A set of statements is logically consistent if they can all be true at the same time. In forming theories about how or why things are as they are, many people seek theories that are logically consistent and reject ones that are logically inconsistent. For example, believing that all fruits are sweet and that tomatoes are fruits is logically inconsistent, so someone who cares about logical consistency would reject this set of beliefs in favor of some alternative–perhaps not all fruits are sweet, or tomatoes are not fruits, or neither claim is true.
There is satisfaction in holding beliefs that are logically coherent, both for factual or scientific claims and for moral stances.1 It’s certainly easier to defend viewpoints when they don’t contradict each other. However, I argue that logical consistency is only one criterion for evaluating an argument, and it is not appropriate for many reasonable circumstances. It is a highly constraining condition to put on the set of beliefs a person can hold. In fact, in an uncertain world, a rational person’s beliefs may be almost certainly logically inconsistent.
Imagine there are three balls in a bag, one green, one red, and one blue. Two of the balls will be drawn at random. For each ball, a risk-neutral gambler is offered a bet that pays them one dollar if the ball is drawn and takes one dollar away if it is not. The gambler decides which of the three bets to take before any balls are drawn.
It should be clear that the gambler should accept all three bets. Each of them has an expected payoff of ⅓ since there is a ⅔ chance of getting paid a dollar and a ⅓ chance of losing a dollar. However, the probability of winning all three bets is 0 since only two balls are drawn. In choosing which bets to take, the gambler cares only about the expected outcome of each individual event given the information available, not about the joint outcome of all three events.2 I argue the same type of reasoning can hold for beliefs about the world more generally, leading a rational individual to have stances on various individual issues that contradict each other jointly.
Consider debates about the minimum wage. Economist James Buchanan once said “The inverse relationship between quantity demanded and price is the core proposition in economic science, which embodies the presupposition that human choice behavior is sufficiently rational to allow predictions to be made. Just as no physicist would claim that ‘water runs uphill,’ no self-respecting economist would claim that increases in the minimum wage increase employment.” Theories of monopsony power had long existed by the time Buchanan said this, first developed by Joan Robinson in 1933, and such theories do predict that increases in a minimum wage can sometimes increase employment. A burgeoning empirical literature now suggests near-zero average employment effects of minimum wages increases, at least close to current minimum wage levels.3
Purely for the sake of argument–both since it is beyond my expertise to evaluate the minimum wage literature and because settling the issue is not central to my point–consider a person choosing a set of beliefs to hold about the two following claims:
- Claim A: price increases cause reductions in quantity demanded
- Claim B: increases to the minimum wage decrease employment
As a gross oversimplification, assume that each of these claims is simply true or false. Specifically, rejecting Claim A requires believing price increases do not cause reductions in quantity demanded, and rejecting Claim B requires believing that increasing the minimum wage does not decrease employment. This rules out more complex beliefs, reducing stances on each issue to a binary yes or no decision.4 A disciple of Buchanan would view both of these claims as true, which would be logically consistent. What should the disciple do when shown the recent empirical evidence about employment effects of the minimum wage?
To maintain logical consistency, they must either reject the claim that price increases cause reductions in quantity demanded, or they must reject the empirical evidence. Maintaining logical consistency constrains the possible beliefs this person can hold regarding claims A and B. In this toy example, it is simply not logically consistent to believe that price increases cause reductions in quantity demanded and simultaneously that increases to the minimum wage do not decrease employment.
Suppose this person is rational–they seek to form the best set of beliefs possible given the information they have. However, they have a strong conviction that claim A is true, that price increases reduce quantity demanded. Given their theoretical reasoning, lived experience, and other empirical work they have read, they do not think the findings about the effect of the minimum wage on employment are enough to shake their conviction on this point. Suppose we approach this person and ask them whether a new increase of the minimum wage in Illinois will decrease employment. What should they respond?
