Logic and implication negation 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.
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.
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 &quot;If I don't have a sibling I don't have a sister&quot;.