PEMDAS and the Categorical Imperative

Garrett Kincaid — Jan. 24, 2021

How is a mnemonic device for the order of arithmetic operations (PEMDAS) related to Immanuel Kant’s central moral command?

I’ll make a defense of the categorical imperative and offer an an alternative application of it that requires one critical caveat (of which Kant probably wouldn’t approve).

Overview of Kantian Ethics

The three main and overarching ethical theories are virtue ethics, consequentialism, and deontology. In an effort to provide a brief overview of Kantian ethics, I’ll compare it to these other theories and hopefully paint a clear picture.

Kant’s moral theory is duty-based, or deontological. Under such a doctrine, the morality of an action — whether it is good or bad — is determined by the nature of the action itself, not by its consequences. Deontology is commonly placed opposite consequentialism, under which the morality of an action is determined by its consequences. The most common subset of consequentialism is utilitarianism, which posits that good acts are those that bring about the most utility for the most people, or “the greatest good for the greatest number.”

Aristotle’s virtue ethics is tangential to consequentialism and deontology, in that it does not deal with the morality of actions but rather with identifying and adhering to virtues (1). Aristotle argues that one should identify desirable characteristics or motives and work to manifest those in one’s actions. Under virtue ethics, an action is good if it serves one or more virtues, and an action is bad if it works counter to any virtue, serving a vice.

One similarity between consequentialism and virtue ethics is that the same action could be deemed either moral or immoral, depending on the circumstances, but deontology does not allow for such a relative classification of morality; instead, it asserts that certain actions are always wrong. For a utilitarian like John Stuart Mill, stealing may be morally permissible if the good it does outweighs the harm it does. For Aristotle, Robin-Hood-type stealing may not be immoral because it would be in the service of the virtue of charity, for instance. For Kant, theft is immoral regardless of the circumstances because it violates a moral duty.

Kantian ethics is deontological, so it has absolute and universal moral commands. Additionally, Kant assumes that morality is something inherent, that it is not a societal structure but a natural by-product of human reason.

The Categorical Imperative

Kant’s universal moral command, which is the subject of this post, is the categorical imperative:

Act as though the maxim of your action were by your will to become a universal law of nature. — Immanuel Kant, Foundations of the Metaphysics of Morals

A maxim is the reason or rationale for a given course of action. The above rule states that one should only act if they could, at the same time, assert that their rationale would be suitable as a universal law. In other words, one should only act in such a way that they would will everyone else to act. This cognitive exercise of “universalizing” is a powerful way to evaluate the morality of a given action or motive.

Here’s an example: Is it morally permissible to leave your shopping cart outside the corral in the grocery store parking lot? Under the categorical imperative, it is clear that it would be immoral. You wouldn’t will it to be law that everyone at every store with shopping carts must leave their carts strewn about the parking lot. Since the maxim of not retuning the cart to the corral can’t be universalized, according to the categorical imperative, then it is immoral to act according to that maxim.

The beauty (and largely the purpose) of this law is for it to be universal and unconditional — not needing any context. For that reason, the scope of this post remains within that of the categorical imperative and means to evaluate whether it is a rule that should guide our actions.

Criticisms of the Categorical Imperative

I will mention two critiques of the categorical imperative here as context for the following section, making a short defense of Kant after each.

First, Arthur Schopenhauer claims that the categorical imperative reduces to egoism (2):

I ought not to injure anyone, so that, since the principle is assumed to be universal, I also may not be injured. This, however, is the only ground on account of which I, not yet possessing a moral principle but only looking for one, can desire this to be a universal law. But obviously in this way the desire for well-being, in other words egoism, remains the source of this ethical principle.

Schopenhauer says that what one considers to be moral, or suitable of becoming a universal law, is only that which benefits oneself — that it’s egocentric. He says that what we tend would assume that what benefits us would benefit others.

But this critique doesn’t take into account two components of Kant’s theory. The first is the idea all humans are ends in themselves, which means that every person should be respected as an end and not treated as merely a means to any other end. Schopenhauer claims that the only two possible motives for action are egoism and sympathy/compassion and advocates for the latter (2). Since Kant’s theory regards everyone as an end in themselves and calls for each person to be treated as such, Kant would assume sympathy to be a natural part of ethical reasoning and one that guides an individual’s classification of maxims under the categorical imperative.

The second component of Kantian ethics that is relevant here is the metaphysical assumption that reason is the source of morality — that morality is inherent. If you accept this part of Kant’s theory, then it would follow that what is deemed through reason to be moral is the same, universally. In Schopenhauer’s criticism, he builds upon the assumption that the moral agent is “not yet possessing a moral principle but only looking for one.” But, to Kant, morality is something that every human possesses, on account of their reason, which refutes Schopenhauer’s criticism.

