Is the logic of self-reported thoughts related to psychological disorders? Depression seems to be a disruption in thought patterns. In particular, it is characterized by repetitive thoughts. Is it possible to characterize psychological diseases by looking at the logical structure of self reported thoughts? 

For example, suppose Bob is depressed. He may be asked to free associate for 10 minutes. Then would counting the number of double implications be a good characterization of depression. This is a statement of the form $A \Longleftrightarrow B$.

So Bob may say things like  "I am depressed because I have no friends" and "I have no friends because I am depressed." A normal person might have more statements of the form $A \implies B$. So he would say things like "I am happy therefore I should go to the gym. Going to the gym will make me stronger" etc.
There's no theoretical reason to believe that depression is characterized by an increase in "double implications."  If anything, the double implication you've posed (depression ⟺ no friends) is a pretty accurate understanding of the reciprocal relationship between the causes and consequences of depression (e.g., Stice, Ragan, &amp; Patrick, 2004; Teasdale, 1983). So if depressed individuals have such an insight, then we might actually expect this to facilitate mood repair and regulation. 

Otherwise, to my knowledge, no studies have examined the logical structure of depressive thoughts. Perhaps you might be aware that depression is characterized by cognitive distortions, but I'm guessing that this isn't the kind of "logic" that you're referring to.