ΣB
ε- ?
I
id -
I
I
?
π ∗ (ΣA ⊙ ΣB)
id
?
A⊙B
?
η - ∗
π Σ(A ⊙ B)
?
π∗I
id
?
I
η-
?
∗
π ΣI
commute, replacing ⊙ and I with either ⊕ and 0 or ⊗ and 1.
Proof. From symmetric monoidlaity of the adjunction.
Lemma 5.3.5. The Frobenius morphisms combine well with associativity, in the sense
that
ΣA ⊗ (B ⊗ C)
α
-
(ΣA ⊗ B) ⊗ C
frob ⊗ id -
frob
Σ(A ⊗ π ∗ B) ⊗ C
frob
?
Σ(id ⊗ β −1 )
α−1
Σ(A ⊗ π ∗ (B ⊗ C)) Σ(A ⊗ (π ∗ B ⊗ π ∗ C)) ?
Σ((A ⊗ π ∗ B) ⊗ π ∗ C)
commutes.
Proof. We shall demonstrate that the clockwise direction around the above diagram is
104
-
µ
(1)
id
Σ
(i
d
id ⊗ Σβ
⊗
(2)
?
∗
µ
∗
Σ(A ⊗ (π B ⊗ π C))
Σ(id ⊗ β
−1
)
?
∗
-
Σα
(6)
id
∗
∗
-
)
⊗ε
α
∗
ΣA ⊗ (Σπ B ⊗ Σπ C)
α
(3)
∗
(5)
Σ(A ⊗ π (B ⊗ C))
-
ε
⊕(
ΣA ⊗ (B ⊗ C)
6
-
?
id
id ⊗ µ
∗
ΣA ⊗ Σ(π B ⊗ π C)
β)
-
id ⊗ ε
∗
ΣA ⊗ Σπ (B ⊗ C)
∗
Σ(A ⊗ (π B ⊗ π C))
Σα
−1
∗
∗
Σ((A ⊗ π B) ⊗ π C)
≥
id
?
∗
-
µ
∗
Σ((A ⊗ π B) ⊗ π C)
∗
∗
Σ(A ⊗ π B) ⊗ Σπ C
(7)
frob
-
µ ⊗ id
(4)
?
∗
∗
(ΣA ⊗ Σπ B) ⊗ Σπ C
id ⊗
ε
-
(id ⊗ ε) ⊗ ε
(8)
?
(ΣA ⊗ B) ⊗ C
≥ frob ⊗ id
-
?
∗
Σ(A ⊗ π B) ⊗ C
a left adjoint to (id ⊗ ε) ◦ µ; by the uniqueness of adjoints it is then equal to frob.
(1) Naturality of µ. (2) Comonoidality of ε w.r.t. ⊗. (3) Comonoidality of Σ w.r.t. ⊗.
(4) Naturality of α. (5) Invertibility of β. (6) Invertibility of α. (7) Definition of frob.
105
(8) Definition of frob and bifunctoriality of ⊗.
∗
Σ(A ⊗ π (B ⊗ C))
ΣA ⊗ (B ⊗ C)
-
α
(1)
id
?
∗
(id ⊗ ε) ⊗ id)
id ⊗
(3)
∗
(ΣA ⊗ Σπ B) ⊗ Σπ C
∗
(ΣA ⊗ Σπ B) ⊗ C
-
∗
Σ(A ⊗ π B) ⊗ C
id
frob
ε
(4)
µ ⊗ id
∗
∗
Σ(A ⊗ π B) ⊗ Σπ C
?
µ
∗
∗
Σ((A ⊗ π B) ⊗ π C)
α−
1
id
µ⊗
(2)
(ΣA ⊗ B) ⊗ C
6
(id ⊗ ε) ⊗ ε)
-
frob ⊗ id
(ΣA ⊗ B) ⊗ C
(5)
α
∗
?
ΣA ⊗ (B ⊗ C)
(6)
?
∗
ΣA ⊗ (Σπ B ⊗ Σπ C)
ε)
ε⊗
(
⊗
(7)
id
id ⊗ ε
∗
ΣA ⊗ Σπ (B ⊗ C)
id ⊗ µ
∗
Σα
∗
ΣA ⊗ Σ(π B ⊗ π C)
−1 )
−1
id ⊗
Σγ
µ
?
Σ(A ⊗ (π∗ B ⊗ π∗ C))
µ
(8)
∗
id
Σ(
⊗γ
Σ(A ⊗ π (B ⊗ C))
(1) Invertibility of α. (2) Definition of frob. (3) Clear. (4) Definition of frob. (5)
Naturality of α−1 . (6) Comonoidality of Σ w.r.t. ⊗. (7) Comonoidality of ε w.r.t. ⊗.
(8) Naturality of µ.
A key assumption in classical model theory is the non-emptiness of domains: this is
necessary to show that ⊤ ⇒ ∃x⊤ and ∀x⊥ ⇒ ⊥. We will not require non-emptiness
of domains, but instead give a characterization of non-empty types in our base category.
Lemma 5.3.6. In any classical doctrine, if there exists a morphism from the terminal
object 1 to X in B, then there exists a morphism t : ⊤ → Σπ ⊤X , over the terminal
object (where pi : X → 1 is the projection from X to the terminal.)
Proof. The required morphism in the fibre over 1 is then given by
⊤
∼
=-
∗
a ⊤X
a∗η
a∗ π ∗ Σπ ⊤X
∼
=-
Σπ ⊤X
where the first isomorphism is a given property of substitution and the last is given by
a∗ π ∗ = (πa)∗ = (id)∗
since 1 is terminal.
There are many equivalent categorical formulations of this condition in a classical
doctrine.
106
Lemma 5.3.7. Given a pair of functors (F, G) with Frobenius strengths, the following
are equivalent:
1. There is a morphism µ◦F : ⊤ → F ⊤ ;
2. There is a morphism µ◦G : G⊥ → ⊥ ;
3. There is a morphism M : GA → F A, for each A ;
4. There is a morphism ǫ◦F : A → F KA, for each A;
◦
5. There is a morphism ηG
: GKA → A, for each A .
Proof. (1 / 2) ⇒ (3)
Observer the following diagram (which might not commute):
◦
id ⊗ µ-
n F (A ⊗ ⊤)
GA ⊗ F ⊤
-
GA ⊗ ⊤
1
λ
−
λ
⊤
F
-
F A (5.4)
λ⊥
λ−
F
1
-
GA
-
G(A ⊕ ⊥)
id ⊕ µ◦Gn FA ⊕ ⊥
F A ⊕ G⊥
(3) ⇒ (1) Witnessed by the the morphism ⊤
(4 / 5) ⇒ (3)
107
ν⊤
-
G⊤
M
-
F ⊤; (3) ⇒ (2) is similar.
Observe the following diagram (which might not commute):
G
◦
ηG
ηF
-
GKF A
-
FA
(5.5)
F
◦
εF
εG
-
GA
-
F KGA
(1) ⇒ (4)
Given by the composition A
λ
-
A⊗⊤
id⊗µ◦⊤
-
A⊗F ⊤
frob
-
F (KA⊗⊤)
F (λ)
-
F KA
Remark 5.3.8. Using (1) ⇒ (4) and (2) ⇒ (5), it is immediate that the diagram 5.4
and the diagram 5.5 are the same diagram.
We will call a pair (F,G) of functors on a linearly distributive category “mix” if there
is a canonical natural transformation from G to F (that is, if 5.4 commutes).
The morphism ε◦Σ can be regarded a non-parameterized version of frobΣ ; to be convincingly so, it should be left adjoint to εΣ
Lemma 5.3.9. In a classical doctrine in which µ◦⊤ is left adjoint to µ⊤ , ε◦Σ is left adjoint
to εΣ
Proof. We establish both directions.
108
The inequality µ ◦ µ◦ ≤ id is established by:
ε
Σa∗ A
-
λ
λ
A
ε
(1)
-
- A⊗⊤
-
⊗ id
Σa∗ A ⊗ ⊤
(2) id ⊗ µ◦⊤
Σλ
id ⊗ µ◦⊤
(3)
?
ε ⊗ id A ⊗ Σ⊤
?
Σa∗ A ⊗ Σ⊤
-
µ
≥
?
(4)
id
Σ(a∗ A ⊗ ⊤)
frob
-
Σ(λ −1
)
?
Σ(a∗ A ⊗ ⊤)
Σ(λ−1 )
-
?
Σa∗ A
(1) Naturality of λ. (2) Axiom of monoidal functors. (3) Bifunctoriality of ⊗. (4)
Definition of frob.
The equality µ◦ ◦ µ = id is established by:
λ
A
λ
A⊗⊤ 6
(3)
id
(1)
-
id
-
id ⊗ µ⊤
6
∗
λ −1
ε ⊗ id (4) Σa A ⊗ Σ⊤
µ⊤
id ⊗
∗
λ−1
Σa A ⊗ ⊤
id
(5)
µ
(6)
?
A ⊗ Σ⊤
frob
?
∗
Σ(a A ⊗ ⊤)
Σλ
-
?
Σa∗ A
ε
?
(7)
id ⊗ µ◦⊤
(2)
A ⊗ Σ⊤ ε ⊗ id
A⊗⊤
A
(1) Clear. (2) Left adjointness of µ◦⊤ . (3) Invertibility of λ. (4) Bifunctoriality of ⊗.
(5) Definition of frob. (6) Axiom of monoidal functors. (7) Naturality of λ−1 .
Definition 5.3.10. A projection π in the base B of a classical doctrine is non-empty
if there is a left adjoint µ◦⊤ : ⊤ → Σπ ⊤X to µ⊤ . We will also refer to the substitutions/quantifications associated to such a morphism as non-empty.
109
5.4 Sharp Classical Doctrines
We have seen in the introductory material that if the domain of a (co)monoidal functor
has a (co)monoid on a given object, the image of that object also has a (co)monoid. In
general, there may be several monoids on a particular object, so we need a notion of
compatibility of (co)monoids with a functor, similar to the condition that a category C
has (co)monoids.
Definition 5.4.1. Given C and D, both symmetric monoidal categories that have symmetric comonoids, and a comonoidal functor K : C → D, we say that D has Kcomonoids if
KA
K∆
-
KA
Khi -
hi
∆
K(A ⊙ A)
κ
-
KI
κI
-
?
KA ⊙ KA
?
I
Lemma 5.4.2. Let K : C → D be a strong symmetric comonoidal functor, and let G
be lax right adjoint to K. Let C also have K-monoids. Then, if ∇, G∇, [], G[], µ and
µI are η strong (where η is the unit of the lax adjunction), [] is ε-strong, and ε is ∇and []-strong, then D has G monoids.
Proof. From lemma 5.3.1, ∇ is ε-strong. . The following diagrams therefore commute:
Kµ
K(GA ⊙ GA)
κ −1
-
(2)
K∇
KGA ⊙ KGA
(1)
ε ⊙ε
-
ε
A⊙A
(3)
∇
(4)
KG∇
?
- A
∇
ε
(5)
?
KGA id
KG(A ⊙ A)
ε
?
KGA
(1) Comonoidality of ε w.r.t ⊙. (2) C has K-comonoids. (3) ∇ is ε-strong. (4) ε is
110
∇-strong. (5) Clear.
KµI
κ −1
-
-
(2)
ε
(1)
KGI
KI
I
[] (4)
K[]
KG[]
[]
?
ε
?
KGA ε
(3)- A
?
(5)
id
KGA
(1) Comonoidality of ε (2) C has K-comonoids. (3) ε is []-strong. (4) [] is ε-strong. (5)
Clear
Applying lemma 5.3.2 (using that ∇, [], µ and µI are η strong), we obtain that the
required diagrams commute.
Lemma 5.4.3. Let K : C → D be a strong symmetric comonoidal functor,let G be lax
right adjoint to K. Let C have K-comonoids, and let ∆ and hi be ε-strong, ε be ∆- and
hi-strong, and ν, νI , ∇ and hi be η-strong. Then the following diagrams commute:
G(A ⊙ A)
-
ν
∆
GA
hi
?
?
GA ⊙ GB
I
111
Ghi G1
-
G∆
-
νI
GA
Proof.
K∆
KGA
∆
ε
-
KG∆
-
?
KG(A ⊙ A)
β
(1)
A⊙A id
(5)
Kν
ε⊙ε
(3)
?
-
ε
K(GA ⊙ GA)
KGA ⊙ KGA
A
(2) ∆
−1
(4)
?
A ⊙ A
ε
?
id
KG(A ⊙ A)
(1) C has K-comonoids. (2) ∆ is ε-strong. (3) ε is ∆-strong. (4) Comonoidality of ε.
(5) Clear.
Khi
hi
(1)
-
-
A
KGhi
(2) hi
ε
?
KG1 -
(3)
? id
I
β
I
id
?
I
K1
−1
id
ε
-
(4)
Kη
(6)
KGA
β −1
(5)
KGβ −I
KI
ε
?
KGK1
(1) C has K-comonoids. (2) hi is ε-strong. (3) ε is hi-strong. (4) Clear. (5) β −1 is
ε-strong. (6) Triangle law.
We may now strengthen the notion of classical doctrine, first to that of ν-sharp classical doctrine, encompassing the results of lemma 5.4.3:
Definition 5.4.4 (ν-sharp doctrine). A classical doctrine is ν-sharp if:
1. For each projection π in the base category, the relevant fibre has π ∗ (co)monoids.
112
2. ∆ and hi be εΠ -strong, ∇ and [] are etaΣ -strong, εΠ is ∆- and hi-strong, ηΣ
is ∇ and []-strong, νΠ , ν⊤ , ∇ and hi are ηΠ -strong, and νΣ , ν⊥ , ∆ and [] are
εΣ -strong.
Recall that we can define a morphism νΣ◦ : ΣA ⊕ ΣB → Σ(A ⊕ B) by
ν ◦ = ∇ ◦ Σ(id ⊕ []B ) ⊕ Σ([]A ⊕ id) ◦ Σ(λ⊥ ) ⊕ Σ(ρ⊥ ),
or, writing ι1 as shorthand for (id ⊕ []) ◦ λ, and similarly for ι2 ,
ν ◦ = ∇ ◦ Σ(ι1 ) ⊕ Σ(ι2 ).
If this morphism is inverse to ν, we may drop naturality of ν from the definition of a
classical doctrine.
Lemma 5.4.5. If ν is invertible (so, in particular, if ν ◦ is inverse to ν, then it is natural.
Proof. Owing to the lax naturality of [] and ∇, we have that
ΣA ⊕ ΣB
Σf ⊕ Σg
Σι1 ⊕ Σι2-
Σ(A ⊕ B) ⊕ Σ(A ⊕ B)
≥ Σ(f ⊕ g) ⊕ Σ(f ⊕ g)
?
ΣC ⊕ ΣD
Σι1 ⊕ Σι2-
?
113
Σ(A ⊕ B)
≥ Σ(f ⊕ g)
Σ(C ⊕ D) ⊕ Σ(C ⊕ D)
so we may construct the following diagram
∇-
∇-
?
Σ(C ⊕ D)
ν
Σ(A ⊕ B)
- ΣA ⊕ ΣB
◦
g
id
ν
⊕Σ
-
Σf
Σ(A ⊕ B)
≤
Σ(f ⊕ g)
⊕g
)
Σ(f ⊕ g)
Σ(f
B
◦
id
ν
ν
?
- ΣC ⊕ ΣD
-
? Σ(C ⊕ D)
We know from the preamble to Theorem 5.3.3 that the opposite direction also holds.
In the setting of a ν-sharp doctrine, there are several equivalent conditions for this:
Lemma 5.4.6. In a sharp doctrine, the following are equivalent:
1. Each instance of ν is ∇-strong, and ιi is ν-strong;
2. Each instance of ν is invertible;
3. The families ν and ν ◦ are natural; and
4. The functor Σ preserves ∇-strong morphisms,and ιi is ν-strong .
Proof. 1) ⇒ 2):
114
We verify νΣ◦ ν = id by
∇
Σ(A ⊕ B) ⊕ Σ(A ⊕ B)
Σι
⊕
1
Σι 2
-
ν⊕
(1)
ι1
ν
(3)
Σ(A ⊕ B)
≥
-
ι1 ⊕ ι2
ΣA ⊕ ΣB
-
-
(ΣA ⊕ ΣB) ⊕ (ΣA ⊕ ΣB)
ν
-
⊕
ι2
mo
(2)
n
∇
(4)
-
-
∇⊕∇
(ΣA ⊕ ΣA) ⊕ (ΣB ⊕ ΣB)
-
?
ΣA ⊕ ΣB
where mon is the obvious morphism constructed from isomorphisms of a monoidal
category. (1) Follows from Lemma 5.4.2. (2),(4) hold from having monoids. (3) holds
since ν is ∇-strong.
Finally, observe that ∇ ◦ ι1 is equal to the identity (an axiom of having monoids).
Similarly, ν ◦ νΣ◦ = id
Σ(A ⊕ B)
Σ(ι1 ⊕ ι2 )
ν-
ΣA ⊕ ΣB
?
Σ(A ⊕ B) ⊕ Σ(A ⊕ B)
ν
(1)
Σ((A ⊕ B) ⊕ (A ⊕ B))
Σι1 ⊕ Σι2
Σ∇
-
(2) ∇
-
?
Σ(A ⊕ B)
(1) Holds since ιi is ν-strong. (2) Holds by Lemma 5.4.3.
Observing that Σ∇ ◦ Σ(ι1 ⊕ ι2 ) = id, we obtain the desired result.
2) ⇒ 3): The above lemma gives naturality of ν, and then clearly nu◦ is also natural.
3) ⇒ 4) :Let f : A → B be ∇-strong. Then we have that the following diagram,
expressing ∇-strength of Σf , commutes:
115
Σf
-
ΣA
-
6
6
Σ∇
Σ∇
∇
ΣB
Σ(f ⊕ f-)
Σ(B ⊕ B)
∇
◦
ν
ν◦
-
Σ(A ⊕ A)
Σf ⊕ Σf
ΣA ⊕ ΣA
-
ΣB ⊕ ΣB
The ν-strength of ιi follows immediately from naturality of µ.
4) ⇒ 1): Recall that
ν = ε ◦ Σβ −1 ◦ Σ(η ⊕ η).
Thus ν is ∇-strong if these constituent morphisms are. We know that ε and Σbeta−1
are nabla-strong, from lemma 5.3.1. The family eta is ∇-strong, from the definition
of a sharp doctrine, and ∇-strong morphisms are closed under ⊕, since ∇ is defined
pointwise over ⊕. Hence, by our assumption, Σ(η ⊕ η) is ∇-strong, and so ν is ∇strong.
Associated to the diagrams of lemma 5.4.2 is another notion of doctrine — we will see
from the proof net models that this is a more natural notion of µ-doctrine:
Definition 5.4.7 (µ-sharp doctrine). A µ-doctrine is µ-sharp if:
1. For each projection π in the base category, the relevant fibre has π ∗ (co)monoids.
2. The morphisms ∇, Π∇, [], Π[], µΠ and µ⊤ are ηΠ strong ,∆, Σ∆, hi, Σhi, µΣ
and µ⊥ are εΣ strong [] is εΠ -strong, hi is ηΣ -strong, εP i is ∇- and []-strong, and
etaΣ is ∆- and hi-strong.
Definition 5.4.8. A doctrine is sharp if it is both ν-sharp and µ-sharp.
116
5.5 Linear functors, duality and M IX
We now consider how to axiomatize duality between existential and universal quantification. In Chapter 3 we recalled the notion of a linear functor[16] : a linear functor
F is a pair of functors (F⊕ , F⊗ ), with F⊕ comonoidal wrt ⊕ and F⊗ monoidal wrt F⊗ ,
and additionally the following natural transformations
m : F⊕ A ⊗ F⊗ B → F⊕ (A ⊗ B)
n : F⊗ (A ⊕ B) → F⊕ A ⊕ F⊗ B
such that a large number of coherence conditions hold.
In the same way as SLDCs examine the duality between ∧ and ∨ in *-autonomous
categories, linear functors examine pairs of De-Morgan dual functors. If we can show
that our quantifiers form a linear functor, we will know that they are equivalent to a
functor on a ∗-autonomous category.
Definition 5.5.1. A classical doctrine is a dual doctrine if every pair (Σ, Π) is a linear
functor.
We establish a characterization theorem, similar to Kelly’s for monoidal adjunctions
[42], for SLDCs; we will give some additional structure on a pair of functors sufficient
to establish that they form a linear functor.
To begin with, assume that C and D are SLDCs, that we have functors F, G : C → D,
and a strong monoidal functor K : D → C, such that
F ⊣ K ⊣ G.
This immediately gives us that F is comonoidal and G monoidal wrt both tensors.
We will say that this adjunction “has Frobenius strengths” if we have morphisms
frobF : F A ⊗ B → F (A ⊗ KB)
. and
frobG : G(A ⊕ KB) → GA ⊕ B
117
and that it “has linear strengths” if we have morphisms
mL : F A ⊗ GB → F (A ⊗ B)
nR : G(A ⊕ B) → F A ⊕ GB.
and morphisms mR and nL given by the commutative structure of the category. (As
with Frobenius strengths, we will suppress the distinction between left and rights
strengths where appropriate.)
Lemma 5.5.2. An adjunction has Frobenius strengths iff it has linear strengths.
Proof. Given Frobenius strengths, we can derive linear strengths:
F A ⊗ GB
frobF
-
F (A ⊗ KGB)
id⊗εG
-
F (A ⊗ B),
G(A ⊕ B)
id⊕ηF
G(A ⊕ KF B)
frobG
GA ⊕ F B.
-
-
And vice-versa:
FA ⊗ B
id⊗ηG
-
F A ⊗ GKB
n
-
G(A ⊕ KB)
m
-
GA ⊕ F KB
F (A ⊗ KB),
id⊕εF
-
GA ⊕ B.
By analogy, we may derive some axioms for them from those for linear strengths. The
first three families of linear functor axioms are equivalent, in a classical category, to
diagrams involving only one of the quantifiers. The behaviour of linear strengths with
respect to units
nR
G(⊥ ⊕ A) - F ⊥ ⊕ GA
µ⊥ ⊕ id
G(λ)
?
λ
GA 118
?
⊥ ⊕ GA
becomes, via
id
ηF ⊕ id
frobR
G
- G(KF ⊥ ⊕ A)
- F ⊥ ⊕ GA
G(⊥ ⊕ A) G
G(K
(γ
id
γ⊥ ⊗
⊥ ⊗
⊕
id)
i
⊥
d)
(1)
(2)
Fγ
R
G(ηF ⊕ id)
frobG
- G(KF K⊥ ⊕ A)
F K⊥ ⊕ GA (7)
µ⊥
G(K⊥ ⊕ A)
ε⊕
(5)
id
G(F εF ⊕ id)
(6)
id
- ?
?
frobR
G
- ⊥ ⊕ GA
(3)
(4)
G(K⊥ ⊕ A)
G(λ)
d)
−1 ⊕ i
λ
G(γ ⊥
-
G(⊥ ⊕ A)
)
G(λ
?
?
- GA
id
GA
(1) ηF -strength of γ⊥ . (2) ηG -strength of γ⊥ . (3) Clear. (4) Invertibility of γ⊥ . (5)
Triangle identity. (6) ηG -strength of εF . (7) Comonoidality of F. the equality
−1
◦ Gλ ◦ (Gγid) : G(K⊥ ⊕ A) → ⊥ ⊕ GA
frobR
G = λ
(5.6)
There are two families of interactions between linear functors and associativity. The
first,
G((A ⊕ B) ⊕ C)
Gα-
nR
G(A ⊕ (B ⊕ C))
nR
?
?
F (A ⊕ B) ⊕ GC
F A ⊕ G(B ⊕ C)
ν ⊕ id
id ⊕ nR
?
(F A ⊕ F B) ⊕ GC
α-
119
?
F A ⊕ (F B ⊕ GC)
gives, via
(5.7)
((1) Monoidality of η (2) Naturality of α (3) η-strength of µ. (4) η-strength of η.)
an axiom for Frobenius strengths, in the bottom right-hand polygon:
120
R
R
R
frobG
= α−1 ◦ (id ⊕ frobG
) ◦ frobG
◦ Gα ◦ G(γ ⊕ id)
-
Gα
G((A ⊕ B) ⊕ C)
G(η ⊕ id)
G(A ⊕ (B ⊕ C))
G(η ⊕ id)
G(
(η
?
⊕
G(KF (A ⊕ B) ⊕ C)
G(
K
µ⊕
id
)
η)
?
