The second idea—the primacy of the sentential—has its roots in the thought that the fundamental uses of a term are in assertion and argument: if we understand the use of a defined term in assertion and argument then we fully grasp the term. The sentential is, however, primary in argument and assertion. Sentential items are here understood to include sentences and sentence-like things with free variables, e. The issues the second idea raises are, of course, large and important, but they cannot be addressed in a brief survey.
Let us accept the idea simply as a given. This idea, when conjoined with the primacy of the sentential, leads to a strong version of the Use criterion, called the Eliminability criterion: the definition must reduce each formula containing the defined term to a formula in the ground language, i. Eliminability is the distinctive thesis of the traditional account and, as we shall see below, it can be challenged.
Note that the traditional account does not require the reduction of all expressions of the extended language; it requires the reduction only of formulas. This is not to deny that no new proposition—at least in the sense of truth-condition—is expressed in the expanded language.
Let us now see how Conservativeness and Eliminability can be made precise. First consider languages that have a precise proof system of the familiar sort. Now, the Conservativeness criterion can be made precise as follows.
The Eliminability criterion can be made precise thus:. Folklore credits the Polish logician S. The criteria of Conservativeness and Eliminability can now be made precise thus:. The syntactic and semantic formulations of the two criteria are plainly parallel.
Indeed, several different, non-equivalent formulations of the two criteria are possible within each framework, the syntactic and the semantic. Observe that the satisfaction of Conservativeness and Eliminability criteria, whether in their semantic or their syntactic formulation, is not an absolute property of a definition; the satisfaction is relative to the ground language.
Different ground languages can have associated with them different systems of proof and different classes of interpretations. Hence, a definition may satisfy the two criteria when added to one language, but may fail to do so when added to a different language. For further discussion of the criteria, see Suppes and Belnap Call two definitions equivalent iff they yield the same theorems in the expanded language.
The normal form of definitions can be specified as follows. The general conditions remain the same when the traditional account of definition is applied to non-classical logics e. The specific conditions are more variable.
An existence and uniqueness claim must hold: the universal closure of the formula. In a logic that allows for vacuous names, the specific condition on the definiens of 7 would be weaker: the existence condition would be dropped. In contrast, in a modal logic that requires names to be non-vacuous and rigid, the specific condition would be strengthened: not only must existence and uniqueness be shown to hold necessarily, it must be shown that the definiens is satisfied by one and the same object across possible worlds.
Definitions that conform to 7 — 9 are heterogeneous; the definiendum is sentential, but the defined term is not. One source of the specific conditions on 7 and 9 is their heterogeneity. The specific conditions are needed to ensure that the definiens, though not of the logical category of the defined term, imparts the proper logical behavior to it. The conditions thus ensure that the logic of the expanded language is the same as that of the ground language.
This is the reason why the specific conditions on normal forms can vary with the logic of the ground language. Observe that, whatever this logic, no specific conditions are needed for regular homogeneous definitions. The traditional account makes possible simple logical rules for definitions and also a simple semantics for the expanded language. In classical logic, all definitions can easily be transformed to meet this condition. The semantics for the extended language is also straightforward.
The semantics of defined predicates and function-symbols is similar. The logic and semantics of definitions in non-classical logics receive, under the traditional account, a parallel treatment. Note that the inferential force of adding definition 10 to the language is the same as that of adding as an axiom, the universal closure of. Moreover, the biconditional can be iterated—e. Finally, a term can be introduced by a stipulative definition into a ground language whose logical resources are confined, say, to classical conjunction and disjunction.
This is perfectly feasible, even though the biconditional is not expressible in the language. In such cases, the inferential role of the stipulative definition is not mirrored by any formula of the extended language. The traditional account of definitions should not be viewed as requiring definitions to be in normal form. The only requirements that it imposes are i that the definiendum contain the defined term; ii that the definiendum and the definiens belong to the same logical category; and iii the definition satisfies Conservativeness and Eliminability.
So long as these requirements are met, there are no further restrictions. The definiendum, like the definiens, can be complex; and the definiens, like the definiendum, can contain the defined term. The role of normal forms is only to provide an easy way of ensuring that definitions satisfy Conservativeness and Eliminability; they do not provide the only legitimate format for stipulatively introducing a term.
Thus, the reason why 4 is, but 6 is not, a legitimate definition is not that 4 is in normal form and 6 is not. The reason is that 4 respects, but 6 does not, the two criteria. The ground language is assumed here to contain ordinary arithmetic; under this assumption, the second definition implies a contradiction.
The following two definitions are also not in normal form:. But both should count as legitimate under the traditional account, since they meet the Conservativeness and Eliminability criteria. It follows that the two definitions can be put in normal form. Definition 12 is plainly equivalent to 4 , and definition 13 is equivalent to 14 :. Nevertheless, the definition has a normal form.
