Can Gödel's incompleteness theorems be applied to ethics?

Question

It seems, that simplistically, that Gödel's incompleteness theorems can be applied to ethics in a very straightforward way by:

• Replacing "True" and "False" with "Right" and "Wrong"
• Assuming real world situations display a minimum amount of complexity - analogous to the "capable of proving statements of basic arithmetic" clause.
• Completeness means that an ethical systems can definitively answer whether any proposition about the world is "Right" or "Wrong"
• Consistency means that an ethical system never leads to propositions that are both "Right" and "Wrong".

We could then use Gödel's proof to conclude that no ethical system can be both complete and consistent: Any consistent ethical system would have to prescribe courses of action that are right, but whose righteousness we cannot prove.

This wouldn't be just a mere exercise in linguistics: It would mean that any consistent set of ethics would have to rely on an outside source (Religious scripture, Social convention, evolutionary and game theoretical considerations,....) for at least some of it's rules?

This is very simplistic, where are the holes in this reasoning? Can a more formal, serious version of this argument be made to show that any ethical system is inherently limited?

Answer 1

There are a few limitations that are worth mentioning:

• Arithmetic is not a trivial thing. In particular, one has to deal with the axiom of induction, which is metaphorically quite similar to a tower of Babel argument. It took a lot of work to develop meanings of logical concepts which could reach to infinity with the finesse mathematics does.
• For many, "right" and "wrong" are fluid concepts, unlike the "true" and "false" of strict mathematical speak. Right and wrong have developed to deal with situations where the information is not always perfectly known, so they have subtle shifts in "flavor" when compared to "true" and "false." There are those, of course, that want such distinct "right" and "wrong," so Godel's theorem may apply to them.
• You need a concept of negation with strict enough properties to permit the diagonalization lemma to take charge.

One thing I do think Godel's theorem does is it shows that there are a class of formal ethical systems which cannot be provably consistent. If someone is peddling their ethics system as being provably consistent, and you can find a way to prove arithmetic in using their ethics system, you demonstrate they must have one of the flaws associated with Godel's incompleteness theorem.

This would be very similar to what Godel did to Russel. He took Russel's system for Principia Mathematica, and stood it on its head, using it to prove its own limitations.

When it comes to ethics systems, I find Tarski's non-definability theorem more useful than Godel's incompleteness theorem. They are cut from the same cloth, as both use the diagonalization lemma to prove a system cannot be provably consistent, but Tarski chose to direct his proof towards any formal language. He proved that it is impossible for a formal language which includes negation to define its own semantics. In particular, it cannot define a concept of "True(x)" where x is a sentence in that language. The particular implications of Tarksi's proof are remarkably similar to that of Godel's Incompleteness theorems, but their focus on formal languages is more effective for showing limits of ethics systems. It basically forces any system of ethics which has negation and is strong enough to admit the diagnoinalization lemma to have its semantics defined by a metalanguage (which itself might be corruptible).

Answer 2

I'd say this is your largest concern:

Assuming real world situations display a minimum amount of complexity - analogous to the "capable of proving statements of basic arithmetic" clause

Real world situations usually display an amazing degree of complexity, unlike basic statements of arithmetic.

Unfortunately, most attempts to extend Gödel's theorems outside of math end up mired in these types of problems. Because these theorems were developed particular to say something about number theory (more broadly, formal systems), it's often difficult for theorists to explain how ethics or other fields might constitute a formal system analogous to mathematics.

Not saying it's impossible but it is unlikely that these kinds of theories work out, mainly because of how specific Gödel's work is to the philosophy of mathematics.

Answer 3

This is an interesting idea. Here are some thoughts.

