ONT Re: Extension x Comprehension = Information
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Note 84
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Passage 1
| We come next to consider inductions. In inferences of this kind
| we proceed as if upon the principle that as is a sample of a class
| so is the whole class. The word 'class' in this connection means
| nothing more than what is denoted by one term, -- or in other words
| the sphere of a term. Whatever characters belong to the whole sphere
| of a term constitute the content of that term. Hence the principle of
| induction is that whatever can be predicated of a specimen of the sphere
| of a term is part of the content of that term. And what is a specimen?
| It is something taken from a class or the sphere of a term, at random --
| that is, not upon any further principle, not selected from a part of
| that sphere; in other words it is something taken from the sphere
| of a term and not taken as belonging to a narrower sphere. Hence
| the principle of induction is that whatever can be predicated of
| something taken as belonging to the sphere of a term is part of
| the content of that term. But this principle is not axiomatic
| by any means. Why then do we adopt it?
|
| CSP, CE 1, pages 462-463.
Passage 2
| To explain this, we must remember that the process of induction is a
| process of adding to our knowledge; it differs therein from deduction --
| which merely explicates what we know -- and is on this very account called
| scientific inference. Now deduction rests as we have seen upon the inverse
| proportionality of the extension and comprehension of every term; and this
| principle makes it impossible apparently to proceed in the direction of
| ascent to universals. But a little reflection will show that when our
| knowledge receives an addition this principle does not hold.
|
| Thus suppose a blind man to be told that no red things are
| blue. He has previously known only that red is a color;
| and that certain things 'A', 'B', and 'C' are red.
|
| The comprehension of red then has been for him 'color'.
| Its extension has been 'A', 'B', 'C'.
|
| But when he learns that no red thing is blue, 'non-blue'
| is added to the comprehension of red, without the least
| diminution of its extension.
|
| Its comprehension becomes 'non-blue color'.
| Its extension remains 'A', 'B', 'C'.
|
| Suppose afterwards he learns that a fourth thing 'D' is red.
| Then, the comprehension of 'red' remains unchanged, 'non-blue color';
| while its extension becomes 'A', 'B', 'C', and 'D'. Thus, the rule
| that the greater the extension of a term the less its comprehension
| and 'vice versa', holds good only so long as our knowledge is not
| added to; but as soon as our knowledge is increased, either the
| comprehension or extension of that term which the new information
| concerns is increased without a corresponding decrease of the other
| quantity.
|
| The reason why this takes place is worthy of notice. Every addition to
| the information which is incased in a term, results in making some term
| equivalent to that term. Thus when the blind man learns that 'red' is
| not-blue, 'red not-blue' becomes for him equivalent to 'red'. Before
| that, he might have thought that 'red not-blue' was a little more
| restricted term than 'red', and therefore it was so to him, but
| the new information makes it the exact equivalent of red.
| In the same way, when he learns that 'D' is red, the
| term 'D-like red' becomes equivalent to 'red'.
|
| