Are the laws of nature hyperintensional? I.e., if *p* and *q* are logically equivalent, could it
be that one of them is a law of nature and the other is not?

I am inclined to think so.

**Argument 1:** The laws of nature in our world do not
make reference to particular substances. But if *p* is a law of nature, then let *q* be the proposition that *p* is true and either Biden is
president or Biden is not presiden. Then *p* and *q* are logically equivalent, but
*q* is not a law as it makes
reference to a particular substance.

**Argument 2:** The laws of nature in our world are
first-order. But any first-order proposition *p* is logically equivalent to the
second-order proposition that *p* is true.

**Argument 3:** Plausibly, the values of fundamental
constants like the fine-structure constant *α* are a part of the laws of nature.
But now imagine that it turns out that the infinitely many significant
digits of *α* express the
infinite list of all arithmetical propositions and their truth values in
some specific simple encoding scheme. There are two possibilities.
Supposing that it is a law of nature *that* the digits of *α* have this curious property, then
after verifying this property for a sufficiently large number of digits,
we could *know* which of the remaining arithmetical propositions
are true simply by measuring *α*
to a high degree of precision. But if the law of nature is simply the
brute fact that the digits are 0.007297352569…, and it just
*happens* that these digits encode arithmetical truths in that
encoding scheme, then we wouldn’t *know* truths by just measuring
*α*. (Compare: Imagine a machine
where you input an arithmetical proposition, and the machine flips a
coin to yield an output of “True” and “False”. Even if we are so lucky
that the machine *always* gives the right answer, that answer
wouldn’t be knowledge. It would be just luck.) This means that there is
a difference between having a law that says that the digits of *α* are determined by the arithmetical
truths according to that encoding scheme and having an infinite law that
simply states the digits, even though the two laws are logically
equivalent (assuming the truths of arithmetic are logically necessary;
if not, replace the truths of arithmetic by any sequence of hard to know
logically necessary truths).

**Argument 4:** Laws of nature figure in explanations,
but explanation is hyperintensional. The correct explanation of why the
apple fell down is not that *F* = *G**m*_{1}*m*_{2}/*r*^{2}
and either Biden is president or Biden is not president, but simply that
*F* = *G**m*_{1}*m*_{2}/*r*^{2}.

**Argument 5:** One of our best accounts of laws of
nature is the Lewis-Ramsey best-systems model. But on that model it is
very natural to identify the laws of nature with the axioms of the best
system, and not just with propositions equivalent to the axioms of the
best system.

**Final note:** I wonder, though, whether there is a
unique proposition that expresses any given law of nature. Is there
really a fact of the matter whether the law is *F* = *G**m*_{1}*m*_{2}/*r*^{2}
or *F* = *m*_{1}*m*_{2}(*G*/*r*^{2})?