• I suppose the propositions of the formal system could be rules of action (for example: (for all x)(if x is a lie, then do not say x)) or states of affairs (x is a lie and I say x) and the deduction rules would preserve rightness between propositions just as logic preserves truth (given right premises we reach a right conclusion). Then possibly, Gödel's theorem would prevent assigning rightness-value to certain generic rules, but not necessarily to specific actions in concrete situations. In any case the system could be sufficient for all practical purposes, just like arithmetic is practicable, and the problematic rules could be rather farfetched.

• standard logic is tailored for truth. Should we use the same "logic" for ethical deduction? If not, would that impact Gödel's result? For example: "or" allows one to form a complex proposition that is true when at least one of its two components are true. What meaning would we attach to an operator that forms a complex proposition that is right when at least one of its two components are right? "Give to charity or kill your neighbour" doesn't seem right...

• Gödel's theorem rests on the fact that a proposition like "this propositions cannot be demonstrated" can be constructed. I wonder if there is an equivalent for an ethical system (something like "demonstrating the rightness of this proposition is wrong"? Sounds strange...) More generally, how could an ethical system be self-reflexive in the way arithmetic is?

• in logic, all propositions are true or false. But are all actions or states of affairs either right or wrong? Can't they be neutral?

• when propositions are not decidable in a formal system, the point is not that you have to rely on a non-mathematical source to know their truth. You can always embed your formal system in a more encompassing one where the proposition is decidable. The encompassing system suffers from the same incompleteness though... The same would go for ethical systems: you can always extend your system in a way or another.

• why would religious scriptures or social conventions count as "external"? Shouldn't we view them as specific axiomatic systems? The ten commandments could be considered kinds of axioms formulated in natural language. The main import would be that we are never strictly sure that the system we adopt is consistent, but again, we can assume it for all practical purposes.

• it seems that we can construct abstract formal systems a priori without referring to the actual world (without talking of physical objects or properties...). The only notions needed are that of object and predicate. What matters is the form of propositions, not their content. Is it possible to have the same for an ethical system? What would be the fundamental notions? Generally, ethical systems refer to human beings, murder, etc. These are not abstract entities like numbers... Actually, ethical systems look more like physical theories than like formal mathematical systems.

In conclusion I think that your suggestion raises too much questions to be conclusive. The only way we could know if Gödel's result would apply is to construct the kind of ethical system you're talking about, if possible.

Answer 4

In my opinion, I don't think there are holes in your statements.

Because by referring to Godel's Incompleteness Theorem, it means that there will always be someone who is truly injustice that cannot be proved injustice within any formal ethical system.

Thus, as you mentioned, we always need to refer to something external to the ethical system in order to prove that injusticeness.

Answer 5

I think the question has just become more relevant with the failure of FTX and possible implications for Effective Altruism, which I had considered the ultimate "rational" ethical system.

But there clearly are ethical systems that are complete and consistent; trivially, nihilism (everything is "right") is one.

It seems to me that an act utilitarianism that assigns a number ("utilons") would be able to include arithmetic and so would be subject to the incompleteness theorem.

Answer 6

"On Rorty’s view, the beliefs worth holding onto are not the ones that mirror the world (this notion, Rorty thought, wasn’t even coherent), but the ones that allow us to cope with it. Accordingly, since we face different struggles than those who came before us, and those who come after us will face different struggles, we cannot cling to any understanding of the world we may have in the hopes we might have finally gotten it right. For Rorty, there is no 'final vocabulary'" from Nāgārjuna, Nietzsche, and Rorty’s Strange Looping Trick

Even basic number theory has been revised. New languages, number systems, modes of life, senses, types of mind, will need to address their own systemising in their own way. What is ethical so depends on our state of knowledge, the time culture and language we live within. For instance, it's widely thought beings without a finite lifespan, or with relatively hugely sophisticated technology will see ethics differently. We rarely get concerned about ant nests destroyed during construction works.

I follow Rorty, in the way outlined in this article thar can be considered an application of strange loops. I see this phrasing as applying directly also to ethics. Ethics are our current heuristics for social wellbeing, for copibg with living together.