Continuum fallacy

When does a snack become a meal?

“Where’s the line between a snack and a meal? A handful of almonds is a snack. One more almond doesn’t turn it into a meal. Neither does one more after that. You can’t name the almond where it changes, so the whole distinction is made up, and this plate of nachos is a snack.”

Each step is true: one almond really doesn’t turn a snack into a meal. But the speaker has moved from “there’s no exact point” to “there’s no difference”, and then used that to call something at the far end of the range a snack. This is the clear-cut case, the one the old name “argument of the beard” describes: the fact that two things are connected by small steps is used to deny that they differ.

“Define messy”

Priya: This function is too long and messy. It needs to be split up.

Sam: Define “messy.” How many lines is too long? Forty? So forty-one lines is messy and forty isn’t? That’s absurd. “Messy” doesn’t mean anything, so it isn’t a real criticism.

Sam never argues that the function is clear. He argues that because “messy” has no exact cutoff, it can’t be used at all. But nearly every useful word for judging quality (“readable”, “slow”, “cluttered”) works this way, and a 400-line function with twelve levels of nesting is messy whether or not anyone can name the line count where messiness begins. Robert Thouless, who gave the fallacy its classic description in 1930, warned that “badgering one’s opponent to define his terms” can itself be “a piece of crooked argumentation”. If Priya picks a number, Sam can mock it as arbitrary; if she doesn’t, he can call the word meaningless.

One more day won’t matter

A freelancer has a tax form due at the end of the month. On the 1st, she thinks: “It makes no real difference whether I start today or tomorrow.” On the 2nd she thinks the same, and on every day after that, until the 31st arrives and the form isn’t done.

Here the small-steps reasoning isn’t used to deny a distinction in words but to justify each step of putting something off. Each day’s thought is close to true in isolation, and the conclusion (“it doesn’t matter when I start”) is false. Dorothy Edgington, quoted in the Stanford Encyclopedia of Philosophy‘s entry on the sorites paradox, calls this the “mañana paradox”. It shows where the continuum fallacy borders on the Slippery slope: a whole series of steps rather than a single pair of neighboring cases.

Trudy Govier set out this reasoning, as the linguistic type of slippery slope, under the name fallacy of assimilation. Walton quotes her schema and finds that it fits the textbook accounts of the argument of the beard fairly well:

The first three premises can all be true. The conclusion doesn’t follow, because small differences that don’t matter one at a time can add up to a difference that does. Every step from one almond to a plate of nachos is tiny; the distance from the first case to the last is not.

• Denying the difference between extremes: the classic form. Because there’s continuous variation between two things, there’s supposedly no real difference between them. Thouless’s version: “Because there is no sharp dividing line, it has been suggested that there is no difference.”
• Rejecting a vague term: because a word has no precise cutoff, it’s treated as meaningless or illegitimate, as in the “define messy” example. Dowden’s list in the Internet Encyclopedia of Philosophy defines the line-drawing fallacy this way: improperly rejecting a vague claim because it isn’t as precise as we’d like.
• “Only a matter of degree”: T. Edward Damer, who calls it the fallacy of the continuum, describes it as “assuming that small differences are always unimportant”, so that contraries connected by small differences are “really very much the same”. He notes that this assumption is often expressed as “it’s only a matter of degree”, and that the fallacy’s ancient name is “the fallacy of the beard”.
• The series form: many small steps taken one after another, as in the tax-form example. This is the form of the ancient sorites paradox, and it overlaps with the conceptual type of Slippery slope.

The sorites paradox. The fallacy borrows the machinery of one of the oldest puzzles in philosophy. The Stanford Encyclopedia of Philosophy‘s entry says the Megarian philosopher Eubulides (4th century BC) is usually credited with the first version: one grain of wheat doesn’t make a heap, no single grain can turn a non-heap into a heap, so no number of grains makes a heap (soros is Greek for “heap”). Words like “bald”, “tall” and “old” can all be run through the same argument. The entry explains that a vague word’s tolerance of small differences is what makes each step seem safe, and that because we use such words successfully all the time, most theorists of vagueness suppose the paradox can be solved, even though they disagree about how. Some (the epistemicists) go further and hold that vague words do have sharp boundaries whose locations we can’t know. You don’t need to settle that debate to spot the fallacy. The paradox is a puzzle about where valid-looking reasoning went wrong; the continuum fallacy accepts the absurd conclusion and uses it.

How it relates to the slippery slope. Sources draw the lines differently. The Stanford Encyclopedia of Philosophy‘s entry on fallacies treats “the beard” and “the heap” as slippery slope arguments that depend on vague terms. Douglas Walton, in a paper devoted to the argument of the beard, distinguishes three things. The heap is a paradox, a logician’s model with a valid form and an absurd conclusion. The sorites slippery slope is an argument used in real disputes, which pushes someone step by step through a whole series of cases toward a conclusion. The argument of the beard needs only two neighboring cases, one on each side of a proposed line, and argues that since they don’t really differ, the line (or the distinction it marks) is illegitimate. On Walton’s analysis, the sorites slippery slope is an extension of the argument of the beard, and both belong to a broader family of arguments that attack a criterion for being vague.

