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Fallacy Friday: Division and Composition

March 5th, 2011 by Matt

Today I want to look at two fallacies, the fallacy of composition and the fallacy of division. These two fallacies are related in that they both mistakenly confuse what is true of the parts with what is true of the whole.

The Fallacy of Composition
M&MsThe fallacy of composition involves mistakenly reasoning that what is true of all the parts of something is therefore true of the whole of that thing. Some obvious examples would be reasoning that because every track on a CD is 10 minutes long the whole CD is ten minutes long; or arguing that because atoms are colourless, and m&ms are made up of atoms, it follows m&ms are colourless. These are examples are fairly obviously cases of invalid inferences.

However, some examples are less obvious. In some versions of the Cosmological Argument for God’s existence, or at least in some apocryphal versions of the argument, it is mistakenly argued that because every item in the universe has a prior cause, the universe as a whole has a prior cause. This is mistaken. If the universe is infinite in time, i.e. it has no beginning, then it is possible for the whole universe to be uncaused while every part is caused by a prior part. Hence, what is true of each individual part of the universe will not be true of the whole. Of course, not all forms of the cosmological argument make this mistake but those that do commit the fallacy of composition.

The Fallacy of Division
Just as the fallacy of composition mistakenly infers what is true of the parts is true of the whole, the fallacy of division mistakenly reasons that what is true of the whole is true of the parts. This would occur, for example, if someone reasoned that because the All Blacks score multiple tries in a game, it follows that every member of the All Blacks scores multiple tries in a game; or that because a particular wall is fragile and breakable, every single brick that makes up the wall is fragile and breakable.

Our Schist Flagstone FireplaceThe problem with both these fallacies is that they mistakenly assume that the whole and the parts that make up the whole have the same properties. Madeleine and I have a beautiful flagstone fire-place in our lounge. It is made up of a series of different sized, shaped and coloured pieces of schist rock, plastered together in flagstone formation to make a single rectangle panel. Merely looking at the fire-place gives a visual illustration that the parts can have different properties to the whole. The individual flagstones are a variety of shapes and not very rectangular yet the whole fire-place is rectangular. Some of these rocks are grey, others are brown,  some have amethyst and teal tones to them but the whole panel is not grey, nor is the whole panel brown, amethyst or teal. This visual picture shows, quite nicely, that what is true of the parts is not always true of the whole and vice-versa.

Conflating Collective & Distributive uses of General Terms
Apart from the straight-forward mistake of confusing what is true of a part with what is true of the whole, there are more subtle versions of this kind of fallacy. The most common is where a person conflates what is true of a group distributively and what is true of it collectively. This is less technical than it sounds; consider the following two propositions which can be sensibly said to be true:

1) Cats have roamed the earth longer than humans.

2) Cats have a natural life span of around 15-20 years.

1) is a statement about cats collectively; it is a statement about cats as a group. This type of animal has been roaming the earth since before humans came on the scene. On the other hand 2) is true of cats distributively; it is a  about a property possessed by every individual cat.

Felis takes a break from all the earth roamingIf a person does not pay attention to the differences between distributive and collective uses of general terms one can make fallacious inferences that may sound correct but which are not. An example will illustrate, our children have a cat called Felis who  is around 5 years old. Suppose, however, that I were to reason:

1) Cats have roamed the earth longer than humans;

2) Felis is a cat;


3) Felis has roamed the earth longer than humans.

This argument superficially looks valid.  Yet  premises 1) and 2) are true and the conclusion is clearly false. Why? Because there is a subtle fallacy in the inference: 3) is true of cats collectively; it is true of cats as a group that they have roamed the earth longer than humans but it is not true of cats distributively – no individual cat has been alive this long. The conclusion  3) follows only from 2) and 1) if 1) was meant to be read in the distributive sense.  The person who makes this inference confuses a collective claim about cats, which is true, with a distributive claim about cats that is false.

You might think all this seems silly, a little unnecessarily technical and somewhat obvious; however, there are examples with important social implications, which often arise in our society. Sometimes racists commit this kind of fallacy; they argue that because a certain ethnic group commits more crimes than Pakehas[1] that it follows that individual persons of that ethnicity commit more crimes than individual Pakeha persons.