A second critique comes from Hegel, another German philosopher, who argues that the categorical imperative is merely a principle of non-contradiction and gives no indication of what maxims one should follow (3).

He claims that the categorical imperative only serves as a guide for action insofar as it declares an action as immoral if the maxim for it can’t be reasonably universalized. One counter to this critique would be that for each maxim that is deemed unfit to be a law of nature, there is an indication of the sort of maxims that would be morally permissible. If you think it immoral to leave your shopping cart outside the corral, then you’d naturally ask whether it is moral to put the cart inside the corral.

If one decides not to act in a certain way, to avoid contradicting the categorical imperative, then one would work to determine an alternative course of action and a maxim that could be universalized.

Knock, Knock

Most commonly, those that altogether dismiss Kant’s theory take issue with his deontological approach of demanding that there are moral absolutes — immutable moral duties. As the staunch deontologist that he was, Kant enumerated many such duties.

Possibly the most frequently cited example of a Kantian moral absolute is a thought experiment involving an axe murderer. A murderer knocks on Kant’s door and asks where Kant’s friend is, intending to harm the friend. Kant maintains that, even under these circumstances, it is not right to lie to the murderer in an effort to protect the friend.

It is because of this claim alone (at least in my experience) that many people, especially those that lean heavily toward consequentialism, outrightly discredit Kant’s moral theory. “How could you allow your friend to be murdered merely for the sake of honesty?” one might ask. For Kant, adhering to the truth in all manners and actions is essential, and he does not see the trade as a lie for a life; he sees a lying as risking the entire foundation of morality.

The French philosopher Benjamin Constant challenged Kant on his moral duty to truth, under the circumstances of this thought experiment, to which Kant replied in his essay “On a Supposed Right to Lie from Altruistic Motives” (5).

In that essay, Kant makes the unequivocal claim that “to be truthful (honest) in all declarations … is a sacred and absolutely commanding decree of reason, limited by no expediency” (4). Kant’s main defense of his position is that lying to the murderer would be acting on supposed consequences of the action and not on the nature of the action itself, which would implicate the lier in whatever proceeded thereafter; whereas, if the lier instead told the truth, he/she would not be responsible for any of the consequences.

If by telling a lie you have prevented murder, you have made yourself legally responsible for all the consequences; but if you have held rigorously to the truth, public justice can lay no hand on you, whatever the unforeseen consequences may be (4).

Kant illustrates the lier’s culpability with the following scenario: you lie, telling the murderer that your friend already left the house, when, unbeknownst to you, your friend had left the house and was then promptly found by the murderer. In that case, Kant says, “You might justly be accused as the cause of [your friend’s] death.”

One might argue that the inverse would also be the case. If you tell the truth and your friend is murdered, then you could be considered the cause of your friend’s death. I disagree with that position, as it is the action of the murderer that is the cause of murder, and I agree with Kant that telling the truth would relive one of any culpability. However, you could lie and be the cause of your friend not being murdered, and it seems like that should be morally permissible. Kant does address this too, saying, “A lie always harms another; if not some other particular man, still it harms mankind generally, for it vitiates the source of law itself” (4).

Now, what is the solution? Kant’s argument is sound and adheres to all other components of his moral theory. The main discrepancy here goes back to the differences between consequentialism and deontology. The greatest weakness of consequentialism — acting in an effort to achieve the most desirable consequences — is that consequences are unknown at the time of the action that is their cause. Consequences are impossible to predict with complete certainty.

Given that there is no way to know whether your friend is inside or outside the house (a little Schrödinger’s cat situation), both lying and telling the truth could result in your friend’s murder. As an answer to this uncertainty, Kant demands that you adhere to truth, as it should never be violated. A consequentialist would probably guess at which of the two options — inside or outside — is the most likely location of the friend and then tell the murderer the opposite. So, it’s a stalemate.

As we move on and try to work towards a solution, let’s take Kant’s speculation out of the equation. Consider the the same thought experiment, but you know exactly where your friend is, where he/she will be, and that the murderer will go where you tell them to look. In that case, shouldn’t it be morally permissible to lie?

Please Excuse My Dear Aunt Sally

As I read Kant, I continue to be impressed by the rigor of his arguments and of how compelling they are. Also, I find the categorial imperative to be a beautiful and incredibly powerful tool for ethical decision-making. With that in mind, I do not wish to refute or try to poke holes in Kant’s theory, as I have found none. I only mean to offer an alternative view of the categorical imperative that may appeal to consequentialist thinkers.

