An exchange on Twitter yesterday led me to this post by Paco Jariego summarising this post by Jay Stanley of the ACLU, in which Stanley discusses the concept of a “Gödel’s incompleteness theorem for the law,” and the problems that this creates when embedding legal principles in Internet of Things devices:

no matter how detailed a set of rules is laid out, no matter how comprehensive the attempt to deal with every contingency, in the real world circumstances will arise that will break that ruleset. Applied to such circumstances the rules will be indeterminate and/or self-contradictory.

Stanley makes some good points on how the law cannot be treated like an algorithm, and how human judgement is always going to be necessary – judgement that is hard to build in to the type of automated, embedded systems that will increasingly surround us. But is this the same as being a “Gödel’s incompleteness theorem for the law”?

Gödel’s incompleteness theorem is one of those concepts that gets widely used as a metaphor – along with those other triumphs of early 20th century mathematics and physics, the theory of relativity and quantum mechanics (Schrödinger’s Cat etc.). In its metaphorical use, Gödel’s incompleteness theorem is usually taken to mean that there are areas of knowledge that are necessarily “fuzzy” (“indeterminate and/or self-contradictory”). However, that misunderstands the significance of what Gödel was saying.

What Gödel’s incompleteness theorem actually asserts is subtly different. An earlier post by Paco Jariego quotes Wikipedia’s summary:

For any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms.

In other words, what Gödel’s theorem states is that for any “axiomatic system”, there are propositions that are *true*, but cannot be *proved* from the axioms. The point that is often overlooked is this: a “Gödel proposition” is not “fuzzy”, or “indeterminate and/or self-contradictory”; it is *true*. In fact, if you can show that a proposition is a Gödel proposition then you have proved that it is *true* – you just haven’t proved it from the axioms you started with.

Another subtlety of Gödel’s result is that you can’t plug the gap in your axiomatic system by appending your Gödel proposition as an extra axiom. Gödel’s theorem will continue to apply to *that* system, so that there must be another proposition which cannot be proved from the expanded set of axioms.

So, what would a “Gödel’s incompleteness theorem for the law” actually look like? It wouldn’t simply mean a legal proposition that is “indeterminate and/or self-contradictory”; it would have to mean a legal proposition that is (in some appropriate sense) *true*, but which cannot be proved from the existing “axioms” of the law.

What are the “axioms” of the law? In English law, we might take this to mean statute law together with the fundamental principles of common law. A “theorem” of the law would then be “proved” when a court applies the existing “axioms” to produce new case law applicable to the circumstances before it. (The point made above about appending propositions to the axioms means we needn’t be too precise in drawing the line between “axioms” and “theorems”, though.)

What Jay Stanley’s post argues (correctly, in my view), is that the above analogy is deeply flawed, because the law *isn’t* an axiomatic system in which “theorems” are “proved” from “axioms” by specific rules of inference. It is a more organic system, in which human judgement is always necessary.

But if we keep running with the analogy for now, then we are left with the question of what would be a “true” legal proposition that cannot be “proved” from the axioms (i.e. existing case law and statutes).

The best example that comes to mind is what we might call “textbook law”: the legal principles that have never actually been litigated (and thus never actually the subject of a definitive judgment), but which are widely accepted on the authority of leading legal textbooks. “Chitty on Contracts says this…” – that type of thing. Although even then, arguably a closer analogy would be to unproved mathematical *conjectures* (such as Fermat’s Last Theorem was, before Andrew Wiles ruined everyone’s fun by proving it).

But if anyone has any better ideas, then the comment box is open…

**Edit:** a tweet from Jay Stanley reminds me that another example I considered was the role of equity, whose historical roots can be seen as lying in “filling in the gaps” left by the common law. However, judges applying equity today – even Lord Denning at his most, ah, “creative” – at least *pretend* to be following established principles and precedent; to be “proving” a “theorem” from existing “axioms”, to use the analogy discussed above, rather than simply discovering new legal “truths”.