Is the psyche a logical structure ?

The physical brain is structured by evolution.  How we think is a mixture of our genetics, our environment, and our education.  We often assume that our cerebral cortex (i.e. our "conscious" brain) runs the show but more and more evidence is coming out that we aren't really in control of our brains as much as we think we are.  We can use logic if we want to but it is by no means required.  For all we know about anatomy we still don't know the exact nature of consciousness or how decisions are actually made.

Short answer - no, not naturally. The mind is ruled by the senses and emotion. Logic, in the form of inductive or deductive reasoning methods, is learned.