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I fail to understand why contraposition works intuitively. Take this sentence for example:

$\text{If I pass my exams then I am a good student.}$

$\text{I pass my exams }\implies\text{ I am a good student.}$

If I know I passed my exams then I know I am a good student but if I didn't pass my exams then I don't really know if I'm a good student or not (since this is implication and not an if-and-only-if statement ) and $F\implies T,F\implies F$ are both true, so either outcomes is possible.

Having said this I don't understand why proof by contraposition works. Proof by contraposition would say:

$\text{I didn't pass my exams }\implies\text{ I am not a good student.}$

But I don't understand why? Having not passed my exams doesn't say anything about whether I am a good student or not, and contraposition seems only obvious to me when :

$\text{I pass my exams }\iff\text{ I am a good student.}$

Then yes, the contrapositive makes intuitive sense to me. But why does it hold for implication as well?

εν οίδα ότι ουδέν οίδα
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    The contrapositive of $;A\to B;$ is $;\neg B\to \neg A;$ , not $;\neg A\to\neg B;$ – Timbuc May 15 '15 at 15:01
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    Actually, by contraposition you should say "I am not a good student $\Longrightarrow$ I didn't pass my exams", not the other way around. – marco trevi May 15 '15 at 15:02
  • Contrapositive: "I am not a good student $\implies$ I don't pass my exams" which is the same as (do the truth table) "I pass my exams $\implies$ I am a good student" – Alec Teal May 15 '15 at 16:09

2 Answers2

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Note the following:

  • $P\to Q\qquad$ [Primary statemet]
  • $Q\to P\qquad$ [Converse]
  • $\neg P\to\neg Q\qquad$ [Inverse]
  • $\neg Q\to\neg P\qquad$ [Contrapositive]

These are the various terms for the different forms above. Oftentimes, one will be required (as a rudimentary exercise) to show that the primary statement is logically equivalent to the contrapositive; that is, $P\to Q\equiv \neg Q\to\neg P$. Hence, oftentimes, if it seems to be quite difficult to prove that $P\to Q$, then one may resort to proving the contrapositive, i.e. $\neg Q\to\neg P$, thus proving $P\to Q$.

When you are dealing with a linguistic problem, as in your case, I think it is often a good idea to express your conditional with propositions:

  • $P:$ I pass my exams.
  • $Q:$ I am a good student.

As you noted, you have a claim of the form $P\to Q$. Thus, an equivalent claim to $P\to Q$ would be $\neg Q\to\neg P$, the contrapositive: "If I am not a good student, then I will not pass my exams." What you have done is expressed the inverse, which is $\neg P\to\neg Q$: "If I did not pass my exams, then I am not a good student." This is not valid because, in general, $P\to Q\not\equiv\neg P\to\neg Q$. I would encourage you to use truth tables to see why. What you are looking for is the contrapositive, and that actually is equivalent to your original statement.

Daniel W. Farlow
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The contrapositive of $A \implies B$ is $\sim B \implies \sim A$.

This has got nothing to do with $B \implies A$ or $\sim A \implies \sim B$, which is the converse of the statement $A \implies B$.

In your case, the contrapositive of the statement

If I pass my exams then I am a good student.

is

If I am not a good student then I will not pass my exam.

The statement you have written, i.e.,

I didn't pass my exams $\implies$ I am not a good student.

is not the contrapositive. It is in fact the converse of your initial statement.

Adhvaitha
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