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Intuitionistic Necessity Revisited
G.M. Bierman
University of Cambridge
V.C.V. de Paiva
y
University of Birmingham
Second Revision: June 1996
First Revision: July 1995
Conference Version: Dec 1992
Dedicated to the memory of Hilfred Chau
Abstract
In this paper we consider an intuitionistic modal logic, which we call
IS4
2
. Our
approach is dierent to others in that we favour the natural deduction and sequent
calculus proof systems rather than the axiomatic, or Hilbert-style, system. Our nat-
ural deduction formulation is simpler than other proposals. The traditional means of
devising a modal logic is with reference to a model, and almost always, in terms of a
Kripke model. Again our approach is dierent in that we favour categorical models.
This facilitates not only a more abstract denition of a whole
class
of models but also
a means of modelling proofs as well as provability.
1 Introduction
Prawitz-style natural deduction is a framework somewhat underestimated by modal logi-
cians. But it is the cornerstone of functional programming, via the Curry-Howard corre-
spondence, which is one of the most exciting applications of logic to date.
Modal logic's intensional notions of necessity and possibility have proved useful in
many areas of computer science; so it would be go o d to extend the Curry-Howard corre-
spondence, with all its functional programming possibilities, to modal logic. This task is
nevertheless dicult in two respects. Firstly mo dal logics are traditionally dened in terms
of classical logic, whereas functional programming corresp onds to intuitionistic logic. Sec-
ondly providing any sort of formulation other than an axiomatic one is dicult for many
of the prop osed modal logics. Indeed providing a natural deduction formulation seems
harder than providing a sequent calculus one.
Addressing the rst diculty has recently become a popular topic, with many authors
trying to understand the notion of a constructive mo dal logic. Once the classical basis is
replaced, a multitude of intuitionistic versions b ecomes p ossible and it is challenging to
justify one choice over another. Most choices are made with reference to the mo del theory,
Address: Gonville and Caius College, Cambridge, CB2 1TA.
y
Address: School of Computer Science, University of Birmingham, Edgbaston, Birmingham, B15 2TT.
1
although almost exclusively in terms of Kripke-style models. Kripke semantics works both
for intuitionistic and modal logic, using separate accessibility relations for each, and the
choices appear in deciding how these relations are to interact.
The approach we persue here is somewhat dierent. We also use models to guide
our work but we prefer categorical ones. One reason is that, unlike the situation for
Kripke semantics, we are interested in modelling not just provability but also the proofs
themselves. This approach is often termed categorical pro of theory, or simply categorical
logic [18]. Category theory provides a language for describing abstractly what is required
of a model or, more precisely, what extra structures are needed for an arbitrary category to
model the logic. Checking that a candidate is a concrete mo del then simplies to checking
that it satises the abstract denition. Thus soundness, for example, need only be checked
once and for all, for the abstract denition. Then all concrete mo dels which satisfy the
abstract denition are also sound. Thus categorical semantics provide a general and often
simple formulation of what it is to be a model. This is of interest because it is often the
case that more traditional models lack any generality or are quite complicated to describe
(or both). In particular categorical semantics enable one to mo del some very p owerful
logics such as impredicative type theories and intuitionistic higher order logic.
This pap er is organised as follows. In
xx
2{3 we give axiomatic and sequent calculus
formulations of
IS4
2
, resp ectively. The theorems proven in these sections are surely known
to those working in this area, although we repeat them here for completeness. In
x
4 we
give our natural deduction formulation and compare it to Prawitz's proposal for a similar
logic. In
x
5 we dene the
2
-calculus, which is given by the Curry-Howard correspondence
from our natural deduction formulation. We also suggest a possible computer science
application for this calculus. In
x
6 we give in detail our categorical analysis of the necessity
modality. We give a sound denition of a categorical model for
IS4
2
.
2 An Axiomatic Formulation of IS4
2
Axiomatic, or Hilb ert-style, formulations are probably the more familiar metho d of den-
ing modal logics. They consist of a series of axioms and a few deduction rules. For
IS4
2
this consists of an axiomatic presentation of intuitionistic logic augmented with three new
axioms (
K
,
T
and
4
) and a new rule,
Nec
. The formulation is given in Figure 1.
It is worth explaining our axiomatic formulation. In giving the
Nec
rule it is vital
to insist that there are no free assumptions, otherwise one could deduce, for example,
A
2
A
. This restriction can be found in all presentations of necessity operators (e.g. [22 ]).