If they force themselves to have logically consistent beliefs, they must answer that the minimum wage increase will decrease employment since this is the only stance that coheres with their stance on Claim A. However, without the constraint of logical consistency, it could be perfectly reasonable and rational to answer that the minimum wage increase would not decrease employment. A rational person can alter their beliefs in the face of contrary evidence on one issue without shifting their beliefs about everything else–believing that a minimum wage increase will most likely not decrease employment does not require entirely upending one’s view about the effect of price on quantity demanded. The rational choice in predicting the effect of the minimum wage is the one that best fits the evidence for the minimum wage, not the one that most suitably reconciles stances across all of the issues this person can possibly have an opinion about.
Perhaps the assumptions in this simple example are excessively unrealistic since it rules out more nuanced beliefs, but the intuition is still instructive for more general and genuine cases. Imagine forming beliefs not about just Claim A and Claim B, but about the infinitely many things we can form beliefs about. Forming logically consistent beliefs across all of the issues in the world given only our limited lived experience is a very difficult problem. Further, it likely requires an elaborately twisted and convoluted set of beliefs, so overfit to the tiny slice of the universe we see that it would be basically useless for making meaningful predictions on any given issue. It may also result in rejection of novel information that defies our priors, leading to excessive dogmatism.
This matters especially when some issues are bigger or more important than others. Consider a new claim:
- Claim C: increases to the minimum wage decrease employment at the McDonald’s on East 52nd street and South Lake Park Ave in Chicago
Suppose now the problem is to choose a set of beliefs to hold only for Claims B and C. If the overall evidence overwhelmingly suggests Claim B is false in general, but the specific evidence for the McDonald’s in Chicago indicates Claim C is true, forcing logically consistent beliefs at worst risks warping beliefs about large and important issues to cohere with evidence for small and trivial issues. In this case, it might require rejecting Claim B entirely because it does not fit with Claim C.
Logical consistency is important for answering specific types of questions, in particular when asked to give the most coherent general worldview that jointly explains several different phenomena. Consider again guessing which balls will be drawn from the bag, but with a different betting scheme. Now the gambler gets paid $2 if they guess exactly which balls will be drawn from the bag and loses $1 otherwise. Obviously, it does not make sense to guess that all three balls will be drawn since that happens with probability 0. In this case, since the problem is to guess the joint outcome for the three balls rather than separately choosing the outcome for each ball individually, choosing a logically inconsistent outcome violates rationality. Similarly, when asked to specify our most coherent complete worldview that reconciles our beliefs about a variety of issues, we should aim for logical consistency, even if we would give different answers if asked to evaluate each issue individually.
I do not think the point I raise in this piece is particularly novel, mathematically5 or philosophically, but it is one I myself only started to consider seriously recently. In my experience at graduate school and especially in undergrad, clean logical reasoning has been the standard for intellectual inquiry. Given the limits it places on beliefs, perhaps it should not be such a vaunted standard after all.
- Ronald Dworkin goes as far as to argue that moral responsibility, which in his view underpins living ethically and meaningfully, derives from holding “various concrete [moral] interpretations [that] achieve an overall integrity so that each supports the others in a network of value that we embrace authentically.” Dworkin, Ronald. Justice for Hedgehogs (p. 101). Harvard University Press. Kindle Edition.
- The gambler cares about the marginal distributions, not the joint distribution
- There are a lot of papers in this space. Here are just a few recent ones: Azar, Huet-Vaughn, Marinescu, Taska, and Von Wachter (2019); Cengiz, Dube, Lindner, and Zipperer (2019); Derenoncourt and Montialoux (2020); Harasztosi and Lindner (2019). Monopsony power may not be the only mechanism for explaining low disemployment effects of minimum wage changes.
- More realistically, there are of course gradations to rejecting a claim; beliefs that price increases usually cause reductions in quantity demanded and beliefs that prices increases never cause reductions in quantity demanded both contradict Claim A.
- For those mathematically inclined, an analogous example is estimating the most probable sequence of latent states in a hidden Markov model. As stated in Bishop (2006, pg. 629), “the problem of finding the most probable sequence of latent states is not the same as that of finding the set of states that are individually the most probable.” In fact the set of states that are individually most probably might represent a sequence that has a joint probability of 0. The tension between the set of beliefs that are jointly the most probable versus the beliefs that are individually the most probable is the same tension raised here.