Impact from a water drop to water causes an upward "rebound" jet surrounded by circular capillary waves. Credit: Fir0002/Flagstaffotos.

The metadefinition of a particular definition is the set of attributes that address and satisfy a number of purposes or functions for that definition.[1] Once specifics are included in the higher level definition, that definition applies to the lower level situation in which the specifics are characteristic. The specifics are relatively arbitrary within the relationship of the metadefinition, which means the term of the metadefinition may be used in a large number of specific subject areas. The metadefinition is often viewed as a higher level definition that may consist of the definition of several objects within that higher level, or a definitional set of descriptive attributes or characteristics that can be used to produce a variety of related definitions.

## Notations

Notation: let the symbol word of choice in a quote indicate emphasis of the 'word of choice in a quote', as independent of the actual quote or author's intent.

## Semantics

"The problem of definition is at the basis of the problem of understanding. Any level of understanding of any natural phenomenon or object entails its description, followed by the comparison (in our brain or in the pages of an Herbarium, or by a computer program) with similar phenomena or objects."[2]

Def. “The earliest steps of the evolution of the codons also suggested two major stages in the origin of life – self-reproduction (exact replication of the ideal RNA duplex in the above theory, one strand of which is repeating triplet GCCn, while another strand is complementary GGCn), and variations (appearance of point-mutated versions of GCC and GGC in the subse- quent replications).”[3] is called life.

The metadefinition layer can be defined as a higher level definition,[4] that may consist of the definition of several objects.[4] When each object is made specific, a specific definition is derived or translated from the metadefinition.

Inducting upward from the definitional layer, the metadefinition of a particular definition is the set of attributes that address and satisfy a number of purposes or functions.[1]

Def. "the set of attributes that address and satisfy a number of purposes or functions"[1] for a particular definition is called a metadefinition for that definition.

A definition serves five functions:

1. a statement of identity,
2. a way to define competitive and cooperative relationships with other terms,
3. a way to end conceptual disputes and thus prepare the way for measurement - its preoperational or premeasurement function,
4. a way to locate a term within a particular context - its orienting or contextual function, and
5. a way to generate new ideas - its generative or revelatory function.[1]

When the concept of the description is relatively arbitrary, the definition containing this concept is in fact a metadefinition.[5] Each arbitrary choice from the metadefinition creates a definition.

## Forms

From the theory of a metadefinition, a constituent phrase can be said to have a metadefinition if and only if the set of attributes that address and satisfy a number of purposes or functions for each descriptive definition is related or relatable either to the initial definition or the metadefinition itself through some algorithm such as translation.

"The purpose of the original research is to produce new knowledge, rather than to present the existing knowledge in a new form (e.g., summarized or classified).[6][7]"

For a word or phrase, once a number of purposes or functions (of the metadefinition) are addressed and satisfied, this set of addressed and satisfied attributes is the definition for the word or phrase.[1] Presenting the existing knowledge of or about the word or phrase in an alternate form (the metadefinition) and summarizing or classifying that existing knowledge into the this form is presenting existing knowledge in an alternate form, perhaps for analysis.

Presenting an alternate form for existing knowledge is not considered a primary source,[6][7] but producing new knowledge about changes in the attributes over time or as a function of other variables is original research.

## Tests

By analogy to the constituency tests of constituent linguistics, tests may be arranged on a scale of reliability, with less-reliable tests treated as useful to confirm constituency to a proposed metadefinition though not sufficient on their own. Failing to pass a test may not always mean that the phrase is not a constituent of the metadefinition. It is best to apply as many tests as possible to a given phrase to prove or rule out its constituency.

As an illustration of an alternate form for existing knowledge, a particular description of a ‘dominant group’, perhaps the initial descriptive definition, should consist of attributes addressing and satisfying the following purposes or functions of the proposed metadefinition:

1. a relation,
2. a population,
3. a basis or ‘criteria for dominance’,[8] and
4. a country,[9] region, territory, or the like (of the dominant group)[9].

But, because this form of the definition probably cannot be expressed in the subject area, for example evolutionary biology where the apparent initial descriptive definition may have originated, it may be a metadefinition of ‘dominant group’ for that subject area rather than a definition. When the specific representations for the enumerated metadefinitional components are filled in from the context by any arbitrary algorithm, this specific constituent phrase ‘dominant groups’[9] is a definition within the subject area of evolutionary biology if and only if it is related or relatable to the metadefinition.

Specifically, one set of choices from context yields

1. relation: dominant species,[9]
2. population: 'forms of life',[9]
3. basis or 'criteria for dominance': 'the common, the widely-diffused, and the widely-ranged species',[9] and
4. country (of the dominant group): 'in their own countries'.[9]

Or, a dominant group of life forms is the common, the widely-diffused, and the widely-ranged species (plural form of species) in their own countries.[9] This particular choice is readily related or relatable by inspection; therefore, this descriptive definition of dominant group as used within the authoritative source[9] is within the set of definitions described by the proposed metadefinition. For the population a better description may be 'species of life', though not specifically used by the source[9].

