Kripke used such reference-fixing stipulations to argue for the existence of contingent a priori truths— 1 being an example. Reference-fixing stipulative definitions can be given not only for names but also for terms in other categories, e.
See Frege for a defense of the austere view that, in mathematics at least, only stipulative definitions should be countenanced. Descriptive definitions, like stipulative ones, spell out meaning, but they also aim to be adequate to existing usage.
When philosophers offer definitions of, e. It is useful to distinguish three grades of descriptive adequacy of a definition: extensional, intensional, and sense. A definition is extensionally adequate iff there are no actual counterexamples to it; it is intensionally adequate iff there are no possible counterexamples to it; and it is sense adequate or analytic iff it endows the defined term with the right sense. When definitions are put to an epistemological use, intensional adequacy is generally insufficient.
For such definitions cannot underwrite the rationality or the aprioricity of a problematic subject matter. Horty offers some ways of thinking about senses of defined expressions, especially within a Fregean semantic theory. An explication aims to respect some central uses of a term but is stipulative on others. The explication may be offered as an absolute improvement of an existing, imperfect concept. The quoted phrase is due to Alan Ross Anderson; see Belnap , A simple illustration of explication is provided by the definition of ordered pair in set theory.
And it can be verified that the above definition satisfies the principle. The definition does have some consequences that do not accord with the ordinary notion. But the mismatch is not an objection to the explication.
What is important for explication is not antecedent meaning but function. So long as the latter is preserved, the former can be let go. It is this feature of explication that led W. The truth-functional conditional provides another illustration of explication. This conditional differs from the ordinary conditional in some essential respects. Nevertheless, the truth-functional conditional can be put forward as an explication of the ordinary conditional for certain purposes in certain contexts.
Whether the proposal is adequate depends crucially on the purposes and contexts in question. That the two conditionals differ in important, even essential, respects does not automatically disqualify the proposal. Ostensive definitions typically depend on context and on experience. Suppose the conversational context renders one dog salient among several that are visible. We can think of experience as presenting the subject with a restricted portion of the world. This portion can serve as a point of evaluation for the expressions in an ostensive definition.
See Gupta for an account of the contribution of experience to the meaning of an ostensively defined term. An ostensive definition can bring about an essential enrichment of a language. Unlike other familiar definitions, ostensive definitions can introduce terms that are ineliminable.
So, ostensive definitions can fail to meet the Eliminability criterion explained below; they can fail to meet also the Conservativeness criterion, also explained below. The capacity of ostensive definitions to introduce essentially new vocabulary has led some thinkers to view them as the source of all primitive concepts. Thus, Russell maintains in Human Knowledge that. Such foundationalist pictures were decisively criticized by Ludwig Wittgenstein in his Philosophical Investigations.
Ostensive definitions are important, but our understanding of them remains at a rudimentary level. They deserve greater attention from logicians and philosophers. The kinds into which we have sorted definitions are not mutually exclusive, nor exhaustive.
A stipulative definition of a term may, as it happens, be extensionally adequate to the antecedent uses of the term. A dictionary may offer ostensive definitions of some words e. An ostensive definitions can also be explicative. Moreover, as we shall see below, there are other kinds of definition than those considered so far. Such definitions can be represented thus:. We are setting aside ostensive definitions, which plainly require a richer representation.
When the defined term is clear from the context, the representation may be simplified to. Not all definitions found in the logical and philosophical literature fit under scheme 2. Partial definitions, for example, fall outside the scheme; another example is provided by definitions of logical constants in terms of introduction and elimination rules governing them.
Nonetheless, definitions that conform to 2 are the most important, and they will be our primary concern. Let us focus on stipulative definitions and reflect on their logic.
Some of the important lessons here carry over, as we shall see, to descriptive and explicative definitions. For simplicity, let us consider the case where a single definition stipulatively introduces a term.
Multiple definitions bring notational complexity but raise no new conceptual issues. What requirements must the definition fulfill? Before we address these questions, let us take note of a distinction that is not marked in logic books but which is useful in thinking about definitions.
In one kind of definition—call it homogeneous definition—the defined term and the definiendum belong to the same logical category. So, a singular term is defined via a singular term; a general term via a general term; a sentence via a sentence; and so on.
Let us say that a homogenous definition is regular iff its definiendum is identical to the defined term. Here are some examples of regular homogeneous definitions:. It is sometimes said that definitions are mere recipes for abbreviations. In the second kind of definition—call it a heterogenous definition—the defined term and the definiendum belong to different logical categories.
So, for example, a general term e. For another example, a singular term e. Heterogeneous definitions are far more common than homogenous ones. In a heterogeneous definition, however, the definiens can easily be complex; for example,. If the language has a device for abstraction—e. Observe that a heterogenous definition such as 4 is not a mere abbreviation. Moreover, if such definitions were abbreviations, they would be subject to the requirement that the definiendum must be shorter than the definiens, but no such requirement exists.
On the other hand, genuine requirements on definitions would make little sense. The following stipulation is not a legitimate definition:. Some stipulative definitions are nothing but mere devices of abbreviation e. However, many stipulative definitions are not of this kind; they introduce meaningful items into our discourse. But what is the source of the difference?
Why is 4 legitimate, but not 6? More generally, when is a definition legitimate? What requirements must the definiens fulfill?
And, for that matter, the definiendum? Must the definiendum be, for instance, atomic, as in 3 and 4? If not, what restrictions if any are there on the definiendum? It is a plausible requirement on any answer to these questions that two criteria be respected. We should not be able to establish, by means of a mere stipulation, new things about, for example, the moon. It is true that unless this criterion is made precise, it is subject to trivial counterexamples, for the introduction of a definition materially affects some facts.
Nonetheless, the criterion can be made precise and defensible, and we shall soon see some ways of doing this. There are complications here, however. For example, a definition of quotient may leave some occurrences of the term undefined e. The orthodox view is to rule such definitions as illegitimate, but the orthodoxy deserves to be challenged here.
Let us leave the challenge to another occasion, however, and proceed to bypass the complications through idealization. Let us confine ourselves to ground languages that possess a clearly determined logical structure e. A variant formulation of the Use criterion is this: the definition must fix the meaning of the definiendum. Note that the two criteria govern all stipulative definitions, irrespective of whether they are single or multiple, or of whether they are of form 2 or not.
The traditional account of definitions is founded on three ideas. The first idea is that definitions are generalized identities; the second, that the sentential is primary; and the third, that of reduction.
These are, put crudely, that i any occurrence of the definiendum can be replaced by an occurrence of the definiens Generalized Definiendum Elimination ; and, conversely, ii any occurrence of the definiens can be replaced by an occurrence of the definiendum Generalized Definiendum Introduction. 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. Why did he go? The reason, cause, motive, purpose, etc.
Never mind the why and wherefore. Because of which; on account of which. He knows of no reason why you shouldn't go. Used to express surprise, impatience, indignation, etc.
For what cause , reason , or purpose. Exclamation of mild surprise. Why is defined as for what purpose, cause or reason. For what purpose, reason, or cause; with what intention, justification, or motive.
Why is the door shut? Why do birds sing? Used as an expletive, to preface a remark. Like other sharp children, Why - Why was always asking metaphysical conundrums. New Word List Word List. Save This Word! See synonyms for why on Thesaurus. We could talk until we're blue in the face about this quiz on words for the color "blue," but we think you should take the quiz and find out if you're a whiz at these colorful terms. Words nearby why whsle.
Words related to why cause , motive , mystery , proof , how.
0コメント