# 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

$$P(x) \rightarrow C(x) :premise$$

$$\neg P(x) :premise$$

$$\equiv C(x) :False\rightarrow anything = False$$

$$or$$ $$\equiv \neg C(x) : Modus Pollens$$

• Dan is definitely not a fast car. (what's $x$?) – Karolis Juodelė Sep 28 '13 at 19:40
• @DonLarynx : Karolis's comment is what I meant when I thought OP was confused in his symbols ; I'd admit I didn't try to figure out an example of the confusion. Thanks for pointing it out Karolis. (I believe the answers helped OP understand better now though! :D ) – Patrick Da Silva Sep 29 '13 at 9:50

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 $$P \rightarrow Q$$ $$\lnot Q$$ $$\therefore \lnot P$$

Not necessarily. Denying the consequent (modus tollens) would yield "If the car is not fast, it is not a porsche". However, you seem to be denying the antecedent, a fallacy.