This involves inflating collective and distributive ways of talking about a group. Racists take a claim which might be true of a ethnic group collectively and infer that it is true of that group distributively – which is what they need to do to justify treating individual members of that group as criminal.

Less noticed is that the same kind of inference can also be used to justify giving certain benefits to particular individuals. When I was at University I attended a meeting where it was argued that every individual Pacific Island student should be entitled to gain a certain benefit on the grounds that Pacific Islanders need more economic help than Pakehas. Again, the arguer infers from something that is true of Pacific Islanders, collectively, to a conclusion which assumes it is true distributively. One only needs to compare the Pacific Islander who is a CEO of a major company with the Pakeha who cleans toilets  for a living to see that the conclusion being drawn is false.

A similar line of argument was sometimes pushed by the student unions at the universities Madeleine and I have studied at. As  a criticism of government policy they would point out that “students owe billions of dollars of student loan debt”. Now, this is true of students collectively, students as a whole do owe a vast sum of money in student loans, but it is false of students distributively, no single student has a personal student loan debt in the billions. Often when the claim was made, it was made to claim or insinuate that individual students were crippled by debt but such reasoning is fallacious. What is true of the class distributively is not necessary true of it collectively and vice versa.

As a final point, someone might want to argue that there are times when making part to whole inferences are valid. Take the following inference, all parts of my car are blue, therefore the whole car is blue. This seems to be a case where inferring from parts to whole would lead to a correct conclusion. Similarly, if my whole car is blue then every part is blue. Again we seem to infer from the whole to the parts in a manner that leads to the correct conclusion.

These examples tell us that sometimes what is true of the parts is true of the whole and vice-versa but note the word sometimes here; sometimes this is true and sometimes it is not. As I noted in Assessing Arguments, a valid argument is one where it is impossible for the premises to be true and the conclusion false. The fact that sometimes an inference gets a true conclusion from true premises and sometimes it does not shows that the argument is invalid.

The fact that sometimes what is true of the parts is true of the whole and vice-versa and sometimes it is not, suggests that when making inferences of this sort one needs to ascertain what kinds of properties this is the case with and what kinds of properties it is not. A valid inference will not merely move from part to whole or whole to part, instead it will argue that properties of a certain type are such that the whole and the part share them and it will add, as a  further premise, that the property in question is of the correct type so that the inference works. But this means the argument is no longer merely a parts to whole argument but that it involves additional premises and hence is not a version of the composition or division fallacy.

[1] For overseas readers, the word Pakeha is the Maori term for people of European descent which is commonly used in New Zealand.

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15 responses so far ↓

  • I was watching a debate/talk with WLC, Alvin Plantinga, Quentin Smith, and another Atheist (forget the name :/) and when Quentin Smith went to give his speech he used the exact fallacy you mentioned above–fallacy of composition–regarding the cosmological argument.

    WLC pushed Smith on that it was wrong but I don’t think WLC came off as powerfully as he was trying to, some people probably left thinking Smith’s account was justified.

  • Do you have a link to that talk Robert?

  • @David

    Starts at about 26 minutes in to get to the part about Quentin Smith.

  • I was discussing this yesterday with my wife. This is frequently the case in statistics, where properties of the group are said to resemble properties of individuals.

    This is clearly seen in dichotomous variables. If 3% of women are pregnant in any year, this does not mean that all women are 3% pregnant.

  • I am less certain about your universe example. If we know that things come into existence only if they have a cause, then by inference, if the universe comes into existence, it has a cause. This does not seem to depend on the composition of the universe.

  • Robert, I have a copy of that video and have watched several times. the other philosopher was Richard Gale ( I am not sure what Gale’s stance is, as he has defended arguments for the existence of some deity in the literature). I think what Craig argues is that QS assumes that “if every event in the universe has a cause the universe itself is uncaused” and then suggests that one argument that might be used to substantiate this claim ( the claim that if we can explain every event in the universe we can explain the whole) commits the fallacy. He also offers several other criticisms of QS’s argument.

  • Bethyada, you miss my point, I was referring to a hypothetical situation where the universe was beginning less. The Kalam cosmological argument utilises the premise that “everything that begins to exist has a cause” It does not require that things which have not began to exist need to be caused.

