I'm facing a logical inference problem and seeking guidance to find a proposition p3 that satisfies certain logical conditions.
Given propositions:
p1 = p or r
p2 = q => !p
p3=?
Given conclusions:
z1 = p and q
z2 = q => r
z3 = !r
I aim to determine a proposition p3 such that all conclusions z1, z2, and z3 are simultaneously true, given the premises p1 and p2.
I've conducted a preliminary analysis and tried using the truth table. I guess I how to try finding the values where z1, z2, and z3 are all true, but in this case there were none. I'd appreciate guidance or additional methods to rigorously derive p3 in a systematic manner or to confirm this solution.
Any insights, suggestions, or logical reasoning strategies that could assist in finding the suitable proposition p3 satisfying all conclusions based on the given premises would be greatly appreciated.
This is not possible. The conclusions z1, z2 and z3 cannot be simultaneously true, because z1 and z2 together imply that $r$ is true, and z3 says that $r$ is false. So z1, z2 and z3 together are a contradiction.
The only way to have a contradiction in the conclusions is to have a contradiction in the premises. And then, any contradiction would do, because you can prove anything from a contradiction. Most likely, the problem is ill-posed or a trick question.