# Is this mathematical proof that moral relativism is contradictory valid?

What I mean is, does the conclusion logically flow from the axioms? If so then I'm happy. However, if there is any criticism of the axioms used, I'd love to see those as well. I'm more interested in any major conceptual errors than I am in simple correction of small typos. I imagine axiom 6 will be the most argued about.

## Symbols Used

• AM = Absolute Morality
• RM = Relative Morality
• NM = Nihilistic Morality
• AMS = Absolute Moral Standards
• ARQ = Absolute Moral Requirement
• OLM = Obedience of Local Morals
• NOL = nothing off limits locally
• NOA = nothing off limits absolutely

## Axioms

• Axiom 1 - If RM then AMS is an empty set
• Axiom 2 - Not(if RM then NM)
• Axiom 3 - If ARQ is a nonempty set, then AMS is nonempty
• Axiom 4 - If (OLM is not an element of ARQ) then NOL
• Axiom 5 - If AMS is empty then NOA
• Axiom 6 - If (NOL and NOA) then NM

## Proof

1. Assume RM
2. (OLM is an element of ARQ) or (OLM is not an element of ARQ)
3. If (OLM is an element of ARQ) then ARQ is nonempty
4. If ARQ is nonempty, then AMS is nonempty by Axiom 3
5. If AMS is nonempty then Axiom 1 is contradicted
6. If (OLM is not an element of ARQ) then NOL by axiom 4
7. By Axiom 1 AMS is empty, and by Axiom 5 NOA
8. Since we have NOL and NOA, by Axiom 6, we have NM
9. By lines 1, 6, 7, and 8, we have If RM then NM. This contradicts Axiom 2.

QED

## Explanation of Proof

Axioms are translated at follows:

1. If moral relativism is true then there are no absolute moral standards
2. It's false that moral relativism being true leads to moral nihilism being true
3. If an absolute moral requirement exists then there exists an absolute moral standard
4. If obedience of local morals is not an absolute moral requirement then nothing is off limits locally
5. If there are no absolute moral standards then nothing is off limits absolutely
6. If nothing is off limits locally and nothing is off limits absolutely then moral nihilism is true

Proof is translated as follows:

1. Assume Moral Relativism is true
2. Either obedience of local morals is an absolute moral requirement or it isn't
3. If it is, we have an absolute moral requirement that exists
4. This means we have an absolute moral standard
5. But this contradicts axiom 1 where we assumed no moral standards exist
6. So we'll assume now that obedience of local morals is not an absolute moral requirement. Axiom 4 implies that now nothing is off limits locally.
7. Since we assumed Moral Relativism was true, we have no moral standards and by axiom 5 nothing is off limits absolutely
8. Now we have nothing being off limits locally or absolutely. By axiom 6 this has led to moral nihilism being true.
9. By lines 1, 6, 7, and 8, we moral relativism leading to moral nihilism. This contradicts Axiom 2.

QED

## Quick summary

Under moral relativism, the obedience of local morals is either an absolute moral requirement or it isn't. If it is, then we have an absolute moral standard, which is against moral relativism's definition. If it's not, moral relativism leads to obedience of local morals being optional, and nothing being off limits locally or globally. This leads to moral nihilism, which we assumed was different than moral relativism.

Source used - APA definition of Moral Relativism - The belief that the morality or immorality of an action is determined by social custom rather than by universal or fixed standards of right and wrong.

• Yes, the axiom set is inconsistent over both Classical and Intuitionistic Logic since the latter still proves ~~(Pv~P). Commented Mar 23 at 18:59
• I do not know the difference between the two logic types but where is ~~(Pv~P) implied? That looks like not not (p or not p). I am not sure how my statements have proven such a thing, or which of the statements I've made would be substituted for P. I'm not even sure how proving that would negate the proof. Commented Mar 23 at 22:26
• I don’t know what you mean by “negate the proof,” but if a set of assumptions A is inconsistent using Classical Logic, then it is also inconsistent in Intuitionism. Commented Mar 24 at 2:13
• This is the recurrent question type (RQT) “Is X valid?”. I invite anyone to discuss or analyze how such questions may be improved at the link provided. Commented Mar 24 at 14:52
• Can you give me more detail on where the inconsistency is, like what line of the proof or what axiom? It's like telling me there's a leak in a 300 foot hose. It's hard for someone like me to find unless I'm given more detail such as it's at the 30 foot mark or the 200 foot mark. I have a hard time finding stuff in the refrigerator. Commented Mar 24 at 22:10

Axiom 2 is nonsense. It is equivalent to ( RM and not NM ). That is, your axiom 2 already asserts RM.

