Download 1. Logics MODALIIIES IN SUBSTRUCTURAL LOGICS Greg

Logique Analy s e 141-142 (1993), 25-38
MODA L IIIE S IN SUBSTRUCTURAL LOGICS
Greg RESTALL
1. Logics
Logic is about valid deduction. One central result in logic is the deduction
theorem.
A, I I
This Bresult
i f ties a fact from the metalanguage (that B follows from A, in the
context
a nof 1) to a fact in the object language ( A ---> B , in the context of 1).
This dconnection
is very important, because it shows how the properties o f
o
the conditional
in
a logic depend on the way premises are collected ton l y
gether
—
represented
here by the comma. Standard brands o f logics, like
i f E
intuitionistic
and
classical
logic, allow fo r all sorts o f rearrangements in
l - A They have structural
premises.
rules like these:
—
> B
F(A, A) I- B
1
Contraction (WI )
W e a k e n i n g (K )
-T
(( A )
11
A Mingle (M)
C o m m u t e d Associativity (CB)
C-)
11A, A) I- B 1 1 ( 2 1 , C), MI - D
1
(
1A
(B) C o m m u t a t i v i t y (CI)
-T)BO , B), O F D Associativity
1
1
B
,
A)I-- C
F
E
(
A(In
, Ba substructural logic, one or more of these rules are rejected. This means
that
, our premise collections have more structure than would otherwise be
(F
the(Acase. The number of premises, their order, their bracketing, or what isn't
B
a premise,
matters. Relevant logics, linear logics, and logics like BCK and
, )A logics are all substructural logics, because these logics keep track of
fuzzy
O,I
premises
in ways that logics with all the structural rules don't.
(FHowever,
even if we reject certain structural rules in general, by adoptC
DB
ing
a substructural logic, we may be interested in cases when substructural
F,
(0
A)
,1
B)D
I
26
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rules do apply. We may not think that all premises commute — but there
may be a small class of premises which do commute, and this may be important. Similarly, we may not be able to contract all premises — but some
might contract validly. So, it is interesting to see how we can keep track of
structural rules with a limited applicability. That is the topic of this note.
We will take our cue from Girard, and add modalities to our logics to regain structural rules [2].
It is easiest to give our account of modalities in substructural logic if we
use an algebraic presentation of the formal systems. This will make the
theorems easier to prove than would otherwise be the case. So it is to this
that we will turn.
Substructural logics (like any) can be modelled by propositional structures. Think of the elements of these structures as propositions, and the operations as ways to form new propositions out of old.
Definition 1 A basic propositional structure is a 5-tuple ( P; • , e,
where
• P is a nonempty set of propositions.
• i s a partial order on P, representing entailment between propositions.
• The binary operation • on P, called fusion. This represents premise
combination. Fusion preserves the entailment ordering. I f a a ' and
b b ' then a • b • b r
• The element e E P is a right identity for fusion. This means that for all
a e P, a •e = a Because of this, e represents logical truth.
•The binary operation o n P residuates fusion. For all a, b, c E P,
a • b c if and only if b 5. a c This means that i s our conditional
operation on P. (Recall the deduction theorem, tying the conditional to
valid deduction. This condition is the deduction theorem in algebraic
form.) If a b then by residuation and identity, e a b So, if a entails b, then the conditional a b is a logical truth. This is a desirable
connection between entailment, logical truth, and the conditional.
If we wish to model conjunction and disjunction, we require that 5_ be a lattice order. In these structures, conjunction is the greatest lower bound, and
disjunction, the least upper bound.
Definition 2 A basic propositional structure with conjunction and disjunction is a basic propositional structure where the underlying entailment ordering is a lattice.
Definition 3 A basic propositional structure with a distributing conjunction
and disjunction is a basic propositional structure where the underlying en-
MODALITIES IN SUBSTRUCTURAL LOGICS
2
7
tailment ordering is a distributive lattice.
I f we wish to model negation, we can add a unary operator — on P, with
suitable conditions.
