Is the logic of self-reported thoughts related to psychological disorders?
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.