Similarly, the traditional account is perfectly compatible with recursive a. In Peano Arithmetic, for example, exponentiation can be defined by means of the following equations:.
Here the first equation—called the base clause—defines the value of the function when the exponent is 0. This is perfectly legitimate, according to the traditional account, because a theorem of Peano Arithmetic establishes that the above definition is equivalent to one in normal form.
But the circularity is entirely on the surface, as the existence of normal forms shows. See the discussion of circular definitions below. It is a part of our ordinary practice that we sometimes define terms not absolutely but conditionally.
We sometimes affirm a definition not outright but within the scope of a condition, which may either be left tacit or may be set down explicitly. For another example, when defining division, we may explicitly set down as a condition on the definition that the divisor not be 0. We may stipulate that. This practice may appear to violate the Eliminability criterion, for it appears that conditional definitions do not ensure the eliminability of the defined terms in all sentences.
Thus 16 does not enable us to prove the equivalence of. Similarly 17 does not enable us to eliminate the defined symbol from. However, if there is a violation of Eliminability here, it is a superficial one, and it is easily corrected in one of two ways. The first waythe way that conforms best to our ordinary practicesis to understand the enriched languages that result from adding the definitions to exclude sentences such as 18 and For when we stipulate a definition such as 16 , it is not our intention to speak about the first cousins once removed of numbers; on the contrary, we wish to exclude all such talk as improper.
Similarly, in setting down 17 , we wish to exclude talk of division by 0 as legitimate. So, the first way is to recognize that a conditional definition such as 16 and 17 brings with it restrictions on the enriched language and, consequently, respects the Eliminability criterion once the enriched language is properly demarcated.
This idea can be implemented formally by seeing conditional definitions as formulated within languages with sortal quantification. So, we may stipulate that nothing other than a human has first cousins once removed, and we may stipulate that the result of dividing any number by 0 is 0. Thus we may replace 17 by. The resulting definitions satisfy the Eliminability criterion.
The second way forces us to exercise care in reading sentences with defined terms. So, for example, the sentence. The above viewpoint allows the traditional account to bring within its fold ideas that might at first sight seem contrary to it. This idea is easily accommodated within the traditional account.
So, the traditional account accommodates the idea that theories can stipulatively introduce new terms, but it imposes a strong demand: the theories must be admissible.
Consider, for concreteness, the special case of classical first-order languages. Furthermore, let us say that. That is, an admissible theory fixes the semantic value of the defined term in each interpretation of the ground language.
This observation provides one natural method of showing that a theory is not admissible:. Padoa This question receives a negative answer for some semantical systems, and a positive answer for others.
The converse fails for, e. For a proof of the theorem, see Boolos, Burgess, and Jeffrey ; see also Beth The idea of implicit definition is not in conflict, then, with the traditional account.
Where conflict arises is in the philosophical applications of the idea. The failure of strict reductionist programs of the late-nineteenth and early-twentieth century prompted philosophers to explore looser kinds of reductionism.
The essentials of the argument are found already in Frege Another example: The reductionist program for theoretical concepts e. The program aimed to reduce theoretical sentences to classes of observational sentences. From the Editors at Merriam-Webster. An Explanation of Why We Sometimes Phrases Related to definition by definition. Style: MLA. Kids Definition of definition. Medical Definition of definition. Get Word of the Day daily email! Test Your Vocabulary.
Test your vocabulary with our question quiz! Love words? Need even more definitions? Homophones, Homographs, and Homonyms The same, but different. Outsets and onsets! B2 to say what the meaning of something, especially a word, is:. In the dictionary , " reality " is defined as "the state of things as they are, rather than as they are imagined to be".
See also well defined. B2 to explain and describe the meaning and exact limits of something:. Your rights and responsibilities are defined in the citizens ' charter. See more results ». It is very difficult to define the concept of beauty. The borderline between friendship and intimacy is often hard to define. Security defined in the broadest sense of the term means getting at the root causes of trouble and helping to reduce regional conflicts.
It's a very hierarchical organization in which everyone's status is clearly defined. Responsibility, I suppose , is what defines adulthood. The outline of the castle on the hill was clearly defined against the evening sky. Showing and demonstrating. Related word definable. English to describe the meaning of something, esp. The dark figures are sharply defined on the white background. Economists normally define a recession as two successive quarters of negative growth.
Send us feedback. See more words from the same century. Accessed 11 Nov. More Definitions for define. Nglish: Translation of define for Spanish Speakers. Britannica English: Translation of define for Arabic Speakers. Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free!
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