What is a false premise?

Solution 1:

The following proof is a valid argument, however, the conclusion of the theorem is clearly false. What went wrong?

Theorem 1 Let $1 = 0$, then all natural numbers are equal to zero.

Proof by induction. Obviously, $0 = 0$. Now, let $k$ be any natural number $\geq 1$. By inductive hypothesis we have $k-1 = 0$. Using our assumption we get $k-1+1 = 0+0$, that is $k = 0$ which concludes the proof.

Some funny examples of this kind happen with loaded questions. For example, if you were to answer the well-known loaded question presented below by "Yes, I have" or "No, I haven't",

                                         Have you stopped beating your wife?

then you would admit that, at some point, you were doing it (and that you actually have a wife). To respond to such a question, one usually points out (in whatever way) that it is based on false premises.

A sound argument is one which is both valid and its premises are true. The above is not sound, because the premise $0 = 1$ is not true. Still, the difference is rather subtle. For example, if the conclusion of the theorem was the whole implication $$(0=1) \implies \forall k\in\mathbb{N}.\ k=0$$ then the only premises are the axioms of logic, natural numbers, etc., and such a theorem would be both valid and sound.

I hope this helps $\ddot\smile$

Solution 2:

Note the very first sentence in the last paragraph: "Viewed as atomic sentences". With this, you're supposed to strip the sentences of their meaning and look at them just as literals (i.e. propositional letters or negations of these).

Thus you can attribute truth values to the propositional letters, (see valuations).

Then the author talks about case 3 which I'm guessing is the valuation $(A,B)\mapsto (F,T)$.
This valuation makes both the premises true and the conclusion true also.

Now the author says that in the real world case 3 can't happen, this means that if you now stop looking at the sentences as propositional letters and give them meaning again, then something impossible happens, namely that Teller has never taught logic (this is somehow known to be false, maybe Teller is the author).

The term false is being used with two different meanings, one of them is the natural language interpretation, the other one is as a valuation.