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.
Best Answer
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.