Why do we need definitions

These resources enable one to define, for example, various unary concepts, which are thereby assured of satisfying the Vicious-Circle Principle.

Quantification over these concepts is thus bound to be legitimate, and can be added to the language. The same holds for propositions and for concepts falling under other types: for each type, a quantifier can be added that ranges over items of that type that are definable using the initial unproblematic resources.

The new quantificational resources enable the definition of further items of each type; these, too, respect the Principle, and again, quantifiers ranging over the expanded totalities can legitimately be added to the language. The new resources permit the definition of yet further items. And the process repeats.

The result is that we have a hierarchy of propositions and of concepts of various orders. Each type in the type hierarchy ramifies into a multiplicity of orders. This ramification ensures that definitions formulated in the resulting language are bound to respect the Vicious-Circle Principle. Concepts and classes that can be defined within the confines of this scheme are said to be predicative in one sense of this word ; the others, impredicative.

For a formal presentation of Ramified Type Theory, see Church ; for a more informal presentation, see Hazen See also the entries on type theory and Principia Mathematica , which contain further references. This definition is essentially circular; it is not reducible to one in normal form. The definition leaves unsettled the status of only two objects, namely, Plato and Aristotle. More generally, there is a strong parallel between the behavior of the concept of truth and that of concepts defined by circular definitions.

Both are typically well defined on a range of cases, and both display a variety of unusual logical behavior on the other cases. Indeed, all the different kinds of perplexing logical behavior found with the concept of truth are found also in concepts defined by circular definitions.

This strong parallelism suggests that since truth is manifestly a legitimate concept, so also are concepts defined by circular definitions such as The paradoxes, according to this viewpoint, cast no doubt on the legitimacy of the concept of truth.

They show only that the logic and semantics of circular concepts is different from that of non-circular ones. This viewpoint is developed in the revision theory of definitions. In this theory, a circular definition imparts to the defined term a meaning that is hypothetical in character; the semantic value of the defined term is a rule of revision , not as with non-circular definitions, a rule of application.

Consider 18 again. Then it is easy to see that the definiens is true precisely of Socrates and Plato. The fundamental idea of the revision theory is to view this rule as a revision rule: the output interpretation is better than the input one or it is at least as good; this qualification will be taken as read.

The semantic value that the definition confers on the defined term is not an extension—a demarcation of the universe of discourse into objects that fall under the defined term, and those that do not. The semantic value is a revision rule.

The revision rule explains the behavior, both ordinary and extraordinary, of a circular concept. The resulting sequence,. For example, the revision rule for 18 generates a revision process that consists of the following revision sequences, among others:.

Observe the behavior of our four ancient philosophers in this process. After some initial stages of revision, Socrates always falls in the revised interpretations, and Xenocrates always falls outside. In this particular example, the behavior of the two is fixed after the initial stage; in other cases, it may take many stages of revision before the status of an object becomes settled.

Objects on which the process does not yield a categorical verdict are said to be pathological relative to the revision rule, the definition, or the defined concept. In our example, Plato and Aristotle are pathological relative to The status of Aristotle is not stable in any revision sequence.

It is as if the revision process cannot make up its mind about him. When an object behaves in this way in all revision sequences, it is said to be paradoxical. Plato acquires a stable status in each revision sequence, but the status he acquires depends upon the initial hypothesis. Revision processes help provide a semantics for circular definitions.

A definition is said to be finite iff, roughly, its revision process necessarily requires only finitely many such stages. Because of the expressive power, the general notion of validity for non-finite circular definitions is not axiomatizable Kremer We can give at best a sound logical calculus, but not a complete one. The situation is analogous to that with second-order logic.

Let us observe some general features of the revision theory of definitions. The introduction and elimination rules hold unrestrictedly, and revision stages are dispensable. The deviations from the traditional account occur only over circular definitions. Sentences containing defined terms are subject to the same logical laws as sentences of the ground language.

No definition, no matter how vicious the circularity in it, entails anything new in the ground language. Even the utterly paradoxical definition. Sentences of the expanded language are not, in general, reducible to those of the ground language. This failure has two sources. First, revision theory fixes the use, in assertion and argument, of sentences of the expanded language but without reducing the sentences to those of the ground language.

