Two Plans in the Semantics for Relevant Logic
Takuro Onishi (Kyoto University)
Why it is interesting
American plan
Four-valued semantics (Dunn 1966, 1976; Belnap 1977)
A
∧
B
is true
⇐⇒
A
is true and
B
is true
A
∧
B
is false
⇐⇒
A
is false or
B
is false
A
∨
B
is true
⇐⇒
A
is true or
B
is true
A
∨
B
is false
⇐⇒
A
is false and
B
is false
∼
A
is true
⇐⇒
A
is false
∼
A
is false
⇐⇒
A
is true
Australian plan
Relational semantics (Routley and Meyer 1972 etc.)
•
Implication: ternary relation
R
on a frame.
x
|
=
A
→
B
⇐⇒ ∀
y
,
z
:
Rxyz
&
y
|
=
A
⇒
z
|
=
B
.
•
Negation: the
Routley star
∗.
x
|
=
∼
A
⇐⇒
x
∗̸|
=
A
.
Strong and weak points
American plan
↑
Classical semantic clause for negation
↓
Not rich enough to incorporate full relevant logic
Australian plan
Routley’s “American Plan”
•
Four-valued relational semantics for relevant logic
-
Twin ternary relations for truth and falsity of implication
-
No Routley star, i.e. classical clause for negation
Routley’s conclusion
Virtue of American plan
... So perhaps a major virtue of the American approach
is that it provides another approach to and explanation
of — what it was set up partly in opposition to, and was
supposed to be getting away from — the star negation rule
and its role in the Australian plan. (136)
Aim
•
Present a simplified Routley’s American plan
•
Make clear how Routley star is introduced on the basis of
American plan
1
Routley’s American plan
2
Star-like relation(s)
3
How we get to the star function
Four-valued frame semantics
x
|
=
+A
∧
B
⇐⇒
x
|
=
+A
and
x
|
=
+B
x
|
=
−A
∧
B
⇐⇒
x
|
=
−A
or
x
|
=
−B
x
|
=
+A
∨
B
⇐⇒
x
|
=
+A
or
x
|
=
+B
x
|
=
−A
∨
B
⇐⇒
x
|
=
−A
and
x
|
=
−B
x
|
=
+∼
A
⇐⇒
x
|
=
−A
x
|
=
−∼
A
⇐⇒
x
|
=
+A
Truth and
falsity
of implication
Twin ternary relations for truth and falsehood of implication:
x
|
=
+A
→
B
⇐⇒ ∀
y
,
z
:
Rxyz
&
y
|
=
+A
⇒
z
|
=
+B
x
|
=
−A
→
B
⇐⇒ ∃
y
,
z
:
Syxz
&
y
|
=
−B
&
z
̸
|
=
−A
Frame for the basic relevant logic
B
Definition
A
B
-frame
is a structure
⟨0, 0
∗,
U
,
V
,
K
,
R
,
S
⟩
such that:
•
K
: non-empty set of states
•
0
∈
U
⊆
K
: set of regular states
•
0
∗∈
V
⊆
K
: set of
coregular
states
•
R
,
S
⊆
K
3Validity
Definition
An inference
A
⊢
B
is
2-valid
in a frame if
x
|
=
+A
implies
x
|
=
+B
and
x
|
=
−B
implies
x
|
=
−A
for any valuation
⟨|
=
+,
|
=
−⟩
and any state
x
∈
K
.
A formula
A
is
2-valid
in a frame if 0
|
=
+A
and 0
∗̸
|
=
−A
for any
valuation
⟨|
=
+,
|
=
−⟩.
Theorem (Routley)
Positive extensions
A
→
B
→
.
B
→
C
→
.
A
→
C
corresponds
{
∃
u
:
Rxyu
&
Ruwz
⇒ ∃
v
:
Ryvz
&
Rxwv
∃
u
:
Sxuy
&
Suwz
⇒ ∃
v
:
Sxzv
&
Svwy
A
→
.
A
→
B
→
B
corresponds
{
Negative extensions
A
→
B
→
.
∼
B
→
∼
A
Built-in contrapostion
Doubling of
→-rules:
x
|
=
+A
→
B
⇐⇒ ∀
y
,
z
:
Rxyz
⇒
{
y
|
=
+A
⇒
z
|
=
+B
y
|
=
−B
⇒
z
|
=
−A
x
|
=
−A
→
B
⇐⇒ ∃
y
,
z
:
Sxyz
&
{
y
|
=
+A
&
z
̸
|
=
+B
or
y
|
=
−B
&
z
̸|
=
−A
•
Contraposition validated trivially
To validate (under some structural properties)
A
→
.
