Logical? The wrong word, as always.
The term "logical" cannot be used to describe a characteristic of human beings.
Instead, what you can say is that people are logical or not logical in how they do what they do when they do something.
So, instead of the word "logical", the real question is whether you think you have a capability for logic.
So, yes, I have a capability for logic and I can prove it by being logical in doing something, for example by understanding what people mean or suggest without saying it; or by behaving in a way which shows that I can make valid inferences from whatever I believe.
I can also infer that other people also have a capability for logic by observing some of the objective facts of their behaviour.
Formal logic is beside the point. You don't ask people to prove that they can run by asking them to run a marathon.
Formal logic isn't logic proper. It may be logical, hopefully, but it isn't in itself logic per se. Rather, formal logic is at best both an expression and a formalisation of logic as a performance of humans and a capacity of the brain.
At worst, formal logic is mathematical logic. You couldn't prove anyone logical by using mathematical logic to prove it.
EB