Information Technology Reference
In-Depth Information
(1987) uses similar examples to motivate his Linear Logic since the contraction
axiom depends on this property and linear logic lacks that. Another example
might be if we think of
a
as the code for a program and we apply that very same
program to itself as input. There is no reason to believe it will turn out that
code as output.
Another important property of
∪
is
a
ⓦ
b
=
b
ⓦ
a
(Commutation). The computer
program example destroys that too.
The third property of
c
(Association). This is a little
harder to dismiss. Even if I have three bills in two different pockets (say
a
in
the left, and
b
and
c
in the right), and then rearrange them (both
a, b
in the
left, and
c
in the right) I still have the same documentation as to my wealth.
But again one can devise a programming counter-example (though interestingly
not when one interprets
a, b, c
all as the programs that they code and views
∪
is
a
ⓦ
(
b
ⓦ
c
)=(
a
ⓦ
b
)
ⓦ
•
as
composition of programs).
Let us reexamine the three ways of combining information that we listed
above. Here is a simplified way to think about them.
For interpretation 1 (Data Combining Interpretation), imagine that
a
=
{
p
}
,b
=
{
q
}
.Then
a
•
b
=
{
p, q
}
.
For interpretation 2 (Program Applied to Data Interpretation), imagine that
a
=
{
p
}
,b
=
{
p
→
q
}
.Then
a
•
b
=
{
q
}
.
For interpretation 3 (Program Combining Interpretation), imagine that
a
=
{
p
→
q
}
,b
=
{
q
→
r
}
.Then
a
•
b
=
{
p
→
r
}
.
The problem put quickly is that I may be in different kinds of mental states as
I acquire new cognitive input. If I am in a merely receptive state and I acquire
two pieces of information, say
p
and
q
, I may merely “file them away” into
a
q
and my mind is
not very active and/or they are deeply buried with other pieces of information.
On the other hand, if my mind is very active (and confident of its powers) I
might conclude
a
•
b
=
{
p, q
}
. This may be true even if I acquire
p
and
p
→
•
b
=
{
q
}
and at the same time discard the premises
p
and
p
q
that led to the information
q.
And of course a similar story may be told
about interpretation 3.
→
9N g on
This is a bit of digression, but it is justified by the fact that treatment of negation,
not just implication, is critical in relevance logic. In classical logic, from the
premise
A
A
any other sentence
B
can be derived as a conclusion, because
in classical logic (
A
∧∼
B
is not a
theorem in the systems R, E, and other relevance logics, because of the Variable
Sharing Property. This has at least as much to do with negation as it does with
implication, and indeed even in the FDE (first-degree entailment) fragment of
E and R this is not provable.
∧∼
A
)
Ↄ
B
is a theorem. In general (
A
∧∼
A
)
→
