I'm studying formal logic and i was wondering if you can check whether I've solved this task correctly.
TASK. Two people are arrested as suspects for a murder case, Stan and Peter. Four witnesses have spoken.
- 1st witness: Stan is not guilty.
- 2nd witness: Peter is not guilty.
- 3rd witness: At least one from the two before mentioned witnesses is correct.
- 4th witness: The third witness doesn't speak the truth.
It turned out that the 4th witness is correct. Who is guilty?
Because it is a pain in the neck to format the truth table by typing it, i wrote it on paper, pictured it and upload it on imgur. Here is the solution for the task:
This is how I've transformed the text into statements
This is the truth table that I've made with formal disjunction
This is an alternative way of solving that task with formal implication
If both of them are killers, that means that both the first and the second witness tell lies, so the third one tells a lie, which works perfectly fine with the system, so yes you are correct.
Specifically, the definition of "at least" is basically "greater than or equal to", so the opposite would basically be everything previously ignored, or "less then".
This isn't really formal proof(or logic), and this probably isn't what you were looking for, but I'm pretty sure your right.