• NOT(if RM then NM) can be rephrased as NOT(if A then B). Or NOT(if it rains, then it's cold). That doesn't imply that it's raining. It just states that rain doesn't imply coldness. Commented Mar 24 at 21:56
• @theboombody: You are simply wrong. Please pick up any proper textbook on basic propositional logic. Commented Mar 26 at 14:35
• Not(if RM then NM) would be equivalent to if RM then not NM. But the conditional doesn't simply vanish because you put a not in front of the whole thing. Or even if Axiom 2 is nonsense as you state, if it is replaced with if RM then not NM, everything else checks out. Commented Jul 10 at 11:46
• I suppose you can remove the conditional by saying (RM XOR NM). But you can't say Not(if RM then NM) is equivalent to ( RM and not NM ). The second statement is much stronger without the presence of the conditional. XOR maintains a conditional of sorts. AND does not. Commented Jul 10 at 14:17
• Your first sentence is dead wrong. Please pick up any proper textbook on basic propositional logic. Commented Jul 11 at 4:54

I didn't bother reading your "mathematical proof" because of the following argument:

Mathematics cannot prove moral laws. Mathematics, in it's most general sense, is the study of number and geometry. This has held true for the last 5 millenia, and will continue to do so.

Hume said, you cannot prove an "ought" from an "is". In other words, logic, mathematics and physics cannot be used to prove any moral law. Moral law is of a nature apart from these. That this holds can be easily proved. Look around you and see what people use mathematics for - they use it for number and geometry. And does anyone use mathematics for ethics, justice and law? No, no-one does apart from a few misguided souls who think that since mathematics is so successful in its own domain, then it ought to be successful in other domains. This is an ought that does not succeed.

• Hume is right. The natural sciences like mathematics can't prove what we should or shouldn't do. But they can still show if something is contradictory. These three phrases don't tell you what's ethical, but they do tell you that someone who possesses these three beliefs is holding contradictory beliefs: 1) It's wrong to cut down trees 2) I cut down trees. 3) I can never be wrong. Commented Mar 24 at 21:58
• @theboombody: Hmm, on that basis we would have to throw out Christianity as they take it as axiomatic that Chrust is wholly divine and wholly mortal. This is contradictory, thus you're throwing out the belief and ethics of at least a billion people. Commented Mar 24 at 22:06
• A billion people can still be wrong. More than that were wrong when we thought time flowed at an absolute rate, the same for everyone everywhere no matter what. That was before special relativity, time dilation, and the twin paradox. Commented Mar 24 at 22:49
• @theboombox: No, most of the people on this earth did not think whether time flowed in an absolute rate or not. They didn't care. It's only physicists that care and they are tiny, tiny minority And if if weren't for clocks they would most likely think that time flows differently at different times. Commented Mar 25 at 1:30
• @theboombody: Plus, you're misconstruing my reply - I'm a muslim, so I don't think Christianity is correct. Nevertheless, I think any kind of ethics has to take into account what they think. I'm saying your logico-fetishism doesn't do this. Nor is it, as I've pointed out in my post, empirically speaking, how people reach moral judgements. Commented Mar 25 at 1:33

If your argument is correct, it shows that you have adopted a set of inconsistent axioms, not that any particular axiom is is wrong. Your plain English summary of the argument in your penultimate paragraph contains a non-sequitur. Moral relativism is distinct from moral nihilism, in that the latter says nothing can be morally right or wrong, whereas the former says that people can adopt different views of what is right or wrong. You think the only alternative to an absolute moral requirement is anything goes, whereas another alternative is a relative moral requirement, ie a requirement held to apply by a particular group of people.

• Can you show me which of the axioms are inconsistent with each other? Commented Mar 23 at 22:40
• Not now because it is past my bed-time, but I will try to think about it in the morning. Night night! Commented Mar 23 at 22:49

There is a problem in the logic of axiom 2. What is the conditional within "not (if RM then NM)" supposed to be? If it is the material conditional, then it entails "RM and not NM". In which case, you are not merely assuming RM for the purposes of reductio but in effect stating both RM and not NM as axioms. You then derive NM from RM, so this proves your axioms to be inconsistent.

