4.2 Implicit, Contextual, Ostensive and Axiomatic Definitions in Logic

Implicit definitions

Most striking in the operation of determination, perhaps, is the variety of those specific forms in which it is practically carried out. The task of this operation, as we have already found out, is simple - to reveal the content of the concept. But the ways in which this is achieved are very, very diverse.

First of all, we need to note the difference between explicit and implicit definitions.

The first have the form of equality, coincidence of two concepts. The general scheme of such definitions: "A is (by definition) B". Here A and B are two concepts, and it does not matter in principle, each of them is expressed in one word or a combination of words. For example, the definitions are obvious: “Abracadabra is nonsense”, “Prolegomena is an introduction”, “A molecule is the smallest particle of a substance that preserves all the chemical properties of this substance”.

Implicit definitions do not have the form of equality of two concepts.

Contextual definitions

Every passage of the text, every context in which the concept we are interested in is in some sense an implicit definition of it. The context puts the concept in connection with other concepts and thereby indirectly reveals its content.

Let us assume that it is not quite clear to us what prowess is, and we would like to get its definition. You can refer to the explanatory dictionary and there find a definition (say, this:

The word “courage” is a harsh necessity, the prudence of our actions, more precisely, even counteractions. Courage from the mind, from manhood. A man, having considered and realized that in certain circumstances of life, defending justice, it is necessary to show high stamina, shows this high stamina, courage. Courage is limited by purpose; purpose is dictated by conscience.

Udal, of course, involves the risk of their own lives, courage.

But, having peered at the notion of “boldness”, we feel that this is inferior bravery. It has a self-pumping, intoxication. If there were competitions for courage, then the prowess to these competitions could not be tolerated, for the prowess would have come if there was enough dope.

Dare requires space, the air of space pumps up with artificial courage, intoxicates. Drunk life is a penny. Dare is panic running forward. Udal cuts left and right. Dare - the ability to chop, all the time moving away from the place where you chopped up you already, so as not to think: did I chop correctly?

But still a beautiful word: dare! Quenches longing for thoughtlessness. "

In this passage, the pet is, of course, an explicit definition of delete. And, nevertheless, it is possible to understand well what constitutes prowess and how it relates to courage, courage.

In the “Dictionary of the Russian Language” by S. I. Ozhegov, “hunting” is defined as “searching, tracking animals, birds for the purpose of killing or catching”. This definition sounds dry and detached. It has nothing to do with hot debates about in which extreme cases it is justifiable to kill or imprison animals and birds. In the short poem “Formula of the Hunt”, the poet V. Burich defines the hunt in this way and his attitude to it:

Horizon line

Birds in the numerator

In denominator

Multiplied Shot Shot

And the variable coefficient of fishing rods

Give the product

Available to every mediocrity.

An avid hunter can say that this figurative characteristic of a hunt is subjective and overly emotional. But, nevertheless, it is clearly richer with colors and details related to the mechanism of hunting, than a dry dictionary definition.

In the context of the word is "alive." Taken out of context and placed in a dictionary, it is like an organism placed in a formalin jar and put on display.

Almost all the definitions we encounter in everyday life are contextual definitions.

Having heard a previously unknown word in a conversation, we ns clarify its definition, and try to establish its meaning on the basis of all that has been said. When we encounter one or two unknown words in a foreign language text, we usually don’t rush to turn to the dictionary if you can still understand the text as a whole and get a rough idea of ​​the meaning of the unknown words.

Contextual definitions always remain largely incomplete and unsustainable. It is not clear how extensive the context should be, having become acquainted with that, we will learn the meaning of the word that interests us. It is also not defined what particular concepts may or should be included in this context. It may well be that there are no keywords that are particularly important for the disclosure of the content of a concept in our chosen context.

No dictionary is able to exhaust the full wealth of the meanings of individual words and all shades of these meanings. The word is learned and assimilated not on the basis of dry and approximate dictionary explanations. The use of words in a lively and full-blooded language, in diverse connections with other words - this is the source of complete knowledge of both individual words and the language as a whole. Contextual definitions, however imperfect they may seem, are a fundamental prerequisite for proficiency in a language.

Ostensive definitions

Another interesting kind of implicit definitions is the so-called ostensive definitions, or definitions by show.

We are asked to explain what a giraffe is. We find it difficult to do this, lead the questioner to the zoo, lead him to a cage with a giraffe and show: "This is the giraffe."

Definitions of this type resemble ordinary contextual definitions. But the context here is not a fragment of a text, but a situation in which an object occurs, denoted by the concept of interest to us. In the case of a giraffe it is a zoo, a cage, an animal in a cage, etc.

Ostensive definitions, as well as all contextual definitions, are distinguished by some incompleteness, inconclusiveness.

Definition by means of a show does not distinguish a giraffe from its environment and does not separate what is common to all giraffes from what is characteristic of a given representative. The individual, the individual, is merged in this definition with the general, so that is characteristic of all giraffes.

A person who was first shown a giraffe may well think that a giraffe is always in a cage, that he is always sluggish, that people constantly crowd around him, etc.

Ostensive definitions - and only they - associate words with things. Without them, language is only verbal lace, devoid of objective, substantive content.

It is possible, of course, to determine by showing not all concepts, but only the simplest, most specific ones. You can present the table and say: "This is the table, and all things similar to it are also tables." But it is impossible to show and see the infinite, abstract, concrete, etc. There is no subject, pointing to which, one could declare: "This is what is designated by the word" concrete "." Here it is necessary not a ostensive, but a verbal definition, i.e. purely verbal definition that does not imply display of the object being defined.

Not everything is ostensibly definable. The show is devoid of unambiguity, does not separate the important from the secondary, and sometimes not at all relevant. All this is true. And, nevertheless, without ostensive definitions, there is no language as a means of comprehending the surrounding world. Not every word can be directly associated with things. But it is important that some kind of indirect connection still exists. Words that are completely detached from the visible, audible, tangible, etc. things are powerless and empty.

Axiomatic definitions

An axiomatic definition is a frequent and important for science case of contextual definitions, i.e. definitions of concepts using axioms.

Axioms are statements taken without proof. The set of axioms of a theory is at the same time both a collapsed formulation of this theory and the context that implicitly defines all the concepts included in it.

How do we know, for example, what is a point, a straight line, a plane? From the axioms of Euclidean geometry. They are the limited text in which these concepts are found and with which we establish their meanings.

To find out what the mass, force, acceleration, etc., are, we turn to the axioms of the classical I. Newton peaks. “The force is equal to the mass multiplied by the acceleration”, “The force of the action is equal to the force of the opposition” - these provisions are not, of course, explicit definitions. But they reveal what force is, indicating the connection of this concept with other concepts of mechanics.

The principal difference between axiomatic definitions and all other contextual definitions is that the axiomatic context is strictly limited and fixed. It contains everything that is necessary for understanding the concepts included in it. It is limited in its length, as well as in its composition. It has everything you need and there is nothing superfluous. Axiomatic definitions - one of the highest forms of scientific definitions of concepts. Not every theory is able to define its original concepts axiomatically. This requires a relatively high level of knowledge development about the studied area. The objects studied and their relationships should also be relatively simple.

The point, line and plane Euclid was able to determine with the help of a few axioms two more than a thousand years ago. But how to characterize with the help of several statements such complex, multi-level and multi-aspect objects as society, history or mind? The axiomatic method here would hardly be appropriate. He would only harden and distort the real picture.