Definition 4 A basic propositional structure with an intuition istic negation
is a basic propositional structure equipped with a unary operation — such
that for each a, b , c e P, a • b – c if and only if c • b 5.–a..
This c o n d it io n p ro v id e s u s w i t h co n tra p o sitio n i n t h e f o r m
a – b = b – a , and double negation introduction, a – – a.
Definition 5 A basic propositional structure with a de Morgan negation is
a basic propositional structure with an intuitionistic negation — such that,
in addition, for each a E P, a = – – a.
We use propositional structures to model valid deduction in the usual way.
Take a propositional language with connectives – ), 0 and t and any o f
A, y and –, as desired. We define the interpretation h(A) o f a formula A in
a propositional structure P recursively, f ro m arbitrary assignments t o
atoms, using these clauses:
/40 = e I( , 40 B)= h(A)• h(B) h( A–) B)= /1(A) h ( B )
k i t A B) = h(A) n h(B) h(A\./ B)= h(A)u h(B) 1
(
Take- a1 class
, 1 ) =X of– propositional
h ( A ) structures. We say that a formula A is
X-valid if for each structure P E X and for each interpretation h into P, we
have e h ( A ) . We write this ' I -I A . ' For largely uninteresting historical
reasons, the class of all propositional structures with a de Morgan negation,
conjunction and disjunction is called LDW, and its cousin which contains
only those structures in which disjunction and conjunction distribute is
called DW.
One interesting propositional structure which illustrates our definition is
13N4. Belnap [I I, Meyer, Giambrone and Brady [3] and Slaney [4] all extol
its virtues. The structure is simple.
28
GREG RESTALL
T
e= B
F
F
B N F
T
B N
F
F
T T N F
T
T
F N
T
B
T
B N F
B
N N F F
N
T N T N
N
F
F
T
F
B A I T'
F
F
F
T
T
T
T
F
Conjunction and disjunction are given by greatest lower and least upper
bounds on the lattice ordering — which is indicated here by the Hasse diagram. This structure has a number of notable features: not least being the
fact that negation has two fixed points, B and N. This shows that a de
Morgan negation need not satisfy the usual classical conditions.
Furthermore, premise combination is associative and commutative, but not
idempotent. So BN4 satisfies some of the structural rules we've seen, but
not all. I t is helpful to catalogue structural rules in algebraic form, like this:
WI
b•aa
M a • a a
B ( a • b) • c a • (b • c)
CB ( a • c) • b a • (b • c)
CI a • l , b • a
We can tack selections from these onto our basic logic to make all sorts of
interesting formal systems. Given a class of propositional structures X, the
class X + Y is the subclass of X of all structures which satisfy condition Y.
Girard's original linear logic (without exponentials) is then given by LDW
+ B + CI. So, in linear logic premise combination is associative and commutative. The contraction-free relevant logic C adds distribution to linear
logic, so it is given by DW + B + CI The relevant logic R is given by DW
+ B + CI + WI. We elide the distinction between classes of propositional
structures and logics, so we will say things like R = C + WI from time to
time.
Now to round this section off we'll prove an important result which
shows that we can restrict our attention to propositional structures of a particular kind, when using them to model a logic.
Definition 6 A propositional structure P is said to be complete if for every
X ç P , the join and meet VX and AK exist.
Fact 1 Any propositional structure can be embedded as a substructure in
MODALITIES IN SUBSTRUCTURAL LOGICS
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a complete propositional structure.
Proof Take a propositional structure P. We take the elements of the completion of P, P* to be the ideals of P. A subset X g P is an ideal if for each
x e X, if y x then y E X too. If P is equipped with disjunction, we require
that x, y E X only if XE X too.
Conjunction in this structure is simply intersection of ideals (the intersection of any set of ideals is an ideal). So, arbitrary meets exist in this
structure. This means that containment, c , is the ordering in the new
structure.
If disjunction is present in P, then the disjunction X u Y is the set
t
tion
z as union of ideals. (If ideals are merely downwardly closed sets, then
the union of two ideals is an ideal.) Arbitrary joins exist in this structure
: I f disjunction is simply union, then ideals are closed under infinite
too.
