People talk about emotions vs logic and really mean to question, "What is the function of emotion in logical argumentation?" [E·L} The answer can of course be addressed about the manner in which we treat ordinary language with a logical symbolism: Symbolize it! Quite simply, detecting the emotive elements to an argument can show additional, suppressed propositions that contribute to the strength/weakness/character of the conclusion to be potentially realized. Let Depending on the context, these elements may be intuitively symbolized. The only problem is that emotion can make ambiguity of a situation; the elements symbolized and conjoined or whatever logical fashion seems to fit the context may not correspond to the inner-workings of the other's thought process. Psychodynamic theory does not answer to this. Only years of intuitive practice can lead to a successful symbolization of the emotive signifiers.
I think we as humans need emotion to help logic because we have to make decisions with lots of unknowns, and what would happen to us without emotions is that we would enter AnalysisParalysis. Emotions are like an internal, seemingly random engine that makes us choose one of the options when logic is unable to choose because all choices seem equal. I think that one of the more important components that a really functioning AI will require is an "emotional" one. Without it, a lot of choices would become infinite loops.
This is true. Neurological research increasingly shows that emotions are vital, and foundational, to cognitive and learning processes. In cases where the emotional centers of the brain are damaged, the individual lacks emotion, but rather than demonstrating a Spock-like capacity for rational logic, his or her decision-making process becomes severely impaired. Simple decisions like whether to sign a form with a pen or a pencil can take hours and be practically undecideable -- much to the frustration of those around. The person with the condition has no emotions and doesn't care, of course.
For an accessible discourse on relationship between emotions and cognition, see "The Emotional Brain" by Joseph E. Ledoux. I've also witnessed first-hand, in someone close to me, the effect of bipolar affective disorder -- which presents arbitrary swings in mood from depression to mania -- on cognition: An otherwise rational, intelligent, and practical person can make profoundly bad decisions under the influence of an inappropriate mood state, even while (at times) being aware that the decisions are a bad idea.