Given the importance of the context for this rule, it is surprising to nd that most authors
disregard the context for the other rules. Here we keep the context explicit in all the rules,
thus in the
Identity
rule we allow an arbitrary weakened context,
viz.
from assuming
; A
we can deduce
A
. The
Axiom
rule says that from any assumptions we can deduce one
of the axioms from the list in Figure 1.
Where it is not obvious by context a deduction in the axiomatic system is denoted by
the annotated turnstile
`
A
. An important property possessed by this formulation, which
is not always the case for modal logics, is the deduction theorem.
Theorem 1
If
; A
B
then there exists a proof of
A
B
.
Proof.
By induction on the structure of the derivation.
2
Axioms:
A
(
B
A
)
(
A
B
C
)
((
A
B
)
(
A
C
))
A
(
B
A
^
B
)
A
^
B
A
A
^
B
B
(
A
C
)
((
B
C
)
(
A
_
B
C
))
A
A
_
B
B
A
_
B
?
A
K
2
(
A
B
)
(
2
A
2
B
)
T
2
A
A
4
2
A
22
A
Rules:
Identity
; A
A
Axiom
where
A
is taken from above.
A
A
B
A
Modus Ponens
;
B
A
Nec
2
A
Figure 1: Axiomatic Formulation of
IS4
2
.
3 A Sequent Calculus Formulation of IS4
2
The sequent calculus formulation presented here is adapted from Curry's bo ok [5] and is
given in Figure 2.
;
are used to represent sequences of formulae and
A; B
for single formulae. The
Exchange
rule simply allows the permutation of assumptions. The
Weakening
rule p er-
mits assumptions to be discarded and the
Contraction
rule allows an assumption to be
duplicated. In what follows the
Exchange
rule is considered to be implicit, whence the
convention that
;
denote
multisets
. Negation is dened, as usual for intuitionistic logic,
as
:
A
def
=
A
?
:
The sequent calculus formulation, where we use the symbol
`
S
to represent a sequent
deduction, is equivalent to the axiomatic presentation given in the previous section.
Theorem 2
`
S
A
i
`
A
A
.
Proof.
By induction on the structure of the derivation. For example consider the following
case. Given a sequent derivation of the form
D
2
A
(
2
R
)
2
2
A
3
Axiom
A
A
B B;
C
C ut
;
C
; A; B;
C
Exchange
; B; A;
C
(
?
L
)
;
?
A
C
Weakening
; A
C
; A; A
C
Contraction
; A
C
; A
C
; B
C
(
_
L
)
; A
_
B
C
A
(
_
R
)
A
_
B
B
(
_
R
)
A
_
B
; A
C
(
^
L
)
; A
^
B
C
; B
C
(
^
L
)
; A
^
B
C
A
B
(
^
R
)
A
^
B
A
; B
C
(
L
)
; A
B
C
; A
B
(
R
)
A
B
; A
B
(
2
L
)
;
2
A
B
2
A
(
2
R
)
2
2
A
Figure 2: Sequent Calculus formulation of
IS4
2
.
Then by induction we have the axiomatic deduction of
D
,
`
A
2
A
. Assume that
2
represents the multiset
f
2
G
1
;:::;
2
G
n
g
. Then we can form the following deduction.
K
K
2
A
===========================
D:T :
n
2
G
1
(
2
G
2
:::
(
2
G
n
A
))
Nec
2
(
2
G
1
(
2
G
2
: : :
(
2
G
n
A
)))
22
G
1
2
(
2
G
2
:::
(
2
G
n
A
))
2
G
1
22
G
1
2
G
1
2
G
1
2
G
1
22
G
1
2
G
1
2
(
2
G
2
: : :
(
2
G
n
A
))
2
G
1
;:::;
2
G
n
1
2
(
2
G
n
A
)
2
G
1
; : : :
2
G
n
1
22
G
n
2
A
2
G
n
22
G
n
2
G
n
2
G
n
2
G
n
22
G
n
M.P.
2
G
1
;:::;
2
G
n
1
;
2
G
n
2
A
where
D:T :
n
represents
n
applications of the Deduction Theorem and
K
denotes a suitable
instance of the
K
axiom from Figure 1.
An imp ortant property of sequent formulations is the so-called cut-elimination theo-
4
rem. Here instances of the
Cut
rule are analysed and replaced with instances on smaller
proofs (the technical details are a little delicate; Gallier [8] gives a nice explanation). The
important new case for our logic is an instance of a (
2
R
;
2
L
)-cut,
viz.
2
A
(
2
R
)
2
2
A
; A
B
(
2
L
)
;
2
A
B
Cut
2
;
B
which is rewritten to
2
A
; A
B
Cut
:
2
;
B
Theorem 3
Given a derivation
of
A
, a derivation
0
of
A
can be found which
contains no instances of the
Cut
rule.
