Answers · The mathematics underneath
Why is an if-then statement true when the condition is false?
In shortAn if-then claim only forbids one case, the condition holding while the result fails, so when the condition is false there is nothing left for it to forbid and it stays true.
Because the claim rules out one situation only: the condition holding while the consequence fails. In every other situation, including every situation where the condition is false, there is nothing for the claim to break.
A shop posts a rule: if an order is over a hundred pounds, shipping is free. That sentence tells you about large orders. It tells you nothing at all about a forty pound order, so no forty pound order, however it is shipped, can contradict it.
| A | B | if A then B | if B then A | |
|---|---|---|---|---|
| over 100, free shipping | 1 | 1 | 1 | 1 |
| over 100, shipping charg | 1 | 0 | 0 | 1 |
| under 100, free shipping | 0 | 1 | 1 | 0 |
| under 100, shipping char | 0 | 0 | 1 | 1 |
Count the rows. There are four possible combinations of over a hundred and free shipping, and exactly one of them, over a hundred with shipping charged, breaks the rule. The other three keep it. So a rule of this shape is not a description of every order; it is a filter that rejects one kind of order and passes everything else through untouched.
The last form is the one to hold in your head. There are two separate ways for the claim to come out true: the consequence holds anyway, or the condition fails. For a forty pound order the condition fails, so the second way applies immediately and the answer does not depend on the shipping at all. Free shipping on a small order is not a loophole in the rule. The rule had nothing to say about small orders and it still has nothing to say.
The guess that fails
The natural guess is that a false condition makes the claim false, or at least doubtful. What that guess actually does is replace the rule with a claim about the consequence, namely that free shipping means the order was over a hundred pounds. That is the reverse, if B then A, and it is a different sentence. It fails on the third row: an order under a hundred pounds given free shipping keeps the original rule and breaks the reverse. So the rule can be true exactly where the reverse is false, which is enough to show they are not the same claim.
Exactly one backwards inference is licensed: if shipping was charged, the order was not over a hundred pounds. That is the rewrite, if not B then not A, and it is true in precisely the same cases as the original. From free shipping, with no further information, nothing follows.
- if A then B, false only when A holds and B fails
- the reverse, an independent claim that must be argued on its own
- swap the parts and negate both, which holds in exactly the same cases as the original
- not, reversing the truth of what follows
The disagreement can be read straight off the table. In the third row, condition false and consequence true, the rule is true because nothing was promised about that order, while the reverse is false because it promised free shipping would imply a large order and here it did not. In the second row the positions flip: the rule is false because the promise is broken, and the reverse is true because its own condition, free shipping, does not arise at all. Neither claim follows from the other in either direction.
This is where the misreading gets expensive. Suppose a condition affects one person in a thousand, and a test catches ninety nine per cent of real cases with a one per cent false positive rate. Test a hundred thousand people. A hundred have the condition, and ninety nine of them test positive. Of the ninety nine thousand nine hundred who do not, one per cent, that is nine hundred and ninety nine, test positive anyway. That gives one thousand and ninety eight positives in total, and only ninety nine of them are real, so a positive result is right about nine per cent of the time. Nothing is wrong with the test. The forward claim, if the condition is present then the test is positive, is true and excellent; the question asked was the reverse, if the test is positive then the condition is present, and nobody established it. Nine hundred and ninety nine of the one thousand and ninety eight positives are the healthy majority leaking through the false positive rate.
When both directions are wanted
The claim stops being the whole story the moment you need the two parts to be true in exactly the same cases. That is the biconditional, if and only if, and it is two claims: the original and its reverse, each requiring its own argument. A number is even if and only if it is twice a whole number is meant to read in both directions, and establishing it means establishing both. Proving one and asserting the other proves half of it.
It is also worth being clear what the vacuous rows do and do not give you. They give you truth, not information. A rule that is true of small orders says nothing about them, and if you want a rule that constrains small orders you have to post the biconditional, or post a second rule. Reading the freedom of the vacuous case as permission to conclude anything is the same error as reading the rule backwards, only louder. The claim is true; it is simply silent.
Worth keeping
- If A then B is false in exactly one situation: A holding while B fails.
- If B then A is a separate claim that can fail where the original holds, so it needs its own argument.
- If not B then not A is true in exactly the same cases as if A then B, which makes it a free substitute when the forward direction is awkward.
Where this goes next
This question is one step of Arguing That It Always Holds, a course that works through its subject from the beginning. The page that teaches this part of it is What an Implication Promises, and it reads the whole way through whether or not you have an account.
What an account changes is the practice: problems standing under every idea, marked with a reason rather than a tick, in a copy of the course that is yours.
Read that pageThe whole courseOther questions people ask about this
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