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The continuum fallacy (also known as the sorites fallacy or the fallacy of grey) is the fallacy of assuming that a continuum of possible states between two states means they are not meaningfully different. It is a form of equivocation: treating as equivalent two things that should not be treated as equivalent in context.
The alternative name "fallacy of grey" comes from the assertion that complex, real world decisions and events are rarely "black and white" but consist of "shades of grey" - the corollary to this being that by picking "shades" you've replaced two positions (colours), black or white, with just one colour; grey. Yet some things can be very, very dark shades of grey, and others can be very, very light shades of grey. The fallacy, therefore, is to assume that there is no difference.
When applied to the realm of expertise, this fallacy is sometimes known as "inflation of conflict." Inflation of conflict occurs when someone claims that because two (or more) experts disagree on one point, no conclusion can be drawn and the entire field or area of research is probably discredited to boot.
The form of the argument is as follows:
- X is one extreme and Y is another extreme.
- There is no definable point where X becomes Y.
- Therefore, there is no difference between X and Y.
This has been applied to the concept of uncertainty; that the statements "the moon is made of cheese" and "the Sun is made of mostly hydrogen and helium" are both uncertain statements, but the magnitude of uncertainty is very different between the two. In general, the fallacy takes situations where no clear cut-off point between X and Y exists, and either a) denies that any distinction really exists and commits an equivocation between both or b) takes advantage of the lack of distinction by arbitrarily shifting the distinction to unrealistic positions.
 Relationship to other fallacies
Equivocation - the act of making a false equivalence between two things - is the broader term for the continuum fallacy. An equivocation fallacy doesn't necessarily conflate two connected ideas, they may actually be very different. Further, equivocation is most commonly seen with words or terms that merely sound similar or have very superfluous connections. Take the concept of evolution by natural selection (from which descriptive ethics - i.e., how moral systems are - can be derived) being used to imply Social Darwinism (which applies the descriptive ethics of evolutionary theory/biology to normative ethics, i.e., how things should be), for instance. A continuum fallacy specifically sets up a continuum of ideas gradually connecting one to the other - moral choices being a very easy one - and conflates the two as being equal. A prime example, albeit absurd, is justifying killing newly born children because abortion is legal, which is discussed a little further below.
The term "(semantic) slippery slope" is sometimes used for the continuum fallacy, though on RationalWiki that name is reserved only for causal slippery slopes. The fallacies are similar: both ignore differences based on a lack of a precise cut-off point between two positions, or on the distinction (allegedly) being subjective and arbitrary at best. A causal slippery slope argument suggests that the consequences of an action will gradually progress from one extreme to another due to the lack of any clear stopping point - however, slippery slope arguments found in the wild are often fallacious because they ignore potential cut off points. For instance, the argument that allowing same-sex marriage will lead to bestiality is flawed due to the existence of informed consent, which marks a clearer distinction between the two that cannot be overridden just by allowing some people to get married.
 Common examples
 When does life begin?
This is possibly the single biggest application of the idea behind the continuum fallacy - because when life begins is, indeed, a very fine continuum. It applies both to the origin of life and also to personhood and abortion. The short answer to both of these is that life does not begin at a single point - hence the continuum. Gestation and foetal development are slow continuous processes, and the human body still develops and grows even after birth. Life beginning is also a continuum, without any magic "spark" that makes something alive - DNA can self replicate and evolve in solution, absent from any other cells, and so is alive by some definitions, but not by others. There is no clear separation in both cases.
Where the fallacious use of this continuum occurs is in the anti-abortion argument that life begins at conception - at the stage where a sperm fuses with an egg. But at this point, the emerging embryo is not too dissimilar to any other stem cell. It lacks the complexity of an adult person, and so abortion at an embryonic stage destroys life only in the same sense scratching your own skin and removing (and killing) some cells does. However, this may be a case where a rationalist taboo on the word "life" would be useful, as while an embryo and a skin cell are both "alive", what is often meant is that at fertilization, a new living organism is formed, which is certainly true. At the other end of this continuum, you can suggest that it's acceptable to kill new born children because they have not yet developed the same memory ability and coordination ability that is associated with an adult human. Though this is not a widely endorsed view (even by extremely literal utilitarians), a similar argument that a child hasn't developed a memory and so will forget any pain is often given in defence of circumcision.
 Science was wrong
The "science was wrong before" gambit relies heavily on this trope - mostly through the concept of "relativity of wrong" (see wronger than wrong). Scientific theories are - from a certain perspective - neither right nor wrong, but merely describe the world and make predictions with various degrees of accuracy. This is a subtle point, and so is easily mangled into the (trivially true) statement that "science is wrong".
For example, Newton's gravitational theories are "wrong" (they're a rough approximation) and Einstein's gravitation is almost certainly "wrong" too (it doesn't easily blend with quantum mechanics) but it would be a spectacular fallacy to suggest that they are equally wrong because there is such a continuous shading between "makes rough predictions" and "makes more accurate predictions" when it comes to scientific theories. Saying that the Earth is flat is wrong, and saying that it's spherical is also wrong - it's an oblate spheroid, roughly - but both statements do not have the same degree of "wrongness" on a continuum.
 Moral relativism
Moral relativism can suffer from this immensely, as it lacks any concept of absolutes in terms of good and evil. For instance, it's trivial to note that Winston Churchill was quite a nasty piece of work when it came to allowing millions to starve to death in Burma, while on the other side Adolf Hitler and most of his inner circle were kind animal lovers. So angelic and demonic caricatures of these people are certainly oversimplified and shades of grey exist; but it doesn't mean that both were equally bad or good. Only one, for instance, actively sought to kill millions of civilians as part of a eugenics programme.