Conditional Statements

One case breaks the promise, and only one.

Hypothesis and conclusion

A conditional statement has the form "if P then Q". The part after "if", P, is the hypothesis: the part you assume. The part after "then", Q, is the conclusion: the part the statement promises. It is written P ⇒ Q and read "P implies Q".

In "if n is a multiple of 4, then n is even", the hypothesis is "n is a multiple of 4" and the conclusion is "n is even". The statement is about every whole number n at once: whenever n is a multiple of 4, n is even.

The same statement can be worded without "if". "Every multiple of 4 is even" and "n is even whenever n is a multiple of 4" both say it. To find the hypothesis, ask which part is being assumed.

The one case that breaks it

A conditional makes a promise, and a promise can be broken in only one way. Take P and Q as two statements, each true or false. Together they give four cases, and the truth table lists them.

In row 1, P and Q are both true, so the promise is kept. In row 2, P is true and Q is false: the hypothesis held and the conclusion did not, so the promise is broken and P ⇒ Q is false. In rows 3 and 4, P is false, so nothing was promised and nothing can be broken. P ⇒ Q counts as true in those two rows.

So P ⇒ Q is false in exactly one case, P true and Q false. That also gives its negation: "if P then Q" is denied by "P and not Q", the hypothesis holding while the conclusion fails.

PQP ⇒ Q1TTT2TFF3FTT4FFT

The truth table of P ⇒ Q, with T for true and F for false. Row 2, where P is true and Q is false, is the only row where P ⇒ Q is false.

UP trueQ true1234

The same four cases as regions. The left circle holds the cases where P is true and the right circle the cases where Q is true, and each number is a row of the table. A statement about every n, such as "if n is a multiple of 4, then n is even", is true when no n lands in the shaded part: inside P and outside Q.

When the hypothesis fails

Test the statement at n = 6. Six is not a multiple of 4, so the hypothesis is false and the statement makes no claim about 6. Nothing it said about 6 can turn out wrong. This is row 3 or row 4 of the table, and the conditional counts as true there.

The same holds for any conditional. "If it rained, the ground is wet" says nothing about a day without rain. On that day the ground may be wet from a sprinkler, or dry, and neither case breaks the statement.

What a counterexample needs

Six is even and is not a multiple of 4. That can look like evidence against the statement, but it does not test the statement at all, because the statement speaks only about multiples of 4. A conditional works in one direction, from the hypothesis to the conclusion.

To break "if n is a multiple of 4, then n is even", you need a number that is a multiple of 4 and is odd. There is none. A multiple of 4 is 4k for some whole number k, and 4k = 2(2k), which is 2 times a whole number, so it is even. The statement is true.

One case where both parts hold proves nothing either. n = 8 is a multiple of 4 and is even, but a single case that fits says nothing about any other n. Only an argument that covers every multiple of 4, like the one above, proves the statement.

The usual mistakes

Treating a false hypothesis as a counterexample. "It did not rain and the ground is wet" does not break "if it rained, the ground is wet", because the hypothesis is false and nothing was promised.

Reading the statement backward. "If n is a multiple of 4, then n is even" does not say that every even number is a multiple of 4, so 6 does not break it.

Calling a statement proved from one case that fits. n = 8 agrees with the statement, and that shows nothing about any other n.

Negating "if P then Q" as "if P then not Q". The negation is "P and not Q": one case where the hypothesis holds and the conclusion fails.

A sign in a baggage hall

In the application below, a sign says that a bag weighing more than 23 kg must carry a HEAVY tag. Only a bag that weighs more than 23 kg and has no tag breaks it, so a supervisor checks only the bags where the missing fact could complete that pair.

Worked example: A HEAVY Tag Sign in a Baggage Hall, and Which Bags to Check

Question A sign in an airport baggage hall reads: "If a bag weighs more than 23 kg, it must carry a HEAVY tag." A supervisor has four records, each showing one fact about a different bag. Bag 1 weighs 27 kg. Bag 2 weighs 15 kg. Bag 3 carries a HEAVY tag. Bag 4 carries no HEAVY tag. (a) Which bags must the supervisor go and look at to be sure that none of the four breaks the sign? (b) A new handler reads the sign as its converse. Which bags would the handler look at, and how many of those checks could reveal a breach of the real sign?

