Can a negation have an inverse?

Question

I have started a middle/highschool level geometry text. The text is warming me up for proofs by going through basic logic, which was introduced alongside Euler diagrams.

I am given the following problem:

"Each of the lettered statements below is followed by some other statements. Identify the relation of each of them to the lettered statement if possible. Write "converse," "inverse," "contrapositive," or "original statement," as appropriate.

[Below:]

"Statement D: Lady kangaroos do not need handbags."

"35) If a kangaroo is not a lady, it needs a handbag."

My question:

I believe that conditional statement D is in the form 'if not a, then b'. So: $ \neg a \Rightarrow b$.

That interpretation has a ...negation or negative at $a$ ? If that is so, I think that problem 35) is presenting the inverse which I thinking as: $\neg (\neg a \Rightarrow b)$, which resolves to be $a \Rightarrow \neg b$.

The only notation I have been exposed to so far is the conditional notation $\Rightarrow$, used so far with $a \Rightarrow b$. I have been shown "not" and assume "not" means "$\neg$". (I am sorry for any symbol confusion. I'm new to logic, stackexchange, and latex.)

Answer 1

Translation from English to symbols is not trivial. You could first deal with the anaphoric pronoun by using predicate symbols and a variable, and then analyze the intended meaning of both statements as follows.

• D: $\text{kangaroo}(x)\implies (\text{lady}(x) \implies \neg \text{needhandbag}(x))$
• 35: $\text{kangaroo}(x)\implies (\neg \text{lady}(x) \implies \text{needhandbag}(x))$

Then you could use shorthand for the atomic statements:

• D: $K\implies (L\implies \neg H)$
• 35: $K\implies (\neg L\implies H)$

Now if you focus on the consequents, you can look at the four possible truth assignments (eight with $K$) and see that there is at least one case where $L\implies \neg H$ comes out true but $\neg L\implies H$ as false: the former is compatible with lord kangaroos not needing handbags. Since D and 35 are not equivalent, "contrapositive" and "original statement" are ruled out. I'm not sure what "converse" and "inverse" are, but given the way the question is worded, one should apply, and you probably know which.

(Having "contrapositive" as one of the alternative answers suggests that the framers of the question may intend a translation to a single implication. I think that would be awkward.)

Answer 2

This question is phrased badly, I think (in the book, not your own.) I interpret it to mean:

P="If a kangaroo is a lady"

Q="it does not need a handbag"

$P \implies Q$.

Being statement D, whereas (35) is:

$\neg P \implies \neg Q$, which is a logical fallacy called denial of the antecedent.

To make (35) a contrapositive, you could interpret statement D as follows:

P="if it does not need a handbag"

Q="it is a lady kangaroo"

$P \implies Q$.

In which case:

"If a kangaroo is not a lady, it needs a handbag" which translates symbollically to:

$\neg Q \implies \neg P$ i.e the contrapositive.

About your question, "negation" makes no claim about the nature of the statement itself. If the first statement is "I am not a man," then the negation of this is "I am a man." In other words, just because a sentence includes the word "not" doesn't make it a negation, it just means it's a statement, either true or false.