4 A Natural Deduction Formulation of IS4
2
In a natural deduction system, originally due to Gentzen [9], but subsequently expounded
by Prawitz [19], a deduction is a derivation of a proposition from a nite set of assump-
tion packets, using some predened set of inference rules. Within a deduction, we may
`discharge' any number of assumption packets. Assumption packets can be given natural
number labels (denoted by
x
) and applications of inference rules can be annotated with
the labels of those packets which they discharge.
The formulation is given in Figure 3. Our formulation diers from others in its simpler
treatment of the modality.
Some care should be taken with the (
2
I
) rule. The semantic braces, [[
]], mean not
only that
all
the assumptions are modal
1
but they are
all
discharged (and re-introduced).
The advantage of this formulation of this rule is that it satises a fundamental feature of
natural deduction in that it is
closed under substitution
. One might have been tempted to
give the rule for (
2
I
) as
2
A
1
2
A
k
B
(
2
I
)
;
2
B
where the assumptions must all b e mo dal but are not discharged and reintroduced, though
clearly this rule is
not
closed under substitution. For example, substituting for
2
A
1
, the
deduction
C
2
A
1
C
(
E
)
2
A
1
we get the following deduction
1
In comparison with our (
I
) rule where the standard notation is taken to mean that only one assump-
tion,
A
, is discharged.
5
?
(
?
E
)
A
[
A
x
]
B
(
I
)
x
A
B
A
B
A
(
E
)
B
A
B
(
^
I
)
A
^
B
A
^
B
(
^
E
)
A
A
^
B
(
^
E
)
B
A
(
_
I
)
A
_
B
B
(
_
I
)
A
_
B
A
_
B
[
A
]
C
[
B
]
C
(
_
E
)
C
2
B
(
2
E
)
B
2
A
1
:::
2
A
k
[[
2
A
x
1
1
2
A
x
k
k
]]
B
(
2
I
)
x
1
;:::;x
k
2
B
Figure 3: Natural Deduction Formulation of
IS4
2
.
C
2
A
1
C
(
E
)
2
A
1
2
A
k
B
(
2
I
)
2
B
which is no longer a valid deduction as the assumptions are not all modal. We conclude
that (
2
I
) should be formulated as in Figure 3, where the substitutions are given explicitly.
2
It is possible to present natural deduction rules in a `sequent-style', where given a
sequent
A
, then represents all the undischarged assumptions and
A
represents the
conclusion of the deduction. This formulation should not be confused with the sequent
calculus formulation, which diers by having operations which act on the left and right of
the turnstile, rather than rules for the introduction and elimination of logical operators.
The `sequent-style' formulation of natural deduction for
IS4
2
is given in Figure 4.
Two imp ortant admissible rules are
B
Weakening
; A
B
and
; A; A
B
Contraction
:
; A
B
2
This rule originates from the natural deduction formulation of intuitionistic linear logic [3].
6
; A
A
?
(
?
E
)
A
; A
B
(
I
)
A
B
A
B
A
(
E
)
B
A
B
(
^
I
)
A
^
B
A
^
B
(
^
E
)
A
A
^
B
(
^
E
)
B
A
(
_
I
)
A
_
B
B
(
_
I
)
A
_
B
A
_
B
; A
C
; B
C
(
_
E
)
C
2
A
1
2
A
k
2
A
1
;:::;
2
A
k
B
(
2
I
)
2
B
2
A
(
2
E
)
A
Figure 4: Natural Deduction formulation of
IS4
2
in sequent-style.
This formulation (where we use the symbol
`
N
to represent a natural deduction) is equiv-
alent to the axiomatic formulation given in
x
2.
Theorem 4
`
N
A
i
`
A
A
.
Proof.
By induction on the structure of the derivation.
4.1 Comparison with Prawitz's Prop osal
In his monograph [19, Chapter VI] Prawitz considers adding both necessity and p ossibil-
ity operators to natural deduction formulations of b oth intuitionistic and classical logic.
Our system is equivalent in terms of provability to the system he calls
I
S4
. Prawitz also
noticed the problem of closure under substitution, but his solution involves a new notion
of \essentially modal" formulae. What this amounts to is a relaxing of the restriction
that all the undischarged formulae are modal, but rather that there is somewhere in the
deduction a
complete set
of modal formulae which could have had deductions substituted
in for them. In tree-form this amounts to the rule (where the complete set of formulae is
in bold face)
7
1
2
A
1
k
2
A
k
B
(
2
I
)
:
2
B
Of course there is the extra work of nding this complete set; and indeed there may b e
more than one (the rather serious proof and model theoretic consequences of this are
discussed in
x
7). We feel that our prop osal is conceptually clearer: the only feature we
use is the discharging of formulae, which is already present.
