Contradiction between vacuously true inferences Does allowing inferences to be vacuously true allow for both a sentence and its negation to be true?
Example:
P: If Joe Biden is two feet tall, then all elephants have wings.
Not P: If Joe Biden is two feet tall, then not all elephants have wings.
Joe Biden is not two feet tall, so it seems to follow that it is vacuously true that all and not all elephants have wings, which would be a contradiction.
So what's going on here? Is there some way in which the contradiction is only ostensible? Does this pose a problem for classical logic, and if so, how is it to be dealth with?
Edit: Problem has been solved.
 A: Your "not $P$" is incorrect. The negation of "If $P$, then $Q$" is "$P$ and not $Q$". So, the negation of your statement $P$ would be "Joe Biden is two feet tall and not all elephants have wings". In this way, these statements are logically opposite to each other.
A: It may be clearer if you see the truth table for inference.
Let X: Joe Biden is 2 feet tall
Y: All elephants have wings.
Then the truth table for $X\implies Y$ is as follows:
\begin{array}{|c|c|c|}
\hline
X&Y&X\implies Y\\
\hline
0&0&1\\
0&1&1\\
1&0&0\\
1&1&1\\
\hline
\end{array}
So to answer your question: "Does allowing inferences to be vacuously true allow for both a sentence and its negation to be true?"
Answer: No, it only allows the implication to be true. So $X\implies Y$ can be true, irrespective of whether $Y$ is true or not.
A: There is no problem here. Neither of your implications, by themselves can be used to infer anything about elephants since the antecedent is false. Rarely if ever in daily life do we concern ourselves with implications that have false antecedents. Mathematicians, however, must confront them proofs, e.g. in proofs about empty sets or proofs by cases, some cases of which are known to be false.
The basic principle can be stated as:

For any logical propositions $A$ and $B$, we have $A \implies [\neg A  \implies B]$ regardless of the truth value of $B$.

It is a tautology of propositional logic.
Here is the truth table:

Source: https://www.erpelstolz.at/gateway/TruthTable.html
Here is a formal proof using a simplified form of natural deduction:
(Screen shot  from my proof checker)

