A slippery slope says that no single step makes a difference, so no number of steps does. Written as forty conditionals chained from a starting case, it is valid, and the lab shows what each of four treatments of vagueness does to the chain: classical logic keeps every conditional and the absurd conclusion, a cutoff falsifies one, a borderline range makes the universal claim super-false, and degrees spread the loss over all forty.
Core ideas
The previous lessons tested arguments for validity and sets for consistency. The slope uses both, and adds a choice about how to treat a vague word.
Each step says that if the property holds at one point it holds one step further down. The start is granted, the steps are granted one by one, and the conclusion follows by modus ponens repeated as many times as there are steps.
The chain is valid in form. An argument is faulty when a premise is false, and the slope’s question is which premise. Four treatments of the vague word answer that differently.
A step that makes no noticeable difference may still make a small one, and small differences add up. The degrees treatment is the one that keeps track of the sum.
The material
One slope, its premises, and what each treatment does to them.
A slippery slope argument has a familiar shape: this step makes no moral difference, and so does the next, and so on, so no step does and we may go all the way. It is used both ways, to warn against a first step and to excuse a last one, and it is often dismissed as a fallacy without saying which kind. The lab writes it out and lets the dismissal be tested.
Let F(k) be “k weeks’ notice is adequate notice”. Forty weeks is plainly adequate, so F(40) is a premise. One week less never makes a moral difference, so for each k from 40 down to 1 there is a premise F(k) → F(k − 1). The conclusion is F(0): no notice at all is adequate.
Under the classical treatment, every conditional is true, the start is true and the conclusion is true. The lab reports forty steps, all forty conditionals true, and the conclusion true. The argument is valid, so someone who finds the conclusion absurd must give up a premise, and the question is which one. That is the whole content of the dispute, and it is why “fallacy” is not the right word for it.
A tolerance conditional says that one small step does not change whether a vague property holds: F(k) → F(k − 1). A slope is a start plus a chain of them.
Three other treatments give a different answer to the question. A sharp cutoff says there is some k where adequacy stops, here 24: F(24) is true and F(23) is false, so exactly one conditional is false and the conclusion is false. A borderline range says the boundary is vague: there is no fact about where adequacy stops, only a set of admissible places to put the line. With the range 16 to 32 the lab lets the cutoff fall anywhere from 17 to 32, so the sixteen conditionals from F(17) → F(16) up to F(32) → F(31) are each true on some sharpenings and false on others, and so neither true nor false outright. No single conditional is false, but the claim that every step is tolerated is false on every sharpening, which the lab calls super-false, and so is the conclusion. Degrees give each conditional a value just under 1, and chaining forty of them guarantees nothing.
Let F(k) be “k is smaller than a million”, started at 10,000. Each step down keeps it smaller, so every tolerance premise is true and the conclusion that 0 is smaller than a million is true. The chain is the same chain. What separates this from the notice case is that here the premises are all true, and the form was never the difference.
The lab models the conceptual slope, in which a property is said to survive each step. It does not model the causal slope, in which allowing one step makes the next more likely to be taken. That is a claim about how people and institutions behave, to be supported with evidence of the kind Science, Induction and Causation discusses. A policy can be on a causal slope without any conceptual one, and the argument against it is then an empirical claim.
Interactive
A chain of tolerance conditionals under four treatments of vagueness
Method
Five steps. The third and fourth are two runs of the same chain under different rules.
Name the property, the case where it plainly holds and the case where it plainly fails, and count the steps between them.
Say what one step changes, and write one conditional per step, in the direction that carries the property from the start to the end.
With every conditional true, the conclusion follows. Confirm that it is the absurd one you started with.
Move the treatment menu to the cutoff, the borderline range and degrees. Read the conditional tile and the conclusion tile each time.
A sharp cutoff gives up one conditional. A range gives up the universal claim. Degrees give up the idea that tolerance is perfect. Say what each costs.
The chain has forty steps, from 40 weeks of notice down to none.
Each treatment keeps the chain valid and answers the question of which premise fails. The classical one answers none, which is why it has to accept the absurd conclusion.
Given a slippery slope, you should be able to write its start, its tolerance conditionals and its conclusion, run the chain under the classical, cutoff, range and degree treatments, and report for each which premise gives way.
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