Each of the attributes of the metadefnition can change over time, subject area, or other variables (such as temperature).

## Applications

A broader definition of the assault cycle (a meta-definition) is needed based on levels and types of patterns.[10]

“It turns out that there are over 100 disciplines that have a unique definition of risk, from the military to software engineers. Not easily discouraged I came to the conclusion that there must be some commonality between the definitions, some meta-definition.”[11]

## Biology

"The pragmatic approach asks whether and how strongly recreational fishing compromises the health and fitness of individual fishes and what can be done to avoid or mitigate such effects. Its implementation rests on accepting recreational fishing as a principally legitimate activity."[12]

"The second approach to fish welfare focuses on suffering and pain in fishes and is usually morally prescriptive. Its central tenet is that some or all recreational fishing practices may be unacceptable unless sufficient benefits to humans are created, which justify the supposedly cruel treatment of the fishes."[12]

"[A] suffering-centred definition of fish welfare currently only serves as a meta-definition of fish welfare because there is no scientific method available that allows quantifying pain and suffering in fishes".[12] "[A] suffering-centred approach to fish welfare tends to promote tension and enduring conflict that cannot be reconciled objectively and thus should be avoided."[12]

All in the debates about life and consciousness are "making use of an implicit and unspoken but intuitively known metadefinition of life to decide the rightness or wrongness of any definition."[13] "It was pointed out by Bertrand Russell that efforts to define such terms ultimately devolve on common usage, from which we must "distill the essence of the word.""[13]

Metadef. self-reproduction and variations,[2] is called life.

## Chemistry

• Metadefinition of a clear liquid:

1. A clear flowing substance
2. keeping or retaining no shape or definite shape
a. except that determined by the containing receptacle
3. composed of molecules
a. not tending to separate from one another like those of a gas

is called a clear liquid.

A clear liquid diet consists of transparent liquid foods such as vegetable broth, bouillon, clear fruit juices, clear fruit ices, popsicles, clear gelatin desserts, and no carbonated drinks. Soda's carbonation expands the gastrointestinal tract. Herein are many clear liquids that fit within the metadefinition.

• Definition of water:

Def. a clear liquid having the chemical formula H2O, required by all forms of life on Earth is called water.

• Definition of water from the metadefinition of a clear liquid:
1. A clear flowing substance
2. keeping or retaining no shape or definite shape
a. except that determined by the containing receptacle
3. composed of H2O molecules
a. not tending to separate from one another like those of a gas

is called water.

• Definition of ethanol from the metadefinition of a clear liquid:
1. A clear flowing substance
2. keeping or retaining no shape or definite shape
a. except that determined by the containing receptacle
3. composed of C2H5OH molecules
a. not tending to separate from one another like those of a gas

is called ethanol.

## Computer sciences

For metamodeling, the metadefinition “defines process concepts such as role, activity and interaction”, to help provide a unified process support framework for global software development.[14]

In the customisation of process modeling languages, process metamodeling, which allows process modelling languages to be specified and implemented in a process support environment, is defined.[14] Therefore, process metamodelling allows users to change to a different process modelling language if necessary.[14]

Metadef. "the notation to which all the external process modelling notations are mapped”[14] is called the metadefinition for process metamodelling.

“[I]t needs to have a much more comprehensive coverage of process issues.”[14] “Given that the meta-definition is an internal representation, readability and simplicity are not the main issue.[14] “[I]t is not possible to include all process modelling concepts in the meta-definition so that any process modelling notation can be mapped and integrated.”[14]

A meta-definition is one of the three parts of a model integrator, the other two parts being a number of model translators and an internal representation.[14]

SPASE is an initial protoype of a unified process support framework.[14] The SPASE meta-definition contains only the following concepts:

1. process,
2. node (or action),
3. condition for action,
4. links to traverse to other actions, and
5. communication.[14]

## Economics

Def. definitions which correspond to intuitive views on the knowledge games we shall call metadefinitions.[15] Metadefinitions and metalemmas are essential in order to prove that the formal theory of the knowledge game is well adapted to its intuitive aims.[15]

"Metadefinition 1. A "knowledge game" G is given by G = (n, (Ii)k, E, S) where n is an entire number. Ik is a sequence of subsets of I = {1,2,...,n} such that for any i in I, there exists infinitely many k such that i is in Ik. E is a given set and S a subset of Ea."[15]

