Slippery slope

If each step in a chain is 80% likely, a chain of four steps is only about 41% likely, yet slippery-slope arguments present the end as certain.

One step, and then the whole slope

The Stanford Encyclopedia of Philosophy describes the slippery slope as an argument that goes from a starting point, through a series of small steps, to an undesirable conclusion, and then rejects the starting point because of that conclusion. A made-up example: “If we let students use calculators in class, they will stop learning arithmetic, then they will fail maths, then they will drop out, so calculators must stay banned.” The argument is only as strong as its weakest link, and each link is asserted rather than shown.

Why the chain is weaker than it looks

Each step in a slippery slope may sound plausible on its own. The trouble is that plausibility shrinks when steps are multiplied. Suppose every step has a fairly high chance, 80%, and the steps do not depend on each other. Two steps together have a chance of 0.8 × 0.8 = 0.64. Four steps give about 0.41, and seven give about 0.21. A conclusion that sounded nearly certain turns out to be less likely than a coin toss. Real steps are rarely as likely as 80%, so the true figure is often lower still.

Slippery slopes also skip the brakes. Between “allow calculators” and “drop out” there are teachers, tests, parents and rules that can stop the slide at any point. An argument that ignores every place where people could decide differently treats a series of choices as if it were a rock rolling downhill.

When a slope is real

The Internet Encyclopedia of Philosophy notes that the slippery-slope form occurs in both good and faulty arguments, and that its quality depends on how likely each step really is. Some chains are well supported: a small unpaid bill really can grow through late fees and interest, and the billing rules say exactly how. The question is whether each step comes with a reason. A related puzzle, which the Stanford Encyclopedia also mentions, comes from words without sharp edges. One grain of sand is not a heap, adding one grain never seems to turn a non-heap into a heap, and yet a million grains clearly are a heap. Puzzles like this one, and the similar one about how many hairs make a beard, show that the lack of a clear boundary does not mean there is no difference between the ends.

Slippery slopes are common in debates about new rules because the bad ending is easy to picture and the steps in between are not. A useful exercise is to write the chain out as a numbered list. Seen on paper, a jump such as “then they will drop out” usually stands out as the step with no support at all.

To answer a slippery slope, ask for the evidence behind each step, point out where the chain could stop, and, if the speaker gives numbers, multiply them.