The fallacy of the continuum is committed when it is inferred that a concept or property can only lie at one extreme or the other, without any possible intermediate values. A very common occurrence of this fallacy is when someone asks the question “but where do you draw the line?” This question can sometimes have relevance, for example in the question of enforcing laws which may require drawing a line between legal and illegal behaviour. Often, however, it is not the case that the ability to draw a clear line between two categories is necessary for those categories or distinctions to be meaningful.
A clear example of this can be found in the distinction “there are some people who are short and others who are tall”. Clearly there is no way to ‘draw a line’, or to make a clean or precise distinction between tall people and non-tall people, or short people and non-short people, but clearly this does nothing to invalidate the fact that short and tall people do exist. The continuum fallacy thus occurs when it is falsely asserted that a property is meaningless or unreal merely because it lies along a graded continuum, or because the claim is somewhat vague. Vagueness does not imply invalidity, and thus arguing that a claim cannot be true because it is imprecise is a fallacy.
Argument of the beard: overview from Logically Fallacious
Open and closed concepts and the continuum fallacy: overview and discussion of the idea behind the fallacy