Discrete Maths Logic: Determine the culprit

[Mampac]

So there was no answer elsewhere on the internet.

A and B are suspected being involved in a crime.
A says: B did it, or we did it together.
B says: A did it, or if I did it then we did it together.
Determine the guilty party. Justify your solution.

Me and my friend are arguing about the answer. I say that since it's an OR, then at least one of each of the people's the propositions is correct.
Say A's proposition "B did it" is correct: then it implies that B's second proposition, "if I did it then we did it together" is also true, which means A's second proposition, "we did it together", is also true. Therefore, they're both guilty.

My friend, however, at this point, argues that B doesn't say "I did it", therefore it's impossible to say who's the culprit.

Could anyone expand on this a bit?

 

[HallsofIvy]

If you are assuming the two statements are true, you should say so!

 

[Romsek]

I get that one or the other did it, both are consistent with the guilty party lying and the not guilty party telling the truth.

Both guilty is not consistent.

 

[Mampac]

I get that one or the other did it, both are consistent with the guilty party lying and the not guilty party telling the truth.

Both guilty is not consistent.

View attachment 23593

omg I didn't even know there exist calculators even for such problems...
could you rephrase it in a mathematical way, though? so as I understood first we assume A is guilty when B's not, then vice versa, but what then?

 

[Romsek]

then assume both are guilty.. oh.. oops... let me redo this