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«Abstract: Practical methods are introduced for the construction of definitions, both for philosophical purposes and for uses in other disciplines. ...»

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Finally, in addition to linguistic and categorial congruence, congruence in variables is required. This means that the same variables should be used in the definiens as in the definiendum.

–  –  –

It could be argued in these examples that the variable is implicit on the side of the definition on which it was not explicitly stated. However, experience shows that it is better to write out Sven Ove Hansson definitions meticulously. The practice, shown in the first example, of writing out variables as letters is often preferable in precise definitional work.

Even if a definition satisfies the three congruences, it may of course be faulty due to unintended or unserviceable differences in meaning between the definiens and the definiendum. Discussions about such differences are often couched in terms of how “wide” or “narrow” the definition is. If the definiens includes something that should be excluded, then the definition is said to be too wide. If it excludes something that should be included, then it is called too narrow. A definition can be too wide and too narrow at the same


A Scandinavian is a person who lives in Sweden. (too narrow) A Scandinavian is a person who lives in Northern Europe. (too wide) A Scandinavian is a person who understands the Swedish language. (both too narrow and too wide)

4. Choosing a primary definiendum An often-neglected aspect of definition work is the choice of an expedient form of the definiendum. It is a common mistake to believe that if we want to define a term from either ordinary or scientific language, then we should take this term as it stands and make it the definiendum of our definition.

The term in question may be more difficult to work with as a definiendum than some other, related term from which it can in its turn be defined. Hence, if we want to define “stability”, it is advisable not to proceed in the format “stability is…”. It is much easier in this case to get started if we define stability as the property of being stable, and then focus our serious definitional work on the term “stable”. A (preliminary) format for that definition is “X is stable if and only if…”.

Many of the terms that we wish to define come in clusters of closely related terms. “Stability” and “stable” belong to one such cluster; “know” and “knowledge” to another, “safety”, “safe”, and “safer” to a third. In serious definition work it is essential to identify How to define – a tutorial 13 the cluster to which the term that we began with belongs. A dictionary is a useful tool for doing this, but it should not be taken for granted that the dictionary provides all the relevant forms.

Hence, in the case of “stable”, the comparative form “more stable than” will not be found in most dictionaries.

After the cluster has been identified, a preliminary analysis should be performed of the interrelations among its elements, in order to determine if and how they can be defined in terms of each other. On the basis of this analysis, one of these concepts can be chosen as the primary definiendum, the definiendum on which the work will be focused.

Using the concept of safety as an example, let us begin by identifying the cluster of concepts to which it belongs. A dictionary will provide us with the words “safety”, “safe”, “safer”, “safest”, “safely”, and “safeness”. We need to investigate the relationships of interdefinability between these words.

“Safely” is an adverb. Like many other adverbs, it can in general be defined in terms of the corresponding adjective. Hence, to say that someone drives safely is equivalent to saying that her driving is safe. On the other hand, not all uses of the adjective “safe” can be expressed in terms of the adverb “safely”. The problematic cases are those in which the adjective is applied to a noun that does not express an activity for which there is a corresponding verb to which the adverb can be applied. A “safe drive” is a an act of driving safely, but when we say that somebody stood at a “safe distance” from the fire, there is no activity corresponding to the noun “distance” to which we can apply the adverb “safely”.

(Admittedly, we can reformulate the whole phrase and speak about “a distance at which one can stand safely”, but this is not satisfactory since no such reformulation covers all uses of the phrase “safe distance”.) Therefore, “safely” is definable in terms of “safe” but not the other way around. This is a good reason for preferring “safe” to “safely” as a definiendum.

“Safety” and the more unusual “safeness” are synonyms (terms for the same concept) and thus trivially interdefinable. Safety Sven Ove Hansson is a property, and must be related to something that possesses this property, hence a preliminary definiendum for safety can have the form “Safety is the property which something X has if and only if…”. The adjective “safe” also denotes a property that must be related to something that has the property, hence a preliminary definiendum for “safe” can be: “Something X is safe if and only if…” We can reasonably take “X has the property of safety” and “X is safe” to be synonymous. This means that “safe” and “safety” are fully interdefinable. Since “safe” gives rise to more straight-forward linguistic constructions, it is the better choice of the two.

However, the adjective “safe” has three forms, namely the absolute “safe”, the comparative “safer” and the superlative “safest”.

It remains to choose which of the three to use as a primary definiendum.

“Safest” is definable in terms of “safer”. The safest car is the car that is safer than all the other cars. Similarly, “safer” can be defined in terms of “safest”: X is safer than Y if and only if X is safest among the two cars X and Y. Hence, the two forms are interchangeable. Since the definition of the comparative in terms of the superlative is much more awkward, “safer” will be used as a representative of these two interchangeable forms.

