Logic and implication negation

Question

I am new to predicate logic and there is something I can't seem to understand.

Let's have an implication: If I have a sister, I have a sibling.

Now, it's negation should be: I have a sister and I do not have a sibling.

But I can't understand why the following is not a valid negation: If I have a sister, I do not have a sibling.

It should be a negation because the first part of the implication happened and the second did not.

Answer 1

A statement $\neg A$ is the negation of a statement $A$ if and only if whenever $A$ is true, $\neg A$ is false and whenever $A$ is false, $\neg A$ is true. So to find out which is the negation of the original statement, you just need to investigate all the possible cases and verify that the two statements have "opposite" truth values.

Remember that "If $A$ then $B$" is true whenever $A$ is false or $B$ is true -- that's just how material implication is defined. The problem is the former case: When "I have a sister" is false, then "If I have a sister, I have a sibling" and "If I have a sister, I don't have a sibling" are both true, so they do not have opposing truth values in all cases.

In contrast, "I have a sister and I don't have a sibling" is false whenever "If I have a sister, I have a sibling" is true (namely in those cases wher "I have a sister" is false or "I have a sibling" is true), and "I have a sister and I don't have a sibling" is true whenever "If I have a sister, I have a sibling" is false (namely in that case when "I have a sister" is true but "I have a sibling" false), as you can easily verify with a truth table.

Answer 2

The structure of your statement is $A\Rightarrow B$ where $A=\text{"I have a sister"}$ and $B=\text{"I have a sibling"}$. Then the negation is $\lnot B\Rightarrow \lnot A$ which is "If I don't have a sibling I don't have a sister".

Answer 3

The statement "If I have a sister, I have a sibling" is true. However. you are negating that statement which means that the statement "I have a sister and I do not have a sibling" should be false, which holds. The statement need not be true in order for it to be a valid negation.

Answer 4

First note that $$P\rightarrow Q\equiv\lnot P\lor Q$$ Here's how to properly negate the above statement $$\lnot(P\rightarrow Q)\equiv\lnot(\lnot P\lor Q)\equiv P\land\lnot Q$$ Now let me show you why the following is not a negation of the original statement $$P\rightarrow\lnot Q\equiv\lnot P\lor\lnot Q$$ Notice that this only negates $Q$, however we are seeking to negate the whole statement. Also note that if $P\equiv\mbox{false}$, then we have $$P\rightarrow Q\equiv P\rightarrow\lnot Q\equiv\mbox{true}$$ Which is clearly not a negation of the original statement. I hope this helps you understand.