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other relevant logics, sometimes called the “relational-operational” semantics,
or as I like to think of it “best of both.” We will not go into the details of
the Fine semantics, but one way that Fine presents it is to compare it to the
Routley-Meyer semantics so there is a binary operation
ⓦ
and a partial-order
≤
so that he can define
x
|
=
A
→
B
iff for every
a
,
b
,if
x
ⓦ
a
≤
b
and
a
|
=
A
then
b
=
B
. If you let your eyes go out of focus a bit, you will see, as Fine suggests,
that this is essentially the Routley-Meyer definition with
x
|
ⓦ
a
≤
b
in place of
Rxab
. Fine sometimes writes
a
≤
x
b
, which would serve perfectly for the notion
of contextual relevance.
8 Comparing the Fine and Urquhart Semantics for
Relevance Logic
Let us consider Fine's and Urquhart's two different ways of defining the truth
of a relevant implication using
•
to combine
information states
(we shall use
this term as an abstraction to cover Urquhart's
pieces of information
and Fine's
theories
):
(F)
x
|
=
A
→
B
iff for
∀
a
,
b
,if
x
•
a
≤
b
and
a
|
=
A
,then
b
|
=
B
(Fine),
(U)
x
|
=
A
→
B
iff
∀
a
,if
a
|
=
A
,then
x
•
a
|
=
B
(Urquhart).
(U) kind of hides the contextual relevance, whereas (F) sticks it in your face.
But in fact they are equivalent. Indeed, (U) is a special case of (F), if we in-
stantiate
b
to
x
a
. But (U) can be seen conversely to imply (F). It suces to
show that the right hand side of (U) implies the right hand side of (F). So let
us assume the right hand side of (U):
•
∀
a, b,
if
a
|
=
A
then
x
•
a
|
=
B
. We will
show the right-hand side of (F):
∀
a
,
b,
if
x
•
a
≤
b
and
a
|
=
A
then
b
|
=
B
.So
for this purpose further assume
x
•
a
≤
b
and
a
|
=
A
.From
a
|
=
A
and the right
hand side of (U), we derive
x
•
a
|
=
B
. But from this and
x
•
a
≤
b
,byusingthe
Hereditary Condition, we can show that
b
=
B
, as needed. Of course, we have
to show the Hereditary Condition, but this is routine.
There are clearly ways of combining pieces of information that do not have
all of the properties of
|
∪
. What are the properties of
∪
? It is well-known that
from an equational perspective these are:
a
ⓦ
a
=
a
(Idempotence)
a
ⓦ
b
=
b
ⓦ
a
(Commutation)
a
ⓦ
(
b
ⓦ
c
)=(
a
ⓦ
b
)
ⓦ
c
(Association)
.
These three properties characterize a semi-lattice, and any semi-lattice is iso-
morphic to a collection of sets closed under
∪
.
Let's start with idempotence:
a
ⓦ
a
=
a
, and consider the inequality half:
a
a
(
Square decreasing
). This is just the Routley-Meyer condition
Raaa
,
but that does not make it sacrosanct. If the pieces of paper I am combining are
dollar bills, there is more information in my hand (“I have two dollars”) when I
show two of them than when I show just one of them (“I have one dollar”). Girard
ⓦ
a
≤