⊕
(2)
id)
(1)
-
G(K(F A ⊕ F B) ⊕ C)
-
G(γ ⊕ id)
G((KF A ⊕ KF B) ⊕ C)
-
Gα
(i d
G
⊕
G(KF A ⊕ (B ⊕ C))
))
id
fr
⊕
ob
R
(η
G
G(KF A ⊕ (KF B ⊕ C))
(4)
fr
ob R
G
frobR
G
frobR
G
(3)
-
id
⊕
(η
G
F A ⊕ G(KF B ⊕ C)
id ⊕ frob
?
F (A ⊕ B) ⊕ GC
-
µ ⊕ id
?
(F A ⊕ F B) ⊕ GC
α
-
?
F A ⊕ (F B ⊕ GC)
-
F A ⊕ G(B ⊕ C))
)
id
⊕
The second,
G((A ⊕ B) ⊕ C)
Gα-
G(A ⊕ (B ⊕ C))
nR
nL
?
?
G(A ⊕ B) ⊕ F C
F A ⊕ G(B ⊕ C)
nR ⊕ id
id ⊕ nL
?
(F A ⊕ GB) ⊕ F C
Gα
-
?
F A ⊕ (GB ⊕ F C)
gives, via
G(id ⊕ η)
?
G((A ⊕ B) ⊕ KF C)
frobL
?
G(A ⊕ B) ⊕ F C
G(η ⊕ id) ⊕ id
-
Gα
G(A ⊕ B) ⊕ C)
G(η ⊕ id)
(1)
-
G((η ⊕ id) ⊕ id)
-
Gα
G(KF A ⊕ B) ⊕ KF C)
G(KF A ⊕ (B ⊕ KF C))
(2)
G(id ⊕ (id ⊕ η))
b
fro
frob
⊕ id
?
G(KF A ⊕ (B ⊕ C))
frobR
(2)
fro
bR
L
?
F A ⊕ G(B ⊕ C)
id ⊕ G(id ⊕ η)
- ?
?
G(KF A ⊕ B) ⊕ F C
R
G(A ⊕ (B ⊕ C))
F A ⊕ G(B ⊕ KF C)
id ⊕ frobL
?
-
α
(F A ⊕ GB) ⊕ F C
?
F A ⊕ (GB ⊕ F C)
((1) Naturality of α (2) η-strength of η)
a third axiom, at the bottom of the above diagram:
α ◦ (frobR ⊕ id) ◦ frobL = (id ⊕ frobL ) ◦ frobR ◦ Gα
Lemma 5.5.3. In a classical doctrine for which
Π(a∗ ⊥ ⊕ A)
frobΠ -
ηΠ ⊕ id
id
?
∗
⊥ ⊕ ΠA
Π(a ⊥ ⊕ A)
µΠ
121
?
∗
Πa ⊥ ⊕ ΠA
(5.8)
commutes, we have
−1
◦ Πλ ◦ (Πγ ⊕ id);
1. frobR
Π = λ
R
−1
2. frobR
◦ (id ⊕ frobR
Π = α
Π ) ◦ frobΠ ◦ Πα ◦ Π(γ ⊕ id); and
L
L
R
3. α ◦ (frobR
Π ⊕ id) ◦ frobΠ = (id ⊕ frobΠ ) ◦ frobΠ ◦ Πα
Proof.
1.
frobR
Π
-
⊥ ⊕ ΠA
Πa∗ ⊥ ⊕ ΠA
⊕
id
)
µ⊥
µΠ
λ
?
⊥
Πγ⊥ ⊕ id
Π
(γ
?
Π(⊥ ⊕ A) ⊕i
d
-
Π(a∗ ⊥ ⊕ A) id
ηΠ
id
⊕
id
Π(a∗ ⊥ ⊕ A)
µ
Π⊥ ⊕ ΠA
Π(λ)
?
ΠA
id
2. This is the dual of Lemma 5.3.5.
122
-
?
ΠA
3.
123
(1)Naturality of frob. (2) Monoidality of C. (3) Part 2. (4) Naturality of γ. (5)
η-strength of σ. (6) Part 2. (7,8) Definition of frobL and frobR . (9) Naturality
of σ. (10) Monoidality of C.
∗
Π(σ ⊕ id)
L
ob
fr
∗
Π(a A ⊕ B) ⊕ C
frobR ⊕ id
frobL
Πσ ⊕ id
(8)
∗
Π((B ⊕ a A) ⊕ a C)
(1)
?
?
(A ⊕ ΠB) ⊕ C
σ
(3)
?
∗
∗
Π(B ⊕ (a A ⊕ a C))
Π(id ⊕ γ)
?
∗
Π(B ⊕ a A) ⊕ C
(ΠB ⊕ A) ⊕ C
d
⊕i
-
Π(α)
∗
frobL ⊕ id
∗
Π(B ⊕ a (A ⊕ C))
frobL
α
-
?
ΠB ⊕ (A ⊕ C))
Πσ
-
Π(id ⊕ σ)
(4)
∗
-
Π(id ⊕ a σ)
α
?
-
Π(α)
frobL
(6)
-
?
ΠB ⊕ (C ⊕ A))
∗
?
∗
∗
frobL
?
∗
Π(B ⊕ a C) ⊕ A
frobL ⊕ id
α
fr
ob
Π((B ⊕ a C) ⊕ a A)
Π(B ⊕ a (C ⊕ A))
(5)
id ⊕ σ
∗
Π(B ⊕ (a C ⊕ a A))
Π(id ⊕ γ)
∗
(10)
∗
Π(a A ⊕ (B ⊕ a C))
(2)
?
∗
-
Πα
∗
Π((a A ⊕ B) ⊕ a C)
-
?
R
(7)
σ
∗
A ⊕ Π(B ⊕ a C)
id ⊕ frobL
(9)
(ΠB ⊕ C) ⊕ A
σ
-
-
?
A ⊕ (ΠB ⊕ C)
For the remaining axioms, we merely convert the linear strengths into statements involving Frobenius strengths:
ΠA ⊗ Π(B ⊕ C)
ν
id ⊗ Π(id ⊕ η)
-
ΠA ⊗ Π(B ⊕ a∗ ΣC)
L
id ⊗ frobΠ
δ
?
?
(ΠA ⊗ ΠB) ⊕ ΣC
Π(A ⊗ (B ⊕ C))
Πδ
ΠA ⊗ (ΠB ⊕ ΣC)
?
Π(id ⊕ η) -
Π((A ⊗ B) ⊕ C)
Π((A ⊗ B) ⊕ a∗ ΣC)
ν ⊕ id
?
frobL
Π Π(A ⊗ B) ⊕ ΣC
(5.9)
and
ΠA ⊗ Π(B ⊕ C)
id ⊗ Π(η ⊕ id)
-
ΠA ⊗ Π(a∗ ΣB ⊕ C)
id ⊗ frobR
Π-
ΠA ⊗ (ΣB ⊕ ΠC)
δ
ν
?
(ΠA ⊗ ΣB) ⊕ ΠC
?
frobL
Σ ⊕ id
Π(A ⊗ (B ⊕ C))
?
(Σ(a∗ ΠA ⊗ B) ⊕ ΠC
Πδ
?
Π((A ⊗ B) ⊕ C)
Σ(ε ⊗ id) ⊕ id
Π(η ⊕ id) -
Π(a∗ Σ(A ⊗ B) ⊕ C)
frobR
Π
?
- Σ(A ⊗ B) ⊕ ΠC.
(5.10)
Taking 5.9 and 5.10 and their duals for each non-empty type, plus naturality of the
linear strengths, gives an equivalent definition of a dual classical doctrine.
This notion of dual doctrine is useful, insofar as it corresponds closely to the strongest
theory on proofs we will introduce in the next section. However, it is difficult in
practice to check for a particular concrete example. In the remainder of this chapter
we will show sufficient conditions for the axioms of a dual doctrine to hold, in a sharp
doctrine.
First, note that in the presence of the axioms of a linear functor, the pair Q = (Σa , Πa )
is canonically mix.
A preliminary lemma:
Lemma 5.5.4. Given a morphism µ◦⊤ : ⊤ → Σ⊤, and a morphism µ◦⊥ : Π⊥ → ⊥,
we may construct another morphism from ⊤ to Σ⊤ by following the steps 2 ⇒ 3 ⇒ 1
124
from Lemma 5.3.7. If µ◦⊥ ◦ µ⊥ = id, and we have the inequality
a∗ µ◦⊤
∗
a⊤
ηΣ
≤
γ⊤
- a∗ Σ⊤
-
?
⊤
then the new constructed morphism is smaller than µ◦⊤
Proof. The morphism from ⊤ to Σ⊤ constructed from mix and µ◦⊥ is given by the
anti-clockwise direction in the diagram below:
-
µ◦
⊤
⊤
Σ⊤
6
λ−1
⊥
(1)
λ⊥
-Σ⊤ ⊕ Π⊥
µ⊤ ⊕
ν⊤
◦
?
(2)
ν
⊤
Π⊤
-
λ⊥
(6)
?
⊕
id ⊕ ν⊤
Π⊤ ⊕ ⊥
Πλ
⊥
∗
Πa ⊤ ⊕ ⊥
µ
⊥ (7)
-
(3)
ηΠ ⊕ id
-
Πa∗ µ◦
⊤ ⊕ ν⊥
(8)
Πγ⊤ ⊕ µ⊤
Πη Σ
≤
⊕i
d
(5)
?
∗
◦
id ⊕ ν⊥
Πa Σ⊤ ⊕ Π⊥
-
µ
Π⊤ ⊕ Π⊥
(4)
µ
(10)
?
-
Π(ηΣ ⊕ id)
Π(⊤ ⊕ ⊥)
?
∗
Π(a Σ⊤ ⊕ ⊥)
-
frobΠ
-
-
6
ν⊥
- ?
(9)
Σ⊤ ⊕ ⊥
id
ν⊤ ⊕ id
-
ηΠ ⊕ id
⊤⊕⊥
-
◦
id ⊕ ν⊥
Σ⊤ ⊕ Π⊥
(1) is given below. (2) Monoidality of Π w.r.t ⊗(3) η-strength of Π (4) Definition of
frob (5) Clear (6) Bifunctoriality of ⊕ (7) Monoidality of Π. (8) is our assumption (9)
Monoidality of Π w.r.t ⊕ (10) Naturality of µ.
125
The cell labelled (1) above, commutes by the following decomposition:
µ◦⊤
⊤
-
Σ⊤
6
λ−1
⊥
λ⊥
?
µ◦⊤ ⊕ id-
⊤⊕⊥
Σ⊤ ⊕ ⊥
- 6
id
µ◦⊤
?
id ⊕ ν⊥◦
id ⊕ ν⊥
Σ⊤ ⊕ ⊥
Σ⊤ ⊕ Π⊥
The top square is an instance of the naturality of λ, and the bottom triangle is the
adjointness of ν and ν ◦ .
This is enough to establish canonicity of Q-mix.
Theorem 5.5.5. Given the conditions above, and their duals, the two directions around
the diagram 5.4 are equal.
Proof. Using Lemma 5.5.4, we demonstrate that the clockwise direction is larger than
the anti-clockwise direction:
id ⊗ µ◦
⊤
-
≤
⊗
-
ΠA ⊗ ⊤
id ⊕ ν⊤
(1)
Πλ⊤
(2)
ν
(4)
ν
?
Π(A ⊗ ⊤)
Πλ
ΠA ⊗ Π(⊤ ⊕ ⊥)
-
Π(id ⊗ λ)
Πλ
(8)
Πδ
?
n
Π((A ⊗ ⊤) ⊕ ⊥)
-
?
n
(5)
?
⊕
ΠA ⊗ (Σ⊤ ⊕ ⊥) (7)
λ
δ
(6)
δ
(10)
-
-
id ⊗ µ◦
⊥
(ΠA ⊗ Σ⊤) ⊕ ⊥
id
(9)
id ⊕ µ◦
⊥
Σ(A ⊗ ⊤) ⊕ Π⊥
Σλ ⊕ id
-
-
id ⊗ (id ⊕ µ◦
⊥)
(ΠA ⊗ Σ⊤) ⊕ Π⊥
m
-
ΠA ⊗ (Σ⊤ ⊕ Π⊥)
?
Π(A ⊗ (⊤ ⊕ ⊥))
Π(λ ⊕ id) (11)
Π(A ⊕ ⊥)
-
id ⊗ n
-
ΠA
-
id ⊗ Πλ
id
- 6
m
λ
?
ΠA ⊗ Π⊤
(3)
λ
ΠA ⊗ Σ⊤
(12)
?
id ⊕ µ◦
⊥
ΣA ⊕ Π⊥
-
λ
m ⊕ id
-
Σ(A ⊗ ⊤)
?
Σ(A ⊗ ⊤) ⊕ ⊥
Σλ ⊕ id
-
(13) Σλ
?
ΣA ⊕ ⊥
(1) Comonoidality of Σ (2) Below (3) Lemma 5.5.4 (4) (Naturality of ν. (5) An axiom
of linear functors 6)Naturality of δ (7) An axiom of SLDCs (8) An axiom of SLDCs
(9) Bifunctoriality of ⊕ (10) Naturality of λ (11) Naturality of n (12) Bifunctoriality
of ⊕ (13) Naturality of λ
126
λ
-
?
ΣA
Similarly, from the dual of Lemma 5.5.4, we may demonstrate that the anti-clockwise
direction is larger than the clockwise.
The coherence has the following consequence:
Lemma 5.5.6. If diagram 5.5 commutes, M is natural.
Proof. One direction around the diagram gives
ηΠ◦ FA
GηF GKF A
GA
≥
Gf
?
′
GA
?
GηF GKF A′
ηΠ◦ - ?′
FA
ε◦ F KGA
Fε FA
GA
≤
Gf
?
GA′
≥ Ff
GKF f
≤ Ff
F KGf
?
ε◦ F KGA′
F ε - ?′
FA
Lemma 5.5.7. The conditions of Lemma 5.5.4 hold of a non-empty quantification in a
dual classical doctrine.
Proof. The only diagram to verify is the following:
a∗ µ◦⊤
∗
-
a∗ Σ⊤
6
id
⊤
a∗
µ◦
-
a⊤
γ
µ⊤
?
⊤
id
∗
γ⊤
≤
a
−
⊤ 1
-
-
a∗ ⊤
η
-
127
a∗ Σ⊤
We have proved
Theorem 5.5.8. In a dual doctrine, the pair (Σa , Πa ) is mix for any non-empty term a.
We seek now a partial inverse to this result. Given a pair of functors Q = (Σa , Πa ),
with Σa ⊣≤ a∗ ⊣≥ Πa , and with a natural transformation M : Πa A → ΣA. , what are
sufficient conditions for that pair to form a linear functor?
The sufficient condition we have is a condition on the monoidal strengths µΠ and
µΣ ; that µΠ is an epimorphism and µΣ a monomorphism. See the next chapter for a
discussion of what this condition might mean from a proof-theoretic perspective. In
addition we assume sharpness of the doctrine, , and some more specific coherence
conditions.
Lemma 5.5.9. Given a morphism M : GA → F A, such that the diagram
GA ⊗ GB
id ⊗ M
-
-
n
ν
GA ⊗ F B
?
G(A ⊕ B)
commutes, the diagrams
128
(5.11)
GA ⊗ (GB ⊕ GC)
id ⊗ µ
?
GA ⊗ G(B ⊕ C)
id ⊗ n
GA ⊗ (GB ⊕ F C)
ν
Gδ
?
?
G(A ⊗ (B ⊕ C))
(GA ⊗ GB) ⊕ F C
ν ⊕ id
Gδ
?
n G(A ⊗ B) ⊕ F C
?
G((A ⊗ B) ⊕ C)
and
GA ⊕ GB
µ
?
n
G(A ⊕ B) - GA ⊕ F B
Gf ⊕ F g
G(f ⊕ g)
?
?
n
G(C ⊕ D) - GC ⊕ F D
commutes.
129
Proof.
-
id ⊗ n
GA ⊗ G(B ⊕ C)
id
ν
⊗
µ
(1)
id
⊗(
i
M
d⊕
GA ⊗ (GB ⊕ F C)
)
-
δ
GA ⊗ (GB ⊕ GC)
δ
?
(3)
?
G(A ⊗ (B ⊕ C))(2)(GA ⊗ GB) ⊕ GC
ν ⊕ id
Gδ
-
id ⊕ M
ν ⊕ id
id
µ
(5)
⊕M
-
n
G((A ⊗ B) ⊕ C)
(5.12)
(4)
?
G(A ⊗ B) ⊕ GC
?
?
(GA ⊗ GB) ⊕ F C
- ?
G(A ⊗ B) ⊕ F C
(1),(5) are the assumptions of the Lemma. (2) See lemma 5.3.3 (3) Naturality of δ (4)
Bifunctoriality of ⊕.
G(A ⊕ B)
n
µ
(1)
-
id
⊕M
GA ⊕ F B
-
GA ⊕ GB
G(f ⊕ g)
(2) Gf ⊕ Gb
?
GC ⊕ GD
µ
? G(C ⊕ D)
(4)
n
(3) Gf ⊕ F g
id ⊕
Corollary 5.5.10. If µ is an epimorphism then the diagram
130
M
-
-
?
GC ⊕ F D
GA ⊗ G(B ⊕ C)
id ⊗ n
GA ⊗ (GB ⊕ F C)
Gδ
ν
?
?
(GA ⊗ GB) ⊕ F C
G(A ⊗ (B ⊕ C))
ν ⊕ id
Gδ
?
G((A ⊗ B) ⊕ C)
?
n G(A ⊗ B) ⊕ F C
commutes
Corollary 5.5.11. If µ is an epimorphism then the linear strength n is natural.
So we have recovered axiom 5.9. We now recover axiom 5.10:
Lemma 5.5.12. If 5.11 holds, and additionally the diagram
GA ⊕ GB
id ⊗ M
-
ν
GA ⊕ F B
m
?
G(A ⊕ B)
?
M F (A ⊕ B)
commutes, the diagram
131
(5.13)
GA ⊗ (GB ⊕ GC)
id ⊗ µ
?
id ⊗ n
GA ⊗ G(B ⊕ C)
GA ⊗ (F B ⊕ GC)
ν
Gδ
?
?
G(A ⊗ (B ⊕ C))
(GA ⊗ F B) ⊕ GC
n ⊕ id
Gδ
?
n F (A ⊗ B) ⊕ GC
?
G((A ⊗ B) ⊕ C)
commutes.
Proof.
-
id ⊗ n
GA ⊗ G(B ⊕ C)
id
⊗
µ
(
id ⊗
(1)
GA ⊗ (GB ⊕ GC)
ν
δ
?
?
G(A ⊗ (B ⊕ C))(2)(GA ⊗ GB) ⊕ GC
ν ⊕ id
Gδ
µ
G((A ⊗ B) ⊕ C)
GA ⊗ (F B ⊕ GC)
-
δ
(3)
-
(id ⊗ M ) ⊕ id
?
(GA ⊗ F B) ⊕ GC
(5.14)
?
G(A ⊗ B) ⊕ GC
?
M
)
⊕ id
(4)
id ⊕
(5)
m ⊕ id
M
-
n
- ?
F (A ⊗ B) ⊕ GC
(1),(5) are the assumptions Lemma.5.12 (2) See lemma 5.3.3 (3) Naturality of δ (4)
Assumption of the current lemma..
Corollary 5.5.13. If µ is an epimorphism, then the diagram
132
GA ⊗ G(B ⊕ C)
id ⊗ n
GA ⊗ (F B ⊕ GC)
Gδ
ν
?
?
(GA ⊗ F B) ⊕ GC
G(A ⊗ (B ⊕ C))
n ⊕ id
Gδ
?
?
n F (A ⊗ B) ⊕ GC
G((A ⊗ B) ⊕ C)
commutes
Lemma 5.5.14. Let the morphism M arise from an adjunction; then the diagram 5.13
commutes if the dual of 5.11 commutes.
Proof. We may decompose the diagram as
id ⊗ M
GA ⊗ GB
-
⊗
⊗
id
M
-
M
M
FA ⊗ FB
m
µΠ
GA ⊗ F A
µΣ
?
G(A ⊗ B)
M
-
?
F (A ⊗ B)
in which the upper triangle is clear, and the right-hand triangle is the dual of 5.11.
Assuming M arises as ηΠ◦ ◦ ΠηΣ , the remaining cell follows from:
133
Gη ⊗ Gη
-
GKF A ⊗ GKF B
◦
ηΠ◦ ⊗ ηΠ
FA ⊗ FB
-
GA ⊗ GB
µΠ
(2)
η◦
?
-
G(KF A ⊗ KF B)
Π
(1)
6
µΠ
Gγ
η)
µΣ
⊗
?
G
(η
GK(F A ⊗ F B)
6
(3)
?
G(A ⊕ B)
(4)
Gη
-
µΣ
ηΠ◦ F (A ⊗ B)
GKF (A ⊗ B)
(1) Naturality of µΠ (2) Monoidality of ηΠ◦ (3) Comonoidality of ηΣ (4) Naturality of
µΣ . A similar diagram commutes if M arises as ΣεΠ ◦ εΣ .
So we have proved:
Theorem 5.5.15. A pair of functors Q = (Σa , Πa ) between two SLDCs, with Σa ⊣≤
a∗ ⊣≥ Πa , a∗ strong monoidal in both tensors, and with a natural transformation
M : Πa A → ΣA, forms a linear functor if the induced families µΣ and µP i are
natural, µP i is an epimorphism, µΣ is a monomorphism, and the diagrams
GA ⊗ GB
id ⊗ M
-
GA ⊗ F B
-
n
ν
?
G(A ⊕ B),
Π(a∗ ⊥ ⊕ A)
frobΠ-
⊥ ⊕ ΠA
ηΠ ⊕ id
id
?
Π(a∗ ⊥ ⊕ A) µΠ
and their duals, commutes.
134
?
Πa∗ ⊥ ⊕ A,
Finally, we give sufficient conditions for 5.11 and its dual commute in a classical
doctrine. Here we use canonicity of the morphism M. We also require a generalization
of the notions of epimorphism and monomorphism to the order-enriched setting.
Definition 5.5.16. A morphism f in a category enriched with an order ≤ is a ≤monomorphism if, whenever g1 ◦ f ≤ g2 ◦ f , we have g1 ≤ g2 f . Similarly for ≤epimorphisms. (We will use the terminology “order-epi” etc where the ordering is
clear by context.
Lemma 5.5.17. The diagram 5.11 is satisfied in a classical doctrine in which µΠ is
order-epi.
Proof. First, we show that id ⊕ η ◦ = frob ◦ µ : GA ⊕ GKF B → GA ⊕ F B, as
both are left adjoint to id ⊕ η. This is clear for id ⊕ η ◦ . It remains to demonstrate for
frob ◦ µ.
One direction is given by the definition of frob; frob ◦ µ ◦ (id ⊕ η) is equal to id. For
the other, consider
µ ◦ (id ⊕ η) ◦ frob ◦ µ.
(id ⊕ η) ◦ frob ◦ µ is smaller than id, by the definition of frob, so
µ ◦ (id ⊕ η) ◦ frob ◦ µ ≤ µ ◦ id.
If µ is order-epi, (id ⊕ η) ◦ frob ◦ µ is smaller than id.
GA ⊕ GB
id ⊕ Gη-
GA ⊕ GKF B
?
G(A ⊕ B)
G(id ⊕ η)
-
?
G(A ⊕ KF B)
Theorem 5.5.18. A sharp doctrine in which
135
GA ⊕ F B
-
b
fro
µ
µ
◦
id ⊕ ηΠ
1. for every non-empty projection, mix between the quantifiers is coherent;
2. µΣ is order-monic and and µP i is order-epi; and
3.
Π(a∗ ⊥ ⊕ A)
frobΠ-
⊥ ⊕ ΠA
ηΠ ⊕ id
id
?
Π(a∗ ⊥ ⊕ A) µΠ
?
Πa∗ ⊥ ⊕ A
commutes ,
is a dual classical doctrine.
5.6 A concrete example
A particularly tractable kind of indexed category is given, for a particular category, by
a family construction over Set. Indeed, in many ways indexed categories are intended
as a generalization of the family construction to other base categories than Set. Given
a category C, F am(C) has base category Set, and the fibre over a set P is the category
C P . Given a function f : P → Q, the functor f ∗ : C Q → C P takes (Aq )q∈Q to
(Af (p) )p∈P .