3
unions.
If disjunction is defined in terms of disjunction in P, the definition
isx a little more complex.
3
y V I x , : i E / 1 = l z : f o r so m e i
E a n_ X
ii E
( t,
Z U . d
U 1
X
x
4
,
E
E
Fusion, the residual, e* and negation ( if present) are defined in similar
ways
X
,
y X• Y=tz :3x 3y ( x EX,y EYandz x oy ) }
EX =
,
Yt z : e
a3 x- 3*Xy=1.z:3x(x e X and z n( x x x) 1
ItdEis simple
e
to check that the conditions on propositional structures are satzX }in, the completion P*, so P* is a propositional structure. The original
isfied
structure
P lives in P* in the guise of its principle ideals. The map
xy
fu: l y : y 5_ x l is an injection of P into P*.
EIf our original structure P satisfied a structural rule, then it is clear that P*
a
a satisfies that rule (check the definition of fusion).
also
ln
.d result shows us that complete propositional structures are enough,
This
O
when
z it comes to modelling logics. The class of validities in all 91 propositional
structures is the same as the class of validities in all complete
tx
91
propositional
structures, because any counterexample to validity in an
h>
e
r
w
i
s
e
,
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incomplete structure survives as a counterexample in the completion of that
structure. Th is fact is important, because we w i l l need to assume that
structures are complete in one result coming up.
Note that in complete propositional structures P, A P x for any x E P
So, complete propositional structures have a least element. In what follows,
we will use '0' to denote the least element in a propositional structure.
Excursus This embedding result also motivates a short polemical point
about conjunction and disjunction. The result has a simple corollary that
any structure without conjunction and disjunction can be extended to include conjunction and disjunction in a natural, painless way. Furthermore,
in this extended structure, conjunction distributes over disjunction. This is
especially important for devotees o f linear logic. In linear logic, additive
conjunction does not distribute over additive disjunction. Our result shows
that this is not a feature of the substructural nature of the logic — rather, it
is a bare fact about the additive connectives. Since it is quite hard to interpret additive disjunction in linear logic, it is quite hard to see what a failure
of distribution amounts to. Unless some interpretation for disjunction can
be found, in such a way that motivates the failure of distribution, there is
little reason to favour linear logic over its distribution-added cousin C. E l
Leaving polemics about distribution aside, we will assume from now, that
all logics come equipped with disjunction and conjunction (whether they
distribute or not) because this will make the modal conditions simpler to
state.
2. Modalities
There are a number of ways to introduce modalities to substructural logics.
This note will focus on extremely popular modalities in the vicinity o f S4
and S5. In our context, an S4 necessity satisfies the following conditions.
T EA 4 A
A
o
4
D(A B ) —› (DA —› EB)
E
I
f
h
A
then
D
A
A
0
A
E
D
B
I f a de
C
and if the S4 necessity also satisfies
A Morgan negation is present,
O
D
A
B
5
o
)
0
A
A to be an S5 necessity. These necessities have interesting properties
it is said
—
>
n
—
c
—
A
MODALITIES IN SUBSTRUCTURAL LOGICS
3
1
in substructural logics.
Fact 2 All S4 necessities satisfy 1- * A Y oB) 44 DA y DB
Proof 1- * A Y oB) n A y oB by T
oI- DA -4 oA v DB, so by N
we have I- oB °
.I-oDA
F —}
o D(oA
r
ty DB)
h Similarly
e
result,
using the lattice properties of y .