The number of players in the game is n, also each i in I represents a player.[15] Ik is the set of players who are asked at any time k, for example, whether they are able to deduce their own value.[15] The set of values E are those which can be attributed to each player (colors, numbers, distinquishing marks, etc.).[15] The set of all the attributions of values to the players (S) are those effectively allowed by the rules of the game.[15]

"Metadefinition 2. A knowledge game G = (n, (Ii)k, E, S) is convergent if for every initial attribution of values a ∈ S, there exists a time k such that some player of Ik is able to deduce his own value."[15]

A formal translation of metadefinition 2 is the following definition: "Definition 3. A game G = (n, (Ii)k, E, S) is decidable if there is an explicit algorithm determining for every a in S, every i in I and for every k such that Ik contains i, whether SkEi(a) is of cardinality 1, and, if that occurs, giving this unique element. G is convergent if it verifies one of the following equivalent properties:

(a) S = ∪kTk
(b) ∩kSk = ∅."[15]

Tk is the set of all initial values "a" for which the game finishes at time k.[15] Cardinality 1 is the winner. And, "giving this unique element" is designating which player is the winner.

Alternative translations of metadefinition 2 can yield, for example, a ranking of players, subject to the algorithm chosen.

## History

Possibly the earliest use of metadefinition as 'meta-definition' is in symbolic logic[16].

Per JSTOR,[17] metadefinition, or meta-definition, also occurs within the subjects of language and rhetoric, social research, management, and probability.

An early use of term dates from 1964 in mathematical logic[18].

In 1971, the term is used in theoretical physics as part of the axiomatic foundations of special relativity.[19]

By 2003 it has found its way into literature with respect to the Heian literary system[20].

## Mathematics

There are many metadefinitions in mathematics in addition to those of game theory.

### Elementary arithmetic

The finitary theory of finite sets can be used to develop a theory of ‘natural number arithmetic’, or ‘elementary arithmetic’ (EA).[21] For a specific EA, depending on its characteristics, in this instance the one described by Mayberry, ‘EA1’,[21] a metadefinition cannot be expressed in this EA.[21] If a definition cannot be expressed in EA1, then relative to EA1 the definition becomes a metadefinition.[21]

### Principal eigenvalue

The meta-definition of principal eigenvalue for boundary value problems of elliptic operators in quasi-nonlinear and fully nonlinear equations could be the following:[22]

Given a zero order and odd operator H with the same homogeneity than the second order elliptic operator F, and given a domain Ω,

λ is a principal eigenvalue if there exists a non trivial solution of constant sign of the problem
${\displaystyle {\begin{cases}F(x,\nabla u,D^{2}u)+\lambda H(u)=0{\text{ in }}\Omega \\u=0{\text{ on }}\partial \Omega .\end{cases}}}$

## Physics

• Definition of a liquid:

Def. a substance that is flowing, and keeping no shape, a substance of which the molecules, while not tending to separate from one another like those of a gas, readily change their relative position, and which therefore retains no definite shape, except that determined by the containing receptacle is called a liquid in physics.

1. A flowing substance
2. keeping or retaining no shape or definite shape
a. except that determined by the containing receptacle
3. composed of molecules
a. not tending to separate from one another like those of a gas

is called a liquid.

## Psychology

"A metadefinition of management development is offered: management development is a metafield that emerged from a range of disciplines (primarily, though not exclusively psychology, social science, and management studies), which either attempts to frame the reality of management or re-frame the reality experienced by managers, with the aim of contributing to the personal resource base of managers, and/or the intellectual capital of organisations".[23]

Metadef. "a metafield that emerged from a range of disciplines (primarily, though not exclusively psychology, social science, and management studies), which either attempts to frame the reality of management or re-frame the reality experienced by managers, with the aim of contributing to the personal resource base of managers, and/or the intellectual capital of organisations"[23] is called management development.

## Technology

"The complementary rule states that upon insertion of a new element to a meta-definition (eg an attribute to a class) all instance objects must be extended with values initialized to the default value of the type, and the respective M0 data elements must be added."[24] The M0 layer of metadata architecture is the information layer consisting of instance, or application, data (e.g. objects and records).[25]

## Hypotheses

1. For metadefinition, the proof of concept is a formalistic expression that allows transmission of the concept from one field to another.

