Fallacy of division

The profitable company

“The company had a record year, so the design team must be doing well too. There’s no need to look at their numbers.”

The company’s profit is a total. It can be high because a few divisions earn a lot while others lose money. Nothing about the total says how it’s spread out. This is the clear-cut case: a property of the whole assigned to a part without looking at the part.

A club that reads a lot

“Our book club got through more than two hundred books last year. Priya’s a member, so she must read a huge amount.”

“The book club read two hundred books” is true of the club collectively: it’s a total across everyone. It isn’t true of each member distributively. Priya may have read four of them. Group terms are often ambiguous in just this way: “the committee is large” describes the committee, not anyone on it, and nobody would conclude that its members are large. The fallacy is more tempting when the property could apply to individuals, as reading a lot can.

The school with high scores

“That school’s students average in the top ten percent on the state exam. Our new hire went there, so he must be a top-ten-percent student.”

An average belongs to the group. It’s compatible with many students scoring far below it, so it can’t tell you where any one student landed. This is the subtlest of the three because averages feel like descriptions of a typical member. At most, the school’s average makes a high score more likely for a student picked at random, and even that depends on how scores are spread out.

As with composition, the inference becomes valid only with a premise saying the property passes down, and that premise is true for some properties (being made of wood, containing no gluten) and false for others (being profitable, being large). Hamblin (1970) points out that composition and division involve the same distinctions, between whole and part and between collective and distributive uses of words, and that most examples of one can be reworded into the other.

• Whole to parts: from a single object to its components. “This is an expensive watch, so the strap must be expensive.”
• Group to members (collective to distributive): from something true of a group taken together to each member. The Stanford Encyclopedia’s entry on ambiguity gives “the politicians lifted the piano”, which can mean they lifted it together or that each one lifted it.
• Average to individual: from a group’s average or rate to a claim about a particular member, as in the school example.

In research, a closely related problem is called the ecological fallacy: inferring something about individuals from data collected about groups, such as regions or schools. A correlation between two measures across regions can be very different from the correlation between the same measures across individuals, a point made influentially by W. S. Robinson in a 1950 paper in the American Sociological Review. The same aggregation problem underlies Simpson’s paradox, where a trend in a combined group reverses within each of its subgroups.

Reasoning from a whole to its parts is fine when the property really does pass down, or when the conclusion is kept to what the group information supports.

• The property is about what the whole is made of, or what it lacks. If a sweater is 100% wool, each thread is wool; if a dish contains no dairy, no part of it does.
• The property applies to every member by definition or by rule. If everyone in the club had to pay dues to join, each member paid dues.
• The group property was measured on each member. “Every employee here has passed a background check” is distributive from the start.
• The conclusion is a probability, not a certainty. A group’s rate is a sensible starting estimate for an unknown member, as long as it’s treated as an estimate and updated with information about the individual.

The test: is this property true of the whole as a total or a unit, or true of each part?

A gluten-free cake

“The bakery labels this whole cake gluten-free, so I can give a slice to my friend with celiac disease.”

This moves from a property of the whole to a part, but “contains no gluten” is a property about what the thing is made of: if the whole cake has none, no slice of it can. The inference is only as good as the label, but it isn’t the fallacy of division.

Planning from a group’s rate

“Most hikers on this trail finish it in under four hours, according to the park’s records. We’re ordinary hikers, so we’ll probably be done before four hours and can plan lunch for after.”

The speaker applies a group statistic to themselves, but claims only that it’s probable, and the property (“finished in under four hours”) belongs to individual hikers: the records count how many of them had it. Nothing true only of the group as a whole is being handed down to its members. This is using a base rate, which is usually good reasoning; ignoring such rates is its own error, Base rate neglect. It would slide toward division if the speaker treated the group’s record as a guarantee: “hikers here finish in under four hours, so we will”.

Everyday language uses the same words for groups and their members. “The team is fast”, “the students are tired”, “the company is struggling” can each describe the group as a whole or each person in it, and the sentence usually doesn’t say which. Once a property has been attached to the group, it’s easy to read it as attached to everyone.

Averages and totals add to the confusion because they’re built from the members, so they feel like facts about the members. But a total says nothing about how it’s divided, and an average says nothing about how far members are from it.

• Ask whether the property is collective or distributive. Is it true of the group as a unit, or of each member? “Large”, “numerous”, “profitable” and “read two hundred books” are usually the first kind.
• Look at the part directly when you can. The design team’s numbers exist; check them.
• Ask how the total or average is spread out. A few extreme members can produce it.
• Don’t swing to the opposite. The inference being unsupported doesn’t show the part lacks the property. Priya may well be a heavy reader; the club’s total just doesn’t show it.