Our concrete example is built from set-indexed families of relations, via the geometry
of interaction construction G used in [22] and described in section 3.2. Each RelP is a
Dummett category, and so G(RelP ) is a classical category. We then show how the set
L
indexed biproduct P can be used to give a functor that denotes quantification.
L
Remark 5.6.1. The indexed category F am(Rel) with
is a doctrine in a broad
L
L
sense of the word; we have that f (i)=j ⊣ fˆ ⊣ f (i)=j , but it would be incorrect to
L
call it a “Dummett doctrine”. The behaviour of
is not of the intuitionistic universal
L
quantifier; in particular, has a Frobenius strength for the universal quantifier, which
is intuitionistically invalid.
Remark 5.6.2. The construction that follows is more general than we need at this
stage: it gives a definition of quantification over arbitrary functions in the base, rather
136
than just for projections, as the former is not substantially harder than the latter. The
model we construct includes enough structure to model equality and, like hyperdoctrines for natural deduction, “generalized quantification” over arbitrary terms; we
will discuss a corresponding syntax and general model for this in Chapter 8.
A few preliminary observations:
Lemma 5.6.3. Given any non-empty set P:
1. RelP has a biproduct;
2. RelP is a Dummett category;
3. G(RelP ) is a classical category, and is equivalent to (G(Rel))P .
Proof. The biproduct and the Dummett properties arise form the pointwise structure
of Rel. To see that G(RelP ) is classical, observe that the trace (Kleene *) on Rel also
lifts to RelP . Finally, the obvious mapping from G(RelP ) to (G(Rel))P ,
−
+
−
((A+
i )i , (Ai )i ) → ((Ai , Ai )i )
extends to an isomorphism of categories.
Theorem 5.6.4. The functor C : Set → Cat, taking a set P to (G(Rel))P , admits a
classical doctrine structure
Proof. Our method of proof will be to establish certain properties of the Dummett
categories RelQ for a set Q, and then lift those properties to the categories G(Rel)Q .
We will need some notation to pick out particular components of a (possibly infinitary)
biproduct over a set P . By analogy with the notation inl/inr for a binary coproduct, we
will write x in i for the object x in the ith component of a product over P .
Given a function f : P → Q, define fˆ : RelQ → RelP as the function that takes
(Aq )q∈Q to (Af (p) )p∈P ,. with the obvious extension to morphisms. This is substitution
(reindexing) in Fam(Rel)
To give a classical doctrine, we require:
137
1. “Substitutions” that preserve the correct structure. Given a function f : P → Q,
the corresponding substitution functor between fibres is as follows:
f ∗ : (G(Rel))Q → (G(Rel))P
−
ˆ + ˆ −
f ∗ : (A+
q , Aq )q∈Q → (f (A ), f (A ))
Since the tensor on (G(Rel))P is also defined pointwise on the pair structure of
its objects, f ∗ preserves the tensor iff fˆ preserves the tensor. We have
ˆ i ⊕ Bi )i ) = (Af (j) ⊕ Bf (j) )j = (Af (j) )j ⊕ (Bf (j) )j
fˆ((Ai )i ⊕ (Bi )i ) = f((A
.
2. Left and right lax adjoints to f ∗ . We begin by establishing
M
⊣ fˆ ⊣
f (i)=j
M
f (i)=j
and then lift this adjunction to the geometry of interaction construction. We will
demonstrate the left hand adjunction; the right hand case is dual.
We claim existence of a co-unit
ηi : Ai →
M
Aj
j | a(j)=a(i)
with
ηi = {(x, x in i)}
and a unit
εi :
M
Ba(j) → Bi
j | a(j)=i
with
εi = {(x in j, x)|a(j) = i}
.
The triangle identities are verified by chasing an object around the following
diagrams:
138
ηfˆA
-
fˆA
fˆ
M
id
fˆA
fˆεA
-
?
ˆ
fA
The relation going clockwise around the diagram is the composition of {(x, x in p)}
and{(x in k, x)|a(j) = a(k)}, so each x in A is related to itself, and only itself.
M
B
L
M M
ηB
fˆ
B
id
εL B
-
M?
B
Going clockwise around the diagram, x inj is related to x inj inj, and then
y ink inl is related to y in k (so x inj inj is related to x inj). Once again, the
composition is the identity.
Finally, we verify that the unit and co-unit are natural transformations. For η,
observe that, for each member p of P , an element x of Ap is related to gp (x)inp,
going both ways around the square:
Ap
η-
M
Aq
(q | a(q)=a(p))
L
gp
?
Bp
M?
η-
(q | a(q)=a(p))
gq
Bq
(q | a(q)=a(p))
Lemma 5.6.5. For any P -indexed family of relations hi : Ai → Bi ,
ε◦
M
ˆ ◦ ε◦ = h
f(h)
f
139
and
M
ˆ ⊆ ε◦ ◦ h ◦ ε.
f(h)
f
◦
where ε is the opposite relation of ε.
L ˆ
Proof. Suppose x is related to y in hi . The relation ε ◦ f f(h)
◦ ε◦ is built as
follows: x is related to xinj, where a(j) = i, xinj is related to yinj, and then yinj
is related to y.
L
L
By contrast, ε◦ ◦h◦ε (which relates f fˆA to f fˆB) is constructed as follows:
xini is related to x, x is related to y, and y is related to y inj (where i and j are
L
unrelated). So x ini is related to y inj, where f fˆ(h) relates x ini only to
y ini
We now extend these definitions to the Geometry of Interaction construction.
We define Σf : G(RelP ) → G(RelQ ) on objects as

Σf ((A+ )p , (A− )p ) = 
M
 
f (p)=q
(A+ ) , 
q
M
f (p)=q

(A− ) 
q
and claim
Σf ⊣≤ f ∗ ⊣≤ Σf .
The unit η of the lax adjunction is given by the following P-indexed family of
diagrams:
A+
p
η-
M
A+
i
(i | f (i)=f (p))
06
0
?
η◦
A−
p
M
(i | f (i)=f (p))
and the co-unit ε is given by
140
A−
i
M
ε - +
Bq
Bq+
6
j | f (j)=q
0
0
M?
ε◦
Bq− Bq−
j | f (j)=q
These squares define lax natural transformations. The composition g ◦ ε , for
instance, is
M
Bq+
ε - +
Bq
6
6
j | f (j)=q
0
g −+
g +−
0
M?
g ++ - +
Cq
?
g −−
Bq− ε◦
Bq− Cq−
j | f (j)=q
which simplifies to
M
g ++ ◦ ε -
Bq+
Cq+
6
j | f (j)=q
◦
ε ◦g
+−
◦ε
g
M?
−+
(g −− ◦ ε)◦
Bq− Cq−
j | f (j)=q
By the properties of ε we have established, this is equal to
M
Bq+
ε◦
L ˆ ++
f (g ) -
6
j | f (j)=q
L ˆ +−
f (g )
M?
Cq+
g −+
Bq−
(ε ◦
L ˆ −− ◦
f(g ))
j | f (j)=q
141
Cq−
The composition ε ◦ Σf ∗ g, is given by
M
L ˆ ++
f (g )
ε◦
Bq+
-
Cq+
6
j | f (j)=q
L ˆ +−
fg
M?
ε◦
Bq−
(ε ◦
L ˆ −− ◦
f (g ))
L ˆ −+
f (g ) ◦ ε◦
Cq−
j | f (j)=q
In the order on morphisms ε ◦ Σf ∗ f is less than f ◦ ε, by Lemma 5.6.5.
The unit inequations are similarly proven.
3. Monoidal strength. This is trivially verifiable;
L
B, and this lifts easily to the GoI category.
L
L
(A⊕B) is isomorphic to A⊕
4. Frobenius morphisms.
In RelQ , we have the morphism
M
M
(A ⊕ fˆB) →
A ⊕ B,
g:
(a inl) inp → (a inp) inl
(b inr) inp → b inr.
The opposite of this morphism, g ◦ , satisfies the (in)equations
g ◦ g ◦ ≤ id,
g ◦ ◦ g = id,
similarly to Lemma 5.6.5.
We can now define frobΣ in G(Rel) as the square of morphisms
M
g M + ˆ +
(A ⊕ f B )
A+ ⊕ B + 6
0
0
?
M
g◦ M − ˆ −
A ⊕ B− (A ⊕ f B )
−
and derive the required properties from those of g.
142
Theorem 5.6.6 (Sharpness). F am(G(Rel)) is a sharp doctrine.
Proof.
1. strong f ∗ -(co)monoids.
This is easily established; for example, the diagram
f ∗A
f ∗∆
-
f ∗ (A ⊗ A)
∆
β
-
?
∗
f A ⊗ f ∗A
β is the identity, and f ∗ ∆ = ∆.
2. Strength
The morphisms ηΠ , εΠ , ηΣ , εΣ , ∆, ∇, [], and hi are all strong in G(Rel). All have
empty vertical components, so they are strong iff the corresponding morphisms
in F am(Rel) are strong; but the relevant transformations in F am(Rel) are natural, so every morphism in F am(Rel) is strong.
Theorem 5.6.7. F am(G(Rel)) is a dual doctrine.
Proof. The transformations µΣ and µΠ are isomorphisms, and so are order-monic and
order-epi.
143
Chapter 6
Modelling classical first-order proofs
6.1 Interpretation of the sequent calculus in classical
doctrines
We present an extension of the notion of a net-theory to the first-order setting. To
capture the dynamic properties of proofs in an enriched-categorical setting, we must
strengthen our notion of proof equality. In particular, we must identify enough proofs
to ensure that our quantifiers are (co)monoidal functors. We also need to specify that
certain applications of the grow rules hold as equalities: specifically, those not involving structural rules or negation.
Definition 6.1.1. We will call a proof net linear–pure if it is formed using only ∧R,
∧L, ∨L, and ∨L and the quantifier rules.
Definition 6.1.2. Given a signature (X , A, T ) a first-order proof theory (hereafter
abbreviated to theory) is a relation 4 on the first-order proof nets over that signature
1. The relation is a (propositional) net theory;
2. The relation 4 is compatible with the quantifier inference rules;
3. Every instance of Cut∃, ∃grow and ∃W Y is in 4;
4. Instances of ∃grow in which both nets are linear–pure hold in both directions;
144
5. Instances of ∃Cut in which both nets are linear–pure and N has only one left
door hold in both directions.
6. The following reductions hold as equalities, where the net M is linear–pure and
the net N has only one left door: ∃expand :
∃xL
φ(x) : L
φ(x) : R
≡
∃xφ : L
∃xφ : R
[x]∃yφ(y) : R
∃xφ : L
.
∃xφ : R
∃merge :
∃x : L
∃y : L
M
ψ:L
∃xψ : L
N
φ(x) : R
φ(x) : L
[y]γ : R
[x]∃yφ : R
∃yφ : R
∃yφ : L
γ:R
∃x : L
[x/y]N
M
≡
ψ:L
φ(x) : R
φ(x) : L
∃xψ : L
[x]γ : R
γ:R
.
The rules ∃expand and ∃merge are enough to give functoriality of the functor Σ defined by
∃xL
M
Σ(M) :=
ψ(x) : R
φ:L
[x]∃yψ(y) : R
∃xφ : L
∃yψ(y) : R
,
and their duals that the corresponding construction Π is a functor.
If, in addition, we want Σ to be comonoidal w.r.t. ⊗ and the universal quantifier
145
monoidal w.r.t. ⊕, we must add the following reduction and its dual, holding as an
equality where N and P have only one right door:
∃comon :
∃x
∃xL
∃z
φ:L
ψ:L
φ:R
ψ:R
[x]∃yφ : R
[x]∃zψ : R
φ∧ψ :L
φ(x) : L
[x]∃yφ ∧ ∃yψ : R
φ′ : R
ψ(z) : L
∃zψ : L
∃yφ : L
ψ′ : R
[z]∃vψ ′ : R
[x]∃wφ′ : R
∃vψ ′ : R
∃wφ′ : R
∃xφ ∧ ψ : L
∃yφ ∧ ∃zψ : R
∃wφ′ ∧ ∃vψ ′ R
∃yφ ∧ ∃zψ : L
.
∃x
≡
φ(x) : L
ψ(x) : L
φ′ : R
[x]∃wφ′ : R
ψ′ : R
[x]∃wψ ′ : R
φ∧ψ :L
→
[x]∃wφ′ ∧ ∃wψ ′ : R
∃wφ′ ∧ ∃wψ ′ : R
∃xφ ∧ ψ : L
.
Definition 6.1.3. A µ-theory is a first-order proof theory in which:
1. Instances of ∃grow in which the net taken into the quantifier box is linear–pure
hold as an equality;
2. Instances of ∃Cut in which the net N is linear–pure and has only one left door
hold as an equality; and
3. The reduction ∃merge holds as an equality when the net N has only one left
door, and the reduction ∃comon holds as an equality when N and P have only
one right door each.
146
Remark 6.1.4. The rule ∃comon hardwires the comonoidality of Σ with respect to ∧
into the theory; this is a little inelegant, and in section 6.4 we will see an alternative
axiomatization of this property.
Given a classical doctrine, C : B → Cat, we interpret proofs in a theory T as follows.
A type X of variables is interpreted by an object ⌊X⌋ of B, and a pair (X, Y ) is
interpreted by ⌊X⌋ × ⌊Y ⌋. A term t with domain X and codomain Y is interpreted by
a morphism ⌊t⌋ : ⌊X⌋ → ⌊Y ⌋ in B. A formula φ(x) over a variable x of type X is
an object of C(⌊X⌋) — the usual interpretation of formulae and proofs in a classical
category carries over to C(⌊X⌋) . What remains is to interpret the additional structure
particular to the first-order calculus.
Given a formula φ(xs, y), whose interpretation lies in C(⌊Xs⌋ × ⌊Y ⌋), the interpretation of ∃yφ(xs) is Σπ ⌊φ(xs, y)⌋, and the interpretation of ∀yφ(xs) is Ππ ⌊φ(xs, y)⌋,
where π is the projection from ⌊Xs⌋ × ⌊Y ⌋ to ⌊Xs⌋ in B. Similarly, the interpretation
of a formula [y]ψ(xs) is π ∗ ⌊ψ⌋.
The interpretation of, for example, an application of ∃xR, is, given by post-composition
of the morphism η : A → π ∗ Σπ A. The interpretation of ∃yL is a little more involved. Suppose that the morphism f is the denotation of the proof Φ of the sequent
[y]Γ, φ ⊢ [y]∆. Then the denotation of the proof ∃yL(Φ) is the following morphism:
⌊Γ⌋ ⊗ ⌊Σπ φ⌋
frob
-
Σπ (π ∗ ⌊Γ⌋ ⊗ ⌊φ⌋)
Σπf
Σπ π ∗ ⌊∆⌋
ε ⌊∆⌋
Theorem 6.1.5 (Soundness). Let ⌊−⌋ be an interpretation of a first-order nets over
(Σ, L) in a classical doctrine C. The judgements M 4 N form a classical first order
proof theory. If C is a µ-doctrine, the theory M 4 N is a µ-theory.
To verify this we must ensure that each cut reduction valid in a net theory holds as
an inequality in a classical doctrine, and that those reductions that hold as equalities
in a theory hold as equalities in the doctrine. We will check this simultaneously for
theories and µ-theories.
Each fibre is a sound model of propositional classical logic (see [23]), so we need only
list the first order reduction rules, and demonstrate their soundness.
147
1. Cut∃
The application of ∃R is expressed in proof nets as a cut against the constant
K ∃.
[x]M
[x]φ(y) : L
φ(x) : R
φ(x) : L
[x]∃yφ : R
[x]∃yφ : L
It therefore suffices to consider the soundness of the following reduction:
[x]Φ(y)
·
·
·
[x, y]Γ, [x]φ(y) ⊢ [x, y]∆
[x]Γ, [x]∃y.φ ⊢ [x]∆
∃yL
φ(x) ⊢ φ(x)
φ(x) ⊢ [x]∃y.φ
[x]Γ, φ(x) ⊢ [x]∆
∃xR
Cut
Φ(x/y)
·
·
→
·
[x]Γ, [x]φ ⊢ [x]∆
Set ⌊Γ⌋ = A, ⌊φ⌋ = B and ⌊∆⌋ = C, set ⌊[x]⌋ = a∗ (more precisely, a : Xs ×
X → Xs is the projection in B) and set f : A⊗a∗B → a∗ C be the denotation of
Φ. We demonstrate here the categorical interpretation of this reduction: reading
clockwise around the diagram gives the redex and anti-clockwise the reduct.
148
a∗ Σa A ⊗a∗ B
id
6
f
a∗
ε
(1)
∗
a
-
η
A⊗a B
⊗
-
(id
⊗
a∗ ((Σa A) ⊗ B))
ǫ)
β
a∗ ΣA ⊗ a∗ Σa∗ B - a∗ (ΣA ⊗ Σa∗ B) (4)
η ⊗ id (2)
∗
β
⊗
a∗ frob
6
η
a∗ µ
(3)
η
-
≤
∗
∗
a Σa (A ⊗ a B)
id
-
?
∗
a Σa (A ⊗ a∗ B)
a∗ f
(5)
?
∗
aC
η
-
id
?
∗
a Σa (a∗ C)
a∗ ε
(6)
-
?
∗
aC
(1) The naturality of β. (2) follows from a triangle law for the adjunction and
the functoriality of ⊗. (3) Follows from the symmetric monoidality of η (4) is a
fundamental property of the model, (5) lax naturality of η (6) Triangle Law.
Notice that the inequalities in this diagram arise in (4) and (5). If B is ⊤ and f
is strong, then this diagram commutes.
2. ∃grow It is sufficient to check the case where the cut formula on the right hand
side of the sequent containing the relevant quantifier application: that is, a cut of
the form
[x]Γ, φ, [x]ψ ⊢ [x]∆
Γ, ∃x.φ, ψ ⊢ ∆
∃xL
Γ′ ⊢ ∆′ , ψ
Γ, Γ′ , ∃x.ψ ⊢ ∆, ∆′
Cut
can always be expressed, using a (denotation preserving) reverse application of
Cut¬, as
149
[x]Γ, φ ⊢ [x]∆, [x]¬ψ
Γ′ , ¬ψ ⊢ ∆′
Γ, ∃x.φ ⊢ ∆, ψ
Γ, Γ′ , ∃x.ψ ⊢ ∆, ∆′
∃xL
Cut
∃x
M
N
ψ:L
∃xψ : L
[x]φ(x) : X
φ:R
φ:L
∃x
M
[x]N
4
ψ:L
[x]φ(x) : X
[x]φ : L
∃xψ : L
Given a denotation f : A ⊗ a∗ D → a∗ (F ⊕ B) for M and a denotation g :
B ⊗ C → E for N (so B is the denotation of the cut formula), the clockwise
150
direction of the following is the redex, and the anti-clockwise the reduct.
(ΣA ⊗ D) ⊗ C
?
Σ((A ⊗ a∗ D) ⊗ a∗ C)
Σ(A ⊗ a∗ D) ⊗ C
id ⊗ ε
(1)
6
µΣ(A ⊗ a∗ D) ⊗ Σa∗ C
(4)
µ
Σf ⊗ id
-
6
-
ε
Σγ ⊗ id (6)
Σ(γ ⊗ id)
(5)
µ
Σ((a∗ F ⊕ a∗ B) ⊗ a∗ C)
-
R
ΣδR
?
Σ(a∗ F ⊕ a∗ (B ⊗ C))
?
Σ(a∗ F ⊕ a∗ E)
R
δR
ν
-
Σa∗ F ⊕ Σ(a∗ B ⊗ a∗ C))
(9)
ν(9)
ν
id ⊕ Σβ
?
id ⊕ µ
(10)
Σa∗ F ⊕ Σa∗ (B ⊗ C)
id ⊕ Σa∗ g
-
(8)
?
(Σa∗ F ⊕ Σa∗ B) ⊗ Σa∗ C
(7)
Σ(a∗ F ⊕ (a∗ B ⊗ a∗ C))
R
δR
?
Σ(a∗ F ⊕ a∗ B) ⊗ Σa∗ C
ν ⊗ id
?
(F ⊕ B) ⊗ C
-
Σ(a∗ (F ⊕ B)) ⊗ Σa∗ C
?
?
Σ(id ⊕ a∗ g)
ε
ε⊗
id ⊗ ε (3)
(2)
Σf ⊗ id
-
Σ(a∗ (F ⊕ B) ⊗ a∗ C)
Σ(id ⊕ β)
ε ⊕ id-
Σ(a∗ (F ⊕ B)) ⊗ C
⊗
Σ(f ⊗ id)
-
ε)
frob
id
-
?
Σ(A ⊗ a∗ D) ⊗ C
(ε
⊕
frob ⊗ id
?
Σa∗ F ⊕ (Σa∗ B ⊗ Σa∗ C)
ε⊕ε
(11)
ε ⊕ (ε ⊗ε)
-
?
F ⊕ (B ⊗ C)
id ⊕ g
-
?
F ⊕E
ε⊕ε
?
Σa∗ F ⊕ Σa∗ E
(1) is an axiom. (2) is by functoriality of ⊗. (3) is clear. (4,5) are naturality of µ. (6) follows from ε being symmetric
monoidal. (7) is an axiom. (8) is naturality of δ. Both (9)’s are naturality of ν. (10) is symmetric monoidality of ε and
functoriality of ⊕. (11) follows from functoriality of ⊕ and lax naturality of ε.
In a µ theory, (11) is an inequality unless g is strong, and an equality otherwise.
Thus, if g is ε-strong (in particular, if it is the interpretation of a linear–pure
proof), this diagram commutes. In a general theory, (4) is also an inequality,
unless f is strong.
151
3. ∃merge We will give an interpretation of the general reduction
∃x : L
∃y : L
M
ψ:L
∃xψ : L
N
φ(x) : R
φ(x) : L
[y]γ : R
[x]∃yφ : R
∃yφ : R
∃yφ : L
γ:R
∃x : L
[x/y]N
M
≡
ψ:L
∃xψ : L
φ(x) : R
φ(x) : L
[x]γ : R
γ:R
and show that the cases where we want this to hold as an equality are such in the
doctrine.
Expressed in the sequent calculus, the reduction is
[x]Γ, ψ(x) ⊢ φ(x), [x]∆
[x]Γ, ψ(x) ⊢ φ(x), [x]∆ [x]Γ′ , φ(x) ⊢ [x]∆′
∃yR
′
[y]Γ , φ(y) ⊢ ∆
[x]Γ, ψ(x) ⊢ [x]∃y.φ, [x]∆
∃xL
Γ, ∃x.ψ ⊢ ∃y.φ, ∆
′
Cut
∃yL
→
[x]Γ, [x]Γ′ ψ(x) ⊢ [x]∆, [x]∆′
Γ′ , ∃y.φ ⊢ ∆′
∃xL
Cut
Γ, Γ′ , ∃x.ψ ⊢ ∆, ∆′
Γ, Γ′ , ∃x.ψ ⊢ ∆, ∆′
Given a denotation f : A ⊗ a∗ B → a∗ D ⊕ E for M and a denotation g :
E⊗a∗ C → F for N, the following diagram is the interpretation of the reduction,
152
(1)
Σ(id⊗β − 1)
6
Σ(A ⊗ (a∗ B ⊗ a∗ C))
?
α−
⊗1
Σ((A ⊗ a∗ B) ⊗ a∗ C)
Σ(f ⊗id)
Σ((a∗ D ⊕ E) ⊗ a∗ C)
Σδ
-
µ
µ
-
ν
-
6
∗
Σ(a D ⊕ (E ⊗ a C))
?
Σ(a∗ D ⊕ E) ⊗ Σa∗ C
(8)
∗
Σa D ⊕ Σ(E ⊗ a C)
-
ν⊗id
(Σa∗ D ⊕ ΣE) ⊗ Σa∗ C
(13)
∗
Σ(a D ⊕ a F )
Σγ
∗
∗
-
⊕i
(ε
d)
d
⊗i
δ
?
-
id⊕µ
?
-
ε⊕(id⊗ε)
∗
Σa D ⊕ (ΣE ⊗ Σa∗ C)
(15)
(14)
≥
-
?
D ⊕ (ΣE ⊗ C)
id⊕frob
id⊕Σg
-
?
D ⊕ Σ(E ⊗ a∗ C)
ε⊕id
ν
?
∗
-
Σa D ⊕ Σa F
?
(D ⊕ ΣE) ⊗ C
(10)
?