## References

1. Stuart Albert (1998). David Allred Whetten, Paul C. Godfrey, ed. The Definition and Metadefinition of Identity, In: Identity in Organizations: Building Theory Through Conversations. Thousand Oaks, California: Sage Publications. pp. 1–13. ISBN 0-7619-0947-8. Retrieved 2011-09-02.
2. Ernesto Di Mauro (February 2012). "Trifonov’s Meta-Definition of Life". Journal of Biomolecular Structure & Dynamics 29 (4): 601-2. PMID 22208252. Retrieved 2014-08-03.
3. Edward N. Trifonov (2011). "Vocabulary of definitions of life suggests a definition". Journal of Biomolecular Structure & Dynamics 29: 259-66. Retrieved 2014-08-03.
4. David P. Hansen, Thure Etzold (November 2001). Thomas Lengauer. ed. Chapter II-1. Integrating and Accessing Molecular Biology Resources, In: Bioinformatics - From Genomes to Drugs. Weinheim, FRG: Wiley-VCH Verlag GmbH & Co. KGaA. doi:10.1002/3527601481.ch9. Retrieved 2011-08-03.
5. Barbour, Robert H. and Hopper, Keith (August 1994). "Survival of the Species vs Survival of the Individual". Computer Science Working Papers 94 (14): 1–7. Retrieved 2011-08-03.
6. David Millet (1981). "Defining the "Dominant Group."". Canadian Ethnic Studies 13 (3): 64–79. Retrieved 2011-07-26.
7. Charles Robert Darwin (1859). On the origin of the species by means of natural selection: or, The Preservation of Favoured Races in the Struggle for Life. London: John Murray. p. 516.
8. Mark S. Carich (1999). "In Defense of the Assault Cycle: A Commentary". Sexual Abuse: A Journal of Research and Treatment 11 (3): 249–51. doi:10.1023/A:1021368510011. Retrieved 2011-08-03.
9. Preston B. Cline (September 2007). "Learning to interact with uncertainty". Inaugural Outdoor Education Australia Risk Management Conference (Ballant, Victoria, Australia): 8.
10. "Contrasting pragmatic and suffering‐centred approaches to fish welfare in recreational angling". Journal of Fish Biology 75 (10): 2448-63. December 2009. doi:10.1111/j.1095-8649.2009.02466.x. Retrieved 2011-09-06.
11. Lawrence L. Horstman and Charles Lee Horstman (2006). Life and Vitality, In: The Lotka Hypothesis Book I, Elements of Consciousness. New York: Vantage Press, Inc.. pp. 53-62. ISBN 0-533-15135-X. Retrieved 2011-09-06.
12. Jin Sa and Elena Maslova (December 2002). A unified process support framework for global software development, In: Proceedings of the Computer Software and Applications Conference. 26th Annual International Conference. IEEE. pp. 1065–70. doi:10.1109/CMPSAC.2002.1045150. Retrieved 2011-08-03.
13. J. M. Lasry, J. M. Morel and S Solimini (1989). "On knowledge games". Revista Matematica de la Universidad Complutense de Madrid 2 (2-3): 187–201. Retrieved 2011-08-03.
14. Hao Wang (June 1950). "Existence of classes and value specification of variables". The Journal of Symbolic Logic 15 (2): 103-12. Retrieved 2011-09-06.
15. Search Results meta-definition. JSTOR is part of ITHAKA, a not-for-profit organization helping the academic community use digital technologies to preserve the scholarly record and to advance research and teaching in sustainable ways. Retrieved 2011-09-06.
16. Petr Hájek (1964). "Die durch die schwach inneren Relationen gegebenen Modelle der Mengenlehre". Mathematical Logic Quarterly 10 (9-12): 151-7. doi:10.1002/malq.19640100905. Retrieved 2011-09-06.
17. Jorge Bosch (May 1971). "On the Axiomatic Foundations of Special Relativity". Progress of Theoretical Physics 45 (5): 1673-88. doi:10.1143/PTP.45.1673. Retrieved 2011-09-06.
18. Rein Raud (2003). Michel Hockx and Ivo Smits, ed. The Heian Literary System: a Tentative Model, In: Reading East Asian writing: the limits of literary theory. New York: RoutledgeCurzon. pp. 92–116. ISBN 0-7007-1760-9. Retrieved 2011-09-06.
19. J. P. Mayberry, Richard Pettigrew (2007). Natural Number Arithmetic in the Theory of Finite Sets. pp. 1–52. Retrieved 2011-08-03.
20. I. Birindelli F. Demengel (September 2010). "Regularity and uniqueness of the first eigenfunction for singular fully non linear operators". Journal of Differential Equations 249 (5): 1089-1110. doi:10.1016/j.jde.2010.03.015. Retrieved 2011-09-03.
21. John Cullen (September 2005). "A Meta-Review of the Management Development Literature". Human Resource Development Review 4 (3): 335-55. doi:10.1177/1534484305278891. Retrieved 2011-09-03.
22. Ilia Petrov, Stefan Jablonski and Marc Holze (2005). "On the notion of consistency in metadata repository systems". Advanced Information Systems Engineering 3520/2005: 193-9. doi:10.1007/11431855_8.
23. Ilia Petrov and Stefan Jablonski (2004). "An OMG MOF based Repository System with Querying Capability – the iRM Project". Proceedings of iiWAS: 153-7. Retrieved 2011-09-06.