4.2 Normalisation
With a natural deduction formulation we can produce so-called detours in a deduction,
which arise where we introduce a logical connective only to eliminate it immediately af-
terwards. We can dene a reduction relation, denoted
;
, (and called
-reduction) by
considering each case in turn. The treatment of the familiar intuitionistic connectives is
entirely standard and the reader is referred to other works [19]. The new case is where
(
2
I
) is followed by (
2
E
). Thus
2
A
1
:::
2
A
k
[[
2
A
1
:::
2
A
k
]]
B
(
2
I
)
2
B
(
2
E
)
B
is reduced to
[[
2
A
1
:::
2
A
k
]]
B:
As is standard, we say that a proof containing no instances of a
-reduction is in
-normal
form
. Our formulation of
IS4
2
has the following prop erty.
Proposition 1
If
A
in
IS4
2
then there is a natural deduction of
A
from
which is
in
-normal form
.
5 Term Assignment for IS4
2
The Curry-Howard correspondence [11] relates constructive logics to typed
-calculi. It
essentially annotates each stage of a deduction with a `term', which is an encoding of the
construction of the deduction so far. Consequently a logic can be viewed as a type system
for a term assignment system. The correspondence also links pro of normalisation to term
reduction.
8
The Curry-Howard corresp ondence can be applied to the natural deduction formulation
to obtain the term assignment system given in Figure 5. It should b e pointed out that
the natural number lab els mentioned ab ove, are replaced by (the more familiar) variable
names. The resulting calculus we call the
2
-calculus.
x
:
A . x
:
A
. M
:
?
(
?
E
)
.
r
A
(
M
):
A
; x
:
A . M
:
B
(
!
I
)
. x
:
A:M
:
A
!
B
. M
:
A
!
B
. N
:
A
(
!
E
)
. M N
:
B
. M
:
A
. N
:
B
(
I
)
.
h
M; N
i
:
A
B
. M
:
A
B
(
E
)
.
fst
(
M
):
A
. M
:
A
B
(
E
)
.
snd
(
M
):
B
. M
:
A
(+
I
)
.
inl
(
M
):
A
+
B
. M
:
B
(+
I
)
.
inr
(
M
):
A
+
B
. M
:
A
+
B
; x
:
A . N
:
C
; y
:
B . P
:
C
(+
E
)
.
case
M
of inl
(
x
)
!
N
k
inr
(
y
)
!
P
:
C
. M
1
:
2
A
1
. M
k
:
2
A
k
x
1
:
2
A
1
;:::;x
k
:
2
A
k
. N
:
B
(
2
I
)
.
box
N
with
M
1
;:::;M
k
for
x
1
;:::;x
k
:
2
B
. M
:
2
A
(
2
E
)
.
unbox
(
M
):
A
Figure 5: Term Assignment for
IS4
2
An important property of our system is that substitution is well-dened in the following
sense.
Theorem 5
If
. N
:
A
and
; x
:
A . M
:
B
then
. M
[
x
:=
N
]:
B
.
Proof.
By induction on the derivation
; x
:
A . M
:
B
.
Before we continue, a quick word concerning the (
2
I
) rule. At rst sight this seems
to imply an ordering of the
M
i
and
x
i
subterms. However, the
Exchange
rule (which does
not introduce any additional syntax) tells us that any such order is really just the eect
of writing terms in a sequential manner on the page.
The reduction rules derived from
x
4.2 can be given at the level of terms. These are
given in Figure 6 where the symbol
;
is used to denote term reduction. We have also used
the shorthand
~
M
in place of the sequence
M
1
;:::M
k
. The last reduction rule corresp onds
to the pro of reduction discussed in
x
4.2.
9
(
x
:
A:M
)
N
;
M
[
x
:=
N
]
fst
(
h
M; N
i
)
;
M
snd
(
h
M; N
i
)
;
N
case inl
(
M
)
of inl
(
x
)
!
N
k
inr
(
y
)
!
P
;
N
[
x
:=
M
]
case inr
(
M
)
of inl
(
x
)
!
N
k
inr
(
y
)
!
P
;
P
[
y
:=
M
]
unbox
(
box
N
with
~
M
for
~x
)
;
N
[
~x
:=
~
M
]
Figure 6:
-reduction rules.