Under the conditions we have just set for the thought experiment, I do think that it should be permissible to lie to the murderer because it guarantees that you would save your friend. That act would also adhere to the categorical imperative, as long as you allow for something that Kant wouldn’t approve of: a hierarchy of maxims, or an ethical order of operations. Enter PEMDAS.

It’s clear that the maxim of “I will lie when it is convenient” can’t be reasonably universalized; one would not will it to be a universal law of nature. On the other hand, the maxim that one should protect their friends and loved ones is something that could reasonably be universalized, according to the categorical imperative.

So, we can reframe the thought experiment to be a competition between these two maxims: 1) abandoning the truth at one’s convenience and 2) acting to protect one’s friend. One of these maxims will be violated, so which one should win out?

One solution is to blend consequentialism and deontology. There are clearly certain actions that have dire consequences, and there are those that are largely inconsequential. A consequentialist would prefer that the categorical imperative take these disproportionate consequences into account, such as the difference between telling lie and saving life. In an attempt to meet somewhere in the middle of these two moral theories, I’m offering an alternative application of the categorical imperative that gives an order of importance to different categories of maxims.

Below is my proposed hierarchy of maxims, mapped to PEMDAS — the order of arithmetic operations.

• Parentheses: Self-Preservation
• Exponents: The Preservation of Others
• Multiplication, Division, Addition, Subtraction: Other Moral Obligations (i.e., not lying)

If you were to apply the categorical imperative with this hierarchical component to our axe murderer thought experiment, the maxim of preserving others would override the moral duty to not lie. This would permit the action of lying to the murder, only when you are certain (or nearly certain) that it would save your friend.

Self-preservation is placed above the preservation of others to allow for actions such as those done in self-defense. If the two were flipped and you were meant to prioritize the wellbeing of others above your own, you may try to attack the axe murderer at the door, sacrificing yourself to save your friend. Since the murderer poses no threat to you, there is no reason to create one. You can protect yourself first, by not inciting the murderer, and then you can preserve others by lying to save your friend.

To expand on this order of operations metaphor, let’s look at each of the three levels of this hierarchy. In a math problem, neither parentheses nor exponents are operations themselves. There may be a whole expression with many operations inside a set of parentheses, and an exponent is only an indication to iteratively perform multiplication.

Here, the mathematical operations are particular actions/duties, and thee parentheses and exponents are motives/intentions for actions.

Self-preservation is not an action itself but a motive for many potential actions. The same is true for preserving others. These two are not actions but maxims — reasons to act — that one should uphold.

The only time a lie would be morally permissible is if

1. The maxim could be universalized, according to the categorical imperative
2. And it would certainly serve either self-preservation or the preservation of others. Otherwise, one should perform the baseline, moral operation of adhering to the truth.

I should clarify, that by “preservation,” I mean guarding someone’s life or dignity, whether it be one’s own or that of another person. This hierarchy would by no means permit actions that are self-serving and have no dire stakes, like what one may want to do to “get ahead” or anything of the sort. The maxims that we are talking about here have to do with severe threats to an individual (like an axe murderer at the door). It is the duty of all people to preserve their life and others’.

This application of the categorical imperative addresses both of the aforementioned criticisms of it. Delineating a hierarchy adds an order of priority, making the moral law more of an indication of what one ought to do — more than just a principle of non-contradiction, as Hegel says This adaptation also addresses Schopenhauer’s concern that the categorical imperative leads to egoism, as the maxims that compel certain moral duties are trumped by the duty to preserve and protect others.

By incorporating the hierarchy of maxims, we can resolve dilemmas such as the one we have covered in the thought experiment. When it comes to a choice between preserving human life or violating another moral duty, the moral duty of preserving life should take precedence. Actions that preserve life should be considered morally permissible both because of their nature and their consequences.

Though I lean toward deontology on the spectrum between it and consequentialism, I do not see either, alone, as sufficient. The answer lies in some blend of the two. The categorical imperative with this added component of a hierarchy of maxims is my attempt to mix black and white so that the categorical imperative can better be applied to our morally gray world.

References

1. “Virtue Ethics,” The Stanford Encyclopedia of Philosophy
2. “The Categorical Imperative” – James Fieser
3. “Kantian Ethics (Criticisms),” Introduction to Philosophy – Lumen
4. “On a Supposed Right to Lie From Altruistic Motives” – Immanuel Kant
5. “Kant and Lying to the Murderer at the Door,“ Journal of Social Philosophy – Helga Varden

I invite and encourage criticisms of this argument, as I would like to continue to refine it, and any feedback would be helpful in doing so. What do you think of the categorical imperative? Is it fit to be used as a moral law? Did you prefer the version with a hierarchy of maxims, or is the three-layer hierarchy too shallow? Let me know!