Definition and meaning

DEFINITION AND MEANING

ANIL MITRA, © APRIL 2014—August 2015

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DEFINITION AND MEANING

Short version

Definition

A definition specifies a concept.

Once specified, it is necessary to establish existence of the object. Generally this is not trivial; there are significant cases—especially in science and mathematics—where existence, if reasonable, is assumed.

Meaning

This concept of definition is based in the concept-object concept of meaning—which is critical to object existence, precision, and consistency (and prevention-elimination of paradox). I.e. this concept of meaning is essential to unpacking and eliminating emptiness, imprecision, and paradox that often stem from assuming that apparently well formed concepts have objects. This is further explained in a discussion labeled Meaning(meaning.html, meaning and method.html).

What is being said is, effectively, that the world itself and in itself has no paradox or error; these arise in knowing the world. The freedom of concept formation, which is essential to the ability and opportunity to know and choose, comes with the price and challenge of error—of inconsistency with fact (» science) and inconsistency among concepts (» logic). Therefore, meaning as defined here is crucial to understanding and using hypothesis and eliminating error.

Note that fact and logic can be brought under one umbrella—fact is consistency among percepts while logic is consistency among concepts; but percepts are kinds of concepts; and science and validity of argument can be seen as incorporating ‘cross consistency’ among percept-concepts and concept-concepts.

Science and logic can be brought under one umbrella. Here it is necessary to remark that science as understood here is a departure from the understanding of science as tentative universal theory. It is expected, of course, that a good scientific theory will outstrip the observations and other ground (previous theories, conflicts from new observations) and have validity beyond current borders of observation. However, it is also expected that full universality will not obtain—for how could a theory of the known project into the possibly limitless unknown? Therefore we restrict the scope of a theory to a know region—with the understanding that the region of the known will expand via mutual expansion via observation and theory. Now, with this understanding, both science and logic involve percept-concept consistency; they are different sides of understanding one concerning particular truth the other universal; and the creation of both is inductive while inference under each is deductive.

Narrative meaning

It is critical to intuitive and analytic understanding that (1) the defined meanings, even of common terms, are always specific and often new and (2) the net meaning of the system of the narrative is greater than and not even implicit in the collection of individual meanings. That is, the net meaning depends on interactive meaning and interpretation.

Examples

Examples are chosen to illustrate the significance of meaning for clear understanding and elimination of paradox.

Long version

Definition

A definition is a specification of a concept.

The statement above defines definition. This form is the form of most definitions in this work; and most statements in this form will be definitions. In a definition the term ‘is’ placed just after the defined term will stand for ‘is defined as’.

It is implicit in definitions that an object corresponding to the concept is intended. From a definition, it does not follow that there is an object and it will therefore be necessary to establish existence of the object. Establishing the object will range from transparent to trivial to requiring care to difficult; there are significant cases—especially in science and mathematics—where existence, if reasonable, is assumed. It is important that when a definition seems plain and showing the existence of the object seems trivial it does not follow that the definition and existence are without significance. Many definitions of the narrative are transparent yet deep in meaning and consequence. What does require care is the selection of terms, transparent or otherwise, that enable building a structure that has significance.

How shall we specify existence of the object? Suppose we have defined the term consciousness. Some thinkers hold that the existence of consciousness is obvious, significant, and causal; however, others hold doubt its existence, hold its nature to be mysterious, or hold it to be non-causal and insignificant like the ‘foam on a wave’; and yet others deny that there is such a thing as phenomenal consciousness. Thus the task of delivering an acceptable demonstration that there is consciousness is not entirely trivial (it seems likely that a non trivial aspect of the demonstration should be a careful specification of what consciousness is and in fielding the objections of those who think that there is no consciousness). In any case, once existence is shown—at least to our satisfaction regarding reason and empirical evidence—we will say something like ‘There is consciousness.’

Here ‘is’ is used to indicate existence. It is important to distinguish the uses of the verb to be ‘is’.