5.1 A Computational Interpretation
As is now well known, the typed
-calculus can be thought of as a prototypical functional
programming language. An alternative view is that it can be thought of as an intermediate
language inside a functional language compiler. (The classic treatment of this is in Peyton
Jones' bo ok [12].) The equational reasoning of the
-calculus enables one to view compiler
optimisations as manipulations of terms of the intermediate language.
Inside a compiler there is a dierence b etween values stored directly in the lo cal stack
and those stored in the heap. Of course in the intermediate language (the
-calculus) such
operational dierences are not distinguished. Certain optimisations in compilers involve
moving between these dierent representations.
It seems that the
2
-calculus is an appropriate language for such distinctions to be
made explicit at the term, and type, level. Thus a value of type
A
is to be considered
a `local' value of (type
A
) and a value of type
2
A
a stored one. The restriction of the
2
R
rule can be interpreted as follows: if a value is to be placed on the heap then it must
only reference values also on the heap (i.e. the free variables should be of type
2
B
).
Manipulations of values to and from the heap are now represented by explicit terms.
This is analogous to Moggi's [16] prop osal of dierentiating, at the term level, between
canonical values and computations.
3
Indeed it would appear that a language combining
both Moggi's ideas and those ab ove, is worthy of further study.
4
6 The Categorical Mo del
The fundamental idea of a categorical treatment of proof theory is that propositions should
be interpreted as the objects of a category and proofs should be interpreted as morphisms.
The proof rules correspond to natural transformations between appropriate hom-functors.
The proof theory gives a number of reduction rules, which can be viewed as equalities
between proofs. In particular these equalities should hold in the categorical model.
Other categorical studies have been carried out, notably by Flagg [7]; Meloni and
Ghilardi [10] and Reyes and Zolfaghari [20]. However these have been mainly concerned
3
Moggi's language, the
computational
-calculus
, can also be seen as a modal logic [2].
4
In fact, this idea is b eing studied (and considerably extended) by P.N. Benton (private communication).
10
with categorical
model
theory, rather than categorical
proof
theory. In particular, they all
assume an isomorphism,
22
A
=
2
A
. In this work we have morphisms in both directions
(as they are provably equivalent) but we have
not
collapsed the model so that they are
isomorphic.
Let us x some notation. The interpretation of a proof is represented using seman-
tic braces, [[
]], making the usual simplication of using the same letter to represent a
proposition as its interpretation. Given a term
. M
:
A
where
M
;
N
, we shall write
. M
=
N
:
A
.
Denition 1
A category,
C
, is said to be a categorical model of a logic/term calculus i
1. For al l proofs
. M
:
A
there is a morphism
[[
M
]]:
!
A
in
C
; and
2. For all proof equalities
. M
=
N
:
A
it is the case that
[[
M
]] =
C
[[
N
]]
(where
=
C
represents equality of morphisms in the category
C
).
Given this denition we simply analyse the introduction and elimination rules for each
connective. Both this and consideration of the reduction rules should suggest a particular
categorical structure to model the connective. The case for intuitionistic logic is well
known; the reader is referred to Lambek and Scott's b o ok [13] for a goo d discussion.
Essentially the categorical model of intuitionistic logic (with disjunction) is a cartesian
closed category (CCC) with coproducts. Hence all we need do here is consider the modality,
which we shall do in some detail. The less-categorically minded reader may wish simply
to skip to Denition 2.
The introduction rule for the modality is of the form
. M
1
:
2
A
1
. M
k
:
2
A
k
x
1
:
2
A
1
;:::;x
k
:
2
A
k
. N
:
B
(
2
I
)
.
box
N
with
~
M
for
~x
:
2
B
To interpret this rule we need a natural transformation with components
:
C
(
;
2
A
1
)
C
(
;
2
A
k
)
C
(
2
A
1
2
A
k
; B
)
! C
(
;
2
B
)
Given morphisms
e
i
:
!
2
A
i
,
c
:
0
!
and
d
:
2
A
1
2
A
k
!
B
, naturality gives
the equation
c
;
(
e
1
;:::;e
k
; d
) =
0
((
c
;
e
1
)
;:::;
(
c
;
e
k
)
; d
)
:
In particular if we have morphisms
m
i
:
!
2
A
i
then we take
c
=
h
m
1
;:::;m
k
i
,
e
i
to be
the
i
-th product projection, written
i
, and
d
to be some morphism
p
:
2
A
1
2
A
k
!
