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A Tripartite Theory of 'Definition' Abstract: This essay analyzes the nature of 'definition' as a definiendum-to-definiens relationship. A 'tripartite theory' of definition is hypothesized. It states that whenever a person defines a definiendum-to-a-definiens, that person can only be interpreted as asserting either a 'reportive definition,' a 'theoretic definition,' or a 'stipulative definition.' In order to verify the truth of the theory, a conceptual investigation about the functional use of definitions in various situations is described by examples. Of special interest are the examples of 'stipulative definition.' As a mathematical anti-realist, I contend that formal systems are largely composed of stipulative definitions that are 'precisely formalized' and 'abbreviatory' in nature. To back up the tripartite theory, I comment about the entries under ‘definition’ in The Cambridge Dictionary of Philosophy (1999). Introduction With a tripartite theory of definition, I hypothesize that whenever a person asserts how a linguistic entity (i.e. word, phrase, symbol, definiendum) has been used, is used, or is going to be used; that person can only be interpreted as asserting a reportive (i.e. lexical) definition, theoretic definition, or a stipulative definition. If this hypothesis is true, we should be able to understand any definition of a definiendum-to-definiens form (in a context) as being one of these three types. If this hypothesis is false, we should be able to find an instance of a linguistic token-to-meaning form that cannot be interpreted as reportive, theoretic, or stipulative. The tripartite theory is not an a priori truth; it is a social scientific conceptual truth that could be disconfirmed with counter examples. The methodology of this essay is that of conceptual analysis. With this method, we will evaluate a theory of 'definition' as a 'best-explanation inference' about the nature
and functional use of the term 'definition. Possessing a concept (such as of 'definition') makes one disposed to have beliefs (or intuitions) about the correct application of the concept in various cases. With conceptual analysis, participants are asked to critically assess their conceptual intuitions (which are subject to clarification). We are concerned with hypotheses and functional explanations about how natural and artificial languages are used in the context of the (implicit and explicit) intentions of users. What is a 'definition' (as a definiendum-to-definiens relationship)? To guide our pre-theoretic intuitions, let's start out with a question. Each of the seven assertions below is an example of a definition. True or false?
- 'Knowledge' means 'cognition, or the fact of knowing something through acquaintance, or range of one's information or understanding, or the sum total of truth, information, and principles acquired by humankind.' (A dictionary).
- 'Water' means 'a clear liquid that falls as rain, and makes up streams, lakes, and seas, and is composed of H2O.' (A man on the street).
- 'Knowledge' means that '1) p is true, 2) S believes p, 3) S believes p upon a set of premises that are relevant for why p should be believed, and 4) there exists no unresolved nor unconsidered undermining evidence that would effectively lead S to doubt or disbelieve p.' (A philosopher).
- 'Water' means 'a substance composed of H2O, which freezes at zero degrees centigrade, and has a high maximum density at 4 degrees centigrade, and a high specific heat.' (A physical scientist).
- I shall at this moment name my new puppy 'Spot.' (A dog owner).
to be the subject of analysis. In physical science, objects such as water, acid, gold, kinetic energy, electron, gene, protein, enzyme, animal species, and plant species are often thought to belong to 'natural kind' categories. In Philosophy, the concepts of knowledge, truth, justification, mentality, cause, law, necessity, identity, explanation, freedom, beauty, goodness, piety, justice, and existence have often been treated as having an objective nature, and capable of theoretic definition. A theoretic definition is correct (i.e. true) if its definiens truly describes instances of the object being defined. Attention to evidence, reasons, and arguments is required to establish the truth of a theoretic definition.
