# Determining the soundness of arguments

I was recently given the following question in an exam.

Determine the soundness of the following argument:

John lives on the same street as Mary.
Mary lives on the same street as Sam.
Therefore, Sam lives on the same street as John.

I understand that an argument is sound when it is valid and all the premises are in fact true.

The problem with the above argument is that I have no way of knowing whether or not the premises are in fact true. I don't know for a fact on which street Mary, John and Sam live. I don't even know who they are.

Looking through past exam papers it seems to be a common trend for the examiner to ask whether or not arguments with premises like these are sound or not.

What is the correct way to determine the soundness of arguments where the truthfulness of the premises fall outside of our range of knowledge?

• Is it actually sound? Try out some examples; suppose two streets - either they are parallel, or not; what happens if Mary is not just a shop assistant in a corner-shop, but lives there too? – Mozibur Ullah Apr 26 '15 at 9:06
• Mozibar makes an interesting point, but I'm not sure you should attend to it too much unless the instructor tends to like tricky wording in natural language that violates the law of identity. – virmaior Apr 26 '15 at 9:07
• As Virmaior points out, the possibility I'm pointing out is unlikely to be what the instructor is testing for. – Mozibur Ullah Apr 26 '15 at 9:16

You are right. The correct answer would be:

This argument is valid. However, for this argument to be sound it is also necessary that its premises are true. Therefore, this argument is sound iff the two premises are true.

NB: see Mozibur Ullah's comment on the question for a way to argue that this argument is invalid (in which case it would directly be unsound); however, in that case I would say the examiner is a little bit pathetic.

I don't know your examiner, but is it possible that this was a trick question?

An argument is sound when it's valid and its premises are true. Therefore, to determine soundness you need to determine validity and truth. To determine validity, you need to answer the question: "would it be possible that the conclusion is false while all premises are true?" If so, then the argument is invalid. If not, the argument is valid. Determining truth... well, let's just say that's more complicated!

If your examiner insists that you can determine the soundness of that argument, you could show him the definitions of soundness and validity, preferably from your book, but if they are not there, from the IEP would be sufficient:

A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Otherwise, a deductive argument is said to be invalid.

A deductive argument is sound if and only if it is both valid, and all of its premises are actually true. Otherwise, a deductive argument is unsound.

Assuming we are working with the definitions of classical logic (which is what you supply), then

There is a way to prove an argument is not sound even without knowledge of the truthfulness of its premises. Specifically this is because All sound arguments are valid arguments.

In other words, you test for validity. And if the argument fails that test, then it is not sound. But you cannot prove soundness itself without the premises also being true...

For instance, given:

``````(1) Either you like apples or you like cherries
(2) Therefore, you like strawberries
``````

you can prove this is not sound, because it is not valid (either using proofs or truth tables).

In the problem you have specified, the argument is valid (with the caveat of a linguistic sleight-of-hand about "same street"), so I would say the best answer is that "the soundness of the argument cannot be proven" (alternately "soundness indeterminate").

Do you know from previous exams what sort of answer is expected or if that sort of answer is possible?