B
,
then by naturality we have
h
m
1
;:::;m
k
i
;
2
A
1
;:::;
2
A
k
(
1
;:::;
k
; p
) =
2
A
1
;:::;
2
A
k
(
m
1
;:::;m
k
; p
)
:
Thus (
m
1
;:::;m
k
; p
) can be expressed as the composition
h
m
1
;:::;m
k
i
; (
p
), where
is a transformation
:
C
(
2
A
1
2
A
k
; B
)
! C
(
2
A
1
2
A
k
;
2
B
)
:
For the moment, the eect of this transformation will be written as (
)
and so we can
make the preliminary denition
11
[[
.
box
N
with
M
1
;:::;M
k
for
x
1
;:::;x
k
:
2
B
]]
def
=
h
([[
. M
1
:
2
A
1
]])
;:::;
([[
. M
k
:
2
A
k
]])
i
; ([[
x
1
:
2
A
1
;:::;x
k
:
2
A
k
. N
:
B
]])
The elimination rule for the modality is of the form
. M
:
2
A
(
2
E
)
.
unbox
(
M
):
A
To interpret this rule we need a natural transformation
:
C
(
;
2
A
)
! C
(
; A
)
:
It follows from the Yoneda Lemma [14, Page 61] that there is the bijection
[
C
op
;
Sets
](
C
(
;
2
A
)
;
C
(
; A
))
=
C
(
2
A; A
)
:
By constructing this isomorphism one can see that the components of are induced by
postcomposition by a morphism
"
:
2
A
!
A
. Thus we make the denition
[[
.
unbox
(
M
):
A
]]
def
= [[
. M
:
2
A
]];
":
From Figure 6 we have the term equality
. M
1
:
2
A
1
. M
k
:
2
A
k
x
1
:
2
A
1
;:::;x
k
:
2
A
k
. N
:
B
.
unbox
(
box
N
with
~x
for
~
M
) =
N
[
~x
:=
~
M
]:
B
Taking morphisms
m
i
:
!
2
A
i
and
p
:
2
A
1
2
A
k
!
B
, say, this term equality
amounts to the categorical equality
h
m
1
;:::;m
k
i
; (
p
)
;
"
=
h
m
1
;:::;m
k
i
;
p:
(1)
We can certainly dene an operation
2
:
C
(
; A
)
! C
(
2
;
2
A
)
;
f
7!
(
"
;
f
)
:
We shall make the simplifying assumption that this operation is a
functor
. However, notice
that if is the object
A
1
A
k
, then
2
will be represented by
2
(
A
1
A
k
), but
clearly we mean
2
A
1
2
A
k
. Thus we shall make the further simplifying assumption
that
2
is a
symmetric monoidal functor
, (
2
;
m
A;B
;
m
1
). This notion is originally due to
Eilenberg and Kelly [6]. In essence this provides a natural transformation
m
A;B
:
2
A
2
B
!
2
(
A
B
)
and morphism
m
1
: 1
!
2
1
which satisfy a number of conditions which are detailed in Appendix A.
Equation 1 gives
12
(
"
A
;
f
)
;
"
B
=
"
A
;
f
for any morphism
f
:
A
!
B
; or, in other words the diagram
2
A
A
2
B
B
-
2
f
?
"
?
"
-
f
commutes. Given the assumption that
2
is a symmetric monoidal functor, this diagram
suggests that
"
is a monoidal natural transformation. Again the unfamiliar reader is
referred to the app endix for denitions.
We have that from the identity morphism
id
2
A
:
2
A
!
2
A
, we can form the canonical
morphism
A
def
= (
id
2
A
)
. Equation 1 gives
A
;
"
2
A
=
id
2
A
:
The categorically-minded reader will recognise this equation as one of the three for a
comonad
. We shall make the simplifying assumption that not only do es (
2
; ";
) form a
comonad but that
is also a monoidal natural transformation. Hence the comonad is
actually a
monoidal
comonad. Thus our denition of a categorical model for
IS4
2
is as
follows.
Denition 2
A categorical model for
IS4
2
consists of a cartesian closed category with
coproducts, together with a monoidal comonad
(
2
; "; ;
m
A;B
;
m
1
)
.
We can now nalise the interpretation of the introduction rule for the modality.
[[
.
box
N
with
~
M
for
~x
:
2
B
]]
def
=
h
[[
. M
1
:
2
A
1
]]
;:::;
[[
. M
k
:
2
A
k
]]
i
;
A
1
A
k
;
m
2
A
1
;:::;
2
A
k
;
2
[[
x
1
:
2
A
1
;:::;x
k
:
2
A
k
. N
:
B
]]
Fact.