∗
-
(ε⊗ε)⊗id
(Σa∗ D ⊕ ΣE) ⊗ C
ε⊕id
∗
ε⊗id
(9)
(id⊕ε)⊗id
id⊗ε
δ
∗
id⊕Σg
Σ(id⊕g)
?
(Σa∗ D ⊕ Σa∗ ΣE) ⊗ C
id
(7)
(12)
?
∗
-
(id⊗Ση)⊗id
(Σa∗ D ⊕ ΣE) ⊗ C
Σf ⊗id
Σa∗ (D ⊕ ΣE) ⊗ C
(4)
ν⊗id
Σ(A ⊗ a∗ B) ⊗ Σa∗ C
(11)
?
(6)
id⊗ε
(3)
ν⊗id
?
Σ(A ⊗ a∗ B) ⊗ C
(5)
?
-
Σγ⊗id
Σ(a∗ D ⊕ a∗ ΣE) ⊗ C
id
?
fr
ob
?
Σ(A ⊗ a∗ (B ⊗ C))
⊗
Σf
id
-
Σ(id⊕η)⊗id
-
(2)
ΣA ⊗ (B ⊗ C)
frob
Σ(a∗ D ⊕ E) ⊗ C
ε
?
-
Σf ⊗id
-
Σ(A ⊗ a∗ B) ⊗ C
id
⊗
α⊗
-
frob⊗id
Σa D ⊕ Σa F
(16)
ε⊕ε
-
?
D ⊕ Σa∗ F
id⊕ε
-
?
D⊕F
with the clockwise direction denoting the redex and anti-clockwise the reduct.
(1) A trivial consequence of the definition of frob. (2) Trivial. (3) Naturality of ν. (4)
Comonoidality of η. (5) Definition of frob. (6) Functoriality of ⊗. (7) Functoriality
of ⊗. (8) Triangle law. (9) Functoriality of ⊕. (10) Naturality of δ. (11) Naturality
153
of µ. (12) From Lemma 5.3.3. (14) Definition of frob. (15) Functoriality of ⊕. (16)
Comonoidality of ε.
(ΣA ⊗ B) ⊗ C
6.2 Completeness
We have established, so far, that given a theory T on first-order classical proofs, the
inequations on the interpretations of those proofs in We establish completeness with
respect to a term model of proof nets, so our first task is to show that each net theory
forms a classical doctrine. We first demonstrate that certain permutation laws hold in
all classical doctrines:
Lemma 6.2.1. The permutation laws ∃ ∧ L
∃xL
∃xL
M
M
≡
φ:L
[x]ψ : L
∃xφ : L
ψ:L
[x]γ : L
φ:L
[x]ψ : L
[x]γ : L
[x]ψ ∨ γ : L
γ:L
ψ∧γ :L
∃xφ : L
ψ∧γ :L
and ∃ ∨ R
∃xL
∃xL
M
M
≡
φ:L
[x]ψ : R
∃xφ : L
ψ:R
[x]γ : R
φ:L
γ:R
[x]ψ : R
[x]γ : R
[x]ψ ∨ γ : R
ψ∨γ :R
∃xφ : L
ψ∨γ :R
are sound as equalities (i.e. their translations hold in any translation of proof nets in
a classical category)
154
Proof. The soundness of ∃ ∧ L holds by the following diagram:
ΣA ⊗ (B ⊗ (C ⊗ D))
frob
?
Σ(A ⊗ a∗ (B ⊗ (C ⊗ D)))
Σ(id ⊗ β)
id ⊗ α
- ΣA ⊗ ((B ⊗ C) ⊗ D)
frob
Σ(id ⊗
a∗ α)
?
- Σ(A ⊗ a∗ ((B ⊗ C) ⊗ D))
Σ(id ⊗ β)
?
Σ(A ⊗ (a∗ B ⊗ a∗ (C ⊗ D)))
?
Σ(A ⊗ (a∗ (B ⊗ C) ⊗ a∗ D))
Σ(id ⊗ (β ⊗ id))
?
Σ(id ⊗ α)Σ(A ⊗ ((a∗ B ⊗ a∗ C) ⊗ a∗ D))
Σ(A ⊗ (a∗ B ⊗ (a∗ C ⊗ a∗ D)))
Σ(id ⊗ (id ⊗ β))
?
The top square holds since α is an isomorphism, by Lemma 5.3.1 and the definition
of frob. The bottom hexagon is the image under Σ of the comonoidal compatibility
of a∗ with α.
The soundness of ∃ ∨ R holds by the following diagram:
ΣA ⊗ Γ
frob
?
Σ(A ⊗ a∗ Γ)
Σf
?
ν
ν ⊕ id
- (Σa∗ ∆ ⊕ Σa∗ A) ⊕ Σa∗ B
Σ((a∗ ∆ ⊕ a∗ A) ⊕ a∗ B) - Σ(a∗ ∆ ⊕ a∗ A) ⊕ Σa∗ B
(ε ⊕
ε) ⊕
ε
Σα
(1)
α
?
?
ν
id ⊕ν
Σ(a∗ ∆ ⊕ (a∗ A ⊕ a∗ B)) - Σa∗ ∆ ⊕ Σ(a∗ A ⊕ a∗ B)
Σa∗ ∆ ⊕ (Σa∗ A ⊕ Σa∗ B)
(5)
Σ(id ⊕ γ −1 )
(3) id ⊕ Σγ −1
?
?
νΣ(a∗ ∆ ⊕ a∗ (A ⊕ B))
Σa∗ ∆ ⊕ Σa∗ (A ⊕ B)
(4)
ε⊕ε
ε ⊕ (ε ⊕ ε)
α
? - ∆ ⊕ (A ⊕ B)
(1) is an axiom of symmetric monoidal functors. (2) follows from naturality of ν. (3)
follows from functoriality of ⊕ and symmetric monoidality of ε
155
-
(∆ ⊕ A) ⊕ B
Lemma 6.2.2. If the net N is linear–pure, the reduction
∃xL
∃xL
M
[x]N
γ : L [x]φ : R
[x]ψ : R
M
4
γ : L
N
[x]φ : R
φ : R
[x]φ ∧ ψ : R
ψ : R
φ∧ψ : R
∃xγ : L φ ∧ ψ : L
∃xγ : L
φ∧ψ : L
holds in both directions.
Proof. Rewriting these proofs, replacing applications of ∧R by cuts against constants,
we can see that the proof
∃xL
N
M
γ:L
K∧
[x]φ : R
φ:R
φ:L
ψ:L
ψ:R
∃xγ : L
φ∧ψ :R
is an instance of the redex of ∃grow. If N is linear–pure, then N cut against the
constant is also linear–pure, and so the reduction preserves the denotation.
Theorem 6.2.3 (Term Model). Every classical net theory forms a classical doctrine
C : Bop → CAT.
We verify the axioms of a classical doctrine in an indexed category constructed from
first-order proof nets over a signature Σ, with respect to some net theory T
6.3 The term model
We consider a term model, CT , in which the base category has objects given by the
sorts of free variables, and morphisms given by terms of the term calculus. The fibre
over each sort X has as objects the formulae of the theory with free variables of sort X,
and a morphism from A to B is a net with a single left door A and a single right door
B.
156
∃xL
M
Σ(M) :=
ψ(x) : R
φ:L
[x]∃yψ(y) : R
∃xφ : L
∃yψ(y) : R
∃xL
εφ :=
[x]φ : L
∃x[x]φ : L
[x]φ : R
φ(x) : R
ψ(x) : R
ψ(x) : L
ηψ :=
[x]∃yψ(y) : R
∃xL
φ(x) : L
[x]ψ : L
φ(x) : R
[x]ψ : R
frob :=
φ(x) ∧ [x]ψ : R
[x]∃yφ(y) ∧ [y]ψR
∃xφ : L
ψ:L
∃xφ : ∧ψL
∃y(φ(y) ∧ [y]ψ)R
Table 6.1: Net figures for the term model
157
The net figures required for classical existential structure are given in Table 6.2 (dually
for the universal quantifier). We first show that unit and co-unit are natural:
Lax naturality for η:
The diagram
∃yL
[x]M
φ(y) : L
ψ(y) : R
[y][x]∃zψ(z) : R
ψ(x) : L
ψ(x) : R
[x]∃yψ(y) : R
[x]∃yψ(y) : R
∃zψ(z) : R
cut reduces, via C UT ∃ (with a possible increase in denotation) to
M
ψ(x) : R
φ(x) : L
[x]∃zψ(z) : R
which, by definition is equal to
M
φ(x) : L
ψ(x) : R
ψ(x) : L
ψ(x) : R
[x]∃zψ(y) : R
Similarly, naturality for ε:
158
The diagram
∃xL
∃yL
[x]M
[y]ψ : L
[y]ψ : R
[x]ψ : R
[x]φ : L
[x]∃y[y]ψ(y) : R
∃x[x]φ : L
ψ(x) : R
∃y[y]ψ : L
∃y[y]ψ(y) : R
cut reduces via ∃M ERGE and C UTA X to
∃xL
[x]M
[x]ψ : R
[x]φ : L
∃x[x]φ : L
ψ:R
.
The net
∃xL
[x]φ : L
∃x[x]φ : L
M
[x]φ : R
φ:L
φ(x) : R
ψ:R
.
also reduces, via ∃G ROW and C UTA X to the same net, but with a possible change in
denotation.
We must now show that the triangle identities hold
159
∃yL
[x][y]φ : L
[x]ψ : L
[x][y]φ : R
[x]ψ : R
[x]∃y[y]ψR
[x]∃y[y]φ : L
[x]φ(x) : R
.
reduces to
[x]ψ : L
[x]ψ : R
[x]φ : L
[x]φ : R
.
via C UT ∃ , but notice that the net inside the quantifier is linear–pure, so this reduction
preserves the denotation. The reduct is equivalent to the identity on a∗ A by C UTA X .
∃xL
∃yL
ψ(x) : L
ψ(x) : R
[x]∃zψ(z) : R
[y]∃zψ(z) : L
[y]∃zψ(z) : R
[x]∃y[y]∃zψ(z) : R
∃xψ(x) : L
∃y[y]∃zψ(z) : R
∃y[y]∃zψ(z) : L
∃zψ(z) : R
.
reduces to
∃xL
ψ(x) : L
ψ(x) : R
[x]∃zψ(z) : R
[y]∃zψ(z) : L
[y]∃zψ(z) : R
∃zψ(z) : R
∃xψ(x) : L
160
.
by ∃M ERGE . This net is, by C UTA X , equivalent to
∃xL
ψ(x) : L
ψ(x) : R
[x]∃zψ(z) : R
[x]∃zψ(z) : R
∃xψ(x) : L
.
which, by ∃A X is the identity on ∃ψ
The comonoidal strength νΣ is given by:
∃x
φ:L
ψ:L
φ:R
[x]∃yφ : R
ψ:R
[x]∃xψ : R
φ∨ψ :L
[x]∃yφ : R
∃xφ ∨ ψ : L
[x]∃xψ : R
.
φ∨ψ :R
We obtain naturality as follows: the net
∃x
N
∃z
P
∃x
φ:L
ψ:L
φ:R
ψ:R
[x]∃yφ : R
[x]∃xψ : R
φ(x) : L
φ∨ψ :L
∃yφ : L
∃yφ : R
∃xφ ∨ ψ : L
ψ(z) : L
∃zψ : L
φ′ : R
[x]∃wφ′ : R
∃wφ′ : R
∃xψ : R
∃yφ ∨ ∃zψ : L
∃yφ ∨ ∃yψ : R
161
ψ′ : R
[z]∃vψ ′ : R
∃vψ ′ : R
∃wφ′ ∨ ∃vψ ′ R
reduces, via C UT ∧and two applications of C UT ∃ , to
∃x
N
P
φ(x) : L
ψ(x) : L
φ′ : R
[x]∃wφ′ : R
ψ′ : R
[x]∃wψ ′ : R
φ∨ψ : L
∃wφ′ : R
∃wψ ′ : R
∃xφ ∨ ψ : L
(6.1)
∃wφ′ ∨ ∃wψ ′ : R
The net
∃y
N
P
∃x
φ(y) : L
′
ψ(y) : L
φ :R
φ′ : L
ψ′ : R
ψ′ : L
φ′ : R
′
[x]∃yφ : R
φ′ ∨ ψ ′ : R
′
φ∨ψ :L
ψ′ : R
[x]∃yψ ′ : R
′
φ ∨ψ :L
[y]∃xφ′ ∨ ψ ′ : R
∃yφ ∨ ψ : L
[x]∃yφ′ : R
∃xφ′ ∨ ψ ′ : L
∃xφ′ ∨ ψ ′ : R
[x]∃yψ ′ : R
∃yφ′ ∨ ∃yψ ′ : R
also reduces, via C UT ∃ , to 6.1.
Finally, we check the Frobenius rules: The net (id ⊗ ε) ◦ µ is given by
∃xL
∃yL
φ:L
[x]ψ : L
φ:R
[x]∃yφ : R
[x]ψ : R
[y]φ : L
[y]φ : R
[x]∃y[y]ψ : R
φ∧ψ :L
∃x[y]φ : L
φ(y) : R
[x]∃yφ ∧ ∃[y]yψ : R
∃xφ ∧ [x]ψ : L
∃yφ : L
∃yφ ∧ ∃y[y]ψ : R
∃yφ ∧ ∃y[y]ψ : L
∃yφ : R
∃yφ ∧ ψ : L
.
162
Since the net denoting (id ⊗ ε) is linear–pure, we can apply ∃G ROW , to obtain
∃xL
∃yL
[x]ψ : L
φ:L
[x]ψ : R
φ:R
[x]∃yφ : R
[x, y]φ : L
[x, y]φ : R
[x]∃y[y]ψ : R
φ∧ψ :L
[x]∃y[y]φ : L
[x]φ(y) : R
[x]∃yφ ∧ ∃y[y]ψ : R
[x]∃yφ : L
[x]∃yφ : R
[x]∃yφ ∧ ∃y[y]ψ : L
∃xφ ∧ [x]ψ : L
[x]∃yφ ∧ ψ : L
.
∃yφ ∧ ψ : L
and then C UT ∧, and C UTA X , to obtain
∃xL
∃yL
φ:L
[x]ψ : L
φ:R
[x]∃yφ : R
[x]ψ : R
[x, y]φ : L
[x, y]φ : R
[x]∃y[y]ψ : R
φ∧ψ :L
[x]∃x[y]φ : L
[x]φ(y) : R
[x]∃yφ ∧ ψ : L
∃yφ ∧ ψ : L
.
∃xφ ∧ [x]ψ : L
The remaining cut can be eliminated by noticing that it is an instance of one of the
triangle identities we established above, under an application of [x]; it is therefore
equal to the identity, and applying C UTA X again we obtain
163
∃xL
[x]ψ : L
φ:L
[x]ψ : R
φ:R
[x]∃yφ : R
φ∧ψ :L
[x]∃yφ ∧ ψ : L
.
∃yφ ∧ ψ : L
∃xφ ∧ [x]ψ : L
Using this net for (id ⊗ ε) ◦ µ, the composition (id ⊗ ε) ◦ µ ◦ frob is given by
∃xL
φ(x) : L
[x]ψ : L
φ(x) : R
[x]ψ : R
φ:L
[y]ψ : L
φ(x) ∧ [x]ψ : R
φ:R
[y]ψ : R
[y]∃xφ : R
[x]∃yφ(y) ∧ [y]ψ : R
φ ∧ [y]ψ : L
[y](∃xφ ∧ ψ) : R
∃xφ : L
ψ:L
∃y(φ ∧ [y]ψ) : L
∃xφ ∧ ψ : L
∃xφ ∧ ψ : R
∃y(φ(y) ∧ [y]ψ) : R
.
We show how to reduce this net, preserving denotation, to the identity.
We apply ∃merge to the remaining cut, to obtain
164
∃xL
φ:L
[x]ψ : L
φ:R
[x]ψ : R
φ:R
[x]ψ : L
φ:L
φ ∧ [x]ψ : R
[x]ψ : R
[x]∃zφ : R
φ ∧ [x]ψ : L
[x](∃zφ ∧ ψ) : R
∃xφ : L
∃zφ ∧ ψ : R
ψ:L
.
∃xφ ∧ ψL
With an application of Cut∧ and two of CutAx, this cut is eliminated to give
∃xL
φ:L
[x]ψ : L
φ:R
[x]ψ : R
[x]∃zφ : R
[x](∃zφ ∧ ψ) : R
∃xφ : L
ψ:L
∃zφ ∧ ψ : R
.
∃xφ ∧ ψL
By 6.2.2, the application of ∧R can be pulled outside of the quantifier box:
∃xL
φ:L
φ:R
[x]∃zφ : R
∃zφ : R
ψ:L
ψ:R
∃xφ : L
∃xφ ∧ ψL
∃xφ ∧ ψR
165
.
Now, via Ax∃ and Ax∧, we see that this net is the identity.
The composition in the opposite order (which we expect to be “smaller” than the identity, with respect to the ordering on proofs) is
∃xL
φ(x) : L
φ:L
[y]ψ : L
φ:R
[x]ψ : L
φ(x) : R
[y]ψ : R
[x]ψ : R
φ(x) ∧ [x]ψ : R
[y]∃xφ : R
[x]∃yφ(y) ∧ [y]ψR
φ ∧ [y]ψ : L
∃xφ : L
[y](∃xφ ∧ ψ) : R
∃y(φ ∧ [y]ψ) : L
∃xφ ∧ ψ : R
ψ:L
∃xφ : ∧psiL
∃y(φ(y) ∧ [y]ψ)R
.
Applying ∃merge to the ∃y box, and then applying Cut∧, yields:
∃yL
∃xL
[y]φ(x) : L
φ:L
[y]ψ : L
φ:R
[y][x]ψ : L
[y]ψ : R
[y]φ(x) : R
[y][x]ψ : R
[y](φ(x) ∧ [x]ψ) : R
[y]∃xφ : R
[x][y]∃z(φ(z) ∧ [z]ψ)R
φ ∧ [y]ψ : L
[y]∃xφ : L
[y]ψ : L
[y]∃z(φ(z) ∧ [z]ψ)R
∃y(φ ∧ [y]ψ) : L
[y]∃z(φ(z) ∧ [z]ψ)R
We consider the empire of φ (before the application of ∧L); this is, after an application
of CutAx:
166
.
∃xL
[y]φ(x) : L
[y][x]ψ : L
[y]φ(x) : R
[y][x]ψ : R
[y](φ(x) ∧ [x]ψ) : R
φ:R
φ:L
[x][y]∃z(φ(z) ∧ [z]ψ)R
[y]∃xφ : R
[y]∃xφ : L
[y]ψ : L
[y]∃z(φ(z) ∧ [z]ψ)R
[y]∃z(φ(z) ∧ [z]ψ)R
.
This is of the form of the redex of Cut∃. Eliminating this cut causes an essential
change in denotation in the case where ψ is non-trivial.
Applying ∧L, we obtain
∃yL
φ(y) : L
[y]ψ : L
φ(y) : R
[y]ψ : R
(φ(y) ∧ [y]ψ) : R
φ ∧ [y]ψ : L
[y]∃z(φ(z) ∧ [z]ψ)R
[y]∃z(φ(z) ∧ [z]ψ)R
∃y(φ ∧ [y]ψ) : L
∃z(φ(z) ∧ [z]ψ)R
.
which, by Ax∃ and Ax∧, is the identity.
The Beck-Chevalley condition is implicit in our variable handling. Clearly, the functor corresponding to each quantifier preserves the ordering on proofs given by cutreduction, since we may reduce a cut under a quantifier.
So we have a classical doctrine structure for each net theory.
Completeness relies on the following technical lemma, which is the equivalent of
lemma 5.5 in [23]:
167
Lemma 6.3.1. Let M be a net in a first order net theory T , with left doors φ1 ...φn and
right doors ψ1 ...ψn . The interpretation CT ⌊M⌋ of M in CT is the equivalence class
w.r.t. ≡ of
M
...
φ1 : L
φn : L
φ1 ∧ · · · ∧ φn : L
...
ψ1 : R ψm : R
ψ1 ∧ · · · ∧ ψm : R
where n, m ≥ 0, of
M
...
ψ1
ψm
ψ1 ∧ · · · ∧ ψm
⊤:L
ψ1 ∧ · · · ∧ ψm
where n = 0, and dually for m = 0.
168
Proof: The proof is by induction over the rules of the calculus, where the only cases which cause any difficulty (and are not already dealt
with in the propositional case) are the quantifier boxes; we consider ∃L.
∃x
M
φ1 : L
∃xφ : L
[x]φ2 : R
ψ2 : R
[x]φn : R
ψm : R
[x]ψ1 : R
ψ1 : R
[x]ψm : R
ψm : R
169
By our induction hypothesis, the denotation of the net M is equivalent to a net with only one left and one right door (the net obtained by
forming the conjunction of the right doors and the disjunction of the left doors.) To this net we apply our categorical interpretation of left
existential quantification:
∃zL
∃xL
M
φ1 : L [z](φ2 ∧ · · · ∧ φn )L
φ1 : R
[z](φ2 ∧ · · · ∧ φn )R
φ1 : L
(φ1 ∧ [z](φ2 ∧ · · · ∧ φn )) : R
[z]∃x(φ1 ∧ [x](φ2 ∧ · · · ∧ φn )) : R
[x]φ2 : R [x]φn : R
[x]φ2 ∧ · · · ∧ φn : R
[x]ψ1 : R
[x]ψm : R
[x]ψ1 ∨ · · · ∨ ψn : R
φ1 ∧ [x](φ2 ∧ · · · ∧ φn ) : R
[x]∃y[y](ψ1 ∨ · · · ∨ ψn ) : R
∃zφ1 : L
∃x(φ1 ∧ [x](φ2 ∧ · · · ∧ φn )) : R
(φ2 ∧ · · · ∧ φn )L
∃φ1 ∧ (φ2 ∧ · · · ∧ φn )L
∃yL
∃x(φ1 ∧ [x](φ2 ∧ · · · ∧ φn )) : R
∃y[y](ψ1 ∨ · · · ∨ ψn ) : R
[y](ψ1 ∨ · · · ∨ ψn ) : L
∃y[y](ψ1 ∨ · · · ∨ ψn ) : L
[y](ψ1 ∨ · · · ∨ ψn ) : R
(ψ1 ∨ · · · ∨ ψn ) : R
170
Two applications of ∃merge and one of Cut∧ results in
∃xL
M
φ1 : L
[x]φ2 : R [x]φn : R
[x]φ2 ∧ · · · ∧ φn : R
∃xφ1 : L
[x]ψ1 : R
[x]ψm : R
[x]ψ1 ∨ · · · ∨ ψn : R
φ2 ∧ · · · ∧ φ n : R
∃xφ1 ∧ φ2 ∧ · · · ∧ φn : R
ψ1 ∨ · · · ∨ ψn : R
which is equal, via ∃∧ and ∃∨ to
∃xL
M
φ1 : L
∃xφ : L
[x]ψ2 : R[x]ψn : R
[x]ψ1 : R
ψ1 : R
ψ2 : R ψm : R
ψ2 ∧ · · · ∧ ψ : L
[x]ψm : R
ψm : R
ψ1 ∨ · · · ∨ ψm : R
∃xφ1 ∧ (φ2 ∧ · · · ∧ φn ) : L
.
The only remaining fact to establish is that the Beck condition holds; since we are (for
the moment) working in a setting without equality on terms, the only pullbacks we
need to deal with are those of the form
Xs × X × Y
πY-
Xs × X
′
πX
πX
?
Xs × Y
πY - ?
Xs
171
which corresponds to asking that ∃x∃yφ is isomorphic to ∃y∃xφ where x and y are
independent of each other. This equality is clear.
Proposition 6.3.2. In every first order net theory T , for any two nets M and N with
matching sequences of doors, it holds that
M ≡N
T
in
CT ⌊M⌋ = CT ⌊N⌋
iff
Proof. By the previous theorem, CT ⌊M⌋ = CT ⌊N⌋ iff M ′ ≡ N ′ , where M ′ is the net
arising from the previous lemma, and similarly N ′ . It can be easily checked that this
holds iff M ≡ N in T .
Completeness follows immediately as for propositional nets.
Theorem 6.3.3 (Completeness). Let T be a first order net theory, and let M and N be
nets of T with matching sequences of doors. If M ≤ N holds in every interpretation
of proofs in a classical doctrine C, then M 4 N is in T .
Finally, we show that the term model is initial among interpretations.
