Vague boundaries

Practices have vague boundaries. Does peeling potatoes belong to cooking, or is it merely a prerequisite that does not belong to the practice itself It is difficult to say, the practice of cooking is not sufficiently well-defined. Another sense of vagueness is that it is unclear how much variation is permissible in the practice so... 

As is well known, Cluster Analysis involves the classification of objects into categories. Since most categories have vague boundaries, and may even overlap, the necessity of introducing fuzzy sets is obvious. A discussion of Fuzzy Clustering must refer to the following issues ... 

Many salts crystallize from aqueous solution not as the anhydrous compound but as a well-defined hydrate. Still other solid phases have variable quantities of water associated with them, and there is an almost continuous gradation in the degree of association or bonding between the molecules of water and the other components of the crystal. It is convenient to recognise five limiting types of interaction though the boundaries between them are vague... 

The boundaries between half-synthetic and natural polymers are sometimes vague. 

In the environment we frequently deal with a more vague kind of boundary. For instance, we have the boundary between the warm, less dense surface layer of a lake... 

Sparse (arid) includes oasis farming and scattered irrigation settlements, in many cases used by pastoralists to supplement their livelihoods. The boundary between pastoral systems and sparse farming systems is vague. 

In Table 3.5 are the data and their ranges as found by Probst and Wohlfahrt. It should be realized that the boundaries between the groups are vague. Also, values of m will vary depending on the manufacturing process of the individual catalyst producers. 

I think such accounts fail utterly, for reasons I have elaborated twice (Lycan, 1987, chap. 8 1996, chap. 4) and cannot even summarize here. I also think that, fortunately, the admission of the referential and quantificational structure itself fails to demonstrate the actual existence of sense-data (which is an even longer story), and that there aren t really any phenomenal individuals but my claim for now is that the visual field at least is experienced as a mosaic of patches or regions each having both shape and color, and that this creates a powerful prima facie case for sense-data. (The region boundaries can, of course, be vague, like the boundaries of a topography of the earth s surface.)... 

The boundaries between toxicity, sufficiency, and deficiency are vague. The amounts vary with the species of plant and animal, vary within the species s growth cycle, and vary with the organism s general health and the supply of the other essential elements. The elements in the crosshatched areas of Fig. 2.1 are the toxic elements of primary concern to government regulatory agencies—Be, Cd, Hg, Pb, and all... 

The analysis is vague only to the extent that nowhere is it obvious that the boundary condition... 

Readers are cautioned against the use of various terms in tribology which are either vaguely defined or not defined at all. These would include such terms as lubricating ability, lubricity, and even boundary lubrication. For example, do boundary lubricating properties refer to effects on friction or effects on wear and damage It makes a difference. It is emphasized once again that friction and wear are different phenomena. Low friction does not necessarily mean low wear. We will see several examples of this common error in the discussion of joint lubrication research. 

Several problems exist that make it difficult to understand and interpret many of the published works on synovial joint lubrication. One example is the widespread use of non-operational and vague terms such as lubricating activity, lubricating factor, boundary lubricating ability, and... 

---