Continuum fallacy

The continuum fallacy (also called the fallacy of the beard^[1], line drawing fallacy, bald man fallacy, fallacy of the heap, and the sorites fallacy) is an informal logical fallacy closely related to the sorites paradox, or paradox of the heap. The fallacy causes one to erroneously reject a vague claim simply because it is not as precise as one would like it to be. Vagueness alone does not necessarily imply invalidity.

The fallacy appears to demonstrate that two states or conditions cannot be considered distinct (or do not exist at all) because between them there exists a continuum of states. According to the fallacy, differences in quality cannot result from differences in quantity.

There are clearly reasonable and clearly unreasonable cases in which objects either belong or do not belong to a particular group of objects based on their properties. We are able to take them case by case and designate them as such even in the case of properties which may be vaguely defined. The existence of hard or controversial cases does not preclude our ability to designate members of particular kinds of groups.

Relation with sorites paradox

Narrowly speaking, the sorites paradox refers to situations where there are many discrete states (classically between 1 and 1,000,000 grains of sand, hence 1,000,000 possible states), while the continuum fallacy refers to situations where there is (or appears to be) a continuum of states, such as temperature – is a room hot or cold? Whether any continua exist in the physical world is the classic question of atomism, and while Newtonian physics models the world as continuous, in modern quantum physics, notions of continuous length break down at the Planck length, and thus what appear to be continua may, at base, simply be very many discrete states.

For the purpose of the continuum fallacy, one assumes that there is in fact a continuum, though this is generally a minor distinction: in general, any argument against the sorites paradox can also be used against the continuum fallacy. One argument against the fallacy is based on the simple counterexample: there do exist bald people and people who aren't bald. Another argument is that for each degree of change in states, the degree of the condition changes slightly, and these "slightly"s build up to shift the state from one category to another. For example, perhaps the addition of a grain of rice causes the total group of rice to be "slightly more" of a heap, and enough "slightly"s will certify the group's heap status – see fuzzy logic.

Examples

Fred can never be called bald

Fred can never be called bald. Fred isn't bald now, however if he loses one hair, that won't make him go from not bald to bald either. If he loses one more hair after that, then this one loss, also does not make him go from not bald to bald. Therefore, no matter how much hair he loses, he can never be called bald.

The heap

The fallacy can be described in the form of a conversation:

Q: Does one grain of wheat form a heap?

A: No.

Q: If we add one, do two grains of wheat form a heap?

A: No.

Q: If we add one, do three grains of wheat form a heap?

A: No.

...

Q: If we add one, do one hundred grains of wheat form a heap?

A: No.

Q: Therefore, no matter how many grains of wheat we add, we will never have a heap. Therefore, heaps don't exist!

Others

Other uses of this fallacy seem to prove that:

• No man has a beard, no matter how long it is (or every post-pubescent male has a beard, no matter how cleanly shaven) because a beard can have varying lengths.
• A room is never either "hot" or "cold", because of the continuum of temperatures.

References

1. ^ David Roberts: Reasoning: Other Fallacies

See also

Thinking portal

• Imprecise language
• Fuzzy logic
• Closed concept