Theorem 6.3.4. Given a theory T on first-order classical proofs, and an interpretation C ⌊ ⌋ of that theory in a classical doctrine, that model factors through CT ⌊ ⌋, the
standard interpretation in the term model; that is, the interpretation of terms in the
base category factors as
FB B
-
BT
6
B
⌊−
⌋
BT ⌊−⌋
X
such that FB is a cartesian functor, and for each X ∈ X , the interpretation of propositions and proofs factors as
CT (BT (X))
FX
-
C(B(X))
-
6
CT ⌊−⌋
C
TX
172
⌋
⌊−
such that the functor FX preserves all the classical category structure of CT (BT (X)),
and such that F is natural.
Proof. For the base category, recall from basic model theory (see ??) the Initial Model
Theorem for horn theories; the term model is initial among models of such theories.
This is, in particular, true of the empty theory, and so the term model is initial as base
category.
The interpretation CT is bijective on objects, (so FX is uniquely determined on objects
for each X), and is surjective on morphisms by lemma 6.3.1 (so FX is uniquely determined on morphisms). F is well defined, since C is a model. A simple calculation in
C yields that FX preserves the correct structure.
Finally, we need that F is natural; that is
CT (X)
FX -
C(X)
B(a)∗
a∗
?
CT (Y )
FY -
?
C(Y ).
This follows trivially from the definition of the interpretation of a∗ .
6.4 Permutations with structural rules and sharp doctrines
The soundness results given thus far assure that any cut reduction given in the proof of
cut-elimination holds as an inequality when interpreted in a classical doctrine. Additionally, we have assumed some reductions to be equalities to obtain a good categorical
setting (that of monoidal functors), and have derived that some permutations of linear
rules and quantifiers hold as equalities.
173
It is now reasonable to ask how this structure interacts with weakening and contraction.
If we assume that every fibre has strong a∗ -(co)monoids, the results of Section 5.4 give
us a number of additional permutations.
Lemma 6.4.1. In a classical doctrine with strong a∗ -(co)monoids, permuting quantification with the action of structural rules on side-formulae is sound as an equality.
Proof. The permutation of existential quantification with contraction
∃xL
∃xL
M
M
≡
φ :L
[x]ψ : Y[x]ψ : Y
[x]ψ : Y
φ : L
[x]ψ : Y
[x]ψ : Y
∃xφ : L
ψ :Y ψ :Y
ψ :Y
∃xφ : L
ψ : Y
has two sub-cases. In the case where the contracted formulae are on the left of the
turnstile, the following diagram establishes soundness
ΣA ⊗ Γ
frob
?
Σ(A ⊗ a∗ Γ)
Σf
?
ν
id ⊕ν
Σ(a∗ ∆ ⊕ (a∗ B ⊕ a∗ B)) - Σa∗ ∆ ⊕ Σ(a∗ B ⊕ a∗ B))
Σa∗ ∆ ⊕ (Σa∗ B ⊕ Σa∗ B))
Σ(id ⊕ ∇)
(1)
?
Σ(a∗ A ⊕ a∗ B)
ν
id ⊕ Σ∇ ( 2)
? - Σa∗ ∆ ⊕ Σa∗ B
id ⊕
∇
ε ⊕ (ε ⊕ ε)
?
∆ ⊕ (B ⊕ B)
(3)
ε⊕
ε
id ⊕ ∇
-
?
∆⊕B
(1) is the naturality of ν. (2) Lemma 5.4.3 (3) ∆-strength of ε.
In the case where the contracted formulae are on the right of the turnstile, we have the
174
following.
id ⊗ ∆
(ΣA ⊗ Γ) ⊗ B
frob ⊗ id
frob ⊗ id
(1)
?
id ⊗ ∆
Σ(A ⊗ a∗ Γ) ⊗ B
?
- Σ(A ⊗ a∗ Γ) ⊗ (B ⊗ B)
(2)
frob
?
∗
- (ΣA ⊗ Γ) ⊗ (B ⊗ B)
∗
Σ((A ⊗ a Γ) ⊗ a B)
frob
Σ(id ⊗ a∗ ∆)Σ(id
⊗∆
?
∗
Σ((A ⊗ a Γ) ⊗ a∗ (B ⊗ B))
)Σ(id ⊗ a∗ β −1 )
-
?
Σ((A ⊗ a∗ Γ) ⊗ (a∗ B ⊗ a∗ B))
Σf
?
Σ∆
(1) follows from functoriality of ⊗. (2) ε-strength of ∆ (3) Follows from having a∗
comonoids.
However, two families of reductions do not even hold in our current theory on proofs:
they involve permuting or removing an instance of a structural rule that acts on the
principal formula of a quantifier box. For these reductions we need the stronger notion
of sharpened doctrine.
∃xL
M
[x]M
≡
[a]φ : X
ψ :L
W∃ :
∃xψ : L
[x]φ : X
φ: X
175
φ :X
∃xψ : L
φ :X
∃x
∃y
∃x
N
M
M
(x/y)N
4
′
φ(x) : L [x]ψ : Y [y]ψ : Y φ(y)
:L
ψ : Y
∃xφ : L
′
φ(x) : L[x]ψ : Y[x]ψ : Y φ(x)
: L
ψ : Y′
∃xφ : L
φ(x) : L
∃xφ : L
C∃ :
∃xφ : L
Lemma 6.4.2. The interpretation in a sharp doctrine of the permutation W∃ (C∃ )
holds as an equality if the net M (nets M and N) has only one left door. Dually for
∀W and ∀C
Proof.
frob
-
Σ(B ⊗ a∗ Γ)
Σ(hi ⊗ id)
-
Σ(1 ⊗ a∗ Γ)
Σ(λ⊗)
Σa∗ Γ
-
ΣB ⊗ Γ
µ
µ
(2)
(3)
?
Σhi ⊗ idΣ1 ⊗ Σa∗ Γ
(4)
hi ⊗
id β ⊗ id
id ⊗ ε
1
id
λ⊗
?
ΣB ⊗ Σa∗ Γ
-
-
?
ΣB ⊗ Γ
?
?
1 ⊗ Σa∗ Γ
Σa∗ ∆
id
hi ⊗ id
?
1⊗Γ
λ⊗
(6)
(7)
ε
⊗
ε
(5)
Σa∗ f
ε
(1)
f
?
-∆
-Γ
(1) is from the definition of frob. (2) is naturality of µ. (3) follows from monoidality
of Σ. (4) Lemma 5.4.2. (5) follows from functoriality of ⊗. (6) is the naturality of λ.
(7) Lax naturality of ε
176
∃C
frob
Σ(∆
)
b
fro
? ∗
-
∗
∗
Σ((A ⊗ A) ⊗ (a B ⊗ a C))
-
id
(5)
?
∗
∗
µ⊗µ
Σ(A ⊗ A) ⊗ Σ(a B ⊗ a C)
-
µ
∗
∗
Σ((A ⊗ a B) ⊗ (A ⊗ a C))
Σ(f ⊗ id)
∗
∗
∗
∗
∗
∗
∗
∗
∗
Σf ⊗ id
∗
∗
∗
ν
∗
Σ(a D ⊕ a F )
ν
∗
∗
-
∗
∗
id ⊗ frob
(9)
id ⊗ frob
-
ν ⊗ id
∗
Σf ⊗ id
∗
-
id ⊕ µ
∗
(Σa D ⊕ Σa E) ⊗ Σ(A ⊗ a C)
∗
∗
Σa D ⊕ (Σa E ⊗ Σ(A ⊗ a C))
ε ⊕ id
?
(ε ⊕ ε) ⊗ id
-
ε ⊕ (ε ⊗ id)
∗
δ
(17)
∗
(18)
∗
(id ⊕ frob)
∗
Σa (D ⊕ F )
(19)
∗
id ⊕ Σg
-
?
-
∗
?
D⊕F
(1) Follows from Lemma 5.4.2. (2) Follows from bifunctoriality of ⊗ .(3) Is part of the
definition of frob. (4)Naturality of µ. (5) See below.(6) Naturality of t. (7) Follows
from coherence for monoidal categories. (8) Clear. (9) Follows from the definition
of frob and functoriality. (10) Naturality of µ. (11)Follows from bifunctoriality of
⊗. (12) Naturality of µ. (13) Follows from bifunctoriality of ⊗ and comonoidality of
ε. (14) is the compatibility of Σ with δ. (15) Naturality of δ. (16) Naturality of ν.
(17) Follows from the definition of frob and bifunctoriality of ⊕. (18)Follows from
bifunctoriality of ⊕. (19)Follows from comonoidality of ε.
Lemma 6.4.3. A special case of this reduction is where the cut introduces a conjunc-
177
?
D ⊕ Σa F
id ⊕ ε
ε
?
∗
D ⊕ Σ(a E ⊗ (A ⊗ a C))
ε ⊕ id
Σγ
?
D ⊕ (E ⊗ Σ(A ⊗ a C))
-
Σa D ⊕ Σa F
?
(D ⊕ E) ⊗ Σ(A ⊗ a C)
(15)
?
∗
?
∗
Σ(a (D ⊕ E)) ⊗ (ΣA ⊗ C)
ε ⊗ frob
∗
?
∗
Σ(A ⊗ a B) ⊗ (ΣA ⊗ C)
id ⊗ frob
δ
Σa D ⊕ Σ(a E ⊗ (A ⊗ a C))
id ⊕ Σg
?
(ΣA ⊗ B) ⊗ (ΣA ⊗ C)
(13)
?
Σ(a D ⊕ a E) ⊗ Σ(A ⊗ a C)
∗
-
(id ⊗ ε) ⊗ (id ⊗ ε)
(11)
?
∗
Σγ ⊗ id
?
∗
Σ(a (D ⊕ E)) ⊗ Σ(A ⊗ a C)
(16)
?
∗
µ
∗
∗
(ΣA ⊗ Σa B) ⊗ (ΣA ⊗ Σa C)
(14)
?
Σ(a D ⊕ (a E ⊗ (A ⊗ a C)))
Σ(id ⊕ g)
-
≥
?
∗
(12)
?
Σ((a D ⊕ a E) ⊗ (A ⊗ a C))
Σδ
µ
id
Σ(A ⊗ a B) ⊗ Σ(A ⊗ a C)
(10)
?
Σ(a (D ⊕ E) ⊗ (A ⊗ a C))
Σ(γ ⊗ id)
(8)
-
ε)
(ΣA ⊗ ΣA) ⊗ (Σa B ⊗ Σa C)
t
∗
µ
⊗
(6)
-
∗
µ⊗µ
Σ(A ⊗ a B) ⊗ Σ(A ⊗ a C)
?
(ε
t
(7)
Σt
⊗
Σ(A ⊗ A) ⊗ Σa (B ⊗ C)
id ⊗ Σβ
µ
-
Σ(A ⊗ A) ⊗ (B ⊗ C)
ε
∗
(4)
?
id ⊗
(3)
µ
-
id
Σ(A ⊗ A) ⊗ (B ⊗ C)
--
(ΣA ⊗ ΣA) ⊗ (B ⊗ C)
d
µ⊗i
(1)
-
(2)
Σ((A ⊗ A) ⊗ a (B ⊗ C))
Σ(id ⊕ β)
⊗ id
?
∗
Σ(A ⊗ a (B ⊗ C))
Σ(∆ ⊗ id)
-
∆ ⊗ id
ΣA ⊗ (B ⊗ C)
tion:
∃x
∃y
∃x
N
M
M
(x/y)N
≡
φ(x) : L
∃xφ : L
[x]ψ : Y
ψ:R
[y]θ : R′
θ : R′
φ(x) : L
[y]φ(y) : L
∃xφ : L
[x]ψ : Y
φ(x) : L
ψ∧θ : R
∃xφ : L
[x]θ : R′ φ(x) : L
[x]ψ ∧ θ : R
ψ∧θ :R
∃xφ : L
In this case, if the nets M and N have only one left door each, the reduction is an
equality.
Proof.
∆
ΣA
Σ∆
?
µ
(1)
Σ(A ⊗ A)
Σ(f ⊗ g)
-
ΣA ⊗ ΣA
-
Σf ⊗ Σg
?
Σ(a∗ (B ⊕ C) ⊗ a∗ (C ⊕ D))
(2)
µ-
?
Σ(a∗ (B ⊕ C) ⊗ a∗ (C ⊕ D))
ε⊗ε
(3)
?
ε
∗
(B ⊕ C) ⊗ (C ⊕ D))
Σa ((B ⊕ C) ⊗ (C ⊕ D))
Σβ
?
δ̂
(4)
?
ε (B ⊗ D) ⊕ (C ⊕ E)
Σa∗ ((B ⊗ D) ⊕ (C ⊕ E))
δ̂
?
(1) Follows from Lemma 5.4.2. (2) follows from naturality of mu. (3) follows from
comonoidality of ε. (4) follows from ε-strongness of δ̂.
Definition 6.4.4. A proof net is pure if it contains only logical axioms and inference
rules (i.e. no non-logical axioms), and no instances of ¬L or ¬R.
Definition 6.4.5. A sharp theory on classical proofs is a µ-theory which the cases in
which ∃grow and ∃cut hold as equalities are extended from linear–pure proofs to pure
proofs.
Theorem 6.4.6 (Soundness). The theory induced by interpreting proofs in a sharp
doctrine is a sharp theory.
178
Proof. By examining the proof of soundness for µ-theories; the denotation of a pure
proof is built from monoidal structure morphisms, δ and (co)monoid structure morphisms, and so is εΣ -strong (and ηΠ -strong).
Theorem 6.4.7. The classical doctrine formed from a sharp theory is a sharp doctrine.
Proof. We must establish two facts: that the doctrine formed has π ∗ -(co)monoids and
that the relevant morphisms are strong. The first is immediate: for example, in
a∗ A
a∗ ∆
-
a∗ (A × A)
∆
γ
-
?
∗
a A ⊗ a∗ A
we have that a∗ A ⊙ a∗ A = a∗ (A ⊙ A) and that γ is the identity.
Strongness follows from the extension of equality-preserving applications of ∃cut to
pure proofs.
The following is now immediate:
Theorem 6.4.8. Sharp doctrines are a sound and complete class of models for sharp
theories.
Corollary 6.4.9. The permutations ∃C and ∃W are admissible in a sharp theory.
179
An alternative notion of sharp theory
Consider the following proof in the presence of a sharp theory:
∃x
∃x
φ(x) : L
ψ(y) : L
φ(x) : L ψ(x) : L
φ(y) : Lψ(y) : L
φ ∧ ψ(x) : L
∃x(φ ∧ ψ) : L
φ ∧ ψ(y) : L
φ(x) : R
ψ(y) : R
[y]∃zφ : R
[x]∃zφ : R
∃x(φ ∧ ψ) : L
∃zφ : R
∃x(φ ∧ ψ) : L
∃zφ : R
∃zφ ∧ ∃zψ : R
By ∃C, this is equivalent to
∃x
φ(x) : L
ψ(x) : L
φ(x) : L ψ(x) : L
φ(x) : Lψ(x) : L
φ ∧ ψ(x) : L
φ ∧ ψ(x) : L
φ ∧ ψ(x) : L
φ(x) : R
ψ(x) : R
[x]∃zφ : R
[x]∃zφ : R
[x]∃zφ∃zψ : R
∃x(φ ∧ ψ) : L
∃zφ∃zψ : R
By rewiring the weakenings, we obtain
∃x
φ(x) : L
ψ(x) : L
φ(x) : L φ(x) : L ψ(x) : L ψ(x) : L
φ ∧ ψ(x) : L
φ ∧ ψ(x) : L
φ ∧ ψ(x) : L
φ(x) : R
ψ(x) : R
[x]∃zφ : R
[x]∃zφ : R
[x]∃zφ∃zψ : R
∃x(φ ∧ ψ) : L
∃zφ∃zψ : R
180
,
and by the propositional rule C∧, we permute the contractions and the conjunction:
∃x
φ(x) : L
ψ(x) : L
φ(x) : L φ(x) : L ψ(x) : L ψ(x) : L
φ(x) : L
ψ(x) : L
φ ∧ ψ(x) : L
φ(x) : R
ψ(x) : R
[x]∃zφ : R
[x]∃zφ : R
[x]∃zφ∃zψ : R
∃x(φ ∧ ψ) : L
∃zφ∃zψ : R
Finally, by rule W C we can remove both contractions and both weakenings, and derive
the usual comonoidal strength:
∃x
φ(x) : L
ψ(x) : L
φ(x) : R
ψ(x) : R
[x]∃zφ : R
[x]∃zφ : R
φ ∧ ψ(x) : L
[x]∃zφ∃zψ : R
∃x(φ ∧ ψ) : L
∃zφ∃zψ : R
This equality is enough to demonstrate the naturality of this strength morphism. The
net
∃u
∃z
φ(x) : L
ψ(y) : L
φ(x) : Lψ(x) : L φ(y) : L
ψ (y) : L
φ ∧ ψ(x) : L
φ ∧ ψ(y) : L
φ(x) : R
[x]∃uφ : R
ψ(y) : R
[y]∃zφ : R
φ(u) : L
∃uφ : L
∃x(φ ∧ ψ) : L ∃x(φ ∧ ψ) : L
∃x(φ ∧ ψ) : L
∃uφ : R
ψ(z) : L
∃zψ : L
∃wφ′ : R
∃zφ : R
∃uφ ∧ ∃zψ : R
∃uφ ∧ ∃zψ : L
181
φ′ : R
[u]∃wφ′ : R
ψ′ : R
[z]∃vψ ′ : R
∃vψ ′ : R
∃wφ′ ∧ ∃vψ ′ R
cut-reduces, via cut∧ and two applications of ∃cut, to
∃x
∃y
φ(x) : L
φ′ : R
ψ(y) : L
[x]∃wφ′ : R
φ(x) : L ψ(x) : L φ(y) : L ψ(y) : L
ψ′ : R
[y]∃vψ ′ : R
∃wφ′ : R
φ ∧ ψ(x) : L
∃vψ ′ : R
φ ∧ ψ(y) : L
∃wφ′ ∧ ∃vψ ′ R
∃x(φ ∧ ψ) : L
∃x(φ ∧ ψ) : L
∃x(φ ∧ ψ) : L
This net is equivalent to
∃x
≡
φ(x) : L
φ′ : R
ψ(x) : L
[x]∃wφ′ : R
ψ′ : R
[x]∃wψ ′ : R
φ∧ψ :L
[x]∃wφ′ ∧ ∃wψ ′ : R
,
∃wφ′ ∧ ∃wψ ′ : R
∃xφ ∧ ψ : L
which is the result of applying ∃merge to
∃y
∃xL
φ′ : L
ψ′ : L
φ′ : R
ψ′ : R
[x]∃yφ′ : R
[x]∃xψ ′ : R
φ∧ψ :L
φ(y) : L
ψ(y) : L
φ′ : R
[x]∃uφ′ ∧ ∃vψ ′ : R
ψ′ : R
phi′′ ∧ ψ ′ : R
φ∧ψ : L
∃xφ′′ ∧ ψ ′ : L
[y]∃xφ′′ ∧ ψ ′ : R
∃yφ ∧ ψ : L
∃xφ′′ ∧ ψ ′ : R
∃uφ′ ∧ ∃vψ ′ : R
.
We have shown:
182
Theorem 6.4.10. A first-order theory T is a sharp theory if:
1. the cases in which ∃grow, ∀grow, ∃cut and ∀cut hold as equalities are extended
from linear–pure proofs to pure proofs; and
2. T includes the laws ∃C and ∀C.
6.5 Permutations between quantifiers
Our final and strongest notion of theory on proofs concerns the interaction between the
two quantifiers, and as such gives rise to proof theoretic duality between them.
The cases we are interested in are those in which every door of both nets involved in
a cut are quantified by one or other of ∃x of ∀x. For example, general applications of
∃grow and ∀grow hold as inequalities :
∃x : L
∃y : L
N
M
ψ :L
∃xψ : L
φ(x) : R
φ(x) : L
[x]∃yφ : R
∃yφ : L
∃yφ : R
[y]γ : R
[x]θ : L
γ :R
θ:L
∃x : L
M
4
ψ:L
∃xψ : L
183
[x/y]N
[x]γ : R
φ(x) : R φ(x) : L
[x]θ : L
θ:L
γ :R
(In particular, cuts involving linear strength such as
∃x
∀z
$\forall x$
$\phi : L$
$\phi : R$
$\ps : Li$
ψ(z) : L
$\ps : Ri$
φ(x) : L
[x]∃wφ′ : R
$[x] \exists x \phi : R$
$ \phi \lor \psi : L $
φ′ : R
[z]∀vψ : R
∃wφ′ : R
$[x] \forall x \phi \lor \psi : L$
$ \forall x \phi \lor \psi : L$
$\exists x \phi : R$
ψ′ : R
∀zψ : L
∃yφ : L
∀zψ ′ : R
$\forall x \psi : Ri$
do not hold as equalities, so linear strengths are not natural.) In the case where [x]θ is
the result of a ∀L rule, we want this reduction to hold as an equality.
Informally, then, we require that all cut reductions of the form
Qn+1 y
Qx
ψ
φ1
[x]Q1 φ1
[x]Q1 φ1
φn
[x]Qn φn
φn+1
Qn φn
Qn+1 φ1
φn+m
[y]Qxψ
[x]Qn+m φn+m
Qn+m φn+m
Qxψ
Qxψ
to hold as equalities. Given the reductions we already admit as equalities, we must add
the following:
∃x : L
∃y : L
N
M
ψ :L
∃xψ : L
φ(x) : R
φ(x) : L
θ:L
[y]γ : R
[y]∀θ : L
[x]∃yφ : R
∃yφ : L
∃yφ : R
γ :R
∀θ : L
∃x : L
M
≡
ψ:L
[x/y]N
φ(x) : R φ(x) : L
θ:L
[x]γ : R
[y]∀θ : L
∃xψ : L
184
∀θ : L
γ :R
∃y : L
∀x
M
ψ :R
∀xψ :: R
N
φ(x) : R
θ:L
φ(x) : L
[x]∃yφ : R
[y]γ : R
[y]∀θ : L
∃yφ : L
∃yφ : R
∀θ : L
γ :R
∀x
M
≡
ψ :R
[x/y]N
φ(x) : R φ(x) : L
θ:L
[x]γ : R
[y]∀θ : L
∀xψ : R
∀θ : L
γ :R
Definition 6.5.1. We will call a theory satisfying these reductions a dual theory.
Adding these laws makes the quantifier boxes an example of linear functor boxes with
extra structure. We may immediately infer:
Theorem 6.5.2. The pair of functors induced by the quantifier boxes forms a linear
functor.
Corollary 6.5.3. The classical doctrine formed from a dual theory is a dual doctrine.
Theorem 6.5.4 (Soundness). The two new reductions are sound in a dual classical
doctrine.
185
Proof. The first reduction follows from:
(ΣA ⊗ Γ) ⊗ ΠC
frobΣ ⊗ id
?
frobΣ
∗
Σ(A ⊗ a Γ) ⊗ ΠC
Σf ⊗ id
-
∗
(1)
?
(2)
Σ(f ⊗ id)
frobΣ
∗
Σ(a ∆ ⊕ B) ⊗ ΠC
-
-
Σ(id ⊗ εΠ )
∗
Σ((A ⊗ a Γ) ⊗ a ΠC)
?
∗
Σf ⊗ id
-
Σ(id ⊗ εΠ )
∗
Σ((a ∆ ⊕ B) ⊗ a ΠC)
∗
Σ((A ⊗ a Γ) ⊗ C)
?
∗
Σ((a ∆ ⊕ B) ⊗ C)
Σ(id ⊕ ηΣ ) ⊗ id
?
∗
∗
Σ(a ∆ ⊕ a ΣB) ⊗ ΠC
ν
⊗
id
Σβ ⊗ id
?
∗
Σa (∆ ⊕ ΣB) ⊗ ΠC
εΣ ⊗ id
(3)
(ε ⊕ id) ⊗ id
δ
-
?
(∆ ⊕ ΣB) ⊗ ΠC
(4)
∗
(Σa ∆ ⊕ ΣB) ⊗ ΠC)
δ
(5)
?
∆ ⊕ (ΣB ⊗ ΠC)
id ⊕ frob
?
ε ⊕ id
∗
Σa ∆ ⊕ (ΣB ⊗ ΠC)
id ⊕ frob
(6)
?
∗
∆ ⊕ Σ(B ⊗ a ΠC)
id
?