    Now suppose the universe was of infinite age and had no beginning, then it would not on this causal principle require a cause. However, every finite event which occurred within the universe, would have a beginning and hence require a cause. So it seems on the very causal principle used by the Kalam argument one could draw a principled distinction between whats causally required for the parts of the universe to exist which does not apply to the whole.

    Of course there are other arguments which are utilised, arguments to the effect that the universe did begin to exist, and others to the effect that even if the universe is without beginning its contingency needs explaining, but that’s not what I was discussing in this post.

  • Yeah, I got that. But you implied that this was related to the fallacy of composition. Perhaps some people argue this way,

    that everything in the universe has a cause and the universe is made up of everything so it also has a cause.

    This I agree would be a sub optimal argument. But I have always understood it by analogy. I don’t doubt the universe had a beginning, but not because of the cosmological argument, rather the fact of the universe’s beginning is part of the argument.

  • Bethyada, I have heard people claim the cosmological argument runs that way. Though I am not aware of any person who actually argued it. There are versions of the argument from cosmological contingency, which are accused of the composition fallacy as well, Russell made this charge as well. In the article I linked to on the cosmological argument, Reichenbach discusses this.

    There is an argument which is a bit different which argues we have a metaphysical intuition, that nothing can come into existence out of nothing and by nothing, and that this intuition is confirmed by our experience because nothing in our experience ever comes into existence out of nothing by nothing, I suspect some people mistakenly have interpreted this argument as a composition fallacy.

  • Fallacy Friday: Division and Composition…

    The Fallacy Friday Podcast is the weekly audio version of Matthew Flannagan’s Fallacy Friday posts over at the MandM blog. Apologetics315 is producing the audio version, released every Friday as well. Today’s episode: Division and Composition (text |…

  • Hey, you are right. Cats have roamed the earth longer than humans. The cat kind was created earlier on the 6th day of creation prior to man. I’m positive they did some roaming before God breathed the breath of life into Adam. 🙂

  • I concede that some versions of the cosmological argument for the existence of God may be guilty of the composition fallacy, but I wonder if such an argument may be one of the exceptions you noted.

    In the case where the blueness of your entire car is inferred from the blueness of every part of your car, it is clear that the property of blueness, where true of every part, must be true of the whole thing. Whereas, say, it would be fallacious to assert that because, from my point of view, the earth is made of relatively flat surfaces, the planet itself must be a relatively flat surface. For in that case we see that the individual parts (the flat surfaces) can be composed in such a way as to make a sphere.

    But take a version of the cosmological argument which says that, because every part of the universe is metaphysically contingent, therefore the universe as a whole must be metaphysically contingent. Here we seem to have a case comparable to the blue car, where it is just as inconceivable to say that every part of the universe could be contingent without the whole universe being contingent as it is to say that every part of the car could be blue without the whole car being blue. One cannot arrange or compose a number of contingent entities in such a way as to make them a single necessary (i.e. non-contingent) entity, just as you could not arrange blue things in such a way as to make a non-blue whole, not without making the parts non-blue.

    I don’t think that positing the beginninglessness of the universe helps matters in this case. If the universe were beginningless, then the number of past events would have to be literally infinite. And all contingent beings, which by definition begin to exist at a specific point in time and do not exist by a necessity of their own nature but rather by an external cause, must depend on this infinitely long causal chain for their existence. The problem is that for any one of these contingent beings to be brought into existence, its prior cause must already have been brought into existence; and if every link in this causal chain is itself contingent, and there is no non-contingent cause standing at the beginning of this series, it follows logically that none of these contingent beings could ever have been brought into existence. To deny this is to deny that these intermediate causes, these contingent beings, are in fact contingent.

    There are other logical and mathematical absurdities which arise from an actually infinite universe, not to mention the scientific evidence for an absolute beginning of the universe, but I leave these in the far more capable hands of philosophers like William Lane Craig, who dealt with these questions in his Reasonable Faith.

  • Travis, yes Bruce Riechenbach has an argument like that in his book on the cosmological argument and also in the Stanford Encyclopaedia article on the Cosmological argument. I was simply using it as an illustration, and my example was about causes not contingency.

  • […] This monumental intellectual and political construction is built upon a simple fallacy, known as the fallacy of composition. It beggars belief that in a world self-proclaimed to be so smart and so rational that the very […]

  • […] m&ms are colourless. These are examples are fairly obviously cases of invalid inferences.(MANDM)” […]