Recall that by condition 2 of our denition of a categorical model (Denition 1)
if two proofs are equal then so are their denotations. In more traditional model-theory
parlance this is a
soundness
theorem. Hence any concrete model satisfying the abstract
conditions of Denition 2 is a sound mo del of
IS4
2
.
7 Prawitz's Formulation and the Categorical Mo del
Although Prawitz's formulation has the appearence of being equivalent to the formula-
tion presented in this paper, in fact it has rather unfortunate pro of and model theoretic
consequences. Consider the following deduction in Prawitz's formulation.
13
222
A
(
2
E
)
(1)
22
A
(
2
E
)
(2)
2
A
(
2
E
)
A
(
2
I
)
2
A
The problem is deciding which formula was the (modal) assumption when the
2
was
introduced (the so-called `complete set' from
x
4.1). In particular two possibilities are (1)
and (2). In our formulation presented in
x
4, these alternatives represent two distinct
derivations,
viz.
222
A
(
2
E
)
22
A
[[
22
A
]]
(
2
E
)
2
A
(
2
E
)
A
(
2
I
)
2
A
and
222
A
(
2
E
)
22
A
(
2
E
)
2
A
[[
2
A
]]
(
2
E
)
A
(
2
I
)
2
A
Prawitz's formulation essentially collapses these two derivations into one. In other words
his formulation forces a seemingly unnecessary identication of proofs. Let us consider
the consequences of this identication with respect to the categorical mo del. The two
derivations above are modelled by the morphisms
"
22
A
;
2
A
;
2
(
"
22
A
);
2
(
"
2
A
):
222
A
!
A
and
"
22
A
;
"
2
A
;
A
;
2
"
A
:
222
A
!
A
respectively. Insisting on these being equal amounts to the equality
"
22
A
;
2
"
A
=
"
22
A
;
"
2
A
:
Precomposing this equality with the morphism
2
A
gives
2
"
A
=
"
2
A
:
It is easy to see that this is sucient to make the comonad
idempotent
, i.e.
2
A
=
22
A
.
It is worth reiterating that our formulation do es
not
impose this identication of proofs
and consequently do es not force an idemp otency.
14
8 Conclusions
In this paper we have considered the prop ositional, intuitionistic modal logic
IS4
2
, and
have given axiomatic, sequent calculus and natural deduction formulations; the corre-
sponding term assignment system as well as a general categorical model.
As menioned in the introduction we place particular importance on the natural de-
duction proof system. In his seminal monograph, Prawitz also considered formulations of
modal operators although he requires extra machinery specically for these mo dalities. At
the level of proofs his formulation introduces seemingly unnecessary identications, which
in the mo del forces an idempotency. Other authors have proposed alternative natural
deduction formulations but again they all require signicant extensions to the essential
nature of natural deduction (for example, by indexing formulae with certain informa-
tion). Examples of other proposals are those of Segerb erg [4, pages 29{30], Benevides and
Maibaum [1] and Mints [15, Pages 221{294].
5
Again we reiterate the conceptual simplicity
of our proposal.
We also prefer the use of categorical mo dels. Unlike other categorical work we have
placed emphasis on modelling the proof theory not just provability. Our resulting model
is considerably simpler than other proposals.
For the future we should like to consider other mo dal logics within our framework. It is
clear that not all of the hundreds of modal logics will t into our framework. However we
do not view this as a weakness of our work. Rather we feel it is important to identify those
modal logics which have interesting pro of theories and mathematically app ealing classes
of models. We should also like to pursue the computational interpretation discussed in
x
5.1.
Acknowledgements
This work was rst presented at the Logic at Work conference in Amsterdam in 1992. The
delay in publication is due to editorial problems of the conference organisers. We should
like to thank Richard Crouch, Rajeev Gore, Martin Hyland, Frank Pfenning and Alex
Simpson for useful discussions. We received nancial supp ort from the CLICS-I I pro ject
to present this work in Amsterdam.
References
[1]
M. Benevides and T. Maibaum
. A constructive presentation for the mo dal con-
nective of necessity.
Journal of Logic and Computation
, 2(1):31{50, 1992.
[2]
P.N. Benton, G.M. Bierman, and V.C.V. de Paiva
. Computational types from
a logical perspective I. Technical Report 365, Computer Laboratory, University of
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[3]
G.M. Bierman
.
On Intuitionistic Linear Logic
. PhD thesis, Computer Labora-
tory, University of Cambridge, December 1993. Published as Computer Laboratory
Technical Rep ort 346, August 1994.
5
Since we originally wrote this paper, the work of Simpson [21] and Pfenning [17] have also come to
our attention.
