## T-Legality – A Modality for Legal Activists

May 7, 2017 – 10:25 pmFrom Megan McArdle

A commenter claims:

Umm, you can make “corporations” (or engineers) give us more fuel-efficient 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 <>)

**L**p =: It’s a law that p

**P**p =: ~**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** ~**P**X *w*/

…

**L**~X *w*

~**L**X *w*/

…

**P**~X *w*

**PR** **P**X *w*/*v*

…

*w*A*v*

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** **L**X *w*\*v*

*w*A*v*

…

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 **L**A -> 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** **L**X *w*

…

X *w*

Or if we declare that the accessibility relation is reflexive (according to the Orthodox strategy:)

**Refl** …

*w*A*w*

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 => |- **L**A

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 -> B) -> (**L**A -> LB) (Distribution of **L** over -> )

**L2:** **L**A -> 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 -> 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** *w*A*v*

*v*A*u*

…

*w*A*u*

Try **L**p -> **LL**p

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 ***w*A*v*

…

*v*A*w*

Try p -> **LP**p

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 <-> B) -> (**L**A <-> **L**B)

**LTS2: L**(A & B) <-> (**L**A & **L**B)

**LTS3: L**A <-> ~**P**~A

**LTS4: L**~A <-> ~**P**A

**LTS5: **~**P**(A v B) <-> (~**P**A & ~**P**B)

**LTS6:** **P**(A v B) <-> (**P**A & **P**B)

**LTS7: ** **L**(A -> B) -> (**P**A -> **P**B)

**LTS8: ** (**L**A v **L**B) -> **L**(A v B)

**LTS9: P**(A & B) -> (**P**A & **P**B)

Actually, I started writing this as a bit of a joke, but I now wonder if there might not be a non-silly way to apply modality to laws.

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