Disjunctive syllogism
from 300 BCE Fourth and fifth of the Stoic indemonstrables
Either the one or the other; not the one; therefore the other — the whole logic of elimination in three lines.
- Active
- from 300 BCE
- Where
- Athens
- Status
- In use
- Tradition
- Forms of inference
What it is
Given that at least one of two things holds, and that one of them does not, the other must. It is the formal shape of ruling things out.
Everything depends on the disjunction being true — on the alternatives really being exhaustive. The rule itself is beyond argument; the list of options almost never is.
How it works
Establish that the alternatives cover the ground. Eliminate all but one. Accept what is left.
The first step is the whole difficulty and is usually skipped. Most bad uses of this rule are not errors of logic but unstated assumptions that the two options offered are the only ones there are.
The form of the inference
- Either P or Q
- Not P
- Therefore Q
The argument
Proposition The fault is in the house wiring, not the street supply.
- The house has no power, and there are only two places the fault can be: the street supply to the property, or the wiring inside it.
- If the street supply had failed, the neighbouring houses on the same transformer would also be dark.
- The neighbours' lights are on.
- So the street supply has not failed.
Therefore The fault is in the house wiring.
The first step is the one to attack, and it is stated rather than assumed here deliberately. If there is a third possibility — a fault in the service line to this property alone, which is neither the street supply nor the house wiring — the argument fails while looking perfectly valid.
What it is good for
Fault-finding, diagnosis and any situation where the possibilities can genuinely be enumerated. It is the most efficient reasoning there is when the list of options is closed and short.
Where it fails
It fails at the disjunction, essentially always. The false dilemma — presenting two options as exhaustive when they are not — is this rule running on a bad first premise, and because the rest of the argument is impeccable the conclusion inherits an unearned authority.
It also fails on inclusive disjunction. If *either P or Q* allows both, then eliminating P gives you Q, but establishing P gives you nothing about Q — and ordinary English *or* switches between the two readings without warning.
Common misconceptions
That eliminating one option confirms the other in the strong sense. It does so only within the list, and the list is a claim about the world that has to be defended like any other.
Also that this is what detectives mean by eliminating the impossible. That formulation assumes the suspects are all known, which is precisely the assumption that fails in real cases.
Where it came from
The fourth and fifth Stoic indemonstrables cover the exclusive and inclusive forms separately, which is a sharper treatment than the single rule usually taught now.
What became of it
Standard in classical logic, and one of the rules given up in some relevance and paraconsistent logics, since it is the step by which a single contradiction can be made to yield anything at all.
Lasting influence
It is the backbone of diagnostic procedure in every field, and the shape of the commonest rhetorical trick in public argument.
Why it still matters
Constant. Every troubleshooting flowchart ever printed is a tree of these.
Put another way
Checking pockets. It works if you know how many pockets there are.
Texts
Sextus Empiricus, Against the Logicians
Preserves the Stoic treatment of both the exclusive and the inclusive forms, from an author who was arguing against the whole enterprise.
Where to start
Learn the rule, then spend a week noticing how often the two options you are offered are not the only two.
The surprising bit
This innocuous rule is the one that lets a contradiction prove anything whatever, which is why logicians who want to tolerate contradictions have to give it up rather than give up anything more obviously suspect.
In their own words
Either the first or the second; but not the first; therefore the second
Who held it
Sextus Empiricus
Connections
- Influenced
- Constructive dilemma
- Parallel to
- Modus tollens
The verdict
Flawless as a rule and treacherous in use, because it puts all the weight on a premise nobody examines.