(
o, Ft ho e( or A
D
— Ai call
d
r* eva proposition
c t[ D
i oB Nn) a, ,necessitive i f and only i f for some A,
We'll
s
o
w
e
o N 44AoA. Iet is simple to show
w
that if o is an S4 necessity, that N is a nea aif and
v only
cessitive
v
hh
ve if I-eN o N . This fact means that necessitives are
o
u )disjunction
r,
closed
under
(where present). Necessitives are also closed unD
B
der fusion.
a
n
d
—
>
Fact 3 I n lo g ic s w i t h f u s io n , a l l S 4 necessities s atis fy
o D(EA0 oB)<—>IDA0DB.
w
i
t
h
Proof Clearly, I- * A 0 DB) —* oA 0 cif F or the converse, w e have
I-4 DB—*(DA—*DA DB). So, I- DDB o ( D A —*DA0 DB). This quickly gives
g oB --> DA
i —*(DA
v 0 DB) and hence I-DA 0 DB—* o(DA 0 oB) as desired. <
e
s
,
In addition, if the modality is S5ish, then necessitives are closed under
negation.
Fact 4 All S5 necessities satisfy I- D—IA 44
Proof Left to right is T
negation
elimination.
o
structures we can model a necessity by adding a unary
, Inapropositional
n d
operator on the structure.
r i g h t
t
o 7 In a propositional structure P, a function I : P P is an S4
Definition
l e operator
interior
f t if it satisfies these conditions.
i
s
ae n
g
i 1(a)5_
v
i
1
4
n
1 ( ya
b
e
n b )
1
4
l
If,
in
addition,
it satisfies
( =
,
1
a
5
( 1
a
) (
o
e
)
=
a
)
nn
I
d
=
(e o
d
u
•I b
l
(
e
1
Ia
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it is said to be an 55-interior operator. A propositional structure with an
interior operator (whether S4 or S5) is said to be a modal propositional
structure.
These conditions on I are forced by the axioms for S4 and S5 necessities.
(For the fusion and disjunction clauses, see Facts 1 and 2). / is called an
'interior' operator on a set of subspaces of a space because of conditions
like 4
modalities,
and the observation provokes an alternate way of presenting the
/
semantics
of
modal operators. Instead of defining an 'interior' operator /,
a n
we can restrict ourselves to the class C of 'open' elements. The objects a
d
such that 1(a)= a.. That is, the necessitives of our earlier discussion.
T
r
Fact
5 In any complete modal propositional structure the class 0' of open
T
elements
contains e and 0, and is closed under fusion, conjunction and disjunction.
h
i
Proof This is simple. T
s
by
1 eg i v e s
h
conjunction,
disjunction and fusion.
lu s
a
T
4
0call a
) class U on a propositional structure a potential open class if it
Let's
s
h
5
contains
e and 0, and is closed under fusion, conjunction and disjunction.
n
e
Then
we
0
, have another fact.
o
cs
o
t
o
Fact
6( A propositional
structure P with a class potential open class C is a
1
0
)
g
modal
propositional
structure
with interior operator I if we define
n
=
o
d
O
.
n 1( a) =V x 5 a and x 0 1
iWt
e
ie
provided
that the infinite joins exist.
u
o
h
a
n proviso is unimportant in practice, because we can assume that
This
n
v
e
n
propositional
structures are complete, because of Fact 1.
s1
o
n
(
e
Proof
t
It is simple to show that I a) E C for each a e P, d that 1(a)= a
i)
and I(a)n5 a for each a E P.
ifor each a EC. It follows that I(I(a)S= 1(a) a
,=
We
have
1(e)=
e
as
e
EC
U
.
Similarly,
as
C
is
closed under disjunction ( if
c
u
e
present)
a
n
d
f
u
s
io
n
,
w
e
h
a
v
e
1(
.1(
a) u 1(b))= 1(a)u 1(b) and
e
/41(a). 1(b)). 1(a)• 1(b).
d
a Finally, if conjunction is present, we can note that 1(a nI)) 1 ( a ) and
i
nI( anb) . 0 ) gives 1(a n17)5_ 1(a)nl(b). Conv ers ely , 1 ( a ) n l ( b )
n
d
t
h
/
e
s
a
h
l
o
g
w
MODALITIES IN SUBSTRUCTURAL LOGICS
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3
a nb and I( a) n 1(b) E 0 gives I ( a) n 1 ( a n b), as desired.
