Modus tollens
from 300 BCE Second of the Stoic indemonstrables
If the second thing would have followed from the first, and the second is not so, then neither is the first — the shape of every test and every falsification.
- Active
- from 300 BCE
- Where
- Athens
- Also
- modus tollendo tollens
- Status
- In use
- Tradition
- Forms of inference
What it is
Modus tollens takes a conditional and the denial of its consequent, and delivers the denial of its antecedent. Given *if P then Q*, and given that Q is false, P must be false too.
Where modus ponens confirms, this refutes, and that asymmetry matters more than it looks. A single failed consequence brings down the whole antecedent, which is why this rule and not its twin is the engine of testing.
How it works
Work out what would have to be the case if the claim were true. Check whether it is. If it is not, the claim is not either.
The skill is in the first step. A claim that has been stated so vaguely that nothing in particular would follow from it cannot be tested this way at all, and this is the honest core of what Popper was pointing at.
The form of the inference
- If P, then Q
- Not Q
- Therefore not P
The argument
Proposition The irrigation pump is not working.
- If the pump were running, the header tank would have filled overnight, since it fills in four hours and the pump runs for eight.
- The tank was checked at dawn.
- It was at the same level as the previous evening.
- So the tank did not fill overnight.
Therefore The pump was not running.
The conclusion is about the pump although nobody looked at the pump. That is the characteristic power of the rule: it reaches back through a consequence to the thing that would have produced it.
What it is good for
Testing. Every experiment, every diagnostic check and every falsification has this shape, and it is the only rule here that lets you learn something decisive about a claim without examining it directly.
Where it fails
Its weakness is that it never tells you which premise failed. A claim rarely predicts a consequence by itself — it does so together with a raft of assumptions about the equipment, the conditions and the background. When the consequence does not appear, modus tollens says only that something in that whole bundle is false, and the arguer usually picks whichever member of the bundle they were least attached to.
This is the Duhem-Quine problem, and it is why a determined theorist can absorb almost any failed prediction by adjusting something else.
Common misconceptions
That it is the same as *denying the antecedent* — from *if P then Q* and *not P*, concluding *not Q*. That is a fallacy, and it is the single most common invalid move in ordinary argument.
Also that a failed prediction refutes the theory. It refutes the conjunction of the theory and everything assumed alongside it.
Where it came from
The second of the five Stoic indemonstrables, alongside modus ponens. As with its twin, the systematic treatment is credited to Chrysippus and survives mainly through hostile sources.
What became of it
Standard equipment in propositional logic, and given a central methodological role in the twentieth century by Popper, who built an entire philosophy of science on it.
Lasting influence
Falsificationism is modus tollens elevated to a research programme. So is every null-hypothesis test, though the statistical version has to be much more careful about what counts as *not Q*.
Why it still matters
It is the shape of every check anybody runs on anything.
Put another way
Testing a fuse by seeing whether the light comes on.
Texts
Karl Popper, The Logic of Scientific Discovery
Not a logic text. It is the argument that this one rule, and not induction, is what science actually runs on.
Where to start
Learn the rule in five minutes, then read Popper, then read Duhem on why it is not as clean as Popper needs it to be.
The surprising bit
It is strictly more useful than modus ponens for learning about the world, because confirmation of a prediction is compatible with the theory being false while a single genuine failure is not.
In their own words
If the first, then the second; but not the second; therefore not the first
Who held it
Sextus Empiricus
Connections
- Influenced
- Reductio ad absurdum
- Parallel to
- Modus ponens
The verdict
The most powerful rule in ordinary reasoning, limited only by never saying which of your assumptions broke.