- A 'stipulative definition' introduces a specialized definiens for a definiendum. This occurs in the following three contexts: (a) the initial naming of an entity where the entity is newly-discovered, newly-introduced, newly-created, or newly-renamed, or (b) in the notational abbreviation of one linguistic expression for another (meaningful) linguistic expression, or (c) in a precise formalization where a reportive definiendum-to-definiens relation is generally affirmed but a definiens alteration (or explication) is proposed for pragmatic, technical, or personal reasons. The evidential support for the tripartite theory of definition is based upon the observations of speech and writing patterns found in natural and artificial languages. The theory should account for definitions that are found in the physical sciences, mathematics, and elsewhere. All other kinds of definitions (e.g. analytic, ostensive, real, nominal, synonymous, recursive, explicit, implicit, precising, persuasive, operational,
essential, disjunctive, verbal, conventional, intensional, extensional, contextual, explicative, functional, conditional, impredicative, partial, axiomatic, constructive, procedural, direct, legislative, discursive, etc.) should be identical to, fall under, be explainable, or refutable under these three primary types. This theory is a hypothesis about the actual limits (and modes) of how persons can intelligibly specify their use of a linguistic symbol. The tripartite theory is very similar to those found in the elementary logic books of Irving Copi & Carl Cohen (2005) and Patrick Hurley (2009). The 'tripartite theory' shouldn't be controversial. My goal is not to introduce an entirely new theory, but to call attention to it. If the tripartite theory is true, then it has importance for explaining and defending an 'anti-realist' philosophy of mathematics, as well as helping resolving some other issues in analytic philosophy, including issues about language. Although there is an ancient distinction between so-called 'real' and 'nominal' definitions, and the concept is intermittently discussed amid various philosophical inquiries, there is an absence of long-term analyses of 'definition' as a unified concept. The only book-length treatment of this topic that I am aware of is Richard Robinson's Definition (1954). The fact that there are few explicit theories of 'definition' is confirmed by several sources. In the December 1993 volume of Philosophical Studies, guest editor Marian David chose 'Definitions' as a topic for submitted articles because despite their important role in analytic philosophy "there is hardly any literature" about definition. In that same volume, Nuel Belnap (1993) is disappointed about not finding substantial modern theories of definition, especially in texts that are histories of logic.
essence must belong to substances. At Topics 101b38, he states that a definition is a phrase signifying a thing's essence (Ross, translator). Aristotle compares other objects that do not have an intrinsic self-unity (e.g. a pile of sand, a rock, a table, a bronze statue) and calls them ‘deficient’ or ‘derivative’ in 'being' compared to substances. Natural kind entities, but not derivative beings, can be said to be the object of a theoretical definition. Hilary Kornblith provides a scientific characterization of a 'natural kind' as a product of 'homeostatic property clusters.' A 'homeostatic relationship' is where a relatively stable state of equilibrium between interrelated physiological factors maintains even in the face of changes in environment. The concepts of homeostatic causal relationships and property clusters are also developed by Richard N. Boyd (1988, 1991). Below is text of Kornblith's (1993) account of physical natural kinds as endorsed here: Natural kinds involve causally stable combinations of properties residing together in an intimate relationship (p. 7). It is nature which divides the world into kinds by creating stable clusters of natural properties residing in homeostatic relationships. Some properties are essential to natural kinds because they are part of this homeostatic cluster or an inevitable part of it; other properties of members of the kind are merely accidental (p. 56, italics added). Examples of entities that purportedly possess essential properties and theoretic definitions are found in physics (electron, kinetic energy, heat, torque, and centripetal force), chemistry (acid, salt, and periodic chart elements), astronomy (black hole, planet), and psychology (intelligence, frustration). More controversially, biological terms are believed by some theorists to be natural kind concepts (gene, mice, marsupial mice, and octopus).