This is a useful fact, because we can define a modal operator on a propositional structure merely by specifying its set of open elements. This way of
specifying a modal operator in a propositional structure is much simpler
than defining a function / from scratch. So, in what follows we will define
our interpretations of modalities by specifying the set of open elements.
This set of open elements provides an interesting structure inside the
larger propositional structure. We can pick out these elements in our language by using the modal operator o Whatever the formula A is evaluated
as, we knowe that DA will be evaluated as an open element. This is where
adding modalised structural rules becomes interesting. We can add a number of axioms or conditions to our logic, saying that the open elements satisfy structural conditions which are not shared by all other propositions.
Here are some axioms and conditions.
WI 0 E I A —› EA 0 EA a a • a
f o r each a e C
K ( D B 0DA—› EA h i • a a
f o r each a, b E C
M
cB n O A 0 DB)0 DC—>EA 0 (DB0 EC) ( a•b) •c a•( b•c ) for each a,b,c E C
o
E
CB0( D
A 0 o A Ed3 o EA a • b 1 3 • a f o r each a,bc e
CI
A
o0 C
)D
These
axioms say that necessitives satisfy conditions which propositions in
A
general,
in our logics, fail. The first axiom, WI , ensures that necessitives
0
—
contract.
I n logics like CK, not all propositidis contract. We 'regain' a
E
measure
)I
of contraction by adding a modality which contracts. Similarly, we
may
E be interested in modalities which give us weakening, or mingling for
B
necessitives.
-A
-a
Definition
8 Given a non-modal logic X, we'll call XS4
C
n
•
E
„logic
t h you
e get when you add an S4 modality satisfying the conditions
a
C
A
loa Adding structural rules to the open elements of a propositional structure
provokes
two questions. Firstly, does it do anything to the non-modal struco
(f of the logic? That is, is the extention conservative?
ture
o
tD
r
o
Definition
B
9 A logic X is a conservative extension of Y if it extends Ys voe
C
cabulary,
but
nothing that is X valid in the old vocabulary of Y isn't also Y
0
valid.
.oa
c
S
C
i)h
m
(a
iaE
l•0
a
c
r)
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Secondly, does the extension do us any good? We will answer these two
questions in the next sections.
3. Conservativeness
Fact 7 XS4 and XS5 are conservative extensions of X.
Proof Suppose A is not X valid. It has a counterexample in the propositional structure P. P can be made into an S4 or S5 modal structure by defining the class 0 of opens on P to be the class of all propositions.
<
Fact 8 Any logic X is conservatively extended by XS4 + C
rules C
istructural
o
.I .D. Ct o t
Proof
C , , ,This time, on any propositional structure P, take the set of opens to
n
beo10, el. This is a sublattice, and it satisfies any added structural rule
,:qp.f o r
a n y
<
.
Fact 9 A n y logic X satisfying K is conservatively extended by
XS4+ C
lo
Proof
. . . C In logics satisfying K, 10, el is closed under negation. Firstly, note
that — 0 = Tand — T = 0 by contraposition, where T is the top element
no
of the lattice. Then it is sufficient to show that given K, e = T . But this is
,
easy.
Given K,b=b•e5_e for all b. So, we have our result, as 10, el is a
f of
o opens satisfying all structural rules.
set
<
r However, the conservative extension results end here. In logics where
—
a e 0 , the class of opens must at least contain e,—e, 0, —O. And in many
logics,
—e is not guaranteed to satisfy structural rules. Let f = —it Then we
n
can collate together a number of facts.
y
s t 10r Any logic X in which le f —› fo f is not conservatively extended
Fact
u XS5
c + WI..
by
t u r
Fact
a l i l Any logic X in which i i l o f — ) f is not conservatively extended
by
r XS5 + M
o o r
u
Fact
X S 125 Any logic X in which k f o f --> fo f is not conservatively exl
tended
+
Kby XS5+ CI o
.oe
Fact
13 Any logic X in which k f 0(f 0 f) - 4 ( f 0 f)—) f is not conserva.s
C
l
o
t
o
C
n
MODALITIES IN SUBSTRUCTURAL LOGICS
3
5
tively extended by XS5+ B
o
These
facts have bite, because they apply to interesting logics. It is not
.
