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## Re: [ontolog-forum] "Ontology-based Standards" mini-series session-4 - T

 To: ontolog-forum@xxxxxxxxxxxxxxxx Tara Athan Sun, 13 Oct 2013 15:19:56 -0400 <525AF25C.5000705@xxxxxxxxxx>
 ```A classical reference regarding concepts in metrology is Krantz et al http://www.amazon.com/Foundations-Measurement-Volume-Representations-Mathematics/dp/0486453146    (01) .    (02) Since it is not downloadable, I'll attempt to give a summary of the points I think are relevant to the development of ontologies of measurement.    (03) The foundational concepts employed in this reference are * a "relational structure", consisting of a set of mathematical entities (such as positive real numbers) and one or more relations among them. * A "numerical assignment" is a homomorphism that maps physical entities and measurement procedures into a relational structure in a way that the properties of the procedures are preserved.    (04) From these foundations, it can be shown that measurement theories corresponding to particular relational structures may be grouped into families according to the "permissible transformations" that may be applied to a numerical assignment that result in another numerical assignment. The familiar Stevens scale types correspond to certain families:    (05) * Ratio scale: entities are 1-dimensional (e.g. real or positive real numbers), and permissible transformation are only multiplication by a positive scalar constant. Extensive measurements such as length fall into this family.    (06) * Log-Interval scale: entities are again 1-dimensional, but typically only the positive real numbers, and permissible transformations include raising to a power in addition to multiplication by a positive scalar constant. Intensive measurements such as absolute temperature and density fall into this family. It's called log-interval because taking the log gives a theory in the "interval-scale" family, and Stevens recognized its existence in his original work, although it is usually not included in elementary treatments today.    (07) and so on. For ordinal-scale measurements, permissible transformations include any strictly monotonic function.    (08) However, there are plenty of measurement theories that do not fit into these families, notably, multi-dimensional theories.    (09) Tara    (010) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J    (011) ```