If-then reasoning has two valid moves and two classic traps. The valid moves are modus ponens (if P then Q, P is true, so Q is true) and modus tollens (if P then Q, Q is false, so P is false). The traps are concluding P from Q, called affirming the consequent, and concluding not-Q from not-P, called denying the antecedent.
The anatomy of an if-then rule
A conditional statement has an "if" part, the antecedent, and a "then" part, the consequent. Take the rule "If a request is over the approval limit, it needs a second signature." The antecedent is "over the approval limit" and the consequent is "needs a second signature".
The rule runs in one direction only. Being over the limit is enough to require the signature, but the signature might be required for other reasons too. Keep that one-way arrow in mind and the rest follows.
The two valid moves
Modus ponens: confirm the "if" part
You are told the rule and you are told the "if" part is true, so the "then" part must be true. "If the door is locked, the alarm is on. The door is locked. So the alarm is on."
This move feels natural, and it is the easiest conditional inference to apply correctly. Questions that use it are testing whether you read the rule carefully, not whether you can see the logic.
Modus tollens: deny the "then" part
Now you are told the "then" part is false. "If the door is locked, the alarm is on. The alarm is off. So the door is not locked."
The reasoning is a short proof by contradiction. If the door were locked, the alarm would be on, and it is not, so the door cannot be locked. Many people find this step harder than modus ponens. In his 1968 experiments on conditional sentences, the psychologist Peter Wason set out to study why the inference from "not Q" to "not P" is so difficult.
The contrapositive: the same rule, turned around
Modus tollens works because "If P then Q" says exactly the same thing as "If not-Q then not-P". The second form is called the contrapositive. "If a parcel is fragile, it is packed in foam" is equivalent to "If a parcel is not packed in foam, it is not fragile".
Two other rewrites look similar but are not equivalent. The converse is "If Q then P" and the inverse is "If not-P then not-Q". They are equivalent to each other, but neither is equivalent to the original rule, and nearly every conditional trap is built from one of them.
The two classic traps
Trap one: affirming the consequent
"If it rained, the pavement is wet. The pavement is wet. So it rained." This argument confuses the rule with its converse. A sprinkler, a burst pipe or a street cleaner can all make a pavement wet.
Fix: when the fact matches the "then" part, stop. The rule says what follows from the "if" part; it says nothing about what else could produce the "then" part.
Trap two: denying the antecedent
"If you submit by Friday, you get a discount. You did not submit by Friday. So you do not get a discount." This argument confuses the rule with its inverse. The same discount might be offered to loyalty-card holders.
Fix: when the fact denies the "if" part, stop. The rule only tells you what happens when the "if" part is true.
Reading the wording correctly
Real questions rarely say "if P then Q" in plain form. These translations cover most of what you will meet.
- "P only if Q" means "if P then Q". "You can enter only if you have a badge" means that entering requires a badge.
- "P if Q" means "if Q then P", the reverse of the previous line.
- "Whenever", "provided that" and "as long as" all introduce the "if" part.
- "Unless" is safest read as "if not". "The report is late unless Jo approves it" means that if Jo does not approve it, the report is late.
- "If and only if" runs in both directions. Here, and only here, the two traps become valid moves.
Everyday speech also muddies things. "If you finish your dinner, you can have dessert" usually implies no dessert otherwise. In a test, do not read that extra meaning into a plain "if".
A two-question check
Write the rule down, then ask two questions about the fact you have been given. Does it confirm the "if" part? Then you may conclude the "then" part. Does it deny the "then" part? Then you may conclude that the "if" part is false. Any other match, confirming the "then" part or denying the "if" part, is a trap, and the conclusion does not follow.
Researchers have studied how people handle conditionals since at least the 1960s, from Wason's experiments to later work by Sandra Marcus and Lance Rips. Philip Johnson-Laird and Ruth Byrne later proposed a detailed theory of how conditionals are understood.
The main practical lesson is stable: the one-way arrow is what people forget.
Practise it
The JobCannon Deductive Reasoning Test includes if-then rules among its "follows / does not follow" questions. For a broader mix that adds analytical puzzles, there is also JobCannon's older logical reasoning test. The questions are written in-house, so read your result as practice feedback.
For the other common slips, see common deductive reasoning mistakes, and for a wider list of fallacies see common logical fallacies.