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A cut-free sequent calculus for distributive lattices with adjoint pairs of modal operators

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A cut-free sequent calculus for distributive lattices withadjoint pairs of modal operators
Mehrnoosh Sadrzadeh (Universities of Oxford & Paris VII)andRoy Dyckhoﬀ (University of St Andrews)
British Logic Colloquium
Annual MeetingNottingham, 4 September 2008
1
Outline of talk
1. Algebras (here, distributive lattices) with adjoint modalities2. Gentzen and Belnap style sequent calculi3. Rules of our calculus, soundness4. Cut elimination argument5. Consequences (completeness, decidability)6. Conclusion and future plans
2
Algebras with adjoint modalities
Classical
algebraic modal logicBoolean Algebra with De Morgan dual operators (
♦
,
)
Non-classical
algebraic modal logic
weaken the base and the duality
Heyting algebra with ’dual’ operatorsComplete lattice with *adjoint* operators
3
Distributive Lattices with Adjoint Modalities
(
DLAM
)Let
A
be a set of
agents
. A
DLAM
over
A
is a bounded dis-tributive lattice
L
with an
A
-indexed family of order-preserving(
o-p
) operators
{
f
A
}
A
∈A
:
L
→
L
, each with an o-p right adjoint
A
:
L
→
L
. Thus, for all
l,l
′
∈
L
, the following hold:
f
A
(
l
)
≤
f
A
(
l
′
)
if
l
≤
l
′
A
(
l
)
≤
A
(
l
′
)
if
l
≤
l
′
f
A
(
l
)
≤
l
′
iff
l
≤
A
(
l
′
)and then (routinely) the following hold (for (
l
i
:
i
∈
I
)
∈
L
):
f
A
(
i
l
i
) =
i
f
A
(
l
i
)
A
(
i
l
i
) =
i
A
(
l
i
)In particular,
f
A
⊥
=
⊥
and
A
⊤
=
⊤
. If the lattice is complete,then the existence of the right adjoints follows routinely providedthe maps
f
A
(exist and) preserve arbitrary joins.
4
Examples
Let
R
be a binary relation on a set
S
; then
R
extends to operators
R
∗
and
R
∗
on the power set
P
(
S
); these are order-preserving, and join (resp. meet) -preserving, with
R
∗
left adjoint to
R
∗
. Thedeﬁnitions are standard: for
X,Y
∈
P
(
S
),
R
∗
(
X
) =
{
y
∈
S
| ∃
x
∈
X. xRy
}
R
∗
(
Y
) =
{
x
∈
S
| ∀
y. xRy
⊃
y
∈
Y
}
.
Let
♦
be the modal operator deﬁned as the De Morgan dual of
in classical modal logic: then, provided the symmetry axiom
B
(i.e.
♦
φ
⊃
φ
) is valid,
♦
is left adjoint to
. More generally,with
given the semantics of
♦
but using the inverse of theaccessibility relation,
is left adjoint to
.
5

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