The theory thus meets the Use criterion, but not the stronger one of Eliminability. Second, in this theory, a definition can add logical and expressive power to a ground language. The addition of a circular definition can result in the definability of new sets. This is another reason why Eliminability fails. It may be objected that every concept must have an extension, that there must be a definite totality of objects that fall under the concept. If this is right then a predicate is meaningful—it expresses a concept—only if the predicate necessarily demarcates the world sharply into those objects to which it applies and those to which it does not apply.

Hence, the objection concludes, no predicate with an essentially circular definition can be meaningful. The objection is plainly not decisive, for it rests on a premiss that rules out many ordinary and apparently meaningful predicates e. Nonetheless, it is noteworthy because it illustrates how general issues about meaning and concepts enter the debate on the requirements on legitimate definitions.

The principal motivation for revision theory is descriptive. It has been argued that the theory helps us to understand better our ordinary concepts such as truth, necessity, and rational choice.

The ordinary as well as the perplexing behavior of these concepts, it is argued, has its roots in the circularity of the concepts. If this is correct, then there is no logical requirement on descriptive and explicative definitions that they be non-circular. See also the entry on the revision theory of truth.

Martin, and the reply by Gupta, in Villanueva See also Shapiro The author would like to thank Ed Zalta and any anonymous editor for helpful suggestions for improving this entry. Some varieties of definition 1. The logic of definitions 2. Some varieties of definition Ordinary discourse recognizes several different kinds of things as possible objects of definition, and it recognizes several kinds of activity as defining a thing.

Thus, Russell maintains in Human Knowledge that all nominal definitions, if pushed back far enough, must lead ultimately to terms having only ostensive definitions, and in the case of an empirical science the empirical terms must depend upon terms of which the ostensive definition is given in perception. See Suppes for a different perspective on conditional definitions. Bibliography Belnap, N. Beth, E. Boolos, G. Carnap, R. Chapuis, A. Charles, D. Chihara, C.

Church, A. Demopoulos, W. Dudman, V. Fine, K. Frege, G. Hermes, F. Kambartel, and F. Kaulbach, Chicago: University of Chicago Press , pp. Bolander, V. Hendricks, and S. Gupta, A.

Hacker, P. Hale B. Hazen, A. Gabbay and F. Guenthner, Dordrecht: Reidel, pp. We won't go into the philosophical debate of whether concepts define words or do words define concepts, a debate that is currently plaguing educators today 4 , as it has been since the times of Plato and Socrates 5.

Nor will we spend time debating about specific definitions of terms, how the definitions morph over time, or the nature of, and demands on, definitions 6. Instead this document will focus on how to fulfill the need to establish and maintain definitions for specific terms within specific subject fields 7. There is an entire international standard dedicated this pursuit 8 , which we have drawn much of our inspiration from. The issue we are attempting to remedy in this document, is the tendency of each subject field to create its own sub-language of specific terms and their definitions.

These terms might be shared with other subject fields, but quite often the definitions for these shared terms are different.

And just as often, those in specific subject fields create new, distinct terms used to describe the same concept as other terms found in other subject fields.

This material may not be published, reproduced, broadcast, rewritten, or redistributed without permission. Use of this site constitutes acceptance of our terms and conditions of fair use. A formal definition is based upon a concise, logical pattern that includes as much information as it can within a minimum amount of space.

For more complex circumstances, it can take years of experience to come up with a simple, useful definition. Few people take the trouble to really, really know their stuff. Not vague, and not convoluted. This is typically a function of precision. This is a good thing, because it then allows you to revise your definition. This allows for healthy, constructive discussions that are focused on outcomes rather than personal egos. Definitions are useful in practically any field.

They help us think and communicate more clearly, which in turn help us understand our businesses our customers, our product, our processes, our value-proposition, etc better. Your customers. We all have an idea of who our customers or users are, and we often mistake the map for the territory. Your product. What is it, exactly? What does it do? Why should anybody want it?