With natural kind concepts attention is paid to the (objective) nature of the phenomena involved. Philosophical concepts such as knowledge, truth, and definition can be conceived of as natural kinds. Paradigm Examples of Natural Kind Concepts (with Theoretic Definitions) (1) 'Electron'-- An electron is associated with quantum numbers, wave-particle duality, indeterminate position-momentum, among others. Buchwald and Warwick (2001) state that the electron's characteristics, charge and mass, have become better known since its discovery in the nineteenth century (pp. 16-17). (2) ‘Knowledge’-- Knowledge is often treated like a natural kind. Epistemic 'personal justification' is sometimes (errantly) treated as a natural kind. (3) 'Truth'-- The 'correspondence theory' is popular: A proposition p is 'true' just in case it corresponds to facts or the world. A p (a belief, proposition, assertion) is true if it corresponds to (or correctly describes) a state of affairs. Another definition not using the term 'correspondence' is from A.N. Prior (1971, pp. 21-22): "To say that S's belief that p is 'true' is to say that one believes that p and (it is the case that) p." (3) Stipulative Definitions A 'stipulative definition' introduces a specialized definiens for a definiendum. There are three subcategories: a) initial naming definitions, b) linguistic abbreviations, and c) formalized definiens for pragmatic, technical, or personal reasons. These examples of definitions should be read carefully. I believe that many issues in all domains of contemporary analytic philosophy are undermined by a lack of understanding of the role of stipulative definitions found in both natural and artificial languages.
(6) In the remainder of this essay, I will abbreviate 'trigeminal neuralgia' as 'TN.' (Source: An article about nerve disorders. The author proposes a short symbol for a longer one to save space and for easier reading). (7) In this contract, the name 'John Smith' designates the term 'lessee.' (Source: An apartment contract where for typographical convenience, and consistency, the predicate 'lessee' is substituted for a proper name). (c) Precisely Formalized Definitions Precise formalized definitions involve terms that have an established use (and typically a reportive definition) but where a definiens alteration is proposed for pragmatic, technical, or personal reasons. The function of a formalization is to modify the definiens of the term (i.e. definiendum) being defined. (1) Pragmatically formalized definitions (5) A person is 'tall' if he or she is 6 feet in height or greater. (Source: A person evaluating how many tall people participate in a basketball league). (6) 'Light' means 'One third fewer calories.' (Source: A definition proposed by the United States Food and Drug Administration with the intent of making the labeling of food more consistent in 1991). (2) Technically formalized definitions (7) An 'analytic sentence' is a sentence that is true solely in virtue of the meaning (or the definitions) of its terms. (Source: A definition originating with Kant). (8) 'Truth' is a property of sentences (in a given formal model) and sentences are truth bearers. (Source: Logician, Alfred Tarski, 1944).
(9) A 'proposition' is an abstract object to which a person is related by a belief, desire, or other psychological attitude, typically expressed in a language containing a psychological verb ('think,' 'deny,' 'doubt,' etc.) followed by a that- clause. The psychological states in question are called propositional attitudes. (Source: The Cambridge Dictionary of Philosophy). (10) An 'intension' is the meaning or connotation of an expression, as opposed to its extension or denotation, which consists of those things specified by the expression. The intension of a declarative sentence is often taken to be a proposition and the intension of a predicate expression (common noun, adjective) is often taken to be a concept. (Source: The Cambridge Dictionary of Philosophy). (3) Personal formalized definitions (11) 'Happiness' is good health and bad memory. (Source: Ingrid Bergman) (12) 'Leadership' is a person's being able to guide or inspire others, to enlist support in the accomplishment of a common task. (Source: Motivational speaker, Mark Shead, 2018). A Comparison with Peter Geach's Similar View About Definitions P.T. Geach (1976) similarly recognizes a difference between real (i.e. theoretic), nominal (i.e. reportive), and proposed (i.e. stipulative) definitions. In the following text, Geach summarizes his intuitions about the concept of definition: It has long been traditional to distinguish between real and nominal definitions. Real definitions aim at marking out a class of things that shall correspond to a natural kind, like gold or acids… We need, then, to recognize the natural kinds of