A
→
B
→
B
,
includes to show for any state
x
,
x
|
=
+A
⇒
x
|
=
+A
→
B
→
B
.
It would involve showing for any
y
,
z
such that
Rxyz
,
y
|
=
−B
⇒
z
|
=
−A
→
B
,
where
z
|
=
−A
→
B
⇐⇒ ∃
v
,
u
:
Svzu
&
{
v
|
=
+A
&
u
̸
|
=
+B
or
v
|
=
−B
&
u
̸
|
=
−Star-like relation
For any
x
and
y
such that
xMy
,
x
|
=
−A
⇐⇒
y
̸
|
=
+A
(
cf.
x
|
=
∼
A
⇐⇒
x
∗̸
|
=
A
)
Review
Suggestion
... So perhaps a major virtue of the American approach
is that it provides another approach to and explanation
of — what it was set up partly in opposition to, and was
supposed to be getting away from — the star negation rule
and its role in the Australian plan. (136)
1
Routley’s American plan
2
Star-like relation(s)
3
How we get to the star function
Star-like relation
For any
x
and
y
such that
xMy
,
x
|
=
−A
⇐⇒
y
̸
|
=
+A
.
A
→
B
→
.
∼
B
→
∼
A
corresponds to
{
Rxyz
⇒ ∃
y
′,
z
′:
yMy
′&
zMz
′&
Rxz
′y
′Syxz
⇒ ∃
y
′,
z
′:
yMy
′&
zMz
′&
Sz
′xy
′A
→
∼
A
→
∼
A
corresponds to
{
Explanatory tasks
Decomposing
M
For any
x
and
y
such that
xMy
,
x
|
=
−A
⇐⇒
y
̸|
=
+A
.
I.e.
(
1
)
x
|
=
−A
=
⇒
y
̸|
=
+A
[
not
x
|
=
−A
and
y
|
=
+A
]
(
2
)
x
̸
|
=
−A
=
⇒
y
|
=
+A
[
x
|
=
−A
or
y
|
=
+A
]
(1)
x
and
y
are
compatible
with each other.
(2)
x
and
y
are jointly
exhautsive
.
Star-like relations
xCy
&
x
|
=
−A
=
⇒
y
̸|
=
+A
(Com)
Two negative modalities
Define
x
|
=
+▷
A
⇐⇒ ∀
y
:
xCy
⇒
y
̸|
=
+A
x
|
=
+Negation as modal operators
x
|
=
▷
A
⇐⇒ ∀
y
:
xCy
⇒
y
̸|
=
A
A
is
impossible
at
x
if it fails at any state
y
compatible with
x
.
(Do˘sen 1986; Dunn 1999; Restall 1999; Berto 2015 etc.)
x
|
=
▶
A
⇐⇒ ∃
y
:
xEy
&
y
̸|
=
A
Criticism against negation as modal operators
Explaining negation in terms of compatibility (or exhaustiveness)?
If there is any kind of explanation as to why these worlds
should be incompatible with ours, we can only see that it
must ultimately appeal to negation.
(De and Omori 2016)
•
Negative modality approach explains the meaning of negation
in terms of
C
and the equivalence:
x
|
=
∼
A
⇐⇒ ∀
y
:
xCy
⇒
y
̸|
=
A
.
The virtue of the American plan
This criticism doesn’t apply to our extended American plan
because negation
∼
and falsity are primitive.
x
|
=
−A
&
xCy
⇒
y
̸
|
=
+A
;
x
̸
|
=
−A
&
xEy
⇒
y
|
=
+A
.
These are explanation of star-like relations, not of negation.
1
Routley’s American plan
2
Star-like relation(s)
3
How we get to the star function
Model with twin star-like relations
Definition
A
B
CE-frame
is a structure
⟨0, 0
∗,
U
,
V
,
K
,
R
,
S
,
C
,
E
⟩, where
⟨0, 0
∗,
U
,
V
,
K
,
R
,
S
⟩
is a
B
-frame and
C
,
E
⊆
K
2, satisfying:
•
xCy
⇒
yCx
;
xEy
⇒
yEx
;
•
xCy
&
x
′⩽
x
⇒
x
′Cy
′;
xEy
&
x
⩽
x
′⇒
x
′Ey
;
•
∃
x
′:
Szx
′y
&
xCx
′⇒ ∃
y
′,
z
′:
yEy
′&
zCz
′&
Rxy
′z
′;
•
∃
x
′:
Rx
′yz
&
xEx
′⇒ ∃
y
′,
z
′:
yCy
′&
zEz
′&
Sz
′xy
′.
