Does this argument prove that 17 is not a number?

Question

I found an example problem in Set Theory: The Structure of Arithmetic by Norman T. Hamilton.

The example comes from a brief section about logo, and it goes as follows:

Consider the sentences

• Three is a purple cow
• If three is a purple cow, then 17 is a number.

The second sentence is true, the first is false. Hence we cannot deduce 17 is a number.

Question: Does it follow that “17 is a number” is false?

I felt as if the answer to this question should be no, because the antecedent is not connected to the conclusion, and whether or not 3 was a purple cow had little bearing on what 17 was.

Answer 1

No conclusion could be made regarding the truth or falseness of the conclusion.

The premises are untrue but the argument is valid (a Modus Ponens form) so we say it is unsound. Soundness necessitates the conclusion being true but this does not mean that the conclusion is untrue because we can easily come up with another argument proving your conclusion. Thus refuting this argument does not refute the conclusion.

Consider someone who makes the following argument:

I argue that as apples are yellow, and if apples are yellow they must exist, so apples must exist. Furthermore, if apples do not exist they can't be eaten and apples can be eaten, so they must exist.

Both arguments are valid (being Modus Ponens and Modus Tollens respectively) but only one is sound, but the conclusion is still true.

Answer 2

The set of statements does not assert that "17 is NOT a number," but the point being made is that the statement that "if three is a purple cow" has no bearing on the claim on 17. The latter statement is not dependent in any way on the previous statement, hence in the truth table we see that "17 is a number" is true regardless of whether "three is a purple cow." Hence in the compound statement "If three is a purple cow, then 17 is a number" is always true, regardless of whether the first part is true or not.

Answer 3

If $p$ is three is a purple cow, and $q$ is 17 is a number, these statements are supposed to be understood as the formal statements $p$ and $p\to\neg q$, respectively. Now recall the truth table for material implication ($\to$): $p\to\neg q$ is true if and only if $p$ is false or $\neg q$ is true. Since in this case $p$ is false, $p\to\neg q$ is automatically true irrespective of whether $\neg q$ is true. In particular, $p\to\neg q$ is true even if $\neg q$ is false, so we cannot legitmately conclude that $\neg q$ is true. Now $\neg q$ is it is not the case that 17 is a number, which is plainly equivalent to ‘17 is a number’ is false, so we cannot conclude that 17 is a number is false.