Hypothetical syllogism
from 300 BCE Stoic in origin; the name is later
If the first leads to the second and the second to the third, then the first leads to the third — and the middle term disappears from the conclusion.
- Active
- from 300 BCE
- Where
- Athens
- Status
- In use
- Tradition
- Forms of inference
What it is
Two conditionals sharing a term can be joined into a third that does not mention it. From *if P then Q* and *if Q then R*, you may write *if P then R*.
What makes it worth naming is the disappearance. The middle term does the work and then drops out of sight, which is how long chains of consequence get compressed into single claims that nobody can any longer inspect.
How it works
Establish a conditional. Establish a second whose antecedent is the first one's consequent. Join them and discard the middle.
The chain can be extended indefinitely, and each link multiplies the chances that one of them is weak. A chain of five conditionals each of which is ninety per cent reliable is not ninety per cent reliable.
The form of the inference
- If P, then Q
- If Q, then R
- Therefore if P, then R
The argument
Proposition If the ferry is cancelled, he will miss the wedding.
- If the morning ferry is cancelled, he cannot reach the mainland before two o'clock.
- If he cannot reach the mainland before two, he misses the last connecting bus.
- If he misses the last connecting bus, he does not arrive before the ceremony begins.
Therefore If the morning ferry is cancelled, he will not arrive before the ceremony begins.
The ferry and the ceremony are four hours and three intermediate steps apart, and the conclusion mentions none of them. That compression is the whole use of the rule, and also where a hidden weak link can sit unexamined.
What it is good for
Planning, risk assessment and any argument about consequences at a distance. It is how a distant outcome is tied to a present decision without walking through the intervening steps every time.
Where it fails
It fails through accumulation. Each conditional carries some chance of being wrong, and the joined conditional carries all of them at once while looking exactly as confident as a single link. Long chains of consequence in policy argument are almost always this rule applied past the point where the premises can bear it.
It also fails in contexts where conditionals are not transitive. Counterfactual conditionals in particular do not chain reliably: *if he had missed the ferry he would have driven* and *if he had driven he would have been late* do not license *if he had missed the ferry he would have been late*.
Common misconceptions
That the slippery slope is a fallacious version of it. Sometimes it is, but a slippery slope argument is fallacious only when the links are not actually there — when they are, it is a valid chain and the objection has to be made link by link.
Where it came from
Third of the Stoic indemonstrables in substance, though the name and the modern statement come much later. Aristotle has a version confined to categorical terms.
What became of it
Standard in propositional logic. Its failure for counterfactuals became a live topic once Stalnaker and Lewis built formal semantics for those in the 1960s and 1970s.
Lasting influence
Every argument that reaches from a decision to a remote consequence uses it, which makes it one of the most consequential rules in public argument.
Why it still matters
Unchanged, and increasingly relevant as policy arguments run longer chains.
Put another way
A chain of people passing buckets. The water arrives, but only as fast and as fully as the weakest passer allows.
Texts
David Lewis, Counterfactuals
Not about this rule, but the book that shows precisely where it stops working, and why.
Where to start
Learn the rule with ordinary conditionals first. The counterfactual complication makes no sense until the simple case is solid.
The surprising bit
The rule is valid for ordinary conditionals and invalid for counterfactual ones, which means the same English word *if* behaves differently depending on which mood it is in.
In their own words
If the first, then the second; if the second, then the third; therefore if the first, then the third
Who held it
Robert Stalnaker
Connections
- Relies on
- Modus ponens
- Parallel to
- Categorical syllogism
The verdict
Perfectly sound and constantly overextended.