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a binary relation
a
≤
b
as
R
0
ab
,andgave
R
properties that assure that
≤
is a
quasi-order (reflexive and transitive).
3
They take a valuation
v
to be a function that assigns to each pair (
p, a
)(
p
an atomic sentence,
x
. They then inductively define
a function
I
that assigns to each pair (
A, x
)(
A
an arbitrary formula) a member
of
∈
K
)amemberof
{
T,F
}
{
T,F
}
. We shall get to this inductive definition, but we shall write
x
|
=
A
rather than
I
(
A, x
)=
T
.
But there is an important restriction. Routley and Meyer require the Hered-
itary Condition on atomic sentences: if
a
=
p
.Itcan
then be shown by induction that this extends as well to compound formulas.
The Hereditary Condition is needed to show that 0
≤
b
and
a
|
=
p
,then
b
|
A
.
This is important because validity on a frame is defined by the condition that
a sentence holds at 0 for all valuations.
4
Routley and Meyer show that a sentence
A
is a theorem of R iff
A
is valid on all frames satisfying the following conditions:
|
=
A
→
p1.
R
0
aa
p2.
Raaa
p3.
∃
x
(
Rabx
and
Rxcd
)
⃒∃
x
(
Racx
and
Rxbd
). (They nicely write this as
R
2
abcd
R
2
acbd
.)
⃒
p4.
R
2
0
bcd
Rbcd
5
⃒
Rac
∗
b
∗
p5.
Rabc
⃒
p6.
a
∗∗
=
a
The Routley-Meyer valuation clauses can now be stated as follows:
(v
p
)
x
|
=
p
iff
x
∈
V
(
p
) (Atomic)
A
iff
not x
∗
|
(v
∼
)
x
|
=
∼
=
A
(Negation)
(v
∧
)
x
|
=
A
∧
B
iff
x
|
=
A
and
x
|
=
B
(Conjunction)
(v
∨
)
x
|
=
A
∨
B
iff
x
|
=
A
or
x
|
=
B
(Disjunction)
(v
→
)
x
|
=
A
→
B
iff
∀
a, b,
if
Rxbc
and
a
|
=
A
then
b
|
=
B
(Relevant implication)
Inthepresentpaperweshallbeexamining an interpretation of relevant im-
plication in terms of, guess what? Relevance! Strangely while this is the most
naive or natural interpretation, it seems not to have been explored or even men-
tioned until now. The whole idea of a
relevant
implication
A
B
is that there
is supposed to be some sort of relevance between the truth of the antecedent
A
and the consequent
B
. What could be more natural than to interpret
Rabc
as that in the context of the information
a,
the information
b
is relevant to the
→
3
They actually use the notation
<
but because the relation turns out to be reflexive
it has become standard to use
≤
.
4
It is easy to miss this important point. Similar points hold for Urquhart's and Fine's
semantics, and I will share with you that when Urquhart first explained his semantics
to me back in 1970, I thought I had found a mistake since it seemed that
A ₒ A
was invalid.
5
There is a typo when this axiom is listed in Routley and Meyer (1973). They have
an “
a
” instead of a “0.” And of course spelled out it means
R
0
bx
and
Rxcd
imply
Rbcd
.