15
[4]
R. Bull and K. Segerberg
. Basic modal logic. In
Handbook of Philosophical
Logic
, pages 1{89. D. Reidel, 1984.
[5]
H.B. Curry
.
Foundations of Mathematical Logic
. Dover, 1976.
[6]
S. Eilenberg and G.M. Kelly
. Closed categories. In
Proceedings of Conference
on Categorical Algebra, La Jol la
, 1966.
[7]
R.C. Flagg
. Church's thesis is consistent with epistemic arithmetic. In
Intensional
Mathematics
, 1985.
[8]
J. Gallier
. Constructive logics part I: A tutorial on proof systems and typed
-
calculi.
Theoretical Computer Science
, 110(2):249{339, March 1993.
[9]
G. Gentzen
. Investigations into logical deduction. In M.E. Szabo, editor,
The
Collected Papers of Gerhard Gentzen
, pages 68{131. North-Holland, 1969. English
Translation of 1935 German original.
[10]
S. Ghilardi and G. Meloni
. Mo dal and tense predicate logic: mo dels in presheaves
and categorical conceptualization. In
Categorical Algebra and its Applications
, volume
1348 of
Lecture Notes in Mathematics
, pages 130{142, 1988.
[11]
W.A. Howard
. The formulae-as-types notion of construction. In J.R. Hindley and
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To H.B. Curry: Essays on combinatory logic, lambda calculus
and formalism
. Academic Press, 1980.
[12]
S.L. Peyton Jones
.
The Implementation of Functional Programming Languages
.
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[13]
J. Lambek and P.J. Scott
.
Introduction to higher order categorical logic
, volume 7
of
Cambridge studies in advanced mathematics
. Cambridge University Press, 1987.
[14]
S. Mac Lane
.
Categories for the Working Mathematican
, volume 5 of
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Texts in Mathematics
. Springer Verlag, 1971.
[15]
G.E. Mints
.
Selected Papers in Proof Theory
. Bibliop olis, 1992.
[16]
E. Moggi
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.
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A. Simpson
.
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. PhD
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16
[22]
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,
50:271{301, 1990.
17
A Monoidal Comonads
In this appendix we simply spell out the conditions implied by requiring that (
2
; ; ";
m
A;B
;
m
1
)
is a monoidal comonad. These notions are due to Eilenberg and Kelly [6].
Firstly requiring that (
2
; ";
) form a comonad amounts to the following two diagrams.
2
A
22
A
2
A
2
A
?
A
"
2
A
-
2
(
"
A
)
id
2
A
@
@
@
@
@
@
@
@
@R
id
2
A
22
A
222
A
2
A
22
A
-
A
-
2
A
?
A
?
2
A
Requiring that (
2
;
m
A;B
;
m
1
) is a monoidal functor amounts to the following four com-
muting diagrams.
1
2
A
2
1
2
A
2
A
2
(1
A
)
6
m
1
id
2
A
-
m
1
;A
?
2
(
snd
A
)
-
snd
2
A
2
A
1
2
A
2
1
2
A
2
(
A
1)
6
id
2
A
m
1
-
m
A;
1
?
2
(
fst
A
)
-
fst
2
A
2
A
(
2
A
2
C
)
2
A
2
(
B
C
)
2
(
A
(
B
C
)
(
2
A
2
B
)
2
C
2
(
A
B
)
2
C
2
((
A
B
)
C
)
6
2
A;
2
B;
2
C
6
2
(
A;B;C
)
-
m
A;B
id
2
C
-
m
A
B;C
-
id
2
A
m
B;C
-
m
A;B
C
2
B
2
A
2
A
2
B
2
(
A
B
)
2
(
B
A
)
-
m
A;B
-
m
B;A
?
A;B
?
2
(
A;B
)
Requiring that
"
is a monoidal natural transformation amounts to the following two com-
muting diagrams.
18
A
B
2
A
2
B
2
(
A
B
)
@
@
@
@
@
@
@
@
@R
"
A
"
B
-
m
A;B
?
"
A
B
2
1 1
1
?
m
1
-
"
1
@
@
@
@
@
@
@
@
@R
id
1
Requiring that
is a monoidal natural transformation amounts to the following two com-
muting diagrams.
22
A
22
B
2
(
2
A
2
B
)
22
(
A
B
)
2
A
2
B
2
(
A
B
)
-
m
2
A;
2
B
-
2
(
m
A;B
)
?
A
B
?
A
B
-
m
A;B
2
1
22
1
1
2
1
-
2
m
1
?
m
1
?
1
-
m
1
19
