Information Technology Reference
In-Depth Information
7 Ways of Combining Information
Here are three natural ways of combining information
a
•
b
:
1. Data Combining Interpretation: the piece of information
a
when combined
with
b
equals (or is included) in
c
. (Urquhart, Fine).
2. Program Applied to Data Interpretation: view information state
a
as “in-
put” (static) and view the information state
b
as a “program” (dynamic).
Information state
c
is a potential result of running the program on that
input. (Dunn).
3. Program Combining Interpretation: view
a
and
b
both as programs, and
view the result of composing these two programs as equal to (or included in)
c
. (Dunn).
The kind of interpretation in 1 comes from Urquhart (1972) who talks of
combining pieces of information. Fine talks of theories instead of pieces of infor-
mation. Interpretations 2 and 3 can be found in various forms in Dunn (2001a,
2001b, 2001c, 2003). Mares (1997) contains another “informational” interpreta-
tion of the ternary accessibility relation.
Imagine pieces of information as piles of paper on your desk. Interpretation
1 has to do with viewing the pieces of paper as containing data and combining
them together into a single pile, and of course, this can be done in different
ways. The simplest being to just treat them as sets and not care about the
order in which they are placed, or whether there are duplicates. Another way
might be to regard them as multisets, and disregard the order while carefully
noting the number of duplicates. Maybe, the order could matter too as with
sequences. And maybe, the way the are grouped into files, say with file folders
could matter. We will not explore all of these here, but list them to provoke
thoughts. For interpretation 2, think of the pieces of paper in the first pile (
a
)as
a kind of program containing instructions about what to do with sentences on
pieces of paper, and the idea is just to apply those instructions to the sentences
in the other pile (
b
). For interpretation 3, the idea is to treat the sentences in
both piles as instructions, and to compose the instructions from the first pile (
a
)
with those in pile (
b
) so as to get new instructions.
Re 1, Urquhart (1972) took the simplest mode of combining pieces of in-
formation. He took pieces of information to be sets of sentence and took the
standard operation of set union to be the way of combining them. He did this
independently about the same time as Routley and Meyer came up with their
ternary relation, and in fact avoided the need for it by defining
x
|
=
A
→
B
iff for every
a
,if
a
|
=
A
then
x
∪
a
|
=
B
. The ternary relation is implicit and
can be defined as
Rxab
iff
x
a
=
b
. Urquhart's way of doing things is often
referred to as the “operational semantics” for relevance logic and is contrasted
with the Routley-Meyer “relational semantics.” There is only one thing wrong
with the operational semantics, and that is that it is not complete for any of
the well-known relevance logics, say R, and in fact it is not nicely axiomatizable
at all, as Fine showed. Fine produced his own semantics for the system R and
∪