The example of consciousness is pertinent to transparency, existence of the object and care in definition. It is precisely the transparency of consciousness that (is one thing that) makes it obvious to some but doubted or refused by others. Those who doubt consciousness for empirical reasons may do so for reasons similar to the reason that atmospheric pressure is not apparent though omnipresent—which was the occasion for the experimental demonstration of it by Otto von Guerick (1602-1686). In fact ‘experience’ as awareness is so fundamental to our being and so basic that it is hardly definable in other terms; this, too, may contribute to the doubt. However, in any system of thought there must be some things that are not defined in other terms. Such things may be defined by pointing them out—i.e., ostensively. Matter-energy is one such entity but in a materialist paradigm, conceptual definition is unnecessary and matter-energy is assumed given. This is a further source of doubt—in a strict materialist paradigm, there may seem to be no place for consciousness or mind. However the real problem as will be seen later lies in the careful definition or understanding of matter: however we see it or define it our positive definition may be good as far as it goes but it is a further assumption, if an implicit one, that matter excludes mind; what we will see later is that if matter is first order existence then mind is second order existence (though not secondary) in the sense of mind as matter-in-relation-to-other-matter. And this gives rise to another point regarding definition and the ground of systems of thought. Since the point to ostension and axiom is to avoid infinite regress the systems often begin with undefined terms and undemonstrated postulates and axioms. This consideration gives rise to the thought that no system has any final foundation. This in term gives rise to the idea of substance as final foundation or the idea of relative foundation in infinite regress. However, these are not the only alternatives. In this narrative we will find an absolute foundation in being but this will be only in some directions of thought; in other directions we will find absolute openness. This is interesting in two ways—firs that we have found such foundation and second that we have found and now know that there are directions in which there can be and in which we should not seek but may rejoice in absent foundation.

It will be useful to say something about the nature of the objects of definitions. In concrete cases the object is in the real world. In abstract cases the nature of the object depends on what ‘theory’ of abstract objects is employed. In formalism the formal definition constitutes the object; in intuitionism the object is intuited and the definition points to the intuited object; in Platonic realism, the object exists in an abstract world; when, as is the case in absence of an overarching view of the world, the notion of abstract objects is not clear the abstract ‘object’ may exhibit more than one of these types; in this narrative we will establish a deep realism—one which may appear naïve if encountered without its foundation—that there is but one world and all objects lie in it (that is the nature of objects is realist and this will allow aspects of formalist, and intuitionist interpretations but not of Platonic realism, but will entail nothing beyond realism, for an implication of the one world conclusion will be that both concrete and abstract objects lie in it and that there is no other Platonic world); and while concepts, therefore, must be objects in themselves, and they may perhaps due to difficulty in locating the object stand in for the object, in no case is the object some spurious ‘mental object’ over and above the concept itself.

Meaning

There is a concept of meaning due to Frege that neither concepts nor objects constitute meaning. Rather it is a concept and its object that constitutes the meaning of the word or concept.

This is essential to unpacking and eliminating emptiness, imprecision, and paradox that often stem from assumption that apparently well formed concepts, iconic or symbolic or mixed, have objects. The separation of concept and object, though inefficient in everyday thought and speech, is important to precision and consistency of understanding. This is further explained in a discussion labeled Meaning (meaning.html, meaning and method.html).

Narrative meaning

It is important to intuitive and analytical understanding that (1) the defined meanings, even of common terms, are always specific and often new and (2) the net meaning of the system of the narrative is greater than and not even implicit in the collection of individual meanings. That is, the net meaning depends on interactive meaning and interpretation.

Examples

I would like to illustrate the use of meaning (1) to clarify the concept and its general significance and (2) to make some points of further particular significance in the scheme of this narrative. A trivial example concerns the problem of the non existent object: if Sherlock Holmes does not exist, what or who is it that does not exist? The resolution is trivial: ‘Sherlock Holmes’ stands for concept (a man who lives at 221 Baker Street, who has a friend named Dr. Watson etc.) and ‘Sherlock Holmes does not exist’ means that there is no object (person) that meets the description supplied in the concept. This resolution is trivial compared to some resolutions in the literature. The first example that now follows illustrates the general point that the naïve assumption that apparently well founded concepts (here linguistic) have objects is mistaken and may lead to paradox.

In everyday speech it is efficient to think that our utterances have meaning. When we have doubts about meaning we refer back to grammar. After all, grammar is supposed to have as one of its functions that it guarantees that our sentences are well formed with regard to meaning. Early in the development of modern logic it was naïvely thought that this was universally true. The discovery of the famous semantic and logical paradoxes therefore came as a shock. However, the attempt to understand the sources of paradox and, then, to eliminate or guard against the paradoxes led to understanding that apparently well formed concepts do not necessarily have—even possible—reference and so to significant progress in the use of language to realistic and logical ends.