  1. 1.The only bag that breaks the sign is one that weighs more than 23 kg and carries no HEAVY tag.

    Sign: if over 23 kg, then HEAVY tagBag 127 kgBag 215 kgBag 3taggedBag 4no tag
    Sign: if over 23 kg, then HEAVY tagBag 127 kgBag 215 kgBag 3taggedBag 4no tag
    Only a bag over 23 kg with no HEAVY tag breaks the sign.

  2. 2.Bag 1 weighs 27 kg, so if it has no tag it breaks the sign: look at its tag. Bag 2 weighs 15 kg, so it follows the sign whatever its tag.

    Sign: if over 23 kg, then HEAVY tagBag 127 kglookBag 215 kgno needBag 3taggedBag 4no tag
    Sign: if over 23 kg, then HEAVY tagBag 127 kglookBag 215 kgno needBag 3taggedBag 4no tag
    Bag 1 might have no tag, so look. Bag 2 follows the sign whatever its tag.

  3. 3.Bag 3 has a tag, so it follows the sign whatever it weighs. Bag 4 has no tag, so if it weighs more than 23 kg it breaks the sign: weigh it. (a) The supervisor must look at bags 1 and 4.

    Sign: if over 23 kg, then HEAVY tagBag 127 kglookBag 215 kgno needBag 3taggedno needBag 4no taglook
    Sign: if over 23 kg, then HEAVY tagBag 127 kglookBag 215 kgno needBag 3taggedno needBag 4no taglook
    (a) Bag 3 follows the sign whatever it weighs, and bag 4 must be weighed: look at bags 1 and 4.

  4. 4.The converse is: if a bag carries a HEAVY tag, it weighs more than 23 kg. It is broken by a tagged bag of 23 kg or less, so the handler weighs bag 3 and looks for a tag on bag 2.

    Sign: if over 23 kg, then HEAVY tagConverse: if HEAVY tag, then over 23 kgBag 127 kglookBag 215 kgno needhandlerBag 3taggedno needhandlerBag 4no taglook
    Sign: if over 23 kg, then HEAVY tagConverse: if HEAVY tag, then over 23 kgBag 127 kglookBag 215 kgno needhandlerBag 3taggedno needhandlerBag 4no taglook
    The converse is broken by a tagged bag of 23 kg or less, so the handler checks bags 3 and 2.

  5. 5.(b) The handler looks at bags 2 and 3. A tagged bag of any weight and a 15 kg bag with or without a tag both follow the real sign, so 0 of the handler's 2 checks can reveal a breach, and bags 1 and 4 go unchecked.

    Sign: if over 23 kg, then HEAVY tagConverse: if HEAVY tag, then over 23 kgBag 127 kglookBag 215 kgno needhandlerBag 3taggedno needhandlerBag 4no taglookThe handler's 2 checks can find 0 breaches
    Sign: if over 23 kg, then HEAVY tagConverse: if HEAVY tag, then over 23 kgBag 127 kglookBag 215 kgno needhandlerBag 3taggedno needhandlerBag 4no taglookThe handler's 2 checks can find 0 breaches
    (b) Neither bag 2 nor bag 3 can break the real sign: 0 of the 2 checks can find a breach.

Answer: (a) Bags 1 and 4; (b) bags 2 and 3, and 0 of those checks can reveal a breach

Common mistakes

  • Looking at bag 3 because it carries a tag. The sign does not say that only heavy bags may be tagged, so a tagged bag of 12 kg follows it.
  • Leaving out bag 4 because the sign mentions the tag only in its conclusion. The contrapositive of the sign is: if a bag has no HEAVY tag, it weighs 23 kg or less, and bag 4 must be weighed to test that.

More mathematical statements problems, worked step by step →

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