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What’s the difference between a negation and a contrapositive?

I keep mixing them up, but it seems that a contrapositive is a negation where the terms’ order is changed and where there is an imply sign.

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Consider the statement: $$P \implies Q.\tag{1}$$ Then, as you know, the contrapositive of (1) is $$\lnot Q \implies \lnot P.\tag{2}$$

Now, note that statements (1) and (2) are *equivalent*. I.e., $$\;P \implies Q\equiv \lnot Q \implies \lnot P.$$

One additional observation: Note that the contrapositive of statement $(2)$ is $(1)$, so a contrapositive need not contain any negation symbol $(\lnot)$ at all!

Negating statement (1) (and hence negating (2)) would be a statement to the effect that it is NOT the case that $P \implies Q$. I.e. the negation of statement (1) is given by

$$\lnot (P \implies Q)\tag{3}$$

and since $(P \implies Q) \equiv (\lnot P \lor Q)$, we can write (3) as follows:

$$\lnot(\lnot P \lor Q)\tag{4}$$

By DeMorgan’s, (4) is equivalent to: $$\lnot(\lnot P) \land \lnot Q\tag{5}$$

which yields

$$(P \land \lnot Q)\tag{6}$$

So (3), (4), (5) and (6) are all equivalent expressions, each negating statement (1).

Put another way, the `contrapositve`

of a statement is `equivalent`

to the statement [both a statement and its contrapositive have the same truth-value], while the `negation`

of the statement negates or `reverses`

the truth-value of the original statement.

One last comment:

When one speaks of the *contrapositive* of a statement, one is usually speaking about the contrapositive of an *implication*, whereas one can negate any logical statement whatsoever simply by enclosing the original statement in parentheses and negating it, as in statement (3) above.

When you negate a true statement you get a false statement, but the contrapositive of a true statement is still a true statement, formally if $P \Rightarrow Q$ is your statement then $\sim Q \Rightarrow \sim P$ is the contrapositive, where $\sim$ denotes logical NOT and $\Rightarrow$ denotes logical implication. You could use the truth table to convince yourself that they are equivalent, i.e., $(P \Rightarrow Q) \Longleftrightarrow (\sim Q \Rightarrow \sim P)$. Consider the following statement: If today is Monday then we are tired. This is the same as: If we are not tired then today is not Monday.

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