If instead you want to understand axiom 2 to mean

``````"it is not necessarily the case that if RM then NM", or
"RM does not entail NM", or
"RM is logically consistent with not NM",
``````

then the argument is not valid. The derivation shows only that the material conditional "if RM then NM" is true, and this is compatible with any of those three readings.

Apart from the problem with the logic, your axiom 4 seems pretty implausible. It implies that unless obedience to local moral requirements is itself an absolute moral requirement, then anything goes. Why would this be so? Surely the requirement to obey local moral rules is itself a local moral rule. If you live in my tribe, you must obey the rules of my tribe. That's a rule of my tribe, but other tribes may have different rules. It doesn't follow that anything goes.

You might perhaps counter with: if every tribe makes it a rule that its members must follow its rules then that itself is an absolute rule. This might be the reasoning behind your axiom 4, but if so, it is playing fast and loose with the concept of moral relativism. An advocate of moral relativism is not committed to claiming that there are no rules or principles that can be agreed upon. Moral relativism is the position that your tribe might have different rules from mine and there is no absolute, neutral ground from which to judge that one of us is right and the other wrong.

• I am pretty much stating not NM as an axiom, yes. But deriving NM from RM based on the axiom of not NM shows that it was wrong to assume RM in the first place. Commented Mar 25 at 17:35
• You bring up a good argument for a weakness in axiom 4. But your last sentence there really doesn't really prohibit moral nihilism from still being true along with moral relativism. If there is no absolute neutral ground to judge, perhaps there are no rules period. It leaves that possibility open. Commented Mar 25 at 17:43
• The problem with the logic is that if axiom 2 is a material conditional, you are stating "not (if RM then NM)" and also deriving from the other axioms "if RM then NM". Hence your axioms are inconsistent. However, it is implausible to interpret axiom 2 as a material conditional. If it is modal or consequential, then there is no inconsistency, but there is no validity to the argument either. It does not prove a contradiction from RM. Commented Mar 25 at 18:51
• I agree that moral relativism is consistent with moral nihilism and does not entail it. Commented Mar 25 at 18:51
• If we define moral nihilism and moral absolutism as follows, where would moral relativism fall? Moral Absolutism = at least one moral principle cannot be destroyed by man. Moral Nihilism = every moral principle can be destroyed by man. Commented Mar 27 at 2:40

It's an interesting approach, with some strengths to it. There's probably two reasons it hasn't been received well:

1. It's fairly widely accepted that pure moral relativism isn't sustainable, so this isn't a novel line of argument.
2. In terms of formalizing this as an argument, there's a significant error, which has led to several answers dismissing you on technical grounds.

The error is an easy and common one, which is to conflate the meanings of the English "if-then" construction with the conditional in propositional logic. They are not the same. That means your Axiom 2 does not express what you intend it to express, and since it is central to your proof, it means the proof doesn't work the way you intended it to work. Since it seems like your project is to formalize the intuition that moral relativism leads to nihilism, the presence of a technical error is not a trivial one.

What you are attempting to translate with Axiom 2 is "It's false that moral relativism being true leads to moral nihilism being true." But "leads to" doesn't translate into propositional logic, which is static. You might think you could reframe it as "It is not possible that moral relativism could be true and moral nihilism be false," which is static, but that uses possibility, which is also not a part of standard propositional logic. You could perhaps get farther with a predicate or a modal logic, but that's not what you're giving us. Once you get into the realm of formal logic, you have to be very careful and exact, because seemingly unimportant distinctions can have huge, and often unexpected or counterintuitive consequences.

