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What does a definition encompass?

  1. Sep 6, 2004 #1
    Let there be an axiomatic system, x. If I ask you "what is x?" am I asking you to tell me the nature of x or to tell me what it is exactly? If the question addresses this, then is it not more accurate to ask "what is the nature of x?"
    Is there a difference?

    IOW, what does a definition encompass? what is the nature of a definition?
     
  2. jcsd
  3. Sep 6, 2004 #2
    A definition involves:

    1) IDENTIFYING THINGS (by naming or describing them)

    2) COUNTING THEM (standing in natural clarfying relation, to themselves and one to another). This is given as x

    3) CACULATING THE INTERNAL RELATIONS OF (1) and (2) given as:


    y = x - 1 * x + 1

    or

    y = x * x - 2

    4) MAKE JUDGEMENTS (on what results from (1), (2) and (3))

    Now, my standard claim is that the result of (3) is the number of internal relations of a given identity claim. So, all there is to a definition is counting things and their relations and making judgements about them. This definition, though formally consistent and wholly adequate, is however in practice an absolute nightmare trying to establish (1) to (3). The calculation is the easiest bit, because if you know the number of things involved, given by x, you will definitely be able to calculate the internal relations using x. The problem is finding x, as is typical in most scientific investigations. No one would appreciate the implication of this until the things that you are counting run into infinities, given that such things are humanly observable in the first place.


    NOTE: I have an attachement of a graphical illustration of the above definition called "THE CALCULUS OF IDENTITY RELATIONS", but the file is bigger than the permitted limit.
     
    Last edited: Sep 6, 2004
  4. Sep 7, 2004 #3

    HallsofIvy

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    What do YOU see as the difference between a "thing" and "the nature of that thing"?
     
  5. Sep 7, 2004 #4
    Definitions also include examples of context in which the term is used. Variables such as X only have demonstrable meaning according to their function in a given context. To say X equals Z and Z equals A is meaningless without a context. All three are obviously letters of the alphabet, and I suppose you could infer that this is what the implied meaning is, but that would simply be a guess. Likewise, to say X=Z is to imply that this is some kind of mathematical formulation, but for all I know it is simply someone who perfers not to spell out the word "equals".

    Again, words and concepts such as "the nature of a thing" only have demonstrable meaning according to their function in a given context. If the dictionary did not also contain examples of the use of words in various contexts, then it would remain meaningless nonsense just as X=Z remains meaningless nonsense without a context.
     
  6. Sep 7, 2004 #5
    Whether you are talking about a 'thing' or the 'nature of a thing', you just cannot avoid dealing with both quantitatively. I know this sounds intellectually spooky; a definition is a class of relations and you cannot make any judgement of a kind until these quantitative relations are passively or explicitely presupposed.

    Given that x is a class of defineable properties (physical or abstract), then;

    x = x is a class of (x, x, xx)

    and this is quantitatively equivalent to:

    x = z = (x, z, zx)

    Our natural language already embodies this, yet we fail to acknowledge it by constantly going around in circles. The implication of this definition is that, even self-identicality in a given identity claim in our natural language also relies on the same mathematics. Language cannot function without quantitative accounts of things, properties and their internal relations.

    Question: How, for example, does the langauge handle the notion of 'Identicality' or 'equality', 'similarity', 'absolute difference or distinction'? I answer that it is the same formal definition or mathematics as so given that permits the language speaker to do so.

    However, I do agree with you that the notions are fundamentally vague by the careless way that we use and understand them. Of all these, the most problematic is the notion of 'equality' that you mentioned. My own studies suggest that this term is often used and understood interchageably with the notion of 'similarity'. We talk about the equality of things even when we imply their mere similarity, and vice versa. This is one most problematic source of misunderstandings and conflicts in all the human interactions the world over.
     
    Last edited: Sep 7, 2004
  7. Sep 9, 2004 #6
    How is the word "asleep" a quantitative statement?
     
  8. Sep 10, 2004 #7
    Duh! Because getting to sleep requires counting sheep! :rofl:
     
  9. Sep 11, 2004 #8
    The term 'asleep' passively presupposes its opposite (awake) as a class of relations and explicitely presuposes itself as a class of relations. And the calculus is the same.

    Worst still, the nightmare begins when the shared-property principle suspends a class of being 'asleep' and a class of being 'dead' in the same logical space. Consider the following sentences, for example:

    1) the dog with closed eyes is aleep

    2) the dog with closed eyes is dead

    3)) the dog that is lying down,face down and with eyes closed is asleep.

