Information Technology Reference
In-Depth Information
is relevant to
y
in context
b
. It is easiest to consider this in the Fine framework.
We have
a
•
b
≤
x
and
x
•
c
≤
d
. We need a
y
such that
a
•
y
≤
d
and
b
•
c
≤
y
.
The obvious answer is that
x
=
a
c
). This last
means that it does not matter given three pieces of information in a given order,
which two we combine first - as long as we preserve their order.
R5: Let us suppose that
a
is absolutely relevant to
a
andthatinthecontext
a, b
is relevant to
c
. It then seems that, in the context
a
,
b
should be relevant
to
c
. Note that canonically the information states
a,b,c,...
are certain sets of
sentences (“prime theories” - but we will not go into the details here - see e.g.,
Dunn (1986)) and that
a
•
b
and that (
a
•
b
)
•
c
≤
a
•
(
b
•
b
, i.e., the theory
a
is included in the
theory
b
. Another way to look at this is that the information in
a
is included
in the information in state
b
. Honesty compels me to raise the question as to
whether this conforms to intuitions we might have about
a
being absolutely
relevant to
b
. Is a subtheory relevant to a theory? I believe the answer is 'yes'.
Consider the example of the role of Peano arithmetic in the more general theory
of say the positive and negative integers. But what about the other way around?
The intuitions here are admittedly slippery, but while it seems to me that the
theory of the natural numbers is “integral” (absolutely relevant) to the theory
of the integers, this is no way holds the other way around.
Put quickly, for sets generally (not just for theories), if
b
≤
b
is just
a
ↆ
c
, then if there is
any change in
b
, say in an extreme case that its members were to cease to exist,
then there is of course a change in
c
. This is a case of Absolute Relevance.
Contextual Relevance will be the way that we interpret the Routley-Meyer
accessibility relation
R
, but we need to be sensitive about how we define it. Union
(i.e.,
a
ↆ
b
) is perhaps the first idea that springs to mind, and if we interpret
Rabc
as
a
∪
c
then this interpretation does satisfy all of the requirements
R1-R5.
9
But there are some definite peculiarities in understanding Contextual
Relevance using union. In order for
a ∪ b ↆ c
,wemusthave
b ↆ c
,andso
strangely Contextual Relevance implies Absolute Relevance, but not vice versa
(since maybe
a
c
). Note also that the context
a ↆ c
, and so the context
a
is
also absolutely relevant to
c
. These are downright strange relationships.
But there are various natural ways to combine information states, and union
is only one of these. Let us denote a combination of the information states
a
and
b
as
a
∪
b
ↆ
b
, and then general idea is that we can interpret the Routley-Meyer
ternary accessibility relation
Rabc
as something like
a
•
•
b
≤
c
,andthereisno
reason to think that any of
a, b, a
∪
b
are included in
c
.
9
Food for thought. Propositions viewed as sets of information states (or possible
worlds) are disjunctive in character. Each member of a proposition
p
can be viewed
as a way that
p
might be true. If proposition
p
is included in proposition
q
,then
p
entails
q
. Information states viewed as sets, say of sentences as with Urquhart, are
conjunctive in character. So if
a ↆ b
,then
b
contains more information that
a
,and
viewing
a
and
b
as theories (or as sets of axioms for a theory), we can speak of
b
“entailing”
a
.
