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Propositional knowledge is often described as knowledge that such-and-such is true. This stands in contrast to other forms of knowledge, such as knowledge how to such-and-such. What term is most appropriate for characterizing "knowledge how to such-and-such?" And, are there similar forms of knowledge that one should be concerned about with respect to knowledge as justified belief?

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Classically, this is procedural or behavioral knowledge. Both propositional and procedural knowledge are forms of 'alethic' knowledge about what does and does not happen in reality. I would contrast it with moral or 'deontic' knowledge about what is expected by other people (including conscience which is about what we expect of ourselves), 'doxastic' knowledge about how we communicate and decide (including 'hedonic' knowledge about what does and does not affect your own homeostasis in your own terms).

The upside of propositional logic is that via modal inflection it can encode most of the other forms of knowledge in ways that allow them to be referenced and evaluated. In the case of procedural knowledge, we can write algorithms and say 'It is effective when faced with this arrangement to invoke this algorithm' as a propositional statement. Likewise most of the other forms of knowledge can be encoded as complex references within a propositional language.

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1. For your first question, suggest a look at Michael Polanyi's tacit knowledge:

Tacit knowledge (as opposed to formal, codified or explicit knowledge) is the kind of knowledge that is difficult to transfer to another person by means of writing it down or verbalizing it. For example, stating to someone that London is in the United Kingdom is a piece of explicit knowledge that can be written down, transmitted, and understood by a recipient. However, the ability to speak a language, use algebra,[1] or design and use complex equipment requires all sorts of knowledge that is not always known explicitly, even by expert practitioners, and which is difficult or impossible to explicitly transfer to other users.

Tacit knowledge is very important in doing experimental science; Polanyi's Personal Knowledge dives deeply into this topic and rips apart the theory of justified belief developed by the logical positivists.

2. For your second question, see my What is an “unarticulated background”?. Before you can talk about justified knowledge, you first need to understand what is even meaningful; as I understand the unarticulated background (or similar terms), it is what provides the meaningfulness. One way you can see this matter is that originally the ancients thought that certain (if not all) celestial objects were embedded in crystal spheres. Centuries later, 'scientists' found themselves unable to believe this, unable to render it intelligible. Their newer unarticulated backgrounds simply could not support the old ideas.

You will also want to look at Alasdair MacIntyre's concept of tradition, which you can find most succinctly in his Epistemological Crises, Dramatic Narrative and the Philosophy Of Science. A nice introduction is given by Richard J. Bernstein in Beyond Objectivism and Relativism:

    To speak of a new model of rationality may be misleading, because it suggests that there is more determinacy than has yet been achieved (or can be achieved). Nevertheless, what is striking is the growing awareness and agreement about the components of an adequate understanding of rationality as it pertains to scientific inquiry. There has been a dramatic shift in what is taken to be the significant epistemological unit for coming to grips with problems o the rationality of science. In the philosophy of science, and more generally in contemporary analytic epistemology, we have witnessed an internal dialectic that has moved from the preoccupation (virtually an obsession) with the isolated individual term, to the sentence or proposition, to the conceptual scheme or framework, to an ongoing historical radiation constituted by social practices—a movement from logical atomism to historical dynamic continuity. Awareness has been growing that attempts to state what are or ought to be the criteria for evaluating and validating scientific hypotheses and theories that are abstracted from existing social practices are threatened with a false rigidity or with pious vacuity and that existing criteria are always open to conflicting interpretations and applications and can be weighted in different ways. The effective standards and norms that are operative in scientific inquiry are subject to change and modification in the course of scientific inquiry. We are now aware that it is not only important to understand the role of tradition in science as mediated through research programs or research traditions but that we must understand how such traditions arise, develop, and become progressive and fertile, as well as the ways in which they can degenerate. (24–25)

Polanyi also writes about tradition; two places where his and MacIntyre's views are contrasted are Alasdair MacIntyre’s Tradition-Constituted Enquiry in Polanyian Perspective and Michael Polanyi, Alasdair MacIntyre, and the Role of Tradition.

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What term is most appropriate for characterizing "knowledge how to such-and-such?"? I’d be inclined to use the phrases “agency”, or “absorbed coping” (H. Dreyfus); or if your fond of prolixity- “…cognition with concrete spatio-temporally extended patterns of dynamic interaction between organism and their environments” (Hutto, Myin 2013, p.8, Radicalising Enactivism)? Also, you might want to check out the work of Hubert Dreyfus and Daniel Hutto in general, but a paper (PDF download) entitled “The Primacy of Phenomenology over Logical Analysis” (H.Dreyfus) springs to mind.

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Knowledge how or the greek term technê, might be fine.

http://plato.stanford.edu/entries/knowledge-how/

SEP also suggests "there's the kind of knowledge you have when it is truly said of you that you know a person—say, your best friend", but doesn't address it in the article linked to.

I would suggest there is a knowledge of being acquainted with. Zen speaks of "direct seeing". Some philosophers also speak of "acquaintance" with qualia.

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