Philosophy & Critical Thinking: Counterexample hunting

What you'll be able to do: Test a philosophical generalisation, a definition, a theory, a principle, against a case built specifically to break it, the way Edmund Gettier broke a definition philosophers had accepted for over two thousand years, in three pages.

The definition that stood for two millennia

The classical analysis of knowledge, standard by the mid-twentieth century: S knows that P if and only if (1) P is true, (2) S believes that P, (3) S is justified in believing that P. In 1963, Edmund Gettier published a three-page paper asking whether justified true belief is knowledge, and gave cases where it plainly wasn't. Here's the standard teaching version:

Smith has strong evidence that Jones owns a Ford. Jones has driven one for years and just showed Smith the ownership papers. From this, Smith validly infers and justifiably believes: "Jones owns a Ford, or Brown is in Barcelona" , tacking on a wild disjunct about a colleague whose location Smith has no idea about. Unknown to Smith, Jones sold the Ford yesterday and is driving a rental, so "Jones owns a Ford" is false. But by pure coincidence, Brown really is in Barcelona. The disjunctive belief is therefore true, Smith believes it, and Smith is justified , all three JTB conditions hold. But Smith doesn't know Brown is in Barcelona. It's true by luck, disconnected from anything Smith's justification actually tracked.

That's the whole force of a good counterexample: one case where every stated condition holds and the intuitive verdict still fails is enough to refute a claimed necessary-and-sufficient analysis, the same logical work one black swan does against "all swans are white." The fifty years of "fourth condition" literature that followed, no false lemmas, causal theories, reliabilism, Nozick's tracking conditions, is the measure of how much one well-built case can generate.

The escalating case: Ship of Theseus

Plutarch records the puzzle (Life of Theseus, 23); Hobbes adds the twist that makes it a proper counterexample-hunting exercise rather than a single puzzle.

Generalisation under test: an object stays numerically identical over time iff it retains all its original parts. Broken immediately, a car is "the same car" after an oil change, and nobody thinks otherwise. Refine: iff it retains spatiotemporal and functional continuity through gradual replacement. Hobbes's counterexample is built specifically to break the refinement: someone collects every discarded original plank as Theseus's ship is repaired, and reassembles them into a second ship. Now there are two candidates for "the real ship", one with continuity of form and function along a continuous path, one with continuity of the original matter, and the refined criterion doesn't say which wins. It doesn't give a determinate answer after all.

That's the shape counterexample hunting actually takes in a live literature: generalisation, counterexample, refinement, new counterexample built for the refinement. It doesn't stop after one round, and treating it as a single knockout blow is a misunderstanding of the method.

How to run it

1. State the generalisation precisely, in "iff" form where possible, it's easiest to break a claim stated as a necessary-and-sufficient condition.
2. Ask for a case that satisfies every stated condition while the intuitive verdict fails (or the reverse) , built specifically to break it, not merely a hard case.
3. Check the counterexample is genuine. Does it really satisfy the stated conditions, or does it cheat by quietly changing the terms?
4. If it survives, refine the generalisation to exclude the counterexample without excluding the ordinary cases it was meant to cover.
5. Ask for a counterexample to the refinement. Repeat until it holds or you've learned exactly where it breaks.

Here is my generalisation, stated as precisely as I can: [X holds iff conditions 1, 2, 3]. Give me a case that satisfies every condition I've listed but where the conclusion intuitively fails, or a case where the conclusion clearly holds but one of my conditions doesn't. Don't soften it into a merely hard case; build it to actually break the definition.

The tell

The "counterexample" you're given is really a hard case where the generalisation still holds, argued at length to sound like a refutation. A real counterexample is a single sentence you either have to accept breaks the rule, or can explain, specifically, why it doesn't count.

Try this today

Take a definition from your current reading, a definition of knowledge, personal identity, justice, whatever the unit is on, and ask for the case built to break it before you accept the definition is finished.