Fallacy of composition

A team of stars

“Every player we signed this season was the top scorer on their old team. We’re going to be the best team in the league.”

Being a good team depends on more than having good players: whether they pass to each other, cover the same positions, or all want the ball. Those are facts about how the parts fit together, and the premise says nothing about them. This is the clear-cut case, and a standard textbook one: a team of good players need not be a good team.

Each one is affordable

“The streaming add-on is only $8 a month. So is the cloud storage, and the premium news app is $12. None of those is a lot of money, so I can afford all of them.”

“Affordable” is judged against a budget, and small amounts add up. Each purchase passes the test on its own, and the bundle may still fail it. The logician Richard Whately described this in 1826: the spendthrift who finds he can afford “this, or that, or the other expense” forgets that “all of them together will ruin him”. This version is easy to miss because each step is checked honestly.

If I stand up, I’ll see better

At a crowded outdoor concert, someone on the lawn stands up and finds the view much better. They tell their friends: “Standing up is the way to see. If everybody on the lawn stood up, we’d all have a better view.”

What’s true for one person acting alone (standing gives a better view) stops being true when everyone does it, because the benefit came from being taller than the people around you. Here the parts don’t just fail to add up; their interaction cancels the effect. Many real cases have this shape: an advantage that exists only relative to others can’t be shared by everyone.

The argument on the left is invalid: the premise can be true while the conclusion is false. It becomes valid with the middle premise added, and then the question is simply whether that premise is true for this property. For “made of metal” or “has mass” it is. For “is light” or “is affordable” it isn’t. Logicians disagree about how much adding that premise explains. C. L. Hamblin (1970) calls it “too easy a move”: used generally, it would turn every fallacy into a formal one. Either way, which properties carry over still has to be worked out case by case.

• Parts to whole: from the parts of one thing to the thing itself. “Every component in this phone is cheap, so the phone is cheap.” (Assembly, design and patents also cost money.)
• Members to group, counted collectively: from what each member does to what the group does in total. A term used distributively is true of each member (“buses are big”); used collectively, it’s true of the group taken as a whole (“buses carry most of the city’s commuters”). Irving Copi’s textbook example: “because a bus uses more gasoline than an automobile, therefore all buses use more gasoline than all automobiles.” There may be far more cars.
• Individual to society: from what’s true for one person or household to what’s true for everyone together, as in the concert example. Economics textbooks have long used this pattern. The 1955 edition of Paul Samuelson’s introductory text lists, among claims that sound paradoxical, “Attempts of individuals to save more in depression may lessen the total of the community’s savings”, often called the paradox of thrift. Whether and when a claim like that holds is a question for economics; the logical point is only that it can’t be settled by what’s true for one household.

The reverse move, from a property of the whole to each of its parts, is the Fallacy of division. It isn’t the same as the “combination” fallacy in Aristotle’s list, which concerned grouping words in a sentence differently (“a man can walk while sitting”), not parts and wholes.

Textbooks don’t agree on where composition belongs. Some treat it as a fallacy of ambiguity, since a word like “strong” may not mean the same thing for a soldier and a regiment; Hurley and Watson group it with division as a fallacy of illicit transference. Maurice Finocchiaro (2013) goes further and argues it isn’t one distinctive kind of argument at all, but a family of oversimplifications in reasoning from individuals to groups.

Reasoning from parts to whole is often correct. The question is always whether this property carries over.

• The property adds up. Weight, cost, length and area accumulate: if each part weighs at least a kilogram, the whole weighs at least that much.
• The property is about what things are made of or what they lack. If every part of a chair is wood, the chair is made of wood; if no ingredient contains nuts, and nothing was added, neither does the dish.
• There’s a reason to think the parts combine without interfering. Evidence about how the parts work together (the players have won together before, the components were tested as a system) can bridge the gap that the parts alone leave open.
• The conclusion is only probable, and said to be. Good ingredients make a good meal more likely. That’s a reasonable expectation as long as it isn’t treated as a guarantee.

The test: would this property survive being combined, and has anyone given a reason to think so?

A wall of heavy bricks

“Each of these reclaimed bricks weighs about six pounds, and the garden wall will take four hundred of them. That’s a heavy wall; the patio slab needs to be able to hold it.”

This has exactly the fallacy’s shape: a property of every part, then the same property of the whole. But weight adds up, so a whole made of heavy parts can’t be light. The contrast shows why the fallacy is about the property, not the shape: “each brick is small, so the wall is small” would fail.

Every ingredient is vegan

“I checked the label on the stock, the beans, the tomatoes and the spices, and none of them has any animal products. So the chili is vegan.”

The speaker concludes something about the whole dish from its parts, and it’s sound, because “contains no animal products” is a property about what the thing is made of. If nothing else went into the pot, the whole can’t contain something none of its parts contain. The reasoning would fail for “each ingredient tastes good, so the chili tastes good”, which depends on how they combine.

Most of the time, wholes really do resemble their parts: a box of red apples is red, a bag of heavy books is heavy. That makes the inference a reliable habit, and it’s easy to apply it without noticing that the property in question is one that depends on arrangement, proportion or interaction.

The error is also hard to see because the premise is usually true and has been checked with care. Every player really is excellent; every expense really is small. The part that’s missing, how the parts fit together, is exactly the part nobody stated. It sits close to a Hasty generalization, but the two are different: a hasty generalization goes from a few members to every member, while composition goes from the members, perhaps all of them, to the group taken as one thing.

• Ask whether the property adds up. Weight and cost do; quality, speed, safety and affordability often don’t.
• Ask what happens when the parts combine. Do they interfere, compete, or depend on being rare?
• Look for the missing premise and check it: “if every part is F, the whole is F”. Is that true for F here?
• Don’t assume the opposite. Showing that the inference is unsupported doesn’t show that the whole lacks the property. The team may still be the best in the league; the roster alone doesn’t show it.