ε ⊕ id
⊕Σ
(i d
Σδ
∗
∗
Σa ∆ ⊕ Σ(B ⊗ a ΠC)
⊗ε
Π)
ε⊕
(7)
-
-
id ⊕ Σ(id ⊗ εΠ )
∗
Σa ∆ ⊕ Σ(B ⊗ C)
?
∗
∗
Σ(a ∆ ⊕ (B ⊗ a ΠC))
id
id ⊕ Σg
(8)
?
∆ ⊕ Σ(B ⊕ C)
ν
(9)
∗
∗
Σa ∆ ⊕ Σa ∆
′
ν
Σ(id ⊕ g)
?
∗
∗
′
Σ(a ∆ ⊕ a ∆ )
Σβ
id
⊕
?
Σ
g
(10)
εΣ ⊕ id
∗
′
εΣ
- ?
∆ ⊕ Σa ∆
∗
Σa (∆ ⊕ ∆ )
′
id ⊕ εΣ
-
?
∆⊕∆
′
(1) is the naturality of frob in the left argument. (2) follows from bifunctoriality of
⊗. (3) is shown below. (4) is an axiom of dual doctrines. (5) follows from naturality of
δ. (6, 7) follow from bifunctoriality of ⊕. (8) follows from naturality of ν. (9) follows
from bifunctoriality of ⊕ (10) follows from comonoidality of εΣ .
186
Diagram(3) above follows from:
-
ν
∗
Σ(a A ⊕ B)
Σ(id ⊕ ηΣ )
Σ ⊕
(1)
?
∗
ηΣ
-
Σ(ηΣ ⊕ id)
∗
Σ(a A ⊕ a ΣB)
∗
∗
∗
Σ(a Σa A ⊕ a ΣB)
id
∗
∗
Σa (Σa A ⊕ ΣB)
Σ(a∗ εΣ ⊕ id)
(3)
∗
Σβ
-
εΣ
(2)
)
∗
Σa A ⊕ ΣB
Σ(
η
ε ⊕ id
(5)
?
∗
∗
Σa∗ (εΣ ⊕ id)
Σ(a Σa A ⊕ a ΣB) (4)
Σβ
-
?
∗
Σa (A ⊕ ΣB)
-
ε
?
A ⊕ ΣB
(1) follows from bifunctoriality of ⊕. (2) is the definition of νΣ . (3) is a triangle
equality of the adjunction. (4) follows from naturality of β. (5) follows from ε-strength
of ε.
The second reduction follows from:
ηΠ
(1)
id
?
∗
∗
Πa (Γ ⊗ ΠC)
Πβ
?
∗
Π(a Γ ⊗ a ΠC)
∗
Π(
⊗
∗
Πf ⊗ id
νΠ
(3)
Πa Γ ⊗ ΠC
εΠ
id ⊗
-
Π(A ⊕ B) ⊗ ΠC
-
Π(id ⊕ ηΣ )
∗
Π(A ⊕ a ΣB) ⊗ ΠC
ηΠ
?
Πa Γ ⊗ Πa ΠC
ν
−1
∗
-
ηΠ ⊗ id
Γ ⊗ ΠC
(2)
-
Π(f ⊗ id)
∗
Π(a Γ ⊗ C)
-
(4)
ν
frobΠ ⊗ id
?
?
Π((A ⊕ B) ⊗ C)
(ΠA ⊕ ΣB) ⊗ ΠC
-
)
(5)
δ
Π (i
d⊗
εΠ
?
)
ΠA ⊕ (ΣB ⊗ ΠC)
id ⊕ frobΣ
δ
Π(f ⊗ id)
?
?
∗
Π((A ⊕ B) ⊗ a ΠC)
∗
(6)
ΠA ⊕ Σ(B ⊗ a ΠC)
id ⊕ Σ(id ⊕ εΣ )
?
∗
Π(A ⊕ (B ⊗ a ΠC))
Π(id ⊕ (id ⊗ εΠ ))
-
Πδ
Π(A ⊕ (B ⊗ C))
-
Π(id ⊕ ηΣ )
∗
Π(A ⊕ a Σ(B ⊗ C))
Π(id ⊕ g)
frobΠ
?
∗
-
Π(id ⊕ ηΣ )
(8)
id
∗
∗
Π(A ⊕ a Σa D)
Π(id ⊕ a∗ εΣ )
-
?
ΠA ⊕ Σ(B ⊗ C)
id ⊕ Σg
(7)
Π(A ⊕ a D)
-
frobΠ
(9)
?
∗
Π(A ⊕ a D)
frobΠ
-
?
∗
ΠA ⊕ Σa D
id ⊕ εΣ
-
?
ΠA ⊕ D
(1) Bifunctoriality of ηΠ . (2) See below. (3) Naturality of ν. 4) Axiom of a dual
doctrine. (5) Bifunctoriality of ⊗ and functoriality of Π. (6) Naturality of δ and func-
187
toriality of Π. (7) Naturality of linear strengths. (8) Triangle identity of the adjunction.
(9) ηΠ -strength of εΣ .
The diagram (2) commutes as follows:
ν
-
Π(A ⊗ B)
id
(1)
6
ν
-
ΠA ⊗ ΠB
-
(2) ΠA ⊗ ΠB
id ⊕ ηΠ
id
⊕
?
ΠA ⊗ Πa∗ ΠB
Π
Πε
Π(id ⊕ εΠ )
-
(3)
ν
-
Π(A ⊗ a∗ ΠB)
(1) Clear. (2) Triangle identity. (3) Naturality of ν.
We have proved:
Theorem 6.5.5. Dual doctrines are a sound and complete class of models for dual
theories.
188
Chapter 7
Categorical semantics and the calculus
of structures
We now revisit the calculus of structures and deep inference. The fact that deep inference proof systems are, in a very simple way, categories without structure, is evident
enough to have been observed independently by several researchers ([64, 35, 52]).
Once this observation has been made, is is obvious to ask what categorical structure
corresponds to the proof-theoretic properties of deep inference systems and their reduction systems (given by so-called splitting techniques). Some progress is made in
[45], but there the system of reductions is cut-reduction on a class of proof nets which,
while related to splitting, is not identical.
After spelling out the connections between deep inference and categorical models,
this chapter gives an new analysis of the medial rule in terms of (co)monoidality, and
relates the classes of models satisfying this (co)monoidality to additive cut-reduction
in the sequent calculus. Finally, we show how the necessary conditions for duality in
the previous chapter relate to reduction in the calculus of structures.
7.1 Providing a semantics for deep inference systems
Placing the definition of a classical category side-by-side with the definition of SKSg,
it is clear that a notion of theory T on proofs in the calculus of structures follows
189
immediately:
Definition 7.1.1. The theory T is a set of expressions Φ ≤ Ψ, where Φ and Ψ are
derivations in the calculus of structures. We write ≡ for the symmetric closure of ≤.
We give the inequations in a shallow form, with the understanding that the theory is
closed under formation of contexts.
We deal first with permutation of non-interfering rules. Given two inference rules p
and q, the following holds:
(A, A′ )
(A, A′ )
p
≡ (B, A′ ) (:= (p, q)),
p
q
(B, B ′ )
(B, B ′ )
(A, B ′ )
q
(7.1)
and similarly for disjunction. We require that T contain the equalities (id, id) ≡ id
and [id, id] = id.
The nesting of derivations and switch is also part of our theory:
(A, [B, C])
(E, [F, G])
[(E, F ), G)]
(p, [q, r])
s
(A, [B, C])
≡ [(A, B), C]
[(E, F ), G]
s
[(p, q), r]
,
(7.2)
Each coherence axiom for SLDCs is instantiated as an equality on proofs; for example
(A, B[C, D])
s
s ≡ (A, [(B, C), D]) .
[(A, B, C), D]
s
[(A, B, C), D]
(A, B, [C, D])
(7.3)
The following rule and its dual govern interactions between negations:
A
(A, t)
=
i↓
(A, [Ā, A])
s ≡
[(A, Ā), A]
A
id
(7.4)
A
i↑
[f, A]
A
=
The remaining equations are given in Table 7.1 and the inequations of the theory are
given in Table 7.2.
The following is now clear:
190
S[[P, Q], [P, Q]]
S[[P, P ], [Q, Q]
S[[P, P ], Q]
S[P, Q]
A
=
↓c≡
S[[P, Q], [P, Q]]
S[P, Q]
↓c
[[A, A], A]
[A, [A, A]]
c↓
↓c
c↓
≡
[A, A]
[A, A]
c↓
c↓
A
A
=
[A, B]
f
[w↓ , w↓ ]
≡
f
w↓ ≡ id
f
f
[A, B]
Table 7.1: Equalities: weakening and contraction
A
A
c↑
(A, A)
(B, B)
(p, p)
≤
B
f
p
f
w↓
A
c↑
(B, B)
B
≤
p
[A, C]
w↓
B
(A, f)
c↑
[A, C]
[A, (C, C)]
c↑ ≤
([A, C], [A, C])
s′
[A, A, (C, C)]
w↑
(A, f)
(A, B)
w↓ ≤
(t, f)
=
f
c↓
[A, (C, C)]
w↓
(A, B)
Table 7.2: Inequalities: weakening and contraction
191
w↓ ≡
A
id
A
c↓
f
i↓
=
[A, A]
A
f
[f, f]
[A, f]
Theorem 7.1.2. SKSg with T forms a classical category.
Remark 7.1.3. Categorically, the equalities of Formalism A are captured by bifunctoriality of disjunction and conjunction. (Deep inference in the calculus of structures
is the ordinary functoriality of the connectives with respect to each argument.) The
equalities of Formalism B are captured by naturality of switch.
As SKSg+T forms a classical category, the usual notion of interpretation gives a functor from the proof nets of LK to SKSg.
Each inference rule in SKSg, when read from top to bottom, is a valid entailment
φ → ψ in classical logic, and there is therefore a cut-free proof of φ ⊢ ψ. Composing
these proofs using the cut rule, we obtain:
Theorem 7.1.4 (From SKSg to LK). For every derivation Φ from A to B in SKSg
there exists an LK proof of the sequent A ⊢ B, with a number of instances of cut equal
to the number of rule applications in Φ minus one.
Unlike the translations between KSg and GS1p, these translations do not respect the
notion of cut as an occurrence of i↑ . The following generalization of the notion of a
normal proof agrees with cut reduction in LK:
Definition 7.1.5. A derivation is normal if it contains no ↑ rule below a ↓ rule.
(This observation was also made recently (and independently) by Brünnler.) This definition agrees with the equivalent notion for LK, in the sense that any normal LK
derivation (or proof net) translates to a normal SKSg derivation (taking care to note
that since we have (implicit) cuts against constants in our proof nets, a normal proof
net is one without essential cuts). The notion of a normal SKSg proof is easily seen
to be a special case of this definition, since any ↑ rule with premise t has conclusion
equivalent to t. Notice also that each equational law in T involves only ↑ or ↓ rules:
That is, application of these rules to a normal derivation yields another normal derivation. Meanwhile, the lax naturalities clearly show that moving an w↑ or c↑ rule above
some other derivation generates some change in denotation: the change corresponds
to a move “closer” to a normal form.
192
7.2 Locality and categorical semantics
The theory on proofs in the calculus of structure given so far is deficient in two respects:
1. It is a theory on the non-local calculus SKSg; and
2. It arises, not from the structure of the calculus, but from a translation into LK.
In this section we will examine what needs to be added to a classical category to yield
a model of SKS. We then examine models of SKSq in classical doctrines.
When we give the rules of a calculus, we wake the tacit assumption that each rule
has a unique (or at least canonical) interpretation. Since in the calculus of structures
each rule is a valid inference, this amounts to asking that each of these inferences
has a canonical representant in the semantics. Thus the assumption for SKS, and the
condition we will require of a classical category for it to model SKS, is that the two
ways of presenting the medial rule in LK/SKSg, by a contraction and four weakenings
on the left
φ:L
φ:L
ψ :L
φ∧ψ :L
τ :L
τ :L
χ:L
τ ∧χ:L
(φ ∧ ψ) ∨ (τ ∧ χ) : L
ψ :L
φ:L
ψ:L
φ∧ψ : L
χ:L
τ :L
φ:R
χ:L
φ∨τ
τ :R
ψ:R
τ ∧χ:L
(φ ∧ ψ) ∨ (τ ∧ χ) : L
(φ ∨ τ ) ∧ (ψ ∨ χ) : R
(φ ∧ ψ) ∨ (τ ∧ χ) : L
[(A, B), (C, D)]
([(A, B), (C, D)], [(A, B), (C, D)])
([A, C], [B, D])
193
c↑
w↑ ×4
χ:R
ψ∨χ:R
or by four weakenings and a contraction on the right
φ:L
ψ :L
φ∧ψ :L
τ :L
χ:L
τ ∧χ:L
φ:R
φ:R
τ :R
φ∨τ : R
ψ:R
ψ:R
χ:R
ψ∨χ:R
(φ ∨ τ ) ∧ (ψ ∨ χ) : R
(φ ∧ ψ) ∨ (τ ∧ χ) : L
τ :R
φ:R
φ∨τ :R
[([A, C], [B, D]), ([A, C], [B, D])]
([A, C], [B, D])
χ:R
ψ∨χ:R
(φ ∨ τ ) ∧ (ψ ∨ χ) : R
(φ ∨ τ ) ∧ (ψ ∨ χ) : R
[(A, B), (C, D)]
τ :R
χ:R
ψ:R
.
w↓ ×4
c↓
are equal.
Additionally, the proof theory inspired by Formalism B suggests that the medial rule
should be natural.
Lemma 7.2.1. If the left and right representations of medial are equal in a classical
category , then that canonical morphism is natural
Proof. In the category of proof nets, call the left medial mL and the right medial medR ;
then medR ◦ (f ⊗ g) ⊕ (h ⊕ k) = medL ◦ (F ⊕ h) ⊗ (g ⊕ k); if medR = mL = med
this is precisely naturality of med.
We will call a classical category C “medial” if medL = medR in C The first observation
we should make is that at least one of our existing, nontrivial, models of classical
sequent proofs (namely G(Rel)) is a medial classical category. Both morphisms are
just the obvious morphism (middle four exchange) from (A ⊕ B) ⊕ (C ⊕ D) to (A ⊕
C) ⊕ (B ⊕ D). We have:
Theorem 7.2.2. There is a nontrivial semantics for SKS derived from classical categories.
Proof. The interpretation is the interpretation of SKS proofs in LK, followed by the
interpretation of LK in G(Rel); the choice of proof net for the medial rule in the first
194
makes no difference, since both are sent to the same morphism in the second. We give
the details of a direct interpretation for clarity.
Assign to each atom a a set, Sa and then define the interpretation of a formula by
induction as follows:
1. ⌊a⌋ = (Sa , ∅)
2. ⌊¬a⌋ = (∅, Sa )
3. ⌊⊤⌋ = ⌊⊥⌋ = (∅, ∅)
4. If ⌊A⌋ = (A+ , A− ) and ⌊B⌋ = (B + , B − ), ⌊(A, B)⌋ = ⌊[A, B]⌋ = (A+ ⊕
B + , A− ⊕ B − )
It is clear how each rule of SKS, considered as a derivation, translates into a morphism
of G(Rel); for example, c↓ translates as
A+ ⊕ A+
({(xinl, x), (xinr, x)
-
A+
6
∅
∅
?
({(x, xinl), (x, xinr) −
A
A− ⊕ A− Note that switch and medial translate as associativity of ⊕ and middle four exchange
respectively. A derivation is interpreted as the composition of its inference steps. Now,
given two derivations Φ and Ψ from A to B, we say that Φ 4 Ψ if ⌊Φ⌋ ≥ ⌊Ψ⌋.
Having established a particular model, we now look for the notion of proof net that
satisfies the same condition.
Recall that we can characterize a categorical product or coproduct for a particular
category C via an adjunction in Cat: specifically, if C has products and coproducts,
there is an adjunction:
+⊣D⊣×
195
where ×, + : C 2 → C are the product and coproduct, and D : C → C 2 is the diagonal
in Cat. Taking, for example, the right-hand adjunction, we recover the usual structure
of a product from the co-unit
Id → ×D,
A 7→ A × A
which gives the diagonal, and the unit
D× → Id,
(A × B, A × B) 7→ (A, B)
which gives the two projections.
If we have, instead of adjunctions, (op)lax adjunctions, we recover the setup in a classical category,
⊕ ⊣≤ D ⊣≥ ⊗,
and also the setup we have seen in the setting of quantifiers. (Observe that C 2 has a
canonical classical category structure given by the obvious pointwise construction.)
Applying what we already now about such adjunctions, it is interesting to see that the
associated Frobenius morphism associated to, for example, ⊕, is a morphism of type
⊕((A, B)) ⊗ C → ⊕((A, B) ⊗ DC); that is, (usual) distributivity of otimes over ⊕:
frob⊕ : (A ⊕ B) ⊗ C → (A ⊗ C) ⊕ (B ⊗ C).
Remark 7.2.3. Notice that regarding (⊕, ⊗) as a linear functor (as is done in [16]),
the linear strength associated to ⊕ is m : (A ⊕ B) ⊗ (C ⊗ D) → (A ⊗ C) ⊕ (B ⊗ D).
This has some claim to be a better holder of the name “linear distribution” than the
usual map δ (from which it is, clearly, derivable), as can be seen if we set C = D.
More usefully, medial arises as comonoidal strength of ⊕ with respect to otimes.1 .
This is a morphism µσ from ⊕((A, B)⊗(C, D)) = (A⊗C)⊕(B ⊗D) to to ⊕(A, B)⊗
⊕(C, D) = (A ⊕ B) ⊗ (C ⊕ D). A little thought will show that, in fact, the monoidal
strength µ⊗ of ⊗ with respect to ⊕ yields a morphism of the same type, and that
µσ =medr and µ⊗ = medL .
When might these two strengths be equal in a category of Robinson’s proof nets? We
may answer this question by reconsidering the subject matter of section 6.4. There we
1
More precisely, the monoidality of ⊕ : (C 2 , ⊗2 ) → (C, ⊗)
196
considered nets
∃x
∃x
φ(x) : L
ψ(y) : L
φ(x) : L ψ(x) : L
φ(y) : Lψ(y) : L
φ ∧ ψ(x) : L
∃x(φ ∧ ψ) : L
φ ∧ ψ(y) : L
φ(x) : R
ψ(y) : R
[y]∃zφ : R
[x]∃zφ : R
∃x(φ ∧ ψ) : L
∃zφ : R
∃x(φ ∧ ψ) : L
∃zφ : R
∃zφ ∧ ∃zψ : R
and
∃x
φ(x) : L
ψ(x) : L
φ ∧ ψ(x) : L
φ(x) : R
ψ(x) : R
[x]∃zφ : R
[x]∃zφ : R
[x]∃zφ∃zψ : R
∃x(φ ∧ ψ) : L
∃zφ∃zψ : R
and deduced that they were equal. Notice that there is nothing special about the ∃ box
except that it is a lax left adjoint, so we may replace it, in particular, with ∨. An ∨-box
would be the following:
∨
M
φ:L
θ1 : X1
φ∨ψ
N
θn : Xn
ψ:L
θ1 : X1
θ1 : X1
θn : Xn
θn : Xn
— the application of the additive ∨L rule.
Remark 7.2.4. In general, a dual premise sequent calculus law for a connective takes
two proofs (a morphism in C 2 ), perhaps constrained by their type if dealing with an
additive rule, and returns a single proof. Viewing the branching of these rules in this
light shows that, semantically, the branching in the sequent calculus/proof nets is this
operation on pairs of morphisms, and not, unlike the comma of a sequent system, an
197
instance of a connective in the meta-theory.
For the sake of readability we replace the two nets inside the ∨-box with a single net
over pairs of formulae:
⊕
(M, N )
(φ, ψ) : L
φ∨ψ
D(θ1 : X1 )
θ1 : X1
D(θn : Xn )
θn : Xn
Now medL is given by
∨
∨
(ψ, τ ) : L
(φ, θ) : L
(φ, θ) : L (ψ, τ ) : L
(ψ, τ ) : R
(φ, θ) : L (ψ, τ ) : L
(φ ∧ ψ, θ ∧ τ ) : L
(φ ∧ ψ, θ ∧ τ ) : L
(φ ∧ ψ) ∨ (θ ∧ τ ) : L
(φ ∧ ψ) ∨ (θ ∧ τ ) : L
(φ, θ) : R
(ψ ∨ τ, ψ ∨ τ ) : R
(φ ∨ θ, φ ∨ θ) : R
ψ∨τ :R
φ∨θ :R
(φ ∧ ψ) ∨ (θ ∧ τ ) : L
(φ ∨ θ) ∧ (ψ ∨ τ ) : R
and medR is given by
∨
(φ, θ) : R
(φ, θ) : L
(ψ, τ ) : L
(φ ∧ ψ, θ ∧ τ ) : L
(ψ, τ ) : R
(φ ∨ θ, φ ∨ θ) : R (ψ ∨ τ, ψ ∨ τ ) : R
((φ ∨ θ) ∧ (ψ ∨ τ ), (φ ∨ θ) ∧ (ψ ∨ τ )) : R
(φ ∧ ψ) ∨ (θ ∧ τ ) : L
(φ ∨ θ) ∧ (ψ ∨ τ ) : R
We know, from Section 6.4, that a sufficient condition for these nets to be equal is that
the following reduction holds:
198
∨
∨
(φ, ψ) : L
D(θ) : R
D(θ) : L
θ:R
θ:R
φ∨ψ
(φ, ψ) : L
φ∨ψ
∨
4
(φ, ψ) : L
D(θ) : R
D(θ) : L
(φ, ψ) : L
(φ, ψ) : L
φ∨ψ
Translated back into ordinary proof nets:
M
φ:L
N
θ:L
ψ:L
O
θ:L
φ:L
θ:L
P
θ:R
ψ:L
θ:R
φ∧ψ :L
φ∧ψ :L
θ:R
φ∧ψ :L
≡
M
φ:L
O
θ:L
φ:L
P
N
θ:R
φ:L
ψ:L
θ:L
ψ:L
θ:R
ψ:L
φ∧ψ :L
(where if the nets M, N, O and P have no left doors other than those shown, the reduction holds as an equality). In other words, the permutation of an additive cut and
an additive ∨L is sound. So adding features from the additive theory of cut-reduction
to the theory of multiplicative cut-reduction gives us completeness for medial classical
categories.
199
7.2.1 The first-order medial rules
We have already seen how the structure of SKSq and its inference rules can inform the
semantics of the sequent calculus. We show now that, with a little alteration we may
now model SKSq in a sharp doctrine whose fibres are medial classical categories; we
will call a doctrine of this type a medial doctrine.
A sharp classical doctrine has a natural transformation µΣ which models m2 ↑ , and a
natural isomorphism νΣ whose inverse models m1↓ . Thus the interpretation of SKSq
proofs in a medial doctrine satisfies the equalities of Formalism B on the rules µΣ and
νΣ .
The rules n↑ and n↓ are modelled by ηΣ and εΠ respectively. We separate the equalities
∀R = ∃R = R
with rules
S [∃xR]
S {R}
S {R}
S [∃xR]
if R does not depend on x
S {R}
p1↓
S [∀xR]
S [∀xR]
p1↑
S {R}
p2↓
p2↑
If we assume that the domain of quantification is non-empty, we may model p1 ↓ by
εΣ , p1↑ by ε◦Σ , and dually.
Finally, recall that we may construct linear strengths n and m from Frobenius strengths;
we model u↑ by m and u↓ by n.
The only clear objection remaining is that this notion of inequality is still not native to
the calculus of structures; in the following section we begin to tackle this issue.
7.3 Towards a native notion of proof equality for SKS
A key property of the splitting proof for SKS is that if the lower-most portion of a
proof is cut free (that is, free of the i↑ rule) it survives the cut-elimination procedure.
200
More specifically, the algorithm proceeds by eliminating the topmost cut, replacing the
cut and the proof above it with a cut-free proof of the same conclusion and leaving the
proof below unchanged. It then looks for what is now the topmost cut, and repeats.
This leads us to simple constraint on a 2-category of proofs that respects splitting:
given two proofs Φ and Ψ of the same formula, and any rule ↓ r in the down fragment
(including switch, medial etc.), if we have a reduction
·
·Ψ
·
R
↓r
S
·
·Φ
·
R
↓r
S
we also have a reduction Φ
-monic.
Ψ. In other words, each rule in the down fragment is
We would like to make a similar statement about rules in the up fragment. There is, as
yet, no symmetric version of splitting; no set of reductions taking a general derivation
to a normal form (in the sense of Definition 7.1.5).
