Hence, Thinking analyzes emotion on a deeply unconscious level. Your mind is conditioned to associate certain stimuli with certain mindstates, if you get pinched, you immediately associate it with pain. Yet someone who's body is numb, does not respond to such a stimulus.
For the sake of another example, when somebody tells you a dramatic story, your mind processes these ideas on an unconscious level. For instance, suppose someone said your mother died in a car accident. In order for you to experience the emotion of angst associated with her death you need to understand the fact of her death. This requires dispassionate judgment, or the cognitive activity which simply recognizes ideas devoid of any emotional implications. This is an instance of Thinking analyzing emotion.
For the sake of the third example, consider the following. When somebody asks you how you feel, you need to reflect on how you mind-state to be able to tell them. That is where analysis comes in, or dispassionate contemplation of your mind-state. If you did not engage in the dispassionate contemplation of your mind-state and simply emote, you would just groan or giggle. But then again, in order to groan or giggle in response to such a question you must dispassionately identify the question, this presupposes the use of Thinking.
Hence, it is simply not possible to understand emotion without engaging in dispassionate contemplation of emotion to some degree at least.
What this shows is not that Feeling uses logic, but that functions are intimately intertwined. Feeling in itself is merely a tendency to process emotion. Thinking is a tendency to contemplate dispassionately. Thinking has a distinct tendency to use logic, yet Feeling only a subtle one. Thinking directly leads one to use logic, Feeling only indirectly so and at the price of self-sacrifice. Feeling (note the getting pinched example) may inspire one to think about the causes of one's situation, however because Thinking is dispassionate, it causes for Feeling to diminish. As for example, it is quite natural for us to be led to have our emotions cool down when we sit down to seriously contemplate something.
Is this not logical?
You could argue that Feeling motivates us to do all things. For instance, even when I use thinking or work through an intellectual problem, I only have the ambition to do so because it feels positive to me. However, the feeling in itself does not get me the answer to my problem, only gives me the fuel to move further. It seems inconceivable to me that the mere tendency to process emotion is what grants one knowledge of a solution with regard to a difficult problem.
*Note how I have defined Feeling and Thinking above.*
At the end of the day we study MBTI to better understand reality of human nature. MBTI is means to the end, not an end in itself. Hence, I have offered a method that is superior to MBTI which shall conduce to us arriving at our foremost objective.