From
Megan McArdle
A
commenter claims:
Umm,
you can make "corporations" (or engineers) give us more
fuelefficient cars simply by increasing fuel efficiency standards. If
they passed a law tomorrow that said all cars sold by 2010 must get 45mpg,
Detroit
could do that pretty easily. They just don't, because they don't have to.
Apparently
making a law that P means that P is the case. It’s an interesting point
of view. Let’s see if we can establish a modal logic for that view of
legislative power.
We
define the modal operators L
(the analogue to []) and P
(like <>)
Lp =: It’s a law that p
Pp =: ~L~p =: it’s
legal that p
Let’s
take the semantic approach by defining the rules of the semantic tableaux
for the appropriate logic.
LN
~PX
w/
...
L~X
w
~LX
w/
...
P~X
w
PR
PX
w/v
...
wAv
X
v
where v is new to this path
The
accessibility relation A is a legal
accessibility relation, and the indexes on the right indicate legal
worlds.PR tells us that if X is not forbidden by law in w
then in some state v, accessible
by lawyers from w X occurs.
LR
LX
w\v
wAv
...
X
v
Which
tells us that if X is mandated by law in w,
then in any state v accessible
by lawyers from w X occurs.
We
know that LA É A, which is a statement of
Legal Reflexivity, won’t be a valid formula in this logic (try the tree
and see,) but if we add the rule (following the Hintikka strategy:)
LT
LX
w
...
X
w
Or if we declare that the accessibility relation is
reflexive (according to the Orthodox strategy:)
Refl
...
wAw
for
any w on this branch
We
also know that there is an axiomatization that will give us just the same
valid formulas in LTS:
We
need a rule of inference of Legal Necessitation on the analogy of
plain Necessitation:
LR1: 
A Þ
 LA
Which,
curiously enough, indicates that anything that can be shown to be a thesis
in the logic must be a Law.
The
axioms required are
L1:
L(A
É B)
É (LA É LB)
(Distribution of L
over É )
L2:
LA É A
(Reflexivity)
L1
itself would give us KL (Lawyers’ K,) and it has the at first
sight odd result of saying that if it is a law that A É
B then if it’s a law that A, it is equally a law that B. But how odd is
that? We could reasonably understand this as a statement of proper legal
interpretation: if the law states that A and the law also states that
whenever A then B, then it is at least implicitly according to law that B.
(Roe vs. Wade seems to have
been decided in this way.)
The
axiom L2 gives us TL, Lawyers’
T. It’s an explicit statement of the claim that we started with.
This
is a fairly weak logic, even amongst
Normal
logics. Do we want to add any other conditions? Do we want to give the
legal accessibility relation symmetry or transitivity? What would these
look like?
LT
to L4?
Add
the property of transitivity to the accessibility relation in LT
Trans
wAv
vAu
...
wAu
Try
Lp É LLp
We find that it closes in L4 but not in LT.
Is
this something we want? That if it’s a law that X then it’s a law that
it’s a law that X? That actually seems quite reasonable, especially if
we interpret ‘being a law’ in the way that we did for the
distributivity axiom. Thus; if it’s a law that X then it’s at least
implicitly according to law that it’s a law that X. Perhaps this is
referring to some sort of constitutional understanding of the law.
LT
to LB?
Add
the property of symmetry to the accessibility relation in LT
Sym
wAv
...
vAw
Try
p É LPp
We find that it closes in LB but not in LT.
Is
this something we want? It’s a bit harder to understand. If something is
occurring then by law there is no law against it. I have to say, that
seems unlikely. So let’s not go so far as LB.
(Nor, of course, can we accept L5,
which we’d get by making the relation an equivalence relation.) So the
accessibility relation can’t be symmetric: if v is accessible to lawyers from w
it doesn’t follow that w is
accessible to lawyers from v.
Even
if we just stick to LT there
are plenty of interesting results for the lawyers to get busy on. The
following are all provable in LTS:
LTS1: L(A
º B) É (LA
º LB)
LTS2: L(A
& B) º (LA & LB)
LTS3: LA
º ~P~A
LTS4: L~A º ~PA
LTS5: ~P(A v B) º (~PA
& ~PB)
LTS6: P(A v B) º (PA
& PB)
LTS7: L(A É B)
É (PA É PB)
LTS8: (LA v LB) É L(A v B)
LTS9: P(A & B) É (PA
& PB)
Actually,
I started writing this as a bit of a joke, but I now wonder if there might
not be a nonsilly way to apply modality to laws.