On the Importance of a Theory of Definition A theory of definition is of great importance in understanding the discipline of mathematics. As stated, as a mathematical anti-realist, I contend that formal systems are composed of stipulative definitions and implicit definitions (i.e. axioms). Stipulations are neither true nor false; but can only be agreed-to. In formal deductive systems we typically find (1) the stipulated introduction of a vocabulary of symbols and definitions about what counts as an individual constant, individual variable, predicate, proper name, sentential connective, punctuation, and quantifier, (2) the stipulated introduction of syntactical formation rules (or grammar) that defines how 'well-formed formulas' are to be constructed out of symbols (i.e. a procedure that determines whether a sentence, as a finite strings of words or symbols, is 'meaningful' or not), (3) a set of stipulated truth- preserving inference rules, and (4) a semantics (e.g. truth-table definitions of connectives, or interpretations using symbolization keys and extensions). I contend that formal systems always stipulate rules in the form of stipulative definitions. This perspective is in opposition to mathematical realism which contends that: (1) there exist mathematical objects, (2) mathematical objects are abstract, and (3) mathematical objects are independent of persons, including their thought, language, and practices. Obviously, an anti-realist theory of metamathematics cannot be presented here, but a theory (or hypothesis) about 'stipulative definition' is a good starting point. Part II: How is the Tripartite Theory of Definition Verified? The tripartite theory is a social scientific hypothesis about the actual modes of how persons can intelligibly specify their use of a linguistic symbol in a definiendum-to-
definiens relationship. The theory maintains that these relationships can be objectively quantified as falling under these three kinds based upon a speaker’s intent and the context of the assertion. All other kinds of definitions are identical to, fall under, be explainable, or refutable under these three primary types. How can this hypothesis be verified? In order to verify the truth of the tripartite theory we will investigate language use in various situations. If one were to find definitions that weren't charitably classifiable in one of these categories, then the theory would be disconfirmed (with an appropriate argument). Let us survey, and paraphrase, eight kinds of definitions under the entry of ‘definition’ contributed by Takashi Yagisawa in The Cambridge Dictionary of Philosophy edited by Robert Audi (1999) and start an investigation with short responses. Real Definition A ‘real definition’ is the specification of the metaphysically necessary and sufficient condition for being the kind of thing a noun (usually a common noun) designates. For example: ‘element with atomic number 79’ for ‘gold.’ Response: A ‘real definition’ is best interpreted as a theoretic definition where an assumed natural kind entity is presumed to have some objective condition(s) that are necessary (essential) and/or sufficient for its instantiation. Nominal Definition A ‘nominal definition’ is the definition of a noun (usually a common noun), giving its linguistic meaning. Typically, it is in terms of macro-sensible characteristics: e.g. ‘yellow malleable metal’ for ‘gold.’ Locke spoke of nominal and real essence.
the tripartite theory, where the definiendum is always the subject of a definition. A ‘definition’ is conceived as a sentence that connects a mark or sound (i.e. a definiendum) to a meaningful definiens. So-called ‘nominal definitions’ are understandable in context as stipulative (e.g. an abbreviation) or reportive (e.g. 'gold' is a precious yellow metal often made into jewelry or coins). Explicit Definition An ‘explicit definition’ is a definition that makes it clear that it is a definition and identifies the expression being defined as such: e.g. ‘Father’ means ‘male parent’; 'For any x, x is father by definition if and only if x is a male parent.' Response: With a tripartite theory, we have just analyzed definitions of this explicit definiendum-to-definiens form. Reportive, theoretic, and stipulative definitions are about linguistic expressions (the definiendum) as the subject, and which are potential subjects of an explicit definition (as reported, theoretic, or stipulated). Implicit Definition An ‘implicit definition’ is a definition that is not an explicit definition. Response: A better explanation is that implicit definitions are those found as axioms in deductive systems. An 'axiom' is composed of undefined primitive terms and is not provable from other propositions (and axioms) within the formal system. The role of an axiom (and its content) within a formal system is to characterize certain primitive (undefined) terms. The undefined terms of an axiomatic system do not have any definite meaning (other than from their occurrence in the axioms) and may be interpreted in any way that is consistent with a given set of axioms. Axioms can be assumed-true only