What it’s called. No single name dominates. Thouless’s “argument of the beard” is the earliest textbook name Walton found, and in a search of well over two hundred logic textbooks he found it treated as a distinct fallacy in only nine. Damer calls it the fallacy of the continuum, a label Walton calls “quite acceptable”. Dowden uses “line-drawing” (and lists the bald man fallacy, the fallacy of the heap and the sorites fallacy as other names for it). This site uses “continuum fallacy”.

The opposite error. Thouless paired the fallacy with its mirror image: making sharp distinctions in speech where none exist in fact, as if every object had to be either black or white. Some textbooks Walton reviews treat the argument of the beard as the opposite of this black-or-white thinking, while others file the two together, so this site doesn’t declare either relationship. They share a hidden assumption: that a distinction is real only if it has an exact boundary. The False dilemma accepts the assumption and insists on the sharp line; the continuum fallacy accepts it and gives up the distinction.

Questions about vagueness and where to draw lines are often exactly the right ones to ask. It isn’t the continuum fallacy when:

• Precision actually matters here. A rule that has to be applied the same way to everyone, a measurement, or a contract term may need an exact line. Walton notes that a request for precision is, in general, “appropriate and useful” in a critical discussion; it goes wrong when pressed to deny a difference that plainly exists.
• The argument is about where the line is, not whether there’s a difference. Asking whether a cutoff should be at 60 or 61, or whether one borderline case falls on one side, accepts the distinction.
• The cases really are neighbors. “There’s no meaningful difference between 59 and 60 points” can be true. It becomes the fallacy only if it’s extended to “so there’s no difference between 20 points and 90”.
• Arbitrariness is itself the problem. A line that is presented as marking something in nature, when it was really chosen for convenience, can fairly be called arbitrary. Walton’s test question for this kind of objection: is arbitrariness, in this case, a sufficient reason to reject the line?
• A difference really is only one of degree. Pointing out that two things lie on the same scale is a correction when someone has treated them as different in kind.

The test: is the argument about where to draw the line, or about whether there’s anything on either side of it?

A cutoff inside the margin of error

A pottery course requires 60 points out of 100 to earn a certificate. A student scores 59. She points out that the glaze section is scored by eye in two-point steps, and asks whether her piece could be rescored by a second instructor.

She is relying on the fact that one point is a tiny difference, and the 60-point line is arbitrary in the sense that 59 or 61 could have been chosen instead. But she isn’t arguing that passing and failing are the same, or that the cutoff is meaningless. Her point is that the scoring method isn’t precise enough to tell a 59 from a 60 reliably, which is a question about where the line is, not whether there’s a difference. Walton discusses a similar grading case: pleading for one more mark is not necessarily fallacious, but concluding that there’s no real difference between grades would be.

“About six hours” isn’t enforceable

A trail race’s rules say runners must reach the halfway checkpoint in “about six hours” or be pulled from the course. At the finish, two runners compare notes: one was stopped at 6 hours 5 minutes, the other waved through at 6 hours 20. They ask the organizers to replace “about” with an exact time.

The runners are objecting to a vague criterion because it’s vague, which is the shape of the continuum fallacy. But here precision actually matters: a cutoff that decides who may continue has to be applied the same way to everyone, and the vagueness is producing unequal treatment. They aren’t claiming that 6 hours and 9 hours are the same. W. Ward Fearnside, quoted by Walton, makes the matching point: the practical affairs of life require drawing lines across continuous scales, and those lines are arbitrary, but “’arbitrary’ in this sense means that the rules are matters of human convenience, not that they are unfair”.

Every individual step really is too small to matter, so each premise feels undeniable, and nobody wants to defend the claim that one almond or one hair changes everything. Any exact line someone proposes can be mocked for the same reason. Thouless noticed that the result is easy to see through when the subject is beards, and much harder when the subject engages our feelings more strongly.

Walton describes the underlying bind: in ordinary argument, people are pulled back and forth between vagueness and precision. Criteria stated in everyday words are vague, and can be attacked as vague. Made precise, say with a number, they can be attacked as arbitrary. Both objections are sometimes fair, which is why the move can pass as rigor.

• Point to the clear cases at each end. Thouless’s advice was to point out that the difference is nevertheless real. A plate of nachos and a handful of almonds are not hard to tell apart.
• Concede the gray zone. Agreeing that there are borderline cases costs nothing, and makes clear that they aren’t the issue.
• If a line is needed, say it’s conventional, and defend it as such. Walton notes that admitting a criterion is arbitrary while still defending it can be a reasonable reply. Speed limits and age requirements work this way.
• Ask which question is on the table: where to draw the line, or whether there’s a difference at all.
• In yourself, notice “it’s only one more” or “it’s just a matter of degree” when you’re making a series of small decisions, and look at where the series ends.