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paragraphs. What are the formal properties of absolute relevance? Our choice
of notation suggests that it should be reflexive, transitive, and not generally
symmetric. It seems entirely plausible that an information state
a
should be
relevant to itself, and that if information state
a
is relevant to information state
b
,and
b
is relevant to information state
c
,then
a
is relevant to
c
.Readersof
detective stories would surely agree on this.
Let us now review the Routley-Meyer conditions on
R
with relevance in mind.
For the sake of clarity, we give equivalents of the Routley-Meyer conditions as
they are in Dunn (1986).
R1.
R
0
aa
Reflexivity
R2.
Raaa
Total Reflexivity
R3.
Rabc
⃒
Rbac
Commutativity
∃
x
(
Rabx
and
Rxcd
)
⃒∃
y
(
Rayd
and
Rbcy
) (which can be written
R
2
(
ab
)
cd
R4.
⃒
R
2
a
(
bc
)
d
)
Associativity
R5.
a
≤
a
and
Rabc
⃒
Ra
bc
Monotony (in first position)
Note that because of Commutativity, we also have:
R5
b
≤
Rab
c
Monotony (in second position)
8
b
and
Rabc
⃒
Let us first see if
is reflexive and transitive, so it matches these properties of
absolute relevance. These are well-known properties of the Routley-Meyer se-
mantics, but we shall derive them anyway because of their importance here ((I
almost said “relevance”) and also strangely hard to find in the literature). Re-
flexivity is immediate, since by the definition of
≤
≤
, it is just R1. For transitivity,
assume (1)
a
≤
b
, i.e.,
R
0
ab
,and(2)
b
≤
c
, i.e.,
R
0
bc
. We must show
a
≤
c
, i.e.,
R
0
ac
. This follows immediately using R5
.
Let us now discuss the Routley-Meyer conditions one by one.
R1: The first question to ask is what is the “0”? Let us call “0” the “null
context.” I assume as a degenerate case that even in a lack of context, an in-
formation state is relevant to itself. In Fine's approach, this amounts to saying
that the null information state 0, when combined with any information state
a
,
is included in the information state
a
.
R2: This states that
a
is relevant to itself when taken as its own context.
This amounts to saying that the information state
a
when combined with itself
gives no more information than just
a
. This is a bit questionable as we shall see.
R3: This states that which of the information states
a
and
b
is taken as
context, and which as input, does not matter. To take Sperber and Wilson's
example of the train, it doesn't matter if I see the train arriving and look at my
watch, or vice versa.
R4:
b
is relevant to
x
(in context
a
)and
c
is relevant to
d
(in context
x
).
Then we must find some input
y
such that
y
is relevant to
d
in context
a
and
c
8
Also if we had fusion
ⓦ
as primitive (instead of defining it as
∼
(
B ₒ∼A
)) we would
have to add R6
:
c ≤ c
and
Rabc
Rabc
.
⃒
