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CONCEPTS & CATEGORIES

a work in progress by Cary Cook

The primary cause of miscommunication in philosophical discussion is not bad logic, but failure to properly identify, define, and catrgorize concepts.  This essay is an effort to diminish this problem.

e.g.  In answering the question, "Is the USA a Christian nation?", people are likely to spend much time quoting founding fathers and never arrive at a conclusive answer.  In fact the question is unanswerable until "Christian nation" is defined sufficiently enough to see if the USA is in or out of the category.
Sufficient definition not only makes the questions answerable, but makes the answer obvious.

A concept is any unit of thought corresponding to a real or imaginary thing.  For any ontologically existing thing recognized by a mind, there exists a concept of that thing in that mind, and a one to one correspondence between them.  The concept in the mind represents the thing outside of the mind.  Some things exist only in minds (e.g. thought, will, emotion), but the concepts of those things are separate from the things themselves.
e.g. The concept, "emotion", is separate from the experience of any particular emotion.

Aristotle failed to recognize the null set as a category and was yet able to formulate a coherent system of logic.  His system is no less correct than Boolean logic, which recognizes the null set.  But Boolean logic is more useful, because Boolean logic can do anything that Aristotelian logic can do, but not the reverse.  Computers can't operate on Aristotelian logic.

A category (or set) is a mental container in which concepts may exist together and be seen as a single unit, and hence a single concept.  Once recognized by a mind, a category becomes a concept in that mind.  A category may contain anywhere from zero concepts (the null set) to an infinite number of concepts (any infinite set).

(Man) is a concept;
(Some men) is a category.
(All men) is a category.
(No men) is a category.

Note that there is only one absolutely null set.  The set (no men) is a null set relative to men, but not necessarily an absolutely null set, because it says nothing about inclusion or exclusion of anything other than men.

The null set does not exist as an ontological entity, but does exist as an epistemological entity.  Likewise zero does not exist as an ontological quantity, but does exist as an epistemological quantity.
Spock was wrong.  An epistemological difference that makes no ontological difference remains an epistemological difference, and is likely to make an ontological difference in the future.  What something is and what it makes are different categories.

Given any sufficiently defined concept and any sufficiently defined category, the concept is either inside or outside of the category.  If the concept is inside of the category in one sense, and outside of the category in another sense, then these two senses represent two different ontological facts, and are best described in two separate statements.

e.g.  The null set is a set in the Boolean sense.  The null set is not a set in the Aristotelian sense.

If some of the concept is in the category and some of the concept is out of the category, then either the concept or the category is not defined sufficiently to determine their relationship.

e.g.  Joe may be both inside and outside of the category of all things in Arizona if he is standing with one foot in New Mexico.  But if Joe is seen as indivisible, and the category of all things in Arizona is sufficiently defined, then Joe is either inside or outside of Arizina.  i.e. If all things in Arizona includes all things partly in Arizona, then Joe is in Arizona.  If all things in Arizona excludes all things not totally in Arizona, then Joe is not in Arizona.

If necessary, a category may be further defined to include or exclude certain percentages of concepts.  But though there will usually be borderline cases, there can also be formulated a definition specific enough that any concept is (and can be known to be) either in or out of any category.  Such is a sufficient definition.

i.e.  A sufficient definition of a given concept relative to a given category is a definition which makes it clear whether that concept is in or out of that category.  A sufficient definition of a given category relative to a given concept does the same.  In determining whether a given concept is in a given category, sufficient definitions of the two need to include all of, and only, those requirements necessary to make the determination.
All definitions are made under the assumption that the terms of which they are composed are understood.  Epistemology begins with assumption.  There is no first concept which is understood prior to definition, but there is necessarily a set of such properly basic concepts out of which all other concepts are formed.  It is not currently known whether properly basic concepts form in a pre-natal mind prior to experience (inate ideas), or if they are learned from repeated recognition.  Nor does it need to be known in order to formulate a coherent and operationally sufficient epistemology.  And operational sufficiency is all that can be required of an epistemology.  We all want, but none of us needs, an objectively justified epistemology.  Every decision making organism, however, needs an operationally sufficient epistemology.  I have never seen an objectively justified epistemology.  Yet if I assert that none is possible, I have self-stultified.

A sufficient definition is not necessarily a complete definition.  A complete definition of a concept includes all requirements necessary to distinguish that concept from everything that is not that concept.

e.g.  A complete definition of a triangle would be a closed figure made of three straight lines.  It is not necessary to say it must have three angles, or any angles, because it will necessarily have three angles.  Another complete definition of a triangle would be a closed figure made with the minimum number of of straight lines, or with the minimum number of angles.  These definitions assume that the definition of a straight line is already known.  You could also define a triangle as a closed figure connecting three points by the shortest possible distances.  In this last case, you must say it is a closed figure, or a Y would fit the definition even better than a triangle.