difficult to show that in linear logic or C, f - - - >f o f . So, these logics are
not conservatively extended by their cousins with a contracting S5 modality. Similarly, in R, k f o f —> f . So, R is not conservatively extended upon
the addition of an S5 modality with weakening or mingling. What is the
significance of these results? It is hard to tell, beyond the fact that it is another reason to prefer S4 over SS. S4 gives us a sublattice of 'opens', closed
under fusion. We have sublattices like these at hand which satisfy every
structural rule, in every propositional structure. S5 forces the sublattices to
be closed under negation, and this seems to be a Bad Thing in general.
Only the case of logics satisfying K stands out as an exception.
So much for tracking the effects of adding modalities on the nonmodal
fragment of logics. Now we will see what these modalities can do for us.
4. Embeddings
The substructure of opens in a modal propositional structure has a number
of interesting properties. It is a sublattice, and it is closed under fusion. If
the modal operator is S5ish, it is also closed under negation. However, in
general, it is not closed under implication, and if the modality is S4ish, it is
not, in general, closed under negation. However, we have a number of interesting results in approximating implication and negation in the substructure of opens.
Fact 14 If 0 is a collection of opens in a propositional structure P, then for
each a, b, c E 0, a- b c if and only if b 1 ( a c ) .
Proof I f b 1 ( a c ) then b a c and hence, a. b c . Conversely, if
a -1
,
From
now, let a D b stand for I(a b ) . This result means that in the
c
context of 0 , D residuates fusion. This is an important fact.
a
n 15 If 0 is a collection of opens in a propositional structure P equipped
Fact
d a de Morgan negation —, then for each a, b, c E 0, a • b 1(—c) if and
with
only
if c - b I(—a),
/
(
Proof:
I f a • b .. I(—c) then — /(—c)• b5_—a, and hence I(-4-c)• b) 1(—a).
b
However, c —/(—c), so c • b --1(—c)• b. But /(c • b)= c • b (as b and c are
)
1
(
a
c
)
.
H
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opens), so c • b I ( c • b) I (-1 (— c )• b) 1 (—a ) as desired.
From now, let — a stand for I(—a). This result means that in the context of
0, — is an intuitionistic negation. This, also, is an important fact.
Now we get to the payoffs. Let's show how we can embed logics with
structural rules in logics which don't have those structural rules (which, in
case you've forgotten, was the motivating thought behind this work). We
need to define the translation A ' o f a formula A. It is defined recursively:
13
(A A B)
3
n v= B)
(A
=
A
D(A 0'= =
n
A
(Av —> =
B
p
n (— A
c•
n(A° —› BE)
)°
Fact 16 D
In -any modal propositional structure P, a formula of the form A '
is always
, interpreted as an open element.
( 1
Proof: Observe
the construction of A ° I t s atomic parts p ' are interpreted
V )
as open elements, and any of the original connectives in A are modified in
A ' to always map into open elements too.
Fact 1 7 A i s X S 5 + q , . . . c
C v„ a—lvalid..
nXo+ —
i d
i
f
a
n
d
Proof:
o
nI f Al ' i sy X S 5 + C „ , — valid t h e n i t i s t r u e i n a l l
iXS5+ C f
an
XS5 + C
Al E
1. . elements.
all
So, A ' will come out true in this structure by our assumption
ithat. C
s
,n. o. . itC is
„ , XS5 + C
terpretation,
n
ss tot rr u —
c because I is the identity operation (as 0 = P). This means that
A
X
+
C
vtu is
a
l
i
uc rt e d
IH
No
le t A b e x + c , . . . c
iu r ow
e w e v
_e
XS5+
n
C
v
,
. valid B u t this is simple. Take a XS5 + C
fs . r ,a Cl i „ d —
ture
in which
A " is invalid. Let h be an interpretation which sends A ' to
—
lA
o
W
e
w
L element x e in P. The class of open elements e in P is a X i van
. Cl i in s
h
a
e.e.wa
structure,
as we have seen, where we take D to be the residual o f fusion.
