Epistemic status: I don’t really understand Kantian deontology.

Summary: According to the rules of Kantian deontology, an action must be impermissible if it has any probability of resulting in an impermissible outcome. But all actions have some probability of resulting in such an outcome. Therefore, all actions are impermissible.

The classic footbridge dilemma, a variation on the trolley problem:

You see a trolley running down a track that has five people working on it. If the trolley hits them, it will kill them. You are standing on a footbridge overlooking the track, and a fat man stands next to you. If you push the fat man off the footbridge, then he will get crushed by the trolley, but his death will save the lives of the other five people. Should you push him?

A Kantian deontologist would say no. According to Kant, you must treat people as ends in themselves, not merely as means; so you should not use the man to save the other five. (For our purposes, this is equivalent to the intuitionist moral claim that killing is categorically wrong. This essay somewhat vacillates between describing Kantian and intuitionist deontology.)

Now consider an alternative, what we can call the footbridge phone booth dilemma:

You see a trolley running down a track that has five people working on it. If the trolley hits them, it will kill them. You are standing on a footbridge overlooking the track, and an opaque phone booth stands next to you. The phone booth may be out of commission, in which case it contains some concrete weighing it down; otherwise, there is a person inside the phone booth. If you push the phone booth off the footbridge and it contains either concrete or a person, it will stop the trolley, but anyone inside the phone booth will die1. Should you do it?

In this case, you might be using a person merely as a means, but you don’t know for sure because you don’t know if anyone is inside the phone booth.

A deontologist can claim one of two things about how to handle this new dilemma:

  1. It is always wrong to push the phone booth as long as there is some nonzero probability that it contains a person.
  2. There is some largest probability p of a person being inside the phone booth such that pushing the booth is permissible; at any higher probability, it is wrong to push the booth off the bridge.

(Or you could claim that pushing the phone booth is always permissible, but that is generally regarded as a consequentialist position, so I will not discuss it.)

Position 1: All actions are impermissible

Perhaps it is always wrong to push the phone booth as long as there is some nonzero probability that it contains a person. That is, you ought not perform an action if that action has a nonzero probability of treating a person merely as a means rather than an end. Or, to use a more intuitive/less Kantian formulation, you ought not perform any action that has a nonzero probability of killing someone. But every action has nonzero probability of killing someone; therefore, all actions are impermissible.

Why is it true that every action has nonzero probability of killing someone? This follows from the fact that we should assign nonzero probability to every proposition.

Why should we assign nonzero probability to every proposition? Well, suppose you believe some proposition has probability 0. That means no amount of evidence or reasoning, no matter how strong, could ever convince you to change your mind. In formal terms, if C is a claim with zero probability of being true, and E is some evidence,

`P(C|E) = P(C) \cdot \frac{P(E|C)}{P(E)}`

P(C) = 0, which means P(C|E) must be 0 as well. For any statement, there must be some evidence that would convince you to update your beliefs. As Dennis Lindley wrote in Making Decisions:

[I]f a decision ­maker thinks something cannot be true and interprets this to mean it has zero probability, he will never be influenced by any data, which is surely absurd. So leave a little probability for the moon being made of green cheese; it can be as small as one in a million, but have it there since otherwise an army of astronauts returning with samples of the said cheese will leave you unmoved.

For more on this, read How to Convince Me That 2 + 2 = 3 and Infinite Certainty from LessWrong.

Alternatively, one might object on the basis of the definition of killing, which, one might argue, requires an intention to cause a person to die. Even if all actions might cause someone to die, that’s not the same as killing, and might not necessarily be impermissible. I would respond by returning to the phone booth dilemma. You do not know whether the phone booth contains a person, but you know it will stop the trolley; your intention is only to save the five workers on the track. In Kantian terms, you do not know whether you are treating someone merely as a means. Nonetheless, Position 1 asserts that pushing the phone booth is wrong. If that can be wrong even though you have no intention to cause someone to die, then it is similarly wrong to take an action that entails causing someone to die.

One could go a step further and claim that causing someone to die is only categorically wrong if the desired outcome of your action will come about only if the person dies—basically falling back to the Kantian notion that it is wrong to treat people merely as means. But it is still true that, for any action you take with a desired outcome, there is some nonzero probability that the outcome will come about only if someone dies. The phone booth dilemma demonstrates there could exist a situation in which you probabilistically cause someone’s death, which is categorically wrong according to Position 1; if the situation can exist, then any action could result in this situation with nonzero (if tiny) probability, and therefore all actions are morally wrong.

