Dear Christian-Emile two quick comments.
1) In the discursive parts, probably you swapped the examples: > 1) A relation R is asymmetric if there are no pair x,y such that x relates > to y and at the same time y relates to x. 'less than' (<) is a good example > of an irreflexive relation. > 2) A relation R is irreflexive if no x is related to itself. 'less than’ (<) > is a good example of a asymmetric relation. While both good examples are correct, they are - in my opinion - quoted in the wrong place. I would expect that when defining asymmetric you would mention an asymmetric relation as example; same for irreflexive. --------------------------------------------------------------------------------------------- 2) I am not so comfortable with the example in the Definition of Asymmetric, for which you quote P46. I checked the scope note of P46 and it does not clarify if the part is a “proper” part: "This property associates an instance of E18 Physical Thing with another instance of Physical Thing that forms part of it. The spatial extent of the composing part is included in the spatial extent of the whole.” (from the P46 scope note) Inclusion (part-ness), in set theory, does not exclude that A is included in A, or in other terms that A is a subset of A (part of A, in common speech), To quote an authoritative source (😀) wikipedia states the following: In mathematics <https://en.wikipedia.org/wiki/Mathematics>, set <https://en.wikipedia.org/wiki/Set_(mathematics)> A is a subset of a set B if all elements <https://en.wikipedia.org/wiki/Element_(mathematics)> of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). So either the scope note of P46 should clarify that the part is a “proper” part; or self-inclusion (“self part-ness") is allowed, but then your example of P46 does not work.I would rather go for this second case, but perhaps common sense is for the other one and I am biased by an ancient and long-lasting mathematical infection. Best, Franco Prof. Franco Niccolucci Director, VAST-LAB PIN - U. of Florence Scientific Coordinator ARIADNEplus Technology Director 4CH Editor-in-Chief ACM Journal of Computing and Cultural Heritage (JOCCH) Piazza Ciardi 25 59100 Prato, Italy > Il giorno 7 set 2022, alle ore 13:08, Christian-Emil Smith Ore via Crm-sig > <crm-sig@ics.forth.gr> ha scritto: > > > Dear all, > Please find my hw below and in the attachment. > > Best, > Christian-Emil > > > > The issue is about how to define asymmetric and irreflexive. > > > Background > > Usually the prefix 'non-' in a compound negates the main part. So > 'non-symmetric' should have the same meaning as 'not symmetric'. > > From Latin the prefix 'in-' has a similar function. So irreflexive means > 'not reflexive'. > > From Greek the prefix 'a-/an-' has a similar function: asymmetric is > a+symmetric (as Ancient Greek ἀσυμμετρία (asummetría), “disproportion, > deformity”, wiktionary.org). > > This is not very helpful since 'non-symmetric', 'asymmetric' and 'not > symmetric' all have the same general meaning. However, this is not the case > for specialized language in a given domain. In set theory the terms > 'asymmetric' and 'irreflexive' have a specialized meaning stronger than just > 'not ...': > > 1) A relation R is asymmetric if there are no pair x,y such that x relates > to y and at the same time y relates to x. 'less than' (<) is a good example > of an irreflexive relation. > > 2) A relation R is irreflexive if no x is related to itself. 'less than’ (<) > is a good example of a asymmetric relation. > > In the formal parts of the definition of CRM we use first order logic and > follow standard definitions in set theory. In CRM 'P is not reflexive' means > that at least one x is not related via P to itself. My suggestion is that we > use 'irreflexive' and 'asymmetric' as in common set theory: > A) reflexive: for a property P with domain and range E, P(x,x) for all > instances x in E. > > B) irreflexive: for a property P with domain and range E, P(x,x) for no > instance x in E. > > C) non-reflexive/’not reflexive’: For a property P with domain and range E, > P(x,x) is not true for one or more instances x in E. > > B implies C, so non-reflexive/’not reflexive’ is weaker. > > > > > Proposal: > > Change from noun to adjective; add two new entries in the term definition > list. > > > asymmetric > asymmetric is defined in the standard way found in mathematics or logic: > A property P is asymmetric if the domain and range are the same class and for > all instances x, y of this class the following is the case: If x is related > by P to y, then y is not related by P to x. An example of a asymmetric > property is E18 Physical Thing. P46 is composed of (forms part of): E18 > Physical Thing. > > irreflexive > irreflexive is defined in the standard way found in mathematics or logic: > A property P is irreflexive if the domain and range are the same class and > for all instances x, of this class the following is the case: x is not > related by P to itself. An example of a irreflexive property is E33 > Linguistic Object. P73 has translation (is translation of): E33Linguistic > Object. > > symmetric > symmetry > Symmetric Symmetry is defined in the standard way found in mathematics or > logic: > A property P is symmetric if the domain and range are the same class and for > all instances x, y of this class the following is the case: If x is related > by P to y, then y is related by P to x. The intention of a property as > described in the scope note will decide whether a property is symmetric or > not. An example of a symmetric property is E53 Place. P122 borders with: E53 > Place. The names of symmetric properties have no parenthetical form, because > reading in the range-to-domain direction is the same as the domain-to-range > reading. > reflexive > reflexivity > > ReflexiveReflexivity is defined in the standard way found in mathematics or > logic: > A property P is reflexive if the domain and range are the same class and for > all instances x, of this class the following is the case: x is related by P > to itself. The intention of a property as described in the scope note will > decide whether a property is reflexive or not. An example of a reflexive > property is E53 Place. P89 falls within (contains): E53 Place. > > > > > > <Issue 597 define irreflexivity and > asymmetry.docx>_______________________________________________ > Crm-sig mailing list > Crm-sig@ics.forth.gr > http://lists.ics.forth.gr/mailman/listinfo/crm-sig
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