Need a second view on interpreting if an argument is invalid

Question

I'm currently trying to solve this problem:

Is the following argument valid or invalid?

My stereo system is faulty: there is no sound coming out of the left speaker. Switching the speaker leads will not bring sound to the left speaker if and only if the left speaker is faulty. If switching the speaker leads causes the right speaker to fail, then there is a fault with either the amplifier or the CD player. Switching the leads from the CD player has no effect if and only if there is no problem with the CD player. I discovered the following: switching the leads to the speakers resulted in both channels failing, and switching the leads from the CD player reversed the problem from the left to the right speaker. Therefore replacing the CD player and the left speaker will solve the problem.

Now, breaking down the statements I get:

• sl: Switching speaker leads result in left channel failure.
• sr: Switching speaker leads result in right channel failure.
• fl: The left speaker is faulty.
• fa: The amplifier is faulty.
• fc: The CD player is faulty.
• ne: Switching the leads from the CD player has no effect.

Now rewriting the situation in symbolic form I believe its:

1. $sl \iff fl$
2. $sr \implies (fa \lor fc)$
3. $\neg{fc} \iff ne$

4. $sl \land sr \land \neg{ne}$

   $\therefore fc \land fl \land \neg{fa}$

Answering the question I determine that the argument is invalid I found a contradiction when:

• sl = true
• fl = true
• sr = true
• fa = true
• fc = true
• ce = false

Now I started to look around the internet and found a solution but it is not the same as mine, and can't understand its solution. Here is the link. It basically had most of the same statements, but instead of having statement ne it has two statements:

• cl: Switching CD leads results in left channel failure
• cr: Switching CD leads results in right channel failure

Then the person goes on and rewrites it in symbolic form as follows:

1. $sl \iff fl$
2. $sr \implies (fa \lor fc)$
3. $cl \iff \neg{fc}$

4. $sl \land sr \land cr$

   $\therefore (fc \lor fl) \land \neg{fa}$

First of all I don't understand why "Switching the leads from the CD player has no effect" is interpreted as "Switching CD leads results in left channel failure".

Second I don't understand the reasoning why the conclusion has to have $fc \lor fl$. Isn't "replacing the CD player and the left speaker" saying that the author is assuming that the speaker AND CD player both don't work AND implicitly saying that the amplifier DOES work.

Am I missing something here?

Answer 1

Ok, a bunch of things going on here.

First, when it says 'switching leads from CD has no effect', they mean to say that this is relative to the original situation, where there is no sound at the left speaker, in other words they say that 'switching the leads from the CD still leads to the left speaker not working'. Thus, your $ne$ is sort of the same as their $cl$.

OK, so you (sort of) agree on the first 3 premises.

How about the fourth? Well, your $ne$ does not quite specify what changes when switchng the leads from the CD does have an effect, while their $cr$ does: it specifies that switching the leads from the CD makes the right speaker not make a sound. And, frankly, I think that is better, since it provides more informatin than 'it is not true that there is no effect'. On the other hand, their symboliztin still misses the important bit of information that switching the leads make the left speaker work again, so I am not happy with their solution either! In fact, given the statement that switching the CD leads would have no effect iff the CD is ok, we should logically be able to infer that there is a problem with the CD player, since there was an effect. And at least your symbolization allows that, but theirs does not.

Now, what about the conclusion? The conclusion claims that the problem will be solved by replacing both the left speaker and the CD player. Now, both of you correctly state that this is saying that the amplifier is not at fault (for, if there was a problem with the amplifier, the problem would not be solved by replacing the speaker and the player).

But, does the fact that the problem goes away when both the speaker and the player are replaced mean that there was a problem with both the speaker and the player? No, because if only one of them was defective, then replacing both of them would solve the problem as well. So this is why they symbolize the conclusion with an or.

Now, you used an and, I suspect, because you are thinking 'well, why would it be claimed that replacing both would solve the problem if they didn't imply that there was a problem with both?'. And that makes sense too! In other words, I can understand both symbolizations here.

Finally, is the argument valid? Well, as stated it is not: if you look at the premises, you will quickly see that you can't infer from them that the amplifier is working properly. So both of you correctly say the argument is logically invalid.

OK, but in my eyes that's still not the end of it. If I was to work with these clues, I would reason as follows: Ok, the fact that I don't have sound at my left speaker means that either there is something wrong with the speaker itself, or with the amplifier, or with the CD player. Ok, let's first test the speaker: I know that the right speaker is making sound, so at least the signal to the right channel is getting relayed through the CD player and amplifier. So, switching the leads to the speaker, and still not hearing anything out of my left speaker means that there is definitely something wrong with my left speaker.

However, having switched the leads to the speaker, I now notice something else as well: my right speaker isn't making any sounds any more! Well, I know that the right speaker works (since it made sound before), and so there is something wrong with either my amplifier or the CD player (in addition to my left speaker). Ok, so let's switch the CD leads (while putting the speaker leads back to what they were). What happens?

Well, to be frank, something very strange: the left speaker now makes sound but the right speaker does not?!? That seems to indicate that the left speaker is ok! Well, let's ignore that for a minute, and let's focus on the claim that switching the CD leads would have no effect iff the CD is ok.

Now, this is actually a weird claim in and of itself. It is not inconceivable, for example, that switching these leads would switch the left and right signal (just as switching the speaker leads do), so having this happen would not at all mean that the CD player is faulty.

OK, let's ignore that as well and just note that there is an effect which, it is claimed, would mean that the CD player is not ok, and hence need to be replaced as well (as the conclusion indeed seems to suggest, at least under your interpretation).

So, in sum, there are lots of weird things going on here, not just in terms of the symboliztion, but also in terms of the claims that are being made about what means what, and finally in terms of what actually happens with this stereo system!

Answer 2

Here's a way to look at it. I'll make the assumptions that faults are not transient and that a component is faulty if and only if it never produces any sound. Further, I'll assume that each channel of the CD player and of the amplifier is a separate component. (There's a total of six components.) What did the three experiments teach me?

• Did I hear sound coming out of both speakers? Yes, the right speaker produced sound in the first experiment, while the left speaker produced sound in the third experiment.
  • Conclusion: both speakers work.
• Did I hear sound coming from both amplifier channels? Yes, in the first experiment the sound I heard came from the right channel, while in the third experiment the sound came from the left channel.
  • Conclusion: both amplifier channels work.
• Did at least one channel of the CD player produce sound? Yes, in the first and third experiment sound was produced.
  • Conclusion: at least one channel of the CD player works.
• However, in the second experiment no sound was heard, which requires at least two faults.
  • Conclusion: My assumptions, reasonable as they seemed, are untenable.