Information Technology Reference
In-Depth Information
The “Routley-Star” has come under a lot of criticism both from those within
and outside of the relevance logic community, and was more of a focus of
Copeland's (1979) critical review than the ternary accessibility relation.
The 4-valued approach (Dunn (1976)) assigns each sentence a subset of the
set of truth values
instead of just a single one of the truth values 1
,
0.
10
There are clearly 4 such subsets, and hence 4 values:
{
1
,
0
}
.
Belnap (1977a, 1977b) labeled these
T,F,N,B
,forTrue,False,Neither,Both.
We can understand these as subsets of
{{
1
}
,
{
0
}
,
{}
,
{
1
,
0
}}
, which is the basic approach of Dunn
(1976). We then can do “double-entry bookkeeping” with
x
{
1
,
0
}
=
1
A
meaning that
the information state
a
is assigning the sentence
A
at least the value 1, and
x
|
=
0
A
meaning that
a
is assigning
A
at least the value 0. We start with a
valuation
v
that assigns to each atomic sentence
p
some subset of
|
{
1
,
0
}
.From
this we can define
V
1
=
{
x
:1
∈
v
(
p
)
}
,
V
0
=
{
x
:0
∈
v
(
p
)
}
. Then:
(v
p
)
x
V
1
(
p
) (Atomic)
x |
=
0
p
iff
x ∈ V
0
(
p
)
|
=
1
p
iff
x
∈
Clauses for
∼
,
∧
,
∨
then are as follows:
(v
∼
)
x
|
=
1
∼
A
iff
x
|
=
0
A
(Negation)
x
|
=
0
∼
A
iff
x
|
=
1
A
(v
∧
)
x
|
=
1
A
∧
B
iff
x
|
=
1
A
and
x
|
=
1
B
(Conjunction)
x
|
=
0
A
∧
B
iff
x
|
=
0
A
or
x
|
=
0
B
(v
∨
)
x
|
=
1
A
∨
B
iff
x
|
=
1
A
or
x
|
=
1
B
(Disjunction)
x
|
=
0
A
∨
B
iff
x
|
=
0
A
and
x
|
=
0
B
.
This in fact gives a complete semantics for First Degree Entailments (FDE),
those formulas of R and E that do not contain nested implications, i.e., formulas
of the form
A
.
Plus, the sharp-eyed reader will have noticed, we need to have two clauses for
relevant implications as well. This gets complicated.
Of course we could just continue and write down:
→
B
where
A
and
B
do not contain
→
(v
→
)
x
|
=
1
A
→
B
iff
∀
a, b
,if
Rxab
and
a
|
=
1
A
then
b
|
=
1
B
x
|
=
0
A
→
B
iff
∃
a, b
,
Rxab
and
a
|
=
1
A
then
b
|
=
0
B
.
This might be fine, except it does not seem to give a completeness theorem for
R. Fortunately, Mares (2004) has found a way to get a variant of it to work. But
it needs the complication of adding a “neighborhood semantics,” much like the
neighborhood semantics of various non-normal modal logics. We leave it as an
open problem whether sense can be made of this addition in terms of relevance.
10
Though an alternative was suggested of viewing a valuation as a relation of a sentence
to 1
,
0 instead of a function taking just one of these.