The sufficiency of a definition is relative to the context in which it is used, and will vary accordingly.

e.g.  Consider equality.  A pie chart can be theoretically divided into any number of precisely equal sections, but such precision is impossible on an actual pie.  No matter how sophisticated the technology, any division of a physical object into parts will result in some margin of error in weight and volume.  Therefore, though equal sections of a pie chart would imply zero margin of error, equal pieces of an actual pie would usually be sufficiently defined by their appearance to a casual observer.

Most of the concepts with which we must deal cannot be completely defined objectively because no objective criteria for determining the boundaries of such concepts is known.

e.g.  beard.
How many hairs are necessary?
How long do the hairs have to be?
What are the boundaries of where the hairs have to grow from?
Does it matter how close together the hairs are?
Are there other criteria to consider?

There will always be borderline beards unless the concept is defined with arbitrary or subjective boundaries and zero tolerance of error.  A complete definition of a beard is possible, but it will necessarily be arbitrary or subjective.  Possibly some concepts have objective boundaries, but they are not known, or cannot be known.

e.g.  living organism
Is self replication a sufficient definitive criterion?
If so, crystals are living organisms.
If not, what definitive criterion is sufficient ?

Can you make a list of criteria which collectively make a complete definition?  If you cannot formulate a definition without borderline cases, then you cannot rightly assert that a complete definition exists.  But unless you can show that borderline cases will always exist, you cannot rightly assert that a complete definition does not exist.

Note that a concept may lack objective boundary criteria for two reasons: either the criteria are ambiguous or vague or both.  Back to the beard example: The criteria are ambiguous because we don't know if we must examine the number, the length, or the location of the hairs, or all three, or those criteria plus some more.  The criteria are vague because even if we decide on which criteria are relevant, we then have to decide on a particular quantity or measurement of each criterion.

A complete definition of a category includes all requirements necessary to determine if any concept is in or out of that category.  Most of the categories with which we deal cannot be completely defined objectively, because there is no knowable objective criteria for determining the boundaries of such categories.

e.g.  Are viruses and nanodes in the category of living organisms?

A sufficient definition of a category includes all requirements necessary to determine if a given concept in or out of that category.  Note that the sufficiency of a category definition is relative to the particular concept to be included in the category or excluded from it.  A sufficient category definition does not need to include all requirements necessary to distinguish everything in that category from everything not in that category.

If a concept cannot be clearly understood, as being inside or outside of a particular category, then either:
1.  one of the two is insufficiently defined, or
2.  one of the two is illegitimate, or
3.  one of the terms is being used in an illegitimate sense.

An illegitimate concept or category is something that appears to be a concept or category, but in fact is not.  Illegitimate concepts or categories can be (but need not be) divided into two kinds: paradoxical and meaningless.

A paradoxical concept violates a law of logic.
A meaningless concept lacks what is necessary to be grasped by a mind.
Can you think of an example of a concept that is paradoxical but not meaninglessness, or meaninglessness but not paradoxical?
Try these:  square circle, infinite number, a member of a null set

If you cannot think of a concept that is paradoxical but not meaninglessness, or meaninglessness but not paradoxical, can you conclude that no such concept exists?  No.  Inability to think of an example of X does not necessarily imply that no examples of X exist.  Only if examples of X are proven logically impossible can one conclude that no examples of X exist.  But if examples of X are not proven logically impossible, and you contend that no examples of X exist, could you be proven wrong?  Only by someone who could produce an example of X.

Some of you may be thinking, "if an illegitimate concept is not really a concept, why call it a concept?"  The answer is not that it is a paradox.  The answer is that such things are referred to so infrequently that there is no point in making up a new term for them.  So for the sake of convenience alone, we will call an illegitimate concept an illegitimate concept and not an illegitimate non-concept or an illegitimate whatever.  There is such a thing as a level of precision that is not worth the trouble.

That which is true of illegitimate concepts is also true of illegitimate categories.

Examples of illegitimate categories:
paradoxical category: the set of all sets
meaningless category: all or some members of the null set
or a relatively null set.

A concept may be purely imaginary, yet still be a legitimate concept.  Unicorn is a legitimate concept, because it can be thought in a mind and, by description, be communicated to another mind.  A concept may be expressed in words, but words do not create concepts.  e.g.  One can say "square circle" as if it stood for an imaginary concept, but it doesn't.  Though "square circle" can be said, such a concept cannot even be imagined, much less communicated.  One may assert a "possible world" in which the laws of logic don't apply.  But such a world cannot be shown to be possible.  Is it therefore legitimate to call it a possible world?

Possibility may be defined such that:
1.  only what is shown to be possible is possible
2.  anything not shown to be impossible is possible

Kinds of concepts: There are two kinds of concepts: real and imaginary. And there are two kinds of concepts: divisible and indivisible. And there are two kinds of concepts: basic and derived, absolute and relative, subjective and objective, abstract and extended, changing and unchanging, eternal and temporal, definite and vague.

How many kinds of two kinds of concepts are there?  That would depend on how many things you choose to conceptualize.  If you want to count specifics like cabbages and non-cabbages, then there are an infinite number of two kinds of concepts - but are there only two kinds?  This is as far as Iíve been able to take this line of thinking without feedback, and therefore where I must end this essay.