    4)) the dog that is lying down,face down and with eyes closed is dead.

    5)) the dog that is lying down, breathing, snoring, face down and with eyes closed is asleep.

    6) the dog that is lying down, motionless,face down, decayed, smelly and witheyes closed is dead.


    Logically, you would say:

    a) x(asleep) is x(asleep)
    b) x(asleep) is y(dead) which is equivalent to saying
    x(asleep) is x(asleep) or;
    y(dead) is y(dead)


    c) x(asleep) is similar to y(dead) (pre-loaded with a negative validator)


    Analysis of this kind is already taken care of in standard logic, but the calculation of internal relations of identity claim is a completely different matter. The formula that I provided above demostrates that even when we do not know the number of things involved, the formula remains fundamentally accurate.

    Hence, with (c) you could make a negatively valid statement as:

    'The dog with closed eyes is asleep and dead'

    Which is eqauivalent to combining sentences (1) and (2) into one sentence. You know as well I do that no one would make a sentence like this in the real world. Yet the shared property principle renders a sentence of this kind negatively valid.

    Question: I habitually call things of this kind 'PHANTOM STRUCTURES'. So, why are things structured like this in the world? Why must a definition be tied to logical and quantitative analysis of identity relations or classes, with pre-loaded 'NEGATIVE VALIDATORS'?
     
    Last edited: Sep 11, 2004
  10. Sep 12, 2004 #9
    You would not appreciate the implications of the above analysis until you face real life situations in the real world. Take, for example, the notion of medical diagnoses. Many doctors take for granted the fact that when they are making diagnoses, they are passively and explicitly analysing and distinguishing classes of relations. When a doctor diagnoses his or her patient for a given disease or illness, he/she takes for granted the fact that he or she is explicitly counting and making a list of symptoms and passively comparing to countless diseases with similar symptons.

    The fact is that doctors do not make diagnoses nominally - that is, by ordinary calling of names of diseases - and the very fact that countless diseases share the same symptoms, makes it even the more reason why the medical doctor concerned needs to be more quantitative, detailed and logical.
     
    Last edited: Sep 12, 2004
  11. Sep 12, 2004 #10
    You bet....it's about counting sheep - ah-ah-ah!
     
  12. Sep 13, 2004 #11
    Sorry for the late reply. I'm only aloud online on weekends since school started.
    First off, I would like to thank you for this wonderful description of language. I really appreciate it.

    As to your question, it I can only answer it indirectly:
    Everything exists relative to its constituents. A "ball" is identified as a "ball" to make a linguistic distinction from another object like a planet. In other words, every word exists to make a relative distinction with another object. I should make it evident that words are products of human innovation, and describe a metaphysical or physical element of the universe. If all words have antonyms, then I believe we may conclude that all physically occuring element must have an inversely proportional partner. This quality is evident when considering supersymmetric particles.
    Further, we may swiftly conclude that words are pre-loaded with negative validators because of the natural need for them.

    I must ask, why do you choose the word "pre-loaded" for specifying the nature of negative validators as elements of a definition? (i.e., considering my aforementioned conclusion)

    BTW, are you a logician, philocrat?
     
  13. Sep 13, 2004 #12
    Can't you post it in sections?
     
  14. Sep 13, 2004 #13
    It's a gif file.....it's just too big. I will compose a text version soon, using MS word. Please be patient. Thanks.
     
  15. Sep 13, 2004 #14

    Well, my own study of singular terms (be they names or descriptions) manifested into the discovery of so many mysterious structures in our natural language and the very world that it describes. Singular terms can be used conveniently as we have always done, but they are impossible of use without quantitative and logical presuppositions, passively or explicitly so, or both. In fact no statement of fact can be made in any discipline without this process. From the above systematic analysis, it looks to me as if every statement of fact is an abbreviation of a deductive argument with full quantitative contents, and that any errors that may arise in such a statement is due to errors in the quantification of all the passively presupposed internal relations within the argument.The equation that I provided above is just to tell the langugae user that whether you know the number of things passively or explicitly presupposed in your statement or not this is how to calculate and take account of all the internal relations involved.This may sound spooky, but that's the way the cookies crumble!


    I am not a logician ...... I just make fun of it!
     
    Last edited: Sep 13, 2004
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