Vagueness: How Do You Stop Sliding Down a Slippery Slope?

Core idea

A five-year-old is a child. If someone is a child, they are still a child one second later. Chain that step 630 million times and the conclusion is that a twenty-five-year-old is a child. Something has gone wrong, and there is little room to manoeuvre. This is a sorites paradox — from the Greek soros, "heap" — invented by Eubulides, who also gave us the liar.

The paradoxes arise whenever a predicate is vague: its applicability is tolerant of very small changes. If "is a child" applies, a one-second change cannot remove it. Almost every everyday predicate — "is red", "is awake", "is happy", even "is dead" — is tolerant in this way, so slippery-slope reasoning is endemic.

Priest's argument: A sorites argument chains many true premisses by modus ponens to a false conclusion, so either some premiss is not true or modus ponens does not preserve truth — classical bivalence cannot hold its ground.

Why it matters

Fuzzy logic: truth comes by degrees

One response is fuzzy logic. Childhood fades out gradually, so the truth value of "Jack is a child" fades from true to false. Measure truth by a number from 0 (complete falsity) to 1 (complete truth). Negation becomes 1 minus the value; a conjunction takes the minimum of its parts; a disjunction the maximum. Each typical sorites conditional then carries a value just short of 1 — say 0.75 — rather than being flatly true.

Either way, the argument fails

Validity now needs an acceptability level, written ε, fixed by context: a conclusion holds if its value is at least ε in every situation where the premisses reach ε. If ε is 1, modus ponens is valid but the 0.75 conditional premisses are unacceptable. If ε is below 1 — say 0.75 — the premisses pass but modus ponens fails, since a chain can drop the value below the line. The paradox collapses on both settings. We are taken in, the topic suggests, because we confuse complete truth with near-complete truth — harmless once, ruinous across a long chain.

Key takeaways

Mental model

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Practical application

Whenever an argument advances by many tiny, individually-undeniable steps, suspect a sorites and check whether "near-certainly true" is being passed off as "certainly true".

Example

A barista pours an espresso shot. "This is a fresh shot" is true the instant it is pulled. One second later it is still fresh — no perceptible change. Repeat the reasoning and a shot pulled an hour ago counts as fresh, which no one would accept. Fuzzy logic reframes it: freshness is not on or off but a value gliding down from 1. A café might fix ε at 0.9 for serving — so a shot may be "fresh enough to serve" while no longer fully fresh. The catch, and the topic's closing crack, is higher-order vagueness: there is no non-arbitrary moment when freshness first slips below exactly 1. We have not removed the boundary problem; we have only moved it.

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