Consider the well known liar paradox ‘This assertion is false’ which is, apparently, true if and only if it is false.

In common use an assertion is an assertion of truth. ‘The sun is shining’ is true if and only if the sun is shining; since this assertion has a definite object in the world its truth can be assessed. What is the object of the apparently paradoxical liar assertion? It is not merely the assertion but the falsity of the assertion.

If we assume that every apparently well formed assertion (or sentence or concept) has an object, the paradox results. But when we try to verify that there is an object of the liar sentence the verification fails. In looking for the intended object of the sun sentence we find there is one. However, we fail to find an intended object to the paradoxical assertion: the truth value of the sentence. Clearly, however, the liar sentence refers to itself and, so, how does it fail to refer? It says of itself that it is false and thus it refers not only to the bare form of the sentence but it also refers to its truth value—it makes an implicit assumption that there is a truth value and it refers to that truth value. If our first thought about the sentence is that it refers to its bare form then we are forced to the conclusion that it is paradoxical and this is the source of the apparent paradox.

Consider ‘This assertion is true’ which is true if and only if it is true. It seems to have meaning but since it is really referring to its truth value it does not follow from its form that it does have a truth value—i.e., it does not follow that it has an object and, indeed, inspection shows that it does not. In both cases (assertion of truth and of falsity) there is no object, i.e. no meaning, and in one case (falsity) assuming that there is an object—that there is meaning—leads to paradox. In fact the other case also has an associated paradox: since inspection shows that there is no object, the assumption that there is an object is already paradoxical. To clearly see the assumption that there is an object in both cases we can write the two assertions as ‘This assertion has a truth value and that value is falsity.’ and ‘This assertion has a truth value and that value is truthfulness.’

Further clarification is possible. In the sun sentence we can look out in the world and see that the sentence has an object. But if there were no suns it would then be true that ‘All suns in the universe are shining.’ and ‘No suns in the universe are shining.’ and this would be apparently paradoxical (which is avoided in logic by stipulating that our ‘universes’ of discourse shall not be empty). Now not all self referring sentences are paradoxical. If we define shortness of sentences as having four words or less then ‘This sentence is short.’ ‘The sentence you are reading is short.’ are both meaningful but the first is true and the second false. It is clear, then, that it is assuming that there is a well defined object when there is not one is the (or a) source of paradox.

Note that although a source—perhaps the essential source—of vagueness and paradox in concept and language use has been uncovered an approach to resolution has not been given. There are in the literature various approaches (Russell’s theory of types, ZFC— Zermelo–Fraenkel set theory with the axiom of choice, Quine’s ‘new foundations’ and others). The principle is to exclude concepts or language that allows too much without excessive restriction and two approaches are restriction and construction. In this narrative will I give a brief constructive example in relation to the concept of ‘all possibilities’.

I chose ‘living fully in the present’ as second example. This will illustrate the significance of meaning for (a) understanding in general and (2) the aim of the narrative.

First enquire of the meaning of ‘living in the present’. Unpack it, first, as a concept ‘living well in the present’. Does the idea entail no thought of the future whatsoever? Perhaps what is desirable is some balance of living in the present and living for the future.

Having arrived via at the present via natural and intelligent adaptation over the past society, culture, and individuals encode not just a moment in time but process. Therefore living in the present and living for the future are not distinct. Understood properly they are in some appropriate balance. What is this ‘appropriateness’?

In a matter for which the future is uncertain too much anticipation may be defeating and so we may choose the attitude to live one day at a time. But living one day at a time in a material sense is, in this case, living spiritually without regard to time.

In general future is not so uncertain: we devote time and intelligence to conceiving, planning, and preparing for it. Even here, though, the best effort emphasizes enjoyment of the task with detachment appropriate to uncertainty of outcome. Living well is an adapting mix of accepting transience and anticipation—this is practical.

This mix also has an ideal side. Materialists may think that their future is their legacy; others may think their identity will merge-or is already merged—with universal Identity and survive the death of body, world, and cosmos. In neither case—the material or the apparently non-material—is living well in transience distinct from living well in permanence.