First I'd like to affirm the importance of the distinction between judgements which are analytic and those which are synthetic (because without it the kind of logic I'm talking about just doesn't exist). And I'd like to throw value judgements into the picture. These distinctions are of importance because of the methodological differentiation which they underpin (see the analytic/synthetic dichotomy).
This picture tells two stories at once on the analytic/synthetic/value judgement trichotomy. The narrow focus story and the broad sweep.
The narrow focus is upon those parts of a discipline which it regards as peculiarly its own, on knowledge that rather than knowledge how. The broad sweep considers also judgements which are borrowed or incidentally introduced, knowledge of methods and techniques, and acquired skills.
The narrow focus is shown by the three horizontal colour bands, blue for analytic, green for synthetic and red for values.
The distinction is precise and the line sharp between mathematics and science.
Elsewhere it is not so clear, the distinction between areas whose judgements are synthetic and those where value judgements are involved is harder to place.
Ethics is concerned with certain kinds of value judgements. Insofar as a value judgement concerns something which is an end in itself, it may be independent of any analytic or synthetic judgements. Where something is valued either in part or wholly as a means to other valued ends, a synthetic understanding of the world, possibly expressed through mathematical models, may be needed to establish the connection with the desired ends. In this way, the study of ethics involves not only value judgements, but also synthetic and analytic judgements.
A utilitarian, in order to make a moral judgment about an action, has to be able to predict the effects of the action on the aggregate future happiness of the world.
If you were really going to do this you would need some pretty good models of the world, and they would be mathematical and hence logical in nature.
Politics is sometimes called "the science of the possible", but this concerns political possibility rather than logical or physical possibility.
At one extreme politics delivers inspiration and leadership, at the other expedience dissimulation and corruption.
Epistemologically, insofar as politics concerns enunciation of truths these consist both of value judgements and synthetic judgements.
In assessing the potential social or other implications of policy options, mathematical models, and hence analytic judgements, will often be appropriate.