With this we have reduced our list to two terms, namely the absolute “safe” and the comparative “safer”. The choice between these two forms will have to depend on whether we are going to treat safety as an absolute or a relative concept. If “safe” is an absolute concept, in the sense that it only comes in two degrees (safe and unsafe), then the relative form “safer” has no use (other than for the trivial observation that X is safer than Y if and only if X is safe and Y is unsafe). If, on the other hand, “safe” is a relative concept, then the term “safer” will be much more useful.

The two words “safe” and “safer” exemplify a relationship that holds for many other such pairs of an absolute and a relative property-concept. (The most philosophically important such pair is “good” and “better”, see Hansson 2001.) We have a scale from the least to the most safe, and the relative concept “safer” can be used to How to define – a tutorial 15 express relative positions on that scale. At some point on the scale, we can insert a limit for the “safe”. Similarly, there is a scale from the destitute to the very rich, with relative positions defined by the term “richer”. Somewhere on that scale the limit for “rich” can be inserted.

It is typical of such scales that the relative concept is more easily defined than the absolute one. The criterion for when one person is richer than another can be determined in a relatively precise and non-arbitrary way, but to determine the level at which a person is “rich” is a more arbitrary undertaking. Similarly, the criteria for when a ski-lift is safer than another ski-lift are relatively easy to decide, but it is much more difficult to decide at what level of fulfilment of these criteria the lift can be said to be “safe” simpliciter.

Another way to express this is that in order to define an absolute property-concept such as “safe” or “rich” we need to determine both the quality and the quantity of the property. In order to define the corresponding relative concept, “safer” respectively “richer”, we only need to determine the quality. Therefore, it is expedient to begin with the relative concept, and define it as precisely as we can before we proceed to deal with the absolute concept. This way of proceeding has the advantage that we can distinguish between those problems in the definition that relate to the qualitative respectively quantitative aspects of the concept.

However, we are still not finished. “Safer” is closely related to two other relative concepts, namely “at least as safe as” and “equally safe as”. It is well-known from the logic of relations that “at least as safe as” is more convenient as a logical primitive than the other two, since they can both be defined in terms of it. Clearly, X is safer than Y if and only if X is at least as safe as Y and Y is not at least as safe as X. Similarly, X is equally safe as Y if and only if X is at least as safe as Y and Y is at least as safe as X. For the sake of simplicity it is therefore preferable to use “at least as safe” instead of “safer” as a primary definiendum for the relative concept. This

amounts to the following preliminary format for the definition:

Sven Ove Hansson

X is at least as safe as Y if and only if... 3

5. Selecting the variables Another reason why the definiendum must be chosen with care is that it is often necessary to add variables to it, that will then reappear in the definiens. Hence, in the definition of “stable”, the variablefree format “stable is…” will be difficult to treat. The essential variable that must be added here is of course the object or entity that is stable. Once the definiendum has been reformulated as “X is stable if and only if…” it will be much easier to start looking for a definiens that does the job.

In most non-philosophical contexts it is advisable to avoid clumsy phrases such as “at a certain point in time and in a certain place”. In philosophical contexts, however, such constructions are often useful, and one should not then avoid them for stylistic reasons. In particular, they are often essential components of a definition, without which it may be impossible to achieve sufficient precision. Furthermore, as already mentioned, it is often convenient to use symbols such as letters to keep track of the variables. This also makes it easier to check that the same variables appear in both the definiens and the definiendum.

–  –  –

It is often far from obvious what variables should be included in a definition. In cases of doubt, it is a good working rule to include rather than exclude a variable. If one finds out later that the variable does no useful work, it can then easily be removed.

For a discussion of how this definiendum can be defined, see Möller et al 2006.

How to define – a tutorial 17 In some cases the identification of the relevant variables in a definition can be philosophically, or even politically, controversial.

Hence, it is a contested issue in moral philosophy if “duty” should

be defined with one or two person variables:

–  –  –

The substantial issue here is whether duties can be impersonal or all duties are owed to some particular person (counterparty), as is presupposed in the second of these definitions.

(Makinson 1986) The choice of variables for “free” is no less controversial.

Here the major alternatives include:

–  –  –

The choice between these alternatives is controversial in political philosophy. The second of these definienda corresponds to the “negative” notion of freedom, according to which freedom consists in the lack of (human-made) obstacles, and the third to the “positive” notion of freedom according to which freedom consists in ability to make and implement one’s own choices. (Berlin 1969) The fourth definiendum is based on Gerald MacCallum’s (1967) attempt at a unified analysis of the notion of freedom. The negative notion of freedom is usually associated with right-oriented and the positive notion with left-oriented politics. MacCallum’s definition has been accused of both a left-wing and a right-wing bias (Gould 1980.

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