Recall from Chapter 2 that there are two definitions of m1↑ in SKSgq, one of which is
contained in the ↑ fragmet and the other in the ↓ fragment. We therefore regard m1 ↑
(andm2↑ ) as being in both the ↑ and ↓ fragments of SKSq.
Theorem 7.3.1. Under the assumption that m2↑ and m2↓ are respectively -epi, and
-monic, the medial classical doctrine formed as above is a dual doctrine.
Proof. This follows directly from Theorem 5.5.18.
Remark 7.3.2. This is a step in the right direction; we derive duality a property that
makes sense with respect to the calculus of structures and splitting. The proof does,
however, still rely on the order-enriched structure of the doctrine. If we wanted to
axiomatize the duality without using the order, we could start by assume coherence of
mix for the quantifiers, and then add the following axioms (and their duals):
201
ΠA ⊕ ΠB
id ⊗ M
-
ΠA ⊕ ΣA
-
n
ν
?
Π(A ⊕ B)
ΠA ⊕ ΠB
M ⊗M
- ΠA ⊕ ΣA
6
ν
n
?
Π(A ⊕ B)
M Σ(A ⊕ A)
In that case we would only need that m2↑ is epi and m2↓ is monic.
202
Chapter 8
Classical doctrines with equality and
axiomatic theories
The apparatus we have set up so far is sufficient to study algebraic signatures including
functions, but so far we have made no mention of equality, nor of proof theory in the
presence of non-logical axioms on the first-order structure. The base, category of a
doctrine, in this case, will be derived from some model, in the traditional sense of
model theory [34].
We turn our attention now to the status of algebraic and other signatures and axiom sets
in our models. As observed by Lawvere [48], equality can be viewed as a left adjoint
to sustitution of the diagonal (and so as existential quanifictaion in some generalised
sense.) We will recall the details of this construction, and extend our soundness and
completeness results to this setting.
Via the usual categorical notion of a relation as a subobject, we then extend this notion to non-algebraic theories. Using this notion we give extended examples of the
proof theory of classes of models, including the proof theory of (modal and intuitionistic) Kripke structures. We also discuss the presence of nonlogical axioms in classical
categories and doctrines.
203
8.1 Nonlogical axioms in sequent calculi
Suppose we want to incorporate a nonlogical axiom, for example
⊢φ
to our sequent calculus, so that the proofs we derive respresent proofs under the assumption that φ holds. The formula φ might be, for example, ∀x.x = x, or an axiom
of group theory.
A good summary of the different techniques used to introduce axioms to a sequent
calculus is given in [53]. Given a set T of axioms, we may produce judgements ⊢T in
the following ways:
1. Add all instances of the axiom;
2. Add so-called “basic sequents”
φ1 , ...φn ⊢ ψ1 , ...φn
with φi , ψi atomic;
3. Consider proofs with a context φ, so a proof of Γ ⊢T ∆ is a proof φ, Γ ⊢T ∆ in
the usual sequent calculus.
4. Add new nonlogical inference rules to the sequent calculus corresponding to the
axiom.
These approaches have different behaviours with respect to cut-elimination; the first
has no cut-eliminitaion in general. The second allows reduction of arbitrary cuts to
“essential cuts”, that is cuts against basic sequents. The third allows cut elimination
(but loses some good structural properties), while the fourth allows cut reduction in
certain circumstances.
Recall that, for example, negation on the left could be added to the sequent calculus for
LB by adding constants ⊢ ¬ψ, ψ for every formula φ (corresponding to item 1 above),
or only for atoms (corresponding to item 2). Alternatively, we may add the usual rules
for negation (corresponding to item 4 above).
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8.2 Equality and algebraic theories
In [48], Lawvere gives a definition for an equality predicate in a hyperdoctrine (ΘX
for each variable type X), in terms of a left adjoint Σ∆ to ∆∗ , where ∆ : X → X × X
is the diagonal in the base.
Definition 8.2.1. For each type X in B, define the equality predicate ΘX as Σ∆ (⊤X ).
This, in a hyperdoctrine for natural deduction, agrees with the standard definition of
equality as a reflexive substitutive relation. We show that this notion of equality does
not rely on the cartesian closed structure of the fibres of a hyperdoctrine.
Remark 8.2.2. The functor ∆∗X should be interpreted as substituting the pair (z, z)
into predicates over (x, y), where x, y and z have type X. Therefore the unit of the
adjunction Σ∆ ⊣ ∆∗X (a morphism ⊤X → ∆∗ ΘX ) demonstrates the reflexivity of ΘX
Lemma 8.2.3 (Subsitutivity of equality). In a classical doctine C, given an object
X ∈ B, and an object A in C(X), we have the follwing situation:
C(X)
C(X × X)
C(X)
ΘX
A
B
X
π1∗
π1 ∗ A ⇒ π2∗ A
π1
X×X
π2∗
π2
A
X
There is a morphism from ΘX to π1∗ A ⇒ π2∗ A
Proof. First note that, oing to the universal property of the product, πi ◦ ∆ = id :
X → X, and so ∆∗ ◦ πi∗ is isomorphic to Id. Expressing the implication in terms of
negation and disjunction, we may construct the following morphism:
⊤X
-
Ā ⊕ A
-
∆∗ π1∗ Ā ⊕ ∆∗ π2∗ A
-
205
∆∗ (π1∗ Ā ⊕ π2∗ A)
-
∆∗ (π1∗¯A ⊕ π2∗ A)
Applying the lax adjunction, we derive the required morphism.
Corresponding to this is a notion of equality in the sequent calculus and proof nets, in
the sense of style 3 above. This notion is well-established; we add constants
Kref
x=x:R
and, where φ is an atomic predicate over X, x
Ksub
[X]x = y : L
[y]φ(x) : L
[x]φ(y) : R
,
correponding to reflexivity and substitutivity of equality. We label the derived law
for symmetry of equality Ksym . The constant Ksub is defined only for atomic predicates; we may extend the substitutivity ofequality to general predicates by structural
induction; for example
Ksub
Ksub
x = y : L [y]φ(x) : L
x=y:L
[x]φ(y) : L
[y]φ ∧ psi(x) : L
x = y : L [y]ψ(x) : L
[x]ψ(y) : L
[x]φ(y) ∧ ψ : R
.
A series of equations on proofs is given in table 8.2. The first, Eq1 , gives that the
two ways of deriving reflexivity of equality are equal, and the second, Eq2 , gives that
vacuous substitution really is vacuous, The third,, Eq3 , says that substituting x for y
and then y for x is vacuous. Alternatively, we may interpret Eq2 as stating that, in the
Kleisli category for the monad (x = y) ∧ −), φ(x) is isomorphic to φ(y).
206
Ksub
Ksub
≡
[z]x = y : L
[x]y = z : L[z]x = y : L [y]x = z : R
[x]y = z : L [y]x = z : R
Eq1
∆∗ Ksub
≡
x = x : Ł φ(x) : L
φ(x) : L
φ(x) : R
φ(x) : R
Eq2
x = x : L φ(x) : L
[X]x = y : L
[y]φ(x) : L
Ksub
Ksym
Ksub
[x]φ(y) : R
[X]y = x : L
[x]φ(y) : L
[X]y = x : R
[y]φ(x) : R
[X]x = y : L
≡
[X]x = y : L
[y]φ(x) : L
[X]x = y : L[y]φ(x) : L
Table 8.1: Additional axioms for equality
207
[y]φ(x) : R
Eq3
We may now define relativized quantifiers for each term t : X → Y :
∃t ψ := ∃x(tx = y ∧ ψ(x)) and
∀t ψ := ∀x(¬(tx = y) ∨ ψ(x)).
We must show now that these are lax left- and right-adjoints to t∗ .
Theorem 8.2.4. Given a first order theory T which includes all instances of the rules
in Table 8.2, ∃t defines a functor Σt which is lax left-adjoint to t∗ and ∀t an oplax right
adjoint.
For example, for ∃t : the net
Kref
φ:L
t∗ (tx = y) : R
φ:R
t∗ ((tx = y) ∧ [y]φ) : R
t∗ ∃x((tx = y) ∧ [y]φ) : R
will serve as a unit for the adjunction and the net
∃x
Ksub
tx = y : L
t∗ φ : L
[x]φ(y) : R
tx = y ∧ t∗ φ : L
∃x(tx = y ∧ [y]t∗ φ) : L
φ(y) : R
as a counit. We relegate the proof of this theorem to Appendix B.
Recall that in model theory a signature is called algebraic if it consist only of function
symbols and constants, and that a theory ( a set of sentences) is algebraic if it makes
use only of the equality relation.
208
Given a particular algebraic theory, we may now constuct a model of the corresponding
proof theory; the base category is constructed from a set-theoretic model of the theory
and the axioms of the theory lift up to the relevant fibres. We give an extended example;
that of the theory of groups.
Example 8.2.5. A group is an algebraic structure with:
1. A set G of elements
2. a constant e;
3. A function (−)−1 : G → G;
4. A function m : G × G → G, and
5. axioms
(a) m(e, x) = x;
(b) m(x, m(y, z)) = m(m(x, y), z);
(c) m(x, x−1 ) = e
A base category B for a doctrine of proofs over this signature and axioms is the Lawvere theory for a group: a category with objects generated by finite products over a
terminal object 1 and an object G, and morphisms generated by projections and morphisms m : G × G → G, (−)−1 : G → G and e : 1 → G, such that:
1. m◦ < id, e >= id : G → G
2. m ◦ (id × m) = m ◦ (m × id) : G × G × G → G
3. m◦ < id, (−)−1 >= e ◦ πG : X → X
Each of these equalities f = g : Gn → G between morphisms of B lifts to a natural
isomorphism f ∗ = g ∗ in the relevant fibre. Over C(Gn ) we have a morphism
⊤nG (= f ∗ ⊤G ) → f ∗ ∆∗ ΘG
. Since ∆ ◦ f = (f × f ) ◦ ∆ = (f × g) ◦ ∆, we derive a morphism
⊤nG → ∆∗ (f ∗ , g ∗)ΘG
209
(i.e., f (x̄) = g(x̄))
We have proved that given a classical doctrine over this base, the base has morphisms
corresponding to the axioms of group theory. A class of models is given by letting B
be set theoretical (so G is the underlying set of a particular group, and the morphisms
e, m and (−)−1 are given by the structure of that group), and the fibre over Gn be
G(Reln ).
8.3 Theories with relations
Recall that a relation between A and B in a category with finite products is a monic
arrow R → A × B. We give conditions for certain common properties of relations:
Definition 8.3.1.
1. Reflexive A relation R ⊆ A × A is reflexive if we there exists
a monic arrow from A to R such that
A-
-
R
∆
(8.1)
?
A×A
2. Symmetric A relation R ⊆ A × A is symmetric if the diagram
r-
A⊗A
σ
R
r
(8.2)
?
A×A
commutes
3. Transitive First, consider the following pullback:
-
T
R×A
r × id
?
A×R
A ×r
210
(8.3)
?
A×A×A
The relation T is the abstract equlivalent of, for a triple (x, y, z), xRy and yRz.
R is transitive iff there is a morphism h such that
h
T
-
R
r
?
A×A×A
< π1 , π3 >
-
(8.4)
?
A×A
Given a base category B with a relation r : R → A × B, Σr φ in C(A × B) should be
interpreted as “ xRy and φ holds of (x, y).” In that case, the object Σr ⊤R should be
interpreted as “xRy”. It is easy to see that reflexivity and transitivity of a relation lifts
to the corresponding property of Σr ⊤R .
Lemma 8.3.2.
1. If R ⊆ A × A is reflexive, there is a morphism ⊤A → ∆∗ Σr ⊤R
2. If R ⊆ A × A is symmetric, there is a morphism Σr ⊤R → σ ∗ Σr ⊤R
3. If R ⊆ A×A is transitive, there is a morphism [x]Σr ⊤R ⊗[z]Σr ⊤R → [y]Σr ⊤R ,
where [x] is shorthand for < π2 , π3 >∗ , etc.
Proof.
1. The morphism is given by
⊤A
-
a∗ ⊤
a∗ η
-
a∗ r ∗ Σr ⊤
since a∗ r ∗ Σr ⊤ ∼
= (r ◦ a)∗ ⊤ and r ◦ a = ∆.
2. Construct the morphism
⊤r
η
-
r ∗ Σr ⊤r
-
(σ ◦ r)∗ Σr ⊤R
-
r ∗ σ ∗ Σr ⊤R
Apply the bijection associated to the adjunction ∃r ⊣ r ∗ , to obtain a morphism
Σr ⊤R → σ ∗ Σr ⊤R .
3. We prove the lemma in two stages. First, we give a morphism from ∃t ⊤T to
[y]∃r Tr , then one from [x]Σr ⊤R ⊗ [z]Σr ⊤R to ∃t ⊤T .
211
To obtain the first step, apply the bijection associated to the adjunction ∃t ⊣ t∗
to the morphism
⊤T
-
h∗ r ∗ ⊤R = (r◦)∗ ⊤R = [< π1 , π3 > ◦t]⊤R
For the second, observe the following:
⊤T ⊗ ⊤T → ⊤T
p∗1 [x]⊤R ⊗ p∗2 [z]⊤R → ⊤T
-
t∗ [y]⊤R
= (1)
p∗2 [x]⊤R ⊗ p∗1 [z]⊤R → t∗ ∃t ⊤T
∃t : R
p∗1 [x]⊤R ⊗ p∗1 [z]⊤R → p∗2 (id × r)∗ ∃t ⊤T
[x]⊤R ⊗ ∃p2 p∗1 [z]⊤R → (id × r)∗ ∃t ⊤T
= (2)
∃p2 L
[x]⊤R ⊗ (id × r)∗ ∃r×id [z]⊤R → (id × r)∗ ∃t ⊤T
∃id×r [x]⊤R ⊗ ∃r×id [z]⊤R → ∃t ⊤T
[x]Σr ⊤R ⊗ [z]Σr ⊤R → [y]Σr ⊤R
= (3)
∃id×r L
= (4)
where
• The step marked “= (1)” follows from ⊤T = p∗1 [x]⊤R ;
• The step marked “∃t : R” is shorthand for postcomposition by the unit of
the adjunction Σt ⊣ t∗ ,
• The step marked “= (2)’ follows from the definition t = (id ⊗ r) ◦ p2 ;
• The step marked “∃p2 : L” is shorthand for applying the functor Σp2 , followed by postcomposition by the counit of the adjunction Σp2 ⊣ p∗2 , and
precomposition by the associated frobenius morphism.
• The step marked “= (3)” follows from the Beck-Chevally condition applied
to the pullback (8.3);
• The step marked “∃(id×r) : L” is similar to “∃p2 L”;
• The step marked “= (4)” follows from the Beck-Chevally condition applied
to the pullback
A×R
π2
id × r
-
R
r
?
A × (A × A)
212
π2 -
?
A×A
So the proof theory inherits the properties of the relations from the model theory.
Modal logic
The proof theory of modal logic is blighted with the following fact: the modal logic
S5 has no known cut-free presentation in the sequent calculus. Suggested solutions to
this problem include moving outside the sequent calculus to hypersequents [4], display
logic [29] or recently the calculus of structures [62], or to remain in the the sequent
calculus and label the sequents with information about the underlying Kripke structure.
These labelled sequent calculi [29] should be though of as embedding propositional
S5 in a fragment of first-order classical logic. We give in the following example a
semantics for normal modal logics based on this intuition.
Example 8.3.3 (Proof theory of Kripke structures). In [39], Jacobs gives an example
of a first-order fibration with a Kripke structure as the base and Boolean algebras as
fibres. We sketch here how that example (which, using Boolean algebras, contains
no information about proofs) to the proof-theoretic setting, using a special kind of
classical doctrine.
A Kripke structure (W, R) consists of a set W of worlds and a relation R ⊆ W ⊗ W
called “accessibility”. Certain modal logics (called normal modal logics) are sound
and complete with respect to classes of Kripke structures; for example, the modal logic
S4 is sound and complete with respect to Kripke structures for which R is reflexive and
transitive, and the logic S5 is sound and complete with respect to Kripke structures for
which R is an equivalence relation.
Satisfiability of purely propositional formuale is given at a particular world by the
usual boolean semantics, and we write (W, R), w |= φ for satisfaction at a particular
world. We say that (W, R) |= φ iff (W, R), w |= φ for every world w ∈ W . We extend
this definition to modal formulae by saying that (W, R), w |= φ if, for every world
v such that wRv, (W, R), v |= φ, and similarly (W, R), w |= ♦φ if φ holds at some
world v with wRv.
Definition 8.3.4. A Kripke-doctrine is a classical doctrine for which the base category
B has an object W , and an accessibility relation r : R → W ⊗ W , such that r ∗ has a
213
left adjoint Σr and right adjoint Πr .
The fibre over W may be thought of as the propositions of modal logic parametrized
over the worlds; if the fibre is simply a boolean lattice ⌊φ⌋ is the set of worlds where φ
holds. In our richer setting, this behaves more like the formulae of a labelled sequent
calculus for modal logics.
We may now interpret the formula φ over the terminal object 1 of B as ∀v.(wRv =⇒
[w]φ(v)), with denotation
⌊φ⌋ = Πr ⌊φ⌋ .
Of course, Kripke structures are used to model not only modal logics but also intuitionistic logic.
Example 8.3.5 (Semantics of Dummett’s calculus). The major difference between
the Kripke-doctrines for modal logic and that for intuitionistic logic is the requirement
of monotonicity for the interpretation; that is, we require the following as an axiom
(for every atom p) in the fibre over R:
⊢ wRv ∧ p(w) ⇒ p(v)
We may now interpret implication as ∀v.(wRv ⇒ (φ(v) ⇒ ψ(v)), with denotation
¯ ⌊ψ⌋)
⌊φ ⇒ ψ⌋ = Πr (⌊φ⌋ ⊕
214
Chapter 9
Conclusions and further work
We have given a sequence, in increasing strength, of notions of equality for proofs in
the first-order classical sequent calculus, and introduced for each a sound and complete
class of categorical models, extending the models of the propositional calculus by
Führmann and Pym [23]. All but the strongest notions do not assume proof-theoretic
duality, and by utilising axioms for a linear functor due to Cockett and Seely [16] we
give a simple necessary condition for duality to hold, which moreover we hope will
make sense for deep inference systems.
We take a moment now to reflect on how choices of approach have effected the results
acquired:
Proof-theoretic duality On first sight, it might seem wasteful to consider a two-sided
rather that one-sided calculus for classical logic, given the inherent dualities. What we
have demonstrated, by initially rejecting a one-sided approach (and in turn a semantics based upon ∗-autonomous categories) is that in the presence of additive structure
and an order enrichment (which we discuss below), duality corresponds to a stronger
theory than is necessary for cut-reduction, and so to identifying more proofs than is
necessary. Our approach was, instead, to identify as few proofs as seemed possible,
and then to observe that by strengthening some cut-reduction steps to equalities we
could recover duality.
Had we begun with ∗-autonomous categories, we would have been led (quite naturally)
to define one functor with a Frobenius strength (modelling the universal quantifier),
215
and derive another from De-Morgan duality. In so doing, we would have missed that
duality also arises naturally from insisting that the monoidal strengths µ are (order-)epi
and monic.
Order enrichment
Lafont’s example gives a compelling reason for introducing an order-enrichment to the
proof theory of propositional classical logic; without it (or some stronger enrichment)
we cannot hope to model both reductions. When extending the models to first-order
quantification, however, we could have chosen to simply have each reduction hold
as an equality, and have no structural reasons not to find models Indeed, the author
suspects that such models exist and there is an example of a possible place to look
below, based on the proof nets of Hughes.
However, working with the order enrichment modelling the first-order as well as propositional cut-reduction steps provides a link between the sequent calculus and deep inference calculi; by considering the form of the medial rules of the calculus of structures, we can see how to model the monoidal strength ν as a natural transformation
without insisting that the unit and counit of the associated adjunction are natural.
We should consider if it is reasonable to model the elimination and permutation of cuts
involving quantifiers as inequalities. One justification is that this decision has allowed
us to construct a model using the Geometry of Interaction, in which a quantifier is an
infinitary connective. In that model, we can see what features cause the increase in
denotation. Consider the inequality ε◦ ◦ ε ≤ id. The left hand side of the inequality is
given by
A
ε◦
-
M
i∈N
A
n
n
...
-
2
...
ε
2
...
i∈N
A
1
...
M
1
,
(where each copy of A is related to each other copy of A) and the right hand side is
given by
216
-
A
i inqN
n
n
...
M
2
...
id
2
...
i∈N
A
1
...
M
1
,
(where the ith copy of A is related only to the ith copy of A)
Intuitively (using the idea of data flow from [22]) we see that the former proof is
different from the latter in terms of which copies of A are connected to which; in the
former, there are “crossed wires”.
This interpretation breaks down, however, when one considers that proofs involving a
variable are not, in the usual classical sequent calculus, thought of as being a family of
proofs over the type of x. Rather, a proof with x a free variable are thought of as being
parametric in x, in the same way as a polymorphic type might be parametric (see [28]).
If that is the case, it is no longer acceptable to think of ∀xA as an infinitary conjunction.
In this case, we would expect to see models where eliminating and permuting cuts
against quantifiers are interpreted as equalities.
However, with the addition of axiomatic theories this reasoning breaks down. The
standard axiom for induction over the natural numbers, given by
·
·Φ
·
⊢ φ(0)
·
·Ψ
·
⊢ φ(n) ⇒ φ(n + 1)
Ind,
⊢ ∀xφ
gives proofs of ∀xφ that are no longer obviously parametric in x, and the ω rule
·
· Φ1
·
⊢ φ(0)
·
·
· Φ2
· Φn
·
·
⊢ φ(1) . . . ⊢ φ(n) . . .
⊢ ∀xφ
ω,
is not parametric at all. We conjecture that the our ordering is necessary to model such
rules.
217
Combinatorial proofs and Herbrand’s theorem
Hughes gives an account [37, 36] of “combinatorial proofs” for classical logic, built as
a fibration, where an MLL proof net lies over a classical proposition, where some of
the doors of the MLL proof net have been identified and other doors do not appear in
the proof net: thus the fibration models the action of contraction and weakening. The
identification must obey a certain graph-theoretic property. These combinatorial proofs
have the advantage of being canonical objects: no quotienting (over e.g. rewiring of
weakenings) is necessary. Thus the fact that an formula is classically provable iff some
other formula is linearly provable can be used to generate a single classical proof space
from many linear proof spaces.
Similarly, Herbrand’s theorem tells us that a first-order formula is classically provable
iff a certain constructed propositional classical formula is provable. If this relationship between first-order and propositional proofs can be given a similar combinatorial
flavour, then we will derive a similar abstract notion of combinatorial proof for firstorder classical logic. For this construction to work, we would need to identify a formula with its prenex normal form; we would therefore acquire a model of first-order
predicate logic for which the reductions involving quantifiers hold as equalities.
Dummett’s calculus, multi-abstraction and parallel lambda calculi
We have shown in this thesis how we can view proofs in Dummett’s calculus for intuitionistic logic as a fragment of classical first-order logic. It is not clear, however, how
to construct models of the calculus independently of this construction. Models would
need to exhibit a variant of a closure operator, whose properties we could extract from
that of the interpretation in first-order classical proofs.
Corresponding to these models, we would like to see the emergence of a lambdacalculus corresponding to Dummett’s calculus. This calculus would need:
1. Multiple conclusions;
2. A primitive (nondeterministic) composition on terms, analogous to the convolution ∗ on a Dummett category, such that the composition of terms s and t reduces
nondeterministically to either s or t; and
3. A variation on lambda abstraction, such that a term of type λx.(t1 , t2 , t3 ...tn ) :
218
A → (B1 , B2 , . . . Bn ) is thought of as a function with nondeterministic output,
which may be of any of the types B1 , B2 , . . . Bn . We call this multi-abstraction.
This work should benefit from and lead into the related field of lambda calculi for
classical logic [55, 24, 67].
Games semantics
A games semantics in currently development for propositional classical logic, due to
Pym, Ritter and Robinson, should extend easily to the first-order logic. Fascinatingly,
the games model lead to a simple notion of combinatorial model which corresponds
to Bibel’s matrix method of proof-search [9]. Observing that the matrix method for
first-order logic corresponds to a highly structured form of Herbrand’s theorem, we
conjecture that a combinatorial model of first-order classical logic would give an abstract interpretation of Herbrand’s theorem.
