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↓=
Γ in
,
Γ
=
ρλ is a Dedekind-McNeile closure operator and in
:
W
→
P
(W)
→{
}
F
={↓
|
∈
}
the inclusion map a
a
.
We denote by
(W)
a
a
W
the closure system
because for each a
∈
W
,
↓
a is a closed set
.
Then
,
the (
F
(W),
⊆
) is a complete
lattice of closed subsets
,
with the meet operator
∩
(
set intersection
),
and the join
∪=↓
∪
operator
=
Γ
, different
from the set union
∪
,
such that the following
isomorphism is valid
:
F
,W
,
↓:
(W,
≤
,
∧
,
∨
,
0
,
1
)
(W),
⊆
,
∩
,
,
{
0
}
with
↓=
↓
=
id
W
and
id
F
(W)
.
Proof
We will show that each closed set
U
∈
F
(W)
is an ideal in
(W,
≤
)
. In fact,
for any
a
a
∈
a
≤
∈
W
,
↓
a
=
Γ in(a)
=
ρλ(
{
a
}
)
={
W
|
a
}
is an ideal. Thus, for
any two
a,b
∈
W
,
1. If
a
≤
b
then
↓
a
⊆↓
b
;
a
∈
a
≤
b
∈
b
≤
↓
∩↓
={
2.
a
b
W
|
a
}∩{
W
|
b
}={
c
∈
W
|
c
≤
a
∧
b
}
=↓
(a
∧
b)
∈
F
(W)
;
↓
a
↓
b
=↓
∪
(
↓
a,
↓
b)
=↓
(
↓
a
∪↓
b)
=↓
(
{
a
∈
W
|
a
≤
a
}∪{
b
∈
W
3.
b
≤
|
b
}
)
=↓
(a
∨
b)
∈
F
(W)
.
The operator
is the join operator in
F
(W)
and hence
↓
a
⊆↓
b
iff
↓
a
↓
b
=↓
b
.
Thus, the original lattice
(W,
≤
)
is isomorphic to the lattice
(
F
(W),
⊆
)
via map
↓
−
1
(Γ in)
−
1
in
−
1
Γ
−
1
a
→↓
a
. It is easy to verify that the inverse of
↓
, i.e.,
=
=
:
W
is equal to the supremum
, i.e.,
=
{
, with
↓
↓
−
1
(U)
P
(W)
→
a
∈
U
}
a
=
a
, and hence
↓=
↓
=
id
W
and
id
F
(W)
are the identity functions for
W
and
F
(W)
, respectively. It is easy to verify that
F
(W)
is a complete lattice, such that
(W)
, its l.u.b. is
F
for every subset
S
⊆
F
S
=
Γ
∪
S
with the bottom element
↓
0
={
0
}
and the top element
↓
1
=
W
, and for any
U
=↓
a,V
=↓
b
∈
F
(W)
,
U
∩
V
=↓
(a
∧
b)
∈
F
(W)
and
U
V
=↓
(a
∨
b)
∈
F
(W)
.
Remark
What is important to notice is that the autoreferential representation, based
only on the lattice ordering
(W,
)
, naturally introduces the modality in a lattice-
based logic. In fact, a modal operator
Γ
≤
:
P
(W)
→
F
(W)
⊂
P
(W)
, based on the
≤
partial order of a complete lattice
(W,
)
, plays the fundamental role for the modal
disjunction
in the canonical autoreferential lattice
F
(W)
. We need it because,
generally, for any two
a,b
∈
W
,
↓
a
∪↓
b/
∈
F
(W)
. Its restriction on
F
(W)
is an
identity
function
id
F
(W)
, which is a selfadjoint (universal and existential) modal
operator.
Notice that an analogous phenomenon is presented in our logic of weak monoidal
topos, as considered at the end of the previous section. Notice also that in his ap-
proach Lewis distinguishes the standard, or
extensional,
disjunction '
∨
c
' from the
intensional
disjunction
“in that at least one of the disjoined propositions is “neces-
sarily true” [
14
, p. 523]. In the same way, he defined also the “strict” logic implica-
tion
φ
∨
⇒
ψ
equal to
(φ
⇒
c
ψ)
where '
⇒
c
' is the classic extensional implication
such that
φ
⇒
c
ψ
is equal to
¬
c
' denotes the classic logical nega-
tion). In what follows, we will see that this property holds also for the implication
¬
c
φ
∨
c
ψ
(here '
