Let me first mention the difference between knowing that and knowing how, affirm that some of the most important kinds of knowledge are of the latter kind, and then for the present confine the discussion to the former.
While considering judgements, as the carriers of knowledge that, I'd like to work with a tri-partite division. I propose to discuss judgements which are analytic, and those which are synthetic, and finally value judgements, considering these three kinds of judgement to be distinct but not necessarily exhaustive. The importance of these distinctions lies in differences in the methods which are appropriate for establishing or for exploiting the different kinds of knowledge (see for example the analytic/synthetic dichotomy).
In assessing claims of each of these kinds one must exercise judgement (which is a kind of intuition) at some point, but the kinds of evidence which are most relevant differ from one kind (of knowledge) to another. In analytic judgements demonstration in a well established formal logic is the best evidence, though the establishment of such a logical system will depend to a great extent upon intuition. In synthetic judgements experimental or observational data play an important role, supplemented by deduction from proposed general empirical models to their specific behaviours, and influenced by emotional matters primarily in consideration of the importance rather than the truth of the synthetic hypothesis. In value judgements emotions are a primary and irreplaceable source, the relevance of deductive and empirical evidence is confined to evaluation of the consequences of possible actions or policies so that one judges with an informed heart the whole matter.
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.
The key features which upset the cosy horizontal classification are:
The picture therefore suggests, through the vertical bands marked by lines and vertical lettering, that each of these disciplines involves all three kinds of knowledge in varying proportions. Thus, in ethics, values predominate, but synthetic and analytic judgements have their place. In logic, the ostensive subject matter is arguably analytic, and lacks any intrinsic value. Chosing which lines of abstract research to progress must nevertheless depend upon some assessment of value which can only be imported, however tenously, from consideration of applications of logic in support of real world values.
It is desirable to understand the strengths and weaknesses of each source of knowledge; to put them in their proper place.