Logical Inference Question, I disagree with the answer

Question

Apologies for the generic title, couldn't find a better one. So the question is as follows:

Given the following statements, determine which of these conclusions follow:

Statements:

• All laptops are computers
• Some phones are computers
• Some desktops are computers
• Some desktops are phones

Conclusions:

1. Some laptops are phones
2. No desktop is a laptop
3. Some desktops are both computers and phones
4. Some phones are desktops alone

The answer is that #4 is the only valid conclusion. I disagree with this, and their reasoning for it is as follows:

#4 is valid because some desktops are phones. Not all are joined together with computers..

Uhm, no? We don't know that not all are joined together with computers. The way I see it there are three possibilities, bold is my addition:

• Case 1: Some desktops are computers, some desktops are phones, no desktops are both computers and phones

• Case 2: Some desktops are computers, some desktops are phones, some desktops are both computers and phones, some desktops are phones alone

• Case 3: Some desktops are computers, some desktops are phones, some desktops are both computers and phones, no desktops are phones alone

Case 3 means that conclusion #4 could be false. That would look something like this (pardon the poor photoshop):

Am i missing something here? Why is the venn diagram above somehow not possible? Statement #4 still holds. Some desktops are phones, but they also happen to always also be computers. We don't know this for sure, but we know that it may be true, which would make Conclusion 4 not necessarily true

Answer 1

This is a horrible problem.

First of all, even before we start solving things note that $4$ is ambiguous: what does "desktop alone" mean? Maybe it means "desktop but not laptop," but "desktop but not laptop or computer" is also within the bounds of imagination. This is a nit-pick, but we are doing logic after all.

Secondly, the answer - as you observe - is wrong. The easiest way to see this in my opinion is to note that no hypothesis gives any negative information. In particular, the hypotheses listed don't rule out the possibility that phones=computers=desktops=laptops, which of course would make any interpretation of $4$ false. Now maybe we're supposed to assume that every computer is either a desktop or a laptop but not both, but that assumption hasn't actually been stated - and as above, this is a terrible thing to do in a class/book about logic.

To be honest, I don't immediately see a non-silly way to construe the problem so that the given "solution" is valid.

(This all assumes the problem has been copied correctly, of course - the obvious potential issue is if you've omitted hypotheses made earlier in the text which have - perhaps tacitly - continued to be assumed.)