under a consistent interpretation (or model) that gives meaning to a formal system. In modern geometry, 'point' and 'line' are not explicitly defined. Similarly, in set theory, the words 'set,' and 'element' (i.e. 'set membership,' 'belongs') are undefined terms. Implicit definitions are stipulative, where a consistency of related concepts is sought. Recursive Definition A ‘recursive definition’ (also called inductive definition) is a definition in three clauses where (1) the expression defined is applied to certain items (the base clause); (2) a rule is given for reaching further items to which the expression applies (the recursive, or inductive clause); and (3) it is stated that the expression applies to nothing else (the closure clause). For example: ‘John’s parents are John’s ancestors; any parent of John’s ancestor is John’s ancestor; nothing else is John’s ancestor.’ By the base clause, John’s mother and father are John’s ancestors. Then by the recursive clause, John’s mother’s parents and John’s father’s parents are John’s ancestors; so are their parents, and so on. Finally, by the last (closure) clause, these people exhaust John’s ancestors. Response: The tripartite theory maintains that all three clauses of a recursive definition are examples of a technically formalized or abbreviatory stipulative definition. Precising Definition A ‘precising definition’ is a definition of a vague expression intended to reduce its vagueness. Two examples: (a) ‘snake of average length’ is precisely defined as a ‘snake longer than half a meter and shorter than two meters,’ and (b) ‘wealthy’ is defined as ‘having assets ten thousand times the median figure.'
Response: Another example; ‘abortion’ is ‘the ruthless murdering of innocent human beings,’ as opposed to ‘a safe and established procedure whereby a woman is relieved of an unwanted fetus.’ Hurley (2009) says that "the purpose of a persuasive definition is to engender a favorable or unfavorable attitude toward what is denoted by the definiendum” (pp. 88-89). The term ‘persuasive definition’ was introduced by Charles Stevenson (1944) as part of his emotive theory of meaning. A persuasive definition is an example of a personally formalized stipulative definition. Definition At the start of defining 'definition' in the Cambridge Dictionary, it is stated: Definition, specification of the meaning or, alternatively, conceptual content, of an expression. For example, 'period of fourteen days' is a definition of 'fortnight.' It is then said that definitions have traditionally been judged by rules like the following, with example definitions omitted here: (1) A definition should not be too narrow, (2) A definition should not be too broad, (3) The defining expression in a definition should (ideally) exactly match the degree of vagueness of the expression being defined (except in a precising definition), (4) A definition should not be circular. If 'desirable' defines 'good' and 'good' defines 'desirable,' these definitions are circular. The author then lists eighteen kinds of definition, including eight kinds of definition that are discussed above. Response: In defining a 'definition' as specifying a meaning or conceptual content, it is indeed vague, and clarifying this vagueness or ambiguity is the central project of the tripartite theory of definition. We have hypothesized that there are seven ways that a definiendum-to-definiens relationship may be specified: (1) reportive, (2)
theoretic, (3) initial naming, (4) abbreviatory, (5) pragmatically formalized, (6) technically formalized, and (7) personally formalized. The example definition of 'fortnight' is an example of a reportive definition of a fixed-definiens concept. The rules that judge what a 'good definition' is taken to be, aren't problematic, but they are not overly informative because of their vagueness. In response to the rule about 'non-circularity,' Hurford et. al (2007) state that "the commonly accepted idea is that the goal of a dictionary is to define everything. One cannot define absolutely everything without a degree of circularity in one's definitions" (p. 196). Circularity and implicit definitions are a part of languages. In all, the tripartite theory of definition is more detailed than the entry for 'definition' in the Cambridge Dictionary of Philosophy. This should be expected for most entries because they are just reported summaries of terms (and concepts). But the vagueness of this definition also reflects an unfortunate situation where there is inadequate attention paid to the concept of definition by scholars. Conclusion With a tripartite theory of ‘definition’ we have recognized three basic kinds of definition. I respectfully ask for potential counterexamples. If there are none, then the tripartite theory is a true account of how persons (in context) may specify their intended use of a linguistic entity in a definiendum-to-definiens relationship. In other essays, I elaborate in detail upon how this theory is important to a philosophy of mathematics, and to other issues in analytic philosophy.