A
B
CE-model
is a
B
CE-frame equipped with a hereditary valuation
⟨|
=
+,
|
=
−⟩
that satisfies for each propositional parameter
p
,
x
|
=
−p
&
xCy
⇒
y
̸
|
=
+p
;
x
̸
|
=
+Model with twin star-like relations
Proposition
For any formula
A
(built by
∼
,
∧,
∨,
→), we have
x
⩽
y
&
x
|
=
+A
=
⇒
y
|
=
+A
;
(Heredity
+)
x
⩽
y
&
x
|
=
−A
=
⇒
y
|
=
−A
;
(Heredity
−)
xCy
&
x
|
=
−A
=
⇒
y
̸|
=
+A
;
(Com)
xEy
&
x
̸
|
=
−A
=
⇒
y
|
=
+A
.
(Exh)
Relationship between
C
and
E
Proposition
In
B
CEmodels, we can assume that:
xCy
&
xEz
⇒
y
⩽
z
(DS1)
without changing the set of 2-valid formulas.
Proof.
If
xCy
and
xEz
, then for any formula
A
,
y
|
=
+A
⇒
x
̸|
=
−A
⇒
z
|
=
+A
;
y
|
=
−A
⇒
x
̸
|
=
+A
⇒
z
|
=
−A
.
When the star function is definable
Proposition (cf. Onishi 2015)
If a
B
CE-model (with DS1) satisfies
∀
x
∃
y
:
xCy
&
xEy
,
(DS2)
we can define the star function
∗
such that
x
⩽
y
⇒
y
∗⩽
x
∗;
x
∗∗=
x
;
x
|
=
+∼
A
⇐⇒
x
∗̸
|
=
+A
(
⇐⇒
x
|
=
+▷
A
⇐⇒
x
|
=
+▶
A
)
;
x
|
=
−Dualist star postulate
The conjunction of (DS1) and (DS2) is equivalent to:
∀
x
∃
y
:
xCy
&
xEy
&
(
∀
z
:
xCw
⇒
w
⩽
y
)
&
(
∀
w
:
xEw
⇒
y
⩽
w
)
(DS)
x
y
=
x
∗Now the model is essentially two-valued
In a
B
CE-frame with (DS1) and (DS2),
Rxyz
⇐⇒
Sz
∗x
∗y
∗.
•
The structure of
S
is totally determined by
R
1
Routley’s American plan
2
Star-like relation(s)
3
How we get to the star function
Built-in Contraposition (at rule-level)
Recall
: An inference
A
⊢
B
is
2-valid
in a frame if
x
|
=
+A
implies
x
|
=
+B
and
x
|
=
−B
implies
x
|
=
−A
1-validities
Definition
An inference
A
⊢
B
is
t
-valid
[
f
-valid
] in a frame if
x
|
=
+A
implies
x
|
=
+B
[
x
|
=
−B
implies
x
|
=
−A
]
for any valuation
⟨|
=
+,
|
=
−⟩
and any state
x
∈
K
.
•
A
⊢
+F
B
:
t
-validity in a frame
F
;
•
A
⊢
−F
B
:
f
-validity in a frame
F
;
•
A
⊢
+B
:
t
-validity for all frames;
•
A
⊢
−Contraposition at validity-simpliciter level
Proposition
In the class of
B
- (or
B
CE-)frames,
A
⊢
+B
⇐⇒
A
⊢
−B
(
⇐⇒
∼
B
⊢
+∼
A
)
.
Proof.
Given a counter-model for one of them, we can construct a
(dualized) counter-model for the other:
Rxyz
7−→
S
dzxy
Sxyz
7−→
R
dyzx
(
xCy
7−→
xE
dy
xEy
7−→
xC
dy
)
x
|
=
+p
7−→
x
̸|
=
−dp
x
|
=
−p
7−→
x
̸|
=
+dp
,
(Failure of) Contraposition at validity-in-a-frame level
Proposition
There is a
B
CE-frame
F
such that
A
⊢
+F
B
but
A
⊢
−
F
B
.
Proof.
In a
B
CE-frame (where
C
=
E
=
∅) like
0
0
0
R
0
0
∗0
∗R
0
0
∗0
S
0
∗0
∗0
∗S
0
0
∗0
∗S
A
⊢
+F
B
→
B
but
A
̸⊢
−