Semantics for deep inference The work in this thesis has shown that the sequent calculus and atomic deep inference systems may have the same notion of proof equality,
if we add additive cut-reduction to the theory of the multiplicative sequent calculus. It
remains to find a notion (or notions) of normalization on proofs in a symmetric deep
inference system, and to examine which equalities of the sequent calculus are compatible with the associated reductions.
219
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225
Appendix A
Monoidal categories
A.1 Adjunctions in monoidal categories
These proof are taken from [51].
A.1.1 Proof of Theorem 2.3.3
The strong monoidal structure of F and the adjunction induce the following morphisms
in C:
η
Gβ −1
G(ε ⊙ ε)
- G(F GA ⊙ F GB)
- G(A ⊙ B)
GA ⊙ GB - GF (GA ⊙ GB)
I
GβI−1 η G1
GF 1
We denote these morphisms as µA,B and µI respectively, and we observe first of all
that the following diagrams commute:
226
follows from:
F (GA ⊙ GB)
β −1
?
F GA ⊙ F GB
F µF G(A ⊙ B)
ε
?
ε⊙ε A⊙B
227
Fη
F Gβ −1
F G(ε ⊙ ε)
- F G(F GA ⊙ F GB)
- F G(A ⊙ B)
F (GA ⊙ GB) - F GF (GA ⊙ GB)
id
ε
ε
ε
?
?
−1
?
β
ε⊙ε
- (F GA ⊙ F GB)
- A⊙B
F (GA ⊙ GB)
and
F µI -
FI
βI−1
F GI
ε
?
id
I
-
?
I
follows from:
FI
F η-
F GF I
F GβI−1
-
id
ε
F GI
ε
-
?
βI−1 - ?
I
FI
We show naturality of µ:
GA ⊙ GB
Gf ⊙ Gg
µA,B
-
G(A ⊙ B)
G(f ⊙ g)
?
GC ⊙ GD
µC,D
-
?
G(C ⊙ D)
by establishing the commutativity of the following diagram in D:
228
Fµ
F (GA ⊙ GB)
id
6
F (GA ⊙ GB)
F (Gf ⊙ Gg)
-
ε
β −1
-
F GA ⊙ F GB
ε ⊙ε
A⊙B
F Gf ⊙ F Gg
f ⊙g
−1
?
?
β
ε ⊙ε
F (GC ⊙ GD) - F GC ⊙ F GD
C⊙D
id
F G(A ⊙ B)
F G(f ⊙ g)
?
?
Fµ
F (GC ⊙ GD)
-
ε
?
F G(C ⊙ D)
This diagram demonstrates the following equality in G:
ε ◦ F (G(f ⊙ g) ◦ µA,B ) = ε ◦ F (µC,D ◦ (F f ⊙′ F g)).
Since ε ◦ F ( ) is one direction of the isomorphism between the homspaces in the
adjunction, if images of two morphisms under the isomorphism are equal, then the
morphisms themselves are equal.
A similar diagram verifies the naturality of µI .
We now verify that µA,B and µI have the required properties to form, with F , a symmetric monoidal functor. We present diagrams in C which justify the commutativity of
the required diagrams in the same way as above.
229
F ((GA ⊙ GB) ⊙ GC)
id
id
?
F (GA ⊙ (GB ⊙ GC))
-
β −1
F GA ⊙ F (GB ⊙ GC)
-
id ⊙ β −1
id ⊙ F µ
F (id ⊙ µ)
230
?
F (GA ⊙ G(B ⊙ C))
-
β −1
?
F GA ⊙ F G(B ⊙ C)
id ⊙ ε
ε⊙
?
-
α
id ⊙ (ε ⊙ ε)
F (µ)
F G(A ⊙ (B ⊙ C))
F GA ⊙ (F GB ⊙ F GC)
ε
-
?
-
?
β
−1
F (GA ⊙ GB) ⊙ F GC
F ((GA ⊙ GB) ⊙ C)
F (µ ⊙ id)
?
?
(A ⊙ B) ⊙ F GC
ε ⊙ id
-
⊙ id
(ε ⊙ ε) ⊙ id
F GA ⊙ (C ⊙ C)
ε
β
−1
(F GA ⊙ F GB) ⊙ F GC
F (G(A ⊙ B) ⊙ GC)
F (µ)
id ⊙ ε
?
A ⊙ (B ⊙ C)
α
-
?
(A ⊙ B) ⊙ C
?
ε
F G((A ⊙ B) ⊙ C)
6
id
id
?
F G(A ⊙ (B ⊙ C))
F G(α)
-
F G((A ⊙ B) ⊙ C)
G behaves well with respect to associativity:
-
Fα
F (GA ⊙ (GB ⊙ GC)
G behaves well with respect to the units:
Fρ
F (GA ⊙ I)
id
?
F (GA ⊙ I)
F (id ⊙ µI )
?
β −1
-
?
ε
id
⊙
?
id
β −1
id ⊙ε
F GA ⊙ F GI
ε⊙
ε⊙ε
F G(A ⊙ I)
F G(ρ)
F GA ⊙ F I
id ⊙ F µI
F (GA ⊙ GI)
F (µ)
β −1-
- F GA
-
?
ρ
F GA ⊙ I - F GA
id
?
- A⊙I
ε
ρ
?
?
- A
ε
F G(A)
and dually for λ.
It remains to show that the unit and co-unit are monoidal natural transformations. First
ε : F G → Id. F G is a monoidal functor, with natural transformations
F GA ⊙ F GB
I
β
-
βI-
FI
F (GA ⊙ GB)
F µI -
Fµ F G(A ⊙ B)
F GI
Monoidality of ε is given by commutativity of the diagrams
F GA ⊙ F GB
β
ε ⊙ε
I
βI
?
F (GA ⊙ GB)
Fµ
A⊙B
?
ε-
F G(A ⊙ B)
?
F µI
A⊙B
I
?
FI
id
id -
?
F GI
id
ε- ?
I
which we established above.
The functor GF is monoidal
GF A ⊙ GF B
I
µ
-
µI-
GI
G(F A ⊙ F B)
GβI -
231
Gβ GF (A ⊙ B)
GF I
A⊙B
η ⊙η
GF A ⊙ GF B
µ
id -
I
µI
?
G(F A ⊙ F B) id
id
?
A⊙B
Gβ
?
η GF (A ⊙ B)
I
?
GI
?
I
GβI
η- ?
GF I
The left hand diagram follows from the equality
ε ◦ F (µ ◦ (η ⊙ η)) = β −1 = ε ◦ F (Gβ −1 ◦ η)
The left hand equality given by:
F (A ⊙ B)
F (η ⊙ η)
-
F (GF A ⊙ GF B)
β −1
β −1
Fη
?
⊙
Fµ
?
F GF A ⊙ F GF B
ε⊙
Fη
F GF (A ⊙ B)
ε
η F
F (A ⊙ B)
id
F Gβ −1
-
- F (A ⊙ B)
ε
ε
- ?
- FA ⊙ FA
id
FA ⊙ FA
F G(F A ⊙ F B)
- F G(F A ⊙ F B)
ε
β −1
?
- FA ⊙ FA
The right hand diagram follows from the equality
ε ◦ F (µ ◦ (µI ) = βI−1 = ε ◦ F (GβI−1 ) ◦ ε)
The left hand equality has already been established. The right hand equality follows
232
from
F GβI−1
- F GI
F
ε
η
-
F GF I
-
id
FI
ε
- FI
βI−1
-
?
I
A.1.2 Proof of Lemma 2.3.15
We will show that these define a commutative monoid on B by exhibiting the required
diagrams:
Associativity follows from:
FA
F∆
-
F (A ⊙ A)
F (id ⊙ ∆)
F (A ⊙ (A ⊙ A))
F∆
?
F (A ⊙ A)
κ
?
?
FA ⊙ FA
F (α)
f (∆ ⊙ id)
-
F (A ⊙ A) ⊙ F A
Left neutrality follows from
233
FA ⊙ FA
id ⊙ κ
?
?
-
id ⊙ F ∆
?
κ F A ⊙ F (A ⊙ A)
F ((A ⊙ A) ⊙ A)
κ
F ∆ ⊙ id-
κ
?
F A ⊙ (F A ⊙ F A)
α
κ ⊙ id
-
?
(F A ⊙ F A) ⊙ F A
id
-
F (I ⊙ A)
id
Fλ - ?
FA
FA
F∆
?
F (A ⊙ A)
κ
F (hi ⊙ id)
-
κ
?
FA ⊙ FA
F hi ⊙ id-
6
λ
?
F1 ⊙ FA
κ ⊙ id
-
(and similarly for right neutrality).
Symmetry follows from:
F
FA
κ
F (A ⊙ A) - F A ⊙ F A
∆ -
F
Fσ
∆
-
σ
?
?
κ
F (A ⊙ A) - F A ⊙ F A
234
FA
I ⊙ FA
Appendix B
Generalized quantifiers
Proof. We establish first that the triangle equalities hold for our lax adjunction. The
first triangle equality is given by the net
∃x
Ksub
Kref
t∗ φ : L
t∗ φ : R
t∗ (tx = y) : R
tx = tx : L
t∗ ((tx = y) ∧ [y]t∗ φ) : R
t∗ ∃x((tx = y) ∧ [y]t∗ φ) : R
[x]t∗ φ : L
[x]t∗ φ(y) : R
tx = tx ∧ [x]t∗ φ : L
t∗ ∃x(tx = y ∧ [y]t∗ φ) : L
t∗ φ : R
.
235
By the rule Eq2 , this is equivalent to
∃x
[x]t∗ φ : L
Kref
t∗ φ : R
t∗ φ : L
t∗ (tx = y) : R
[x]t∗ φ : L
tx − tx : L
[x]t∗ φ : R
t∗ ((tx = y) ∧ [y]t∗ φ) : R
tx = tx ∧ [x]t∗ φ : L
t∗ ∃x((tx = y) ∧ [y]t∗ φ) : R
t∗ ∃x(tx = y ∧ [y]t∗ φ) : L
t∗ φ : R
.
By the rule C UT ∃ , this reduces to
t∗ φ : L
Kref
t∗ φ : L
t∗ φ : R
tx = tx : R
tx − tx : L
t∗ φ : L
t∗ φ : R
(tx = y) ∧ [x]t∗ φ : R
tx = tx ∧ t∗ φ : L
.
Expanding the cut, using C UT ∧, we obtain
[x]t∗ φ : L
Kref
tx = tx : R
tx − tx : L
t∗ φ : L
t∗ φ : R
t∗ φ : L
t∗ φ : R
,
which, by C UT ⊤, is equivalent to the identity on t∗ φ.
236
The second triangle is given by the net
∃w
[y]Kref
[y]φ : R
[y]∆∗ t∗ (tx = y) : R
∃v
[y]∆∗ t∗ ((tx = y) ∧ [y]φ) : R
tw = y : L [y]φ : L
tw = y : L
Ksub
[y]t∗ ∃x((tx = y) ∧ [y]φ) : R
[v]∃x(tx = y ∧ [y]φ) : L
tw = z ∧ [y]φ : L
∗
tw = y ∧ [y]t ∃x((tx = y) ∧ [y]φ) : R
[y]t ∃x(tx = y ∧ [y]φ) : L
tv = y : L
[w]∃v(tv = y ∧ [y]t∗ ∃x((tx = y) ∧ [y]φ)) : R
∃w(tw = y ∧ [z]φ) : L
∗
∗
tv = y ∧ [y]t ∃x((tx = y ∧ [y]φ) : L
∗
∃v(tv = y ∧ [y]t ∃x((tx = y) ∧ [y]φ)) : R
∃v(tv = y ∧ [y]t∗ ∃x((tx = y) ∧ [y]φ)) : R
∃x(tx = y ∧ [y]φ) : L
.
That net reduces, via ∃M ERGE , to
∃w
[y]Kref
[y]φ : R
[y]∆∗ t∗ ((tw = y) ∧ [y]φ) : R
tw = y : L [y]φ : L
tw = y : L
tw = y ∧ [z]φ : L
Ksub
[y]t∗(tw = y) : R
[w]∃x(tx = y ∧ [y]φ) : L
tw = y : L
[w, y]t∗ ∃x(tx = y ∧ [y]φ) :: L
[y]t∗ existsx((tx = y) ∧ [y]φ) : R
tw = y ∧ [w, y]t∗ ∃x((tx = y) ∧ [y]φ) : R
tw = y ∧ [w, y]t∗ ∃x(tx = y ∧ [y]φ) :: L
∃w(tw = y ∧ [z]φ) : L
∃x((tx = y ∧ [y]φ) : L
.
Expanding the logical cut by C UT ∧, we obtain
∃w
[z]Kref
[z]φ : R
[z]∆∗ t∗ (tx = y) : R
[z]∆ ∗ t∗ ((tx = y) ∧ [y]φ) : R
tw = z : L [z]φ : L
tw = z : L
Ksub
[w]∃x(tx = y ∧ [y]φ) : L
z = tw : L
[z]t∗ ∃x(tx = y ∧ [y]φ) :: L
[z]t∗ ∃x((tx = y) ∧ [y]φ) : R
tw = z ∧ [z]φ : L
∃w(tw = z ∧ [z]φ) : L
∃x((tx = y ∧ [y]φ) : L
To proceed, we must now use that Ksub is defined inductively on the structure of the
relevant predicate; in this case we must expand Ksub applied to a quantified formula as
237
.
follows:
∃w
∃x
Ksub
[y]Kref
[y]φ : R
[w, x]∃k((tk = y ∧ [y]φ: L
[y](tx = tw ∧ t∗ [y]φ: L)
tw = y : L
[w]∃k((tk = y ∧ [y]φ: L
tw = y : L
[y]∆∗ t∗ ((tx = y) ∧ [y]φ) : R
tw = y : L [y]φ : L
[w](tx = y ∧ [y]φ): L
[x]tw = y : L
[y]t∗ (tx = y) : R
[y]t∗ ∃x((tx = y) ∧ [y]φ) : R
[y]t∗ ∃x(tx = y ∧ [y]φ: L)
tw = y ∧ [y]φ : L
∃w(tw = y ∧ [y]φ) : L
∃x(tx = y ∧ [y]φ)L
.
We may now apply C UT ∃ , to obtain
∃w
[y]Kref
[y]φ : R
Ksubs
[y]∆∗ t∗ (tx = y) : R
tw = y : L
[y]∆∗ t∗ ((tx = y) ∧ [y]φ) : R
(tx = y ∧ [y]φ): L
∗ ∗
[y]∆ t ((tx = y) ∧ [y]φ) : R
tw = y : L [y]φ : L
tw = y : R
[w]∃k((tk = y ∧ [y]φ: L
tw = y ∧ [y]φ : L
∃w(tw = y ∧ [y]φ) : L
∃x(tx = y ∧ [y]φ)L
.
Once again, we must expand the occurence of Ksub
∃w
Ksubs
Ksubs
[y]Kref
[y]φ : R
[y]∆∗ t∗ (tx = y) : R
[y]φ : R
tw = y : L
[y]∆∗ t∗ ((tx = y) ∧ [y]φ) : R
tw = y : L [y]φ : L
tw = y : L
[y]φ : R
tw = y : R
∗ ∗
[y]∆ t (tx = y) : R
tw = y : L
(tw = y ∧ [y]φ): L
[y]∆∗ t∗ ((tx = y) ∧ [y]φ) : R
tw = y : R
[w]∃k((tk = y ∧ [y]φ: L
tw = y ∧ [y]φ : L
∃w(tw = y ∧ [y]φ) : L
∃x(tx = y ∧ [y]φ)L
238
.
Since the application of Ksub to [y]φ is vacuous, we may apply Eq2 , to obtain
∃w
Ksubs
[y]Kref
[y]φ : R
[y]∆∗ t∗ (tx = y) : R
[y]φ : L
tw = y : L
[y]φ : L
tw = y : L
[y]∆∗ t∗ ((tx = y) ∧ [y]φ) : R
tw = y : R
[y]∆ t (tx = y) : L
(tw = y ∧ [y]φ): L
tw = y : L
tw = y : L [y]φ : L
[y]φ : R
∗ ∗
[y]∆∗ t∗ ((tx = y) ∧ [y]φ) : L
tw = y : R
[w]∃k((tk = y ∧ [y]φ: L
tw = y ∧ [y]φ : L
∃w(tw = y ∧ [y]φ) : L
∃x(tx = y ∧ [y]φ)L
Rewiring the weakening, we obtain
Ksubs
∃w
tw = y : L
[y]Kref
[y]φ : R
[y]∆∗ t∗ (tx = y) : R
tw = y : L
tw = y : L
[y]φ : L
[y]φ : R
tw = y : R
∗ ∗
[y]∆ t (tx = y) : L
[y]∆∗ t∗ ((tx = y) ∧ [y]φ) : R
tw = y : L
tw = y : L [y]φ : L
(tw = y ∧ [y]φ): L
[y]∆∗ t∗ ((tx = y) ∧ [y]φ) : L
tw = y : R
[w]∃k((tk = y ∧ [y]φ: L
tw = y ∧ [y]φ : L
∃w(tw = y ∧ [y]φ) : L
∃x(tx = y ∧ [y]φ)L
and we can remove the weakening and contraction by WC.
The subnet
Ksubs
[y]Kref
[y]∆∗ t∗ (tx = y) : R
[y]∆∗ t∗ (tx = y) : L
tw = y : L
239
tw = y : R
.
is, by Eq2 , equivalent to the identity on tw = y. Making this reduction, we obtain
∃w
tw = y : L [y]φ : L
tw = y : R
tw = y ∧ [y]φ : L
[y]φ : R
tw = y ∧ [y]φ : R
[w]∃w(tw = y ∧ [y]φ) : R
∃w(tw = y ∧ [y]φ) : L
∃w(tw = y ∧ [y]φ) : R
,
which is, by ∧A X and ∃A X , equivalent to the identity.
Next, we check the existence of a Frobenius strength. The strength is given by
∃x
Ksubs
Ksym
y = tx : R
y = tx : L
tx = yL
frobt :=
[y]t∗ ψ : R
[y]φR
tx = yL
[y]φL
tx = yL
[y](φ ∧ t∗ ψ)R
tx = y : R
[x]ψ : L
tk = y ∧ ([y]φ : ∧t∗ ψL
tx = y ∧ [y]φL
[x]∃k(tk = y ∧ ([y]φ : ∧t∗ ψ)L
∃x(tx = y ∧ [y]φ)L
ψ:L
∃k(tk = y ∧ ([y]φ : ∧t∗ ψ)L
;
∃x(tx = y ∧ [y]φ) : ∧ψL
we check that it is a left adjoint to
∃k
Ksubs
tk = yL
tk = yL
[k]ψ : R
tk = yL [y]φL
∗
[y]t ψL
tk = yR
[y](φ : ∧t∗ ψ)L
[y]φR
tk = y ∧ [y]φR
tk = y ∧ ([y]φ : ∧t∗ ψ)L
[k]∃z(tz = y ∧ [y]φ)R
∃z(tz = y ∧ [y]φ)R
ψ:R
∗
∃k(tk = y ∧ [y](φ : ∧t ψ))L
∃z(tz = y ∧ [y]φ) : ∧ψR
240
.
The composition
∃x
Ksubs
∃k
Ksubs
Ksym
y = tx : yR= tx : L
[y]t∗ ψ : R
[y]φR
tx = yL
tx = yL
tx = yL
[y]φL
[x]ψ : L
tk = yR
tk = y ∧ ([y]φ : ∧t∗ ψ)L
[x]∃k(tk = y ∧ ([y]φ : ∧t∗ ψ)L
ψ :L
[y]t∗ ψL
[y](φ : ∧t∗ ψ)L
tk = y ∧ ([y]φ : ∧t∗ ψL
∃x(tx = y ∧ [y]φ)L
[k]ψ : R
tk = yL [y]φL
[y](φ ∧ t∗ ψ)R
tx = y : R
tx = y ∧ [y]φL
tk = yL
tk = yL
[y]φR
tk = y ∧ [y]φR
[k]∃z(tz = y ∧ [y]φ)R
∃k(tk = y ∧ ([y]φ : ∧t∗ ψ)L
∃z(tz = y ∧ [y]φ)R
ψ : R
∃k(tk = y ∧ [y](φ : ∧t∗ ψ))L
∃x(tx = y ∧ [y]φ) : ∧ψL
∃z(tz = y ∧ [y]φ) : ∧ψR
is equivalent, via C UT ∃ and C UT ∧, to
∃x
Ksubs
Ksubs
Ksym
y = tx : yR= tx : L
[y]t∗ ψ : R
[y]φR
tx = yL
tx = yL
tx = yL
tx = yL
tx = yL
[x]ψ : R
tx = yL [y]φL
[y]φL
[x]ψ : L
[y]t∗ ψL
tx = y : R
tx = yR
tx = y ∧ [y]φL
[y]φR
tx = y ∧ [y]φR
[k]∃z(tz = y ∧ [y]φ)R
∃x(tx = y ∧ [y]φ)L
ψ :L
∃z(tz = y ∧ [y]φ)R
∃x(tx = y ∧ [y]φ) : ∧ψL
∃z(tz = y ∧ [y]φ) : ∧ψR
In this net, we substitute tx for y in ψ, and then y for tx in ψ; by Eq3 it is equivalent to
∃x
[x]ψ : L
tx = yL
ψ : R
[x]ψ : R
tx = yL
tx = yL
[y]φL
[x]ψ : L
tx = y ∧ [y]φL
tx = yR
[y]φR
tx = y ∧ [y]φR
[x]∃z(tz = y ∧ [y]φ)R
∃x(tx = y ∧ [y]φ)L
ψ :L
∃z(tz = y ∧ [y]φ)R
∃x(tx = y ∧ [y]φ) : ∧ψL
ψ : R
∃z(tz = y ∧ [y]φ) : ∧ψR
241
.
.
Rewiring the weakening which intriduces tx = y, we obtain
∃x
tx = yL
tx = yL
[x]ψ : R
tx = yL
tx = yL
[y]φL
tx = yR
[x]ψ : L
tx = y ∧ [y]φL
[y]φR
tx = y ∧ [y]φR
[x]∃z(tz = y ∧ [y]φ)R
∃x(tx = y ∧ [y]φ)L
ψ:L
∃z(tz = y ∧ [y]φ)R
∃x(tx = y ∧ [y]φ) : ∧ψL
ψ:R
∃z(tz = y ∧ [y]φ) : ∧ψR
and we may then remove the weakening and contraction (by WC) to obtain a net which
is equivalent to the identity.
The case for the opposite composition is similar, but involves a change in denotation
similar to that seen in the equality-free case.
Finally, we must check that the Beck condition holds for pullbacks of the form
X
hid, ti
-
X ×Y
t × id
t
?
Y
∆-
?
Y ×Y
We show one case: the pullback induces an isomorphism Σt hXid, ti∗ ∼
= ∆∗ Σt×id
242
One direction is obvious from the adjunction: we have a morphism
∃w
(tw = y) : L (tw = y) : L
(tw = y) : L (tw = y) : L
tw = y ∧ tw = y : L
[y]φ(w, tw)) : L
(tw = y) : L φ(w, tw) : L
tw = y ∧ tw = y ∧ [y]φ(w, tw)) : L
[w]∃(x, z)([z]tx = y ∧ [x]z = y ∧ [y]φ(x, z)) : L
(tw = y) ∧ φ(w, tw)) : L
∃w((tw = y) ∧ φ(w, tw)) : L
∃(x, z)([z]tx = y ∧ [x]z = y ∧ [y]φ(x, z)) : L
,
and the other is given by the net
∃z
∃x
Ksub
Ksym
[z]tx = y : [Lz]tx = y : L
[z]y = tx : R[z]y = tx : R
Ksub
[z]tx = y : L
z=y:L
[z]φ(x, y) : R
[z]φ(x, y) : R
[z]φ(x, tx) : R
[z](tx = y) : R
[z]tx = y ∧ [x]z = y : [y]φ(x,
L
z) : L
[z]((tx = y) ∧ φ(x, tx) :)R
[z]tx = y ∧ [x]z = y ∧ [y]φ(x, z) : L
[x, z]∃w((tw = y) ∧ φ(w, tw)) : R
∃x([z]tx = y ∧ [x]z = y ∧ [y]φ(x, z)) : L
[z]∃w((tw = y) ∧ φ(w, tw)) : R
∃(x, z)([z]tx = y ∧ [x]z = y ∧ [y]φ(x, z)) : L
∃w((tw = y) ∧ φ(w, tw)) : R
.
The proof that these two morphisms are inverse is tedious, and similar to others in this
appendix.
243