▶ CLASSICAL LOGIC·5 MIN

DISJUNCTION ELIMINATION

When you know it's A or B, and you can rule out one, the other is forced.

What this is

Disjunction elimination is the formal name for the move 'either P or Q. Not P. Therefore Q.' Sherlock Holmes used it constantly: when you've eliminated the impossible, whatever remains, however improbable, must be the truth.

The practice trains you to USE disjunction elimination — but more importantly, to notice when you're tempted to use it on disjunctions that aren't actually exhaustive.

Steps

1.Pick a situation where you're trying to figure out what's going on.
2.List the candidate explanations as a disjunction: 'It's either A or B or C.'
3.First check: is the list exhaustive? Are there explanations missing? (This is where most informal disjunction elimination fails — the 'A or B' was actually 'A or B or C or D' the whole time.)
4.If the list is exhaustive, work through each: what would have to be the case for it to be true? What evidence rules it out?
5.Rule out what you can rule out. Whatever remains is your tentative answer.
6.Then test the remainder: even if you've eliminated the others, does this one fit the evidence well?

▸ AFTER

Where in life do you use 'it must be either A or B' without checking whether C, D, and E are also possible?

Reflections you write below are saved to your trajectory — Claude reads the prose and adds a small dimensional shift to your map, the same way it does for daily dilemmas and diary entries.

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More on this practice

The move 'either P or Q; not P; therefore Q' has two ancestries. As a formal rule it sits among Chrysippus's Stoic indemonstrables (the fifth), and logicians call this particular shape the disjunctive syllogism. Its most quoted expression, though, is Sherlock Holmes's: 'when you have eliminated the impossible, whatever remains, however improbable, must be the truth' — a line Conan Doyle gave him as early as The Sign of Four.

The rule is valid, which is exactly what makes it dangerous in casual use. The logic is airtight given the disjunction — but the disjunction itself is an assumption, and it's usually where the reasoning breaks. Holmes can only land on the improbable remainder if his list of possibilities was genuinely complete. In real life, 'it's either A or B' is constantly a disguised 'it's either A or B or some C I haven't thought of,' and the whole elimination collapses the moment the unlisted option turns out to be the truth.

There's a second, quieter trap: the kind of 'or.' Logicians distinguish the inclusive or (at least one, maybe both) from the exclusive or (exactly one). Disjunctive syllogism is safe either way, but everyday reasoning slides between them and sometimes rules out a possibility that was never actually excluded. So the practice has two halves: use the rule when the alternatives are truly exhaustive, and — far more often — catch yourself when they aren't.

Common pitfalls

Assuming the disjunction is complete. 'It's either A or B' is the step that fails most often; the real list was longer the whole time.
Eliminating an option on weak grounds just to force a clean answer. The remainder is only as solid as the ruling-out that produced it.
Forgetting to confirm the survivor. Even after eliminating the rest, check that the last option actually fits the evidence rather than merely being what's left.

A worked example

Your bike is gone from the rack. You reason: either it was stolen, or a friend borrowed it, or I left it somewhere else. You rule out 'borrowed' (no one has the lock code) and lean toward 'stolen.' Then you stop and check the list for completeness — and remember a fourth option you'd omitted: the building moved bikes during the rack repair you got an email about. You check the relocated rack. There it is. The disjunctive syllogism was running perfectly; it just had the wrong menu, and the answer was an item that was never on it.

Thinkers in this lineage

Chrysippus — Catalogued the disjunctive syllogism among the Stoic indemonstrables in the 3rd c. BCE.
Arthur Conan Doyle — Gave Holmes the popular formulation — eliminate the impossible, and the remainder, however improbable, is the truth.
John Venn — Whose diagrams make vivid the inclusive-vs-exclusive 'or' distinction the rule depends on.

Where to read further

Stoic Logic
Benson Mates · 1953
The scholarly account of Chrysippus's logic and the indemonstrable inference forms.
The Sign of Four
Arthur Conan Doyle · 1890
Where Holmes states the elimination principle that dramatizes the rule's appeal and its hazard.

Pairs well with

Kindred practices

Differential diagnosis — Medicine's disciplined version — list the possible causes, then test to eliminate, while guarding against the one you forgot to list.
Process of elimination — The everyday cousin — only as reliable as the completeness of the options you started with.

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