User:Tsinoyboi/JodyQAnswer

C is a modus tollens

Example modus tollens:
 * $$p \to q\land\neg q\vdash\neg p$$
 * $$p \to q\vdash\neg q\to \neg p$$

If Jody going paying ticket is $$p\!$$, and Jody going to jail is $$q\!$$, the logic follows:

Assume $$\neg p$$ if Jody doesn't pay her ticket, then $$\to q$$ she will go to jail. Assuming this is true


 * $$\neg p\to q$$ or $$\mbox{if } p \mbox{ is false, then }q \mbox{ must be true}\!$$,

If $$p\!$$ is not true $$q\!$$ must be true.

Then if $$q\!$$ is false, then $$p\!$$ must be true.

So, since $$p\!$$ is true, then $$q\!$$ can be either true or false.

So it follows:


 * $$\neg p\to q\vdash\neg q\to p$$

Therefore, $$\neg q\to p$$ "If Jody doesn't go to jail, then she paid her ticket must" be true.

Furthermore, for clarity/redundancy and possibly more confusion:

If "if $$p\!$$ is not true, then $$q \!$$ must be true" is true, then "if $$q\!$$ is not true, then $$p\!$$ must be true" must be true, and $$p\!$$ and $$q\!$$ are both are not true, then "if $$p\!$$ is not true, then $$q \!$$ must be true" must not be true.

1. if (if not p then q) then (if not q then p)

2. not p and not q

therefore,

3. not (if not p then q)

$$(\neg p\to q)\to(\neg q\to p)\land\neg p\land \neg q\vdash \neg (\neg p\to q)$$

So:

If it's true that if Jody doesn't pay her tickets, then she will go to jail, then if she doesn't go to jail, then she paid her tickets. Jody didn't pay her tickets and didn't go to jail. Therefore, it is not true that if Jody doesn't pay her tickets, then she will go to jail. This is more akin to possibilities in real life, but it's actually irrelavent to the question since it's not assuming that the prior statement was true.

--Tsinoyboi