Truth-Table Highlighter — P → Q and ¬Q → ¬P Row by Row

The clearest way to be sure that an implication and its contrapositive are logically equivalent is to stare at the truth table and watch them agree in every row. Not just once — four times, in every possible combination of P and Q.

This page is that stare, made interactive. Drag the slider through the four rows of the truth table. Each row highlights the value of P, the value of Q, the computed P \Rightarrow Q, and the computed \lnot Q \Rightarrow \lnot P. The last two columns will match — row for row, with no exceptions. That match is the theorem.

What the truth table says

An implication P \Rightarrow Q is false in exactly one case: when P is true and Q is false. In all other rows, it is true. That is the rule you met in Logic and Propositions.

The contrapositive \lnot Q \Rightarrow \lnot P uses the same rule: it is false in exactly one case — when \lnot Q is true and \lnot P is false. "\lnot Q true" means Q is false; "\lnot P false" means P is true. So the contrapositive fails in exactly the same row where the original fails — when P is true and Q is false.

Why that matters: an implication is fully characterised by when it fails. Two implications that fail in exactly the same rows are logically indistinguishable. They are the same statement in different clothing.

Walk through every row

Row:

Four rows, four combinations of $P$ and $Q$. Row 2 is the only row where the implication fails — when $P$ is true and $Q$ is false. It is also the only row where the contrapositive fails. In the other three rows, both are true. Row-for-row agreement is the whole proof of logical equivalence.

What the highlighted agreement means for proofs

Every time you prove \lnot Q \Rightarrow \lnot P, you have simultaneously established P \Rightarrow Q. Not "implied it." Not "approximately proved it." Established it, because the two statements describe the same promise — one in the forward direction, one in the flipped direction.

This is why Proof by Contrapositive is a legitimate technique. You are not proving a weaker or related statement and hoping the original follows. You are proving the same statement, stated with P and Q swapped and negated. The truth-table match above is the formal guarantee that this rewrite is lossless.

A concrete example row by row

Take P = "n is a multiple of 4" and Q = "n is even." The implication "P \Rightarrow Q" says every multiple of 4 is even — which is true.

Row 1 (P T, Q T): n = 8. Multiple of 4, even. Implication holds. Contrapositive? \lnot Q is "n is odd" — false for n = 8, so \lnot Q \Rightarrow \lnot P is vacuously true. Both true.
Row 2 (P T, Q F): impossible to find a real n here (a multiple of 4 is always even). But if you could, the implication would fail, and so would the contrapositive. Both false.
Row 3 (P F, Q T): n = 6. Not a multiple of 4, but even. Implication is vacuously true. \lnot Q is "odd" — false, so contrapositive is vacuously true too. Both true.
Row 4 (P F, Q F): n = 7. Not a multiple of 4, odd. Implication is vacuously true. \lnot Q is "odd" — true; \lnot P is "not a multiple of 4" — true; contrapositive holds. Both true.

Every row agrees.

Why this is not just symbol-pushing

It is tempting to dismiss the equivalence as a formal accident of how we defined implication. But it is more than that. The implication P \Rightarrow Q and the contrapositive \lnot Q \Rightarrow \lnot P are two descriptions of the same logical prohibition: never P true and Q false. Both sentences exclude that one situation and allow everything else. You can state the prohibition forwards or backwards; the prohibition is the same.

That single-prohibition view is also why the converse and inverse are not equivalent to the original: they describe a different prohibition — "never Q true and P false" — which is a completely different rule. See Converse, Inverse, Contrapositive — The Four-Form Carousel for that comparison.