Position 2: Deontology reduces to consequentialism

Suppose the phone booth has some probability of containing a person. Let p be the largest such probability at which it is permissible to push the phone booth off the bridge.

Let’s talk about my favorite theorem: the Von Neumann-Morgenstern utility theorem, or VNM for short. VNM states that if we accept four axioms (Completeness, Transitivity, Continuity, and Independence), then there exists a utility function such that we best satisfy our values by maximizing the expected value of that function. Three of these axioms are uncontroversial2, but deontology traditionally entails rejecting the axiom of continuity, which states that for three outcomes L, M, and N:

If \(L \le M \le N\), then there exists a probability p such that \(pL + (1 - p)N \sim M\)

(where the ~ symbol indicates that you are indifferent between the two choices.)

Deontologists typically would claim that it is possible for L to be categorically worse than M, such that it is never worth accepting any probability of L, no matter how small. (This is consistent with Position 1 above.)

In the case of the phone booth dilemma, assume the following. L = pushing the phone booth when it contains a person N = pushing the phone booth when it contains concrete M = not pushing the phone booth at all.

From this, we can see that accepting Position 2 requires accepting Continuity. If we accept the other three VNM axioms, then the theorem holds: morality entails maximizing the expected value of some utility function.

Note that, despite the terminology, just because we have a utility function doesn’t mean utilitarianism is true. Utilitarianism necessitates a particular type of utility function where utility is defined as the aggregate well-being minus suffering of all creatures. But VNM does entail consequentialism, because the only thing we care about is consequences—specifically, consequences as measured by the VNM utility function.


When we attempt to resolve the footbridge phone booth dilemma from a deontological perspective, we must take one of two positions. Position 2 reduces to consequentialism, so ultimately Position 1 is the only claim a deontologist can make.

But if this position does not allow us to take any action that has a nonzero probability of causing an impermissible outcome (such as killing someone)3, and we can never be absolutely certain that an action will not result in such an action, then all actions are impermissible.

The footbridge phone booth dilemma demonstrates the impermissibility of all actions for Kantian deontology as well as intuitionist “killing is wrong” deontology. The same reasoning applies to any form of ethics that creates categorical prescriptions. This even applies to mixed forms of consequentialism and deontology. For example, if we say that you ought to save as many lives as possible, but with the restriction that you must never cause anyone to die, we still run into the problem where any action has some probability of resulting in a death. For any ethical system that asserts that an outcome is categorically wrong, all actions are impermissible, because all actions have some probability of causing such an outcome.

A system that declares all actions impermissible cannot prescribe actions, and thus fails as an ethical theory.


  1. We must stipulate that a phone booth without concrete or a person in it is not heavy enough to stop the trolley. If the phone booth by itself can stop the trolley, then we are not using the person inside the booth as a means to stop the trolley, so the act of pushing might not be wrong according to Kant. Whether it is wrong doesn’t matter for our purposes; what we care about is that in the scenario, the (potential) person in the phone booth is being treated merely as a means. 

  2. I know people who reject Independence of Irrelevant Alternatives (IIA), which is conceptually related to but not the same as the VNM Axiom of Independence. You can reject the former while still accepting the latter. (IIA is irrelevant to VNM because VNM assumes that you have a fixed set of choices.) I have never heard of anyone seriously rejecting any of the VNM axioms other than Continuity, although admittedly I don’t know as much as I could about the subject. 

  3. The distinction between deontology and consequentialism often is described as follows: deontology concerns actions/intentions, while consequentialism cares about consequences It may seem strange that I am talking about outcomes, but it still makes sense to speak of impermissible outcomes under deontology. For a deontological theory to make claims about the permissibility of actions, it must examine outcomes. Moral rules such as “do not kill” make absolute prescriptions on actions with respect to the outcomes that those actions produce. An action is wrong if it results in the outcome of a person dying (and when the circumstances of the action fit what we mean by “killing,” as distinct from merely causing someone to die). When I say “impermissible outcome” I mean an outcome such that, if an action would produce that outcome with probability 1, the action is impermissible.