Is there a hierarchy of natural laws?

Question

All matter attracts all other matter.

and

Of a statement and its negation, precisely one is true.

can both with justification be called "natural laws", as they hold true in all cases in our Universe.

However, the first one seems somewhat coincidental, and we can easily imagine (even simulate, using computers) a universe in which that "law" does not hold, e.g. where matter repels other matter rather than attracting it. It would be a very different universe for sure, and likely quite hostile to life, but nevertheless imaginable.

The second law, however, is not so easily altered. A world where it does not hold cannot be simulated by a computer, since the computer's logic fundamentally depends on that law.

What precisely is the nature of this difference? Why are "natural laws of the mind" more universal than "natural laws of matter"? Or does it merely appear so, since we use minds to reason about them?

Answer 1

Forget about the "natural" thing for a second. Let's look at your sentences again:

1. All matter attracts all other matter;

2. ¬(φ ∧ ¬φ).

The first is a physical principle and it does seem to be true of our universe, so we may go ahead and call it "natural". The second is a logical principle, which, having nothing to do with our universe is trivially going to be true of it. (2) may be applied to states of affairs in our universe, but that simply means that (2) has an application in the real world, not that it is about the natural world (not uncontroversial; cf. Pincock's The Applicability of Mathematics).

Their difference, following Carnap, I would characterize as follows.

Definition 1. A sentence φ is a physical law iff the interpretation of the physical constants occurring in φ makes φ true.

Definition 2. A sentence φ is a logical law iff the interpretation of the logical constants occurring in φ makes φ true.

An interpretation of the physical constants is a model that satisfies the laws of physics. An interpretation of the logical constants is a model that satisfies the axioms of logic. The assumption is that there is a standard interpretation for these constants (you may disagree; cf. MacFarlane's Logical Constants). That's the motivation behind the definitions. Let's apply Definition (2) to your sentence (2):

Claim. ¬(φ ∧ ¬φ) is a logical law, i.e., when '¬' and '∧' are interpreted, ¬(φ ∧ ¬φ) becomes true.

Proof. Consider the standard truth-tables f_¬ (unary) and f_∧ (binary) for '¬' and '∧' respectively. Truth-table f_¬ maps its single argument to True if it's False and to False if it's True. Truth-table f_∧ maps its two arguments to True if they're both True; otherwise it returns False. We can evaluate the claim by evaluating the truth-value of f_¬(φ ∧ ¬φ) ≡ f_¬(f_∧(φ, ¬φ)) ≡ f_¬(f_∧(φ, f_¬(φ))). Now, whatever φ is, φ and f_¬(φ) are going to have different truth-values, so f_∧(φ, f_¬(φ)) ≡ ⊥. All that remains to check is the truth-value of f_¬(⊥), which is obviously ⊤.

This shows that (2) is true solely by virtue of the interpretation of the logical constants and is thus, by Definition (2) a logical law. I'm confident that the physical truth of (1), if indeed it is a physical law, can be proved solely on the basis of the interpretation of the physical constants that occur in its definition.

This, of course, is only one way of looking at things. For a general background, I recommend following Mauro's suggestions in the comments above. It's important to become aware that there are disagreements among respected philosophical logicians about what exactly logical constants and logical principles are. There are also different ways of looking at the physical/logical distinction, and any good metaphysics or philosophy of science compilation will contain relevant standard texts on that.

Answer 2

It is clear that there should be such a hierarchy, and the real question would be if we have more than a hand-wavy way of establishing and reasoning about it.

Already the traditional foundation of mathematics in some flavor of set theory is such that even the simplest mathematical statements are only in principle expressible fully formally; and when it comes to traditional mathematical foundations of physics this is so very much primitive or else non-existent that any discussion about what its hierarchy of additional axioms is appears to be lacking substance in its topic. For instance in the other comments we hear about the idea of propositions becoming physical laws if interpreted with physical constants inserted they become true, but that seems more a design criterion for what a formal incarnation of laws of physics should be, than what they actually are in the textbooks.

There is an attempt to improve on this state of affairs by producing an actual and useful foundation of physics in intuitionistic logic, or rather in intuitionistic type theory, viz topos theory. This goes back to William Lawvere, who has articles titled for instance "Toposes of laws of motion" where an actual axiomatization of physics along such lines is attempted/proposed. From there to statements about the universal attractiveness of gravity it is still some way, though.

Since the foundations of mathematics itself saw a kind of revolution by (intuitionistic) homotopy type theory in which it is possible to actually and usefully formalize non-trivial parts of modern mathematics, it is possible to re-examine Lawvere's apporach and ask for an actual and useful formalization of modern physics by adding a hierarchy of axioms to homotopy type theory. It turns out that there is a simple but powerful set of axioms that goes at least a long way towards this goal, which, following Lawvere, are called axioms of "cohesion".

With these axioms added to the bare minimum foundations of homotopy type theory, it turns out that there is an actual and useful formalization of classical/pre-quantum local field theory. I discuss this in some detail in an article Classical field theory via Cohesive homotopy types that goes with a talk at the Conference on Type Theory, Homotopy Theory and Univalent Foundations in Barcelona in 2013. More emphasis of how this proceeds by adding layers of foundational axioms to the bare minimum foundations is in section 2 "Modal type theory" of the companion article Quantization via Linear homotopy types.

Regarding the question about the law (of Einstein gravity) that all matter (and indeed all energy) gravitationally attracts, there are some comments on how laws of generally covariant Einstein gravity arise in such foundations of physics towards the end of Quantum gauge field theory in Cohesive homotopy type theory with Mike Shulman, following the discussion on the nLab at general covariance -- Formalization in homotopy type theory.

From here to formally proving the universal law of attraction is still some way to go, but at least it is clear already that and how this is a statement sitting in a hierarchy of statements which starts with some substrate of logical laws at the very bottom and increasingly picks up laws of more physical nature as one adds more of the system of adjoint modalities that characterizes cohesion.

Answer 3

Yes, there can be a hierarchy of laws in each science, starting from the axioms to subsequent propositions logically proceeding from them. So there can be a hierarchy of natural laws as well as others. But your definition of "natural" laws as "holding true in all cases of universe" is arguable, because the term "nature" doesn't denote such a meaning and also because some natural laws don't universally hold true.

But as for laws that can be proven to hold universally true are laws of logic, such as the second law you have mentioned (which is not a natural law) and is by coincidence among the self-evident first principles which form the axioms of logic itself. The axioms of logic are universally true. Among other first principles is the law of identity. These are also considered as "laws of thought" to which all other scientific laws are reducible. Laws of logic thus comprise axioms common among every scientific field, from math to physics and metaphysics. Thus validity of all deductions and inductions made in fields of science depend on laws of logic. So it makes up the highest level of hierarchy of sciences.