• Reason #1 you give makes a lot of sense, and you are right about it not being novel, but I believe there is a surprisingly large number of people who continue to believe pure moral relativism is sustainable. But do you think that axiom 2 could simply be changed to NOT NM? I figure that might improve the consistency without altering the conclusion. I have no training in modal logic or anything like that. Commented Mar 26 at 0:59
• @theboombody - Honestly, I would just strike Axiom 2 altogether, you generally want to start from the minimum possible set of axioms. Your strategy should be to assume RM and derive NM, which will establish your desired conclusion of RM -> NM. Proof by contradiction isn't really a great strategy in this case.// It would be worth you spending some time studying or reviewing basic modern propositional logic, you clearly have an affinity for formal logic, but could probably use some more practice with it. Commented Mar 26 at 14:19
• I think the problem is there are various definitions for moral relativism and I'm using one that not everyone uses. Because either man creates ALL morality or man doesn't create ALL morality. I think some moral relativists believe man creates all morality and others believe man creates part and nature creates part in some kind of weird hybrid where nature and man hand off morality to each other like a football. The weird hybrid was what I was attempting to address. It makes no sense and it's contradictory, or at the very least anarchistic. Commented Mar 26 at 20:42
• The problem is that you're presenting a formal argument that has a serious technical error. It really has nothing to do with the content. The structure of your argument is flawed, as pretty much everyone who has answered has told you. Either the structural problems need to be addressed, or you should drop the pose of this being a formal argument and present it informally. Commented Mar 27 at 3:53
• Well, if you define moral nihilism as the state where EVERY moral principle can be destroyed by man, and moral absolutism as the state where there is at least ONE moral principle that cannot be destroyed by man, THEN you get a case where moral relativism is going to fall either on one side or the other. That's pretty much where I wanted to go anyway. That may be easier to put into symbolic logic without error, but I don't think symbolic logic is all that useful for communication purposes. Commented Mar 27 at 11:24

The feedback from others here has been very helpful.
I now present a revised argument for further feedback. But I'll refrain from putting it in symbolic notation this time.

## Definitions

Moral Absolutism = at least one moral principle cannot be destroyed by man.

Moral Nihilism = every moral princple can be destroyed by man.

Moral Princple X = it's moral to obey local social order

## Given Axiom

Local social order is allowed to destroy every moral princple except Moral Princple X.

## Conclusion

Given these constraints, local social order is following moral absolutism.

## Extended Conclusion

If moral relativism is defined by its constraints on what it can destroy, it is following moral absolutism.

• StackExchange does not use a system of presenting revisions in answers. Commented Mar 27 at 13:16

Shoot, I forgot that I could try using Microsoft Bing's AI Copilot. I was curious to see what it would say. So I typed in the proof and axioms with the symbols without telling it what the symbols meant to see its response to my question, "Is the following proof with the given axioms mathematically consistent?" Here was the response:

"Yes, the proof is mathematically consistent given the axioms. Here’s why:

The proof uses a method called proof by contradiction (also known as reductio ad absurdum). This method involves assuming the opposite of what you want to prove and then showing that this assumption leads to a contradiction.

In this case, the proof starts by assuming RM (Line 1). It then considers two possibilities: either OLM is an element of ARQ or it is not (Line 2).

For the first possibility, if OLM is an element of ARQ, then ARQ is nonempty (Line 3). According to Axiom 3, if ARQ is nonempty, then AMS is nonempty (Line 4). However, this contradicts Axiom 1, which states that if RM then AMS is an empty set (Line 5).

For the second possibility, if OLM is not an element of ARQ, then NOL (Line 6). According to Axiom 1, AMS is empty, and by Axiom 5, NOA (Line 7). Since we have NOL and NOA, by Axiom 6, we have NM (Line 8). However, this contradicts Axiom 2, which states that not(if RM then NM) (Line 9).

Therefore, both possibilities lead to a contradiction, which means the original assumption (RM) must be false. This is the essence of proof by contradiction, and it shows that the proof is mathematically consistent with the given axioms (Line 10). QED."

Of course it's still possible the axioms are not correct, or that AI is making a mistake.

• Caution: Large language model AIs are excellent at creating logical and mathematical proofs that seem correct but are not. They can use all the right words, phrases, and sentences that belong in the right sort of proof, but fail to connect them to each other correctly. (In this case, I do agree your argument and its rephrasing are both sound, ignoring validity of the axioms.) Commented Mar 25 at 17:47
• There may be a point to what you say there. Particularly if the phrases used for the variables contain logical words like AND and NOT. Commented Mar 25 at 17:55
• Posting AI responses is against SE's terms and conditions, precisely because they are so often plausibly wrong. Commented Mar 25 at 18:56
• Chris Sunami, really? That's interesting. Learn something every day I guess. Commented Mar 25 at 19:54
• Despite AI being prone to error, I believe that was only one consideration in not allowing its use in the terms and conditions. I imagine people were probably blindly using it to answer questions without giving them any thought, and hoping to inflate their reputation points that way. Sort of like using it to write your term paper in school. In any case, I put a variation of the proof into AI again, this time removing one of the NOTs in a random spot, to see if it would still say the proof was valid. It did not. So it definitely doesn't give its seal of approval on every question. Commented Apr 19 at 18:49