dd
n
t
E
s
t
o
r
u
tt c a
a n d e f i n ec a- n in t e rp re t a t io n h ' i n t o 0 b y s e t t i n g
We
aA
s
h
o
khP' (p )= l(h
(p
))=
/
sh
w
a
e(bc
dvt t e s eh
a
e
ithp t)r. e I n
ta h i s
dh
ein An . t e r p r
e
seX' t a t i o n
se+i
,
a
thCs
'
(
m
o
MODALITIES IN SUBSTRUCTURAL LOGICS
3
7
tion, and so, we have a counterexample to A in this structure. But we assumed that A has no counterexamples in X + C
couldn't
have a counterexample to A ' i n P. This means that A i s
I
XS5
+
C
_ C , , s t r u c t u r e s ,
sn 1 8o A i s X S 4 w+ c eno—valid i f a n d o n l y i f A ' i s
Fact
,
X + C „ — valid, provided X is negation free.
3
— Just like the proof in the previous fact. We can get away with using
Proof.
v S4
a style
l modality because we have no need to show that the structure C
an
of
i opens
d
is closed under negation. What about negation and S4 modalities?
Is
a there anything else we can say? There is, but it is incomplete. For now, if
Xs is a logic, let JX be the corresponding logic given by 'liberalising' the
negation laws to allow an intuitionistic negation. Then, we have the followd result.
e
ing
s i
r e19 A is XS4 + c
Fact
if
X + C , , — valid
nd A
a .' is f—
v a l i d
Proof Just as before, but we note that the negation operator—on C satisfies
( w h e r
the conditions for an intuitionistic negation, by Fact 15. So, the resulting
esubstructure C inside an original modal stucture P is a JX + C „ structure.
X
h It is aalot harder
s to prove the converse, because we would have to show
that
a every J X + q... C„ structure can be embedded within a XS4 C
This
1
0
dstructure.
e is difficult, because we need to construct a de Morgan negation
and
a
modality
from 'thin air' while ensuring that they interact in the
.M. . C o
r
desired way (satisfying
—a = I (—a) for each a E
g
a
n
and
and for other techniques.
C
) .oneT I must
h i sleave ifor sanother occasion,
a
n
d i fe f i gc u l t
a
t5. Conclusion
at s i k o ,
n
)
We have learned a number of things in these few pages. We have shown
how to model modalities in a general substructural setting, and we have
seen the way that adding structural rules to the necessitives or 'opens' may
leave the underlying non-modal structure unscathed — or it may change it.
Finally, we've seen how adding structural rules gives us a translation from
systems with structural rules in general, into systems which contain those
structural rules in a qualified form. This has opened a way to see Girard's
results in modalities in linear logic in a much wider setting.*
38
G
R
E
G
RESTALL
6. Note
*Thariks to Pragati Jain, Allegra Bencivenni and an anonymous referee for
comments helping me to clarify my presentation. After writing and submitting this paper, I came across Kosta Dosen [2], which covers similar
ground (but not exactly the same ground) to this paper. Anyone interested
in these issues should carefully study that paper.
Australian National University
REFERENCES
[1] Nuel D. Belnap, Jr. "A Useful Four-Valued Logic," pages 8-37 in J.M.
Dunn and G. Epstein, editors Modern Uses of Multiple-Valued Logics, D.
Reidel, D o
[2]vKosta Dosen. "Modal Translations in Substructural Logics," Journal of
Philosophical
r d r e c h t Logic 21 (1992) 283-336.
[3], Jean-Yves Girard. "Linear Logic," Theoretical Computer Science 50
(1987)1-101.
9 7 K. Meyer, Steve Giambrone and Ross T. Brady. "Where
[4]1Robert
7 .
Gamma
Fails," Studia Logica 43 (1984) 247-256.
[5] John K. Slaney. n'he Implications of Paraconsistency," Proceedings of
the 12th International Joint Conference on Artificial Intelligence, Vol. 2
(1991) 1052-1057.