Why is Q → P a logical consequence of ¬(P → Q ) I don't want to ask my professor about this because I'm awful at this and he's not the, uhh, patient type of professor to say the least.
Anyhow, it was my understanding that ¬(P → Q) and (¬P → ¬Q) mean two different things.
And that Q → P is equal to (¬P → ¬Q).
Yet one of the answers to a question says that Q → P is a logical consequence of ¬(P → Q), which I just don't understand at all.
To confuse matters further, afaik ¬(p → q) ⟺ p ∧ ¬q is correct, so I guess I'm just lost and missing something here in the definition of logical consequence?
Any help would be much appreciated.
P.S. This is for a Reasoning class that's part of a Philosophy minor, but I figured this is probably the best place to ask!
 A: Suppose $Q$.
Suppose $P$. We can conclude $Q$. Therefore, $P \to Q$. But we also know that $\neg (P \to Q)$. Contradiction. Everything follows from a contradiction; therefore, we can conclude $P$.
Thus, we have shown that $Q \to P$. $\square$
Phrasing the proof differently, we can show $\neg Q$ is a logical consequence of $\neg (P \to Q)$. Since $Q$ is false, $Q \to P$ is automatically true.
A: (This answer posted after a previous one was accepted.)
Don't feel bad. It's a tricky little exercise in so-called vacuous truth. The last $P$ could have been any proposition whatsoever, say $R$ as in this truth table.

Sourse: https://www.erpelstolz.at/gateway/TruthTable.html
This result can also be formally obtained using a form of natural deduction and proof by contradiction as follows (screenshot from my proof checker)

A: To say that $$B \text{ is a logical consequence of } A$$ is to say that $$A \text{ logically entails } (\models)\: B.$$
Claim: $$¬(P → Q) \quad \models \quad Q → P$$
A proof: \begin{align}&¬(P → Q)
\\\equiv\:& ¬(¬P \lor Q)
\\\equiv\:& P \land ¬Q
\\\models\:& ¬Q \lor P
\\\equiv\:& Q → P.
\end{align}
A: In addition to Dan Christensen's answer, it's interesting to note that
$\neg(P\rightarrow Q) \rightarrow (R \rightarrow P)$
is also a tautology.
So the second Q could have been any proposition whatsoever, like R.
