Is this argument correct? A Porsche is a fast car. Dan's car is not a Porsche. Therefore, Dan's car is not fast. Let P(x) be a Porsche Let C(x) be a fast car Let x be Dan $$P(x) \rightarrow C(x) :premise$$ $$\neg P(x) :premise$$ $$\equiv C(x) :False\rightarrow anything = False$$ $$or$$ $$\equiv \neg C(x) : Modus Pollens$$

We need for $C(x)$ to denote the car $x$ is a _fast_ car.

I'm not clear what you're trying to conclude, but we cannot conclude $\lnot C$. We cannot conclude anything about how fast Dan's car only from the knowledge that his car is _not a Porche._

We know Porches are fast, but other makes and models may also be fast, perhaps faster! And Dan may have a fast, "non-Porche" car.

In general: From $$P \rightarrow Q$$ $$\lnot P$$ we **cannot** conclude $\lnot Q$. That's a fallacy in reasoning: sometimes called "denying the antecedent".

The error is that:

* It's a misapplication of modus ponens, which tells us that $$P\rightarrow Q$$ $$P$$ $$\therefore Q$$
* or a misapplication of modus tollens which tells us that $$P \rightarrow Q$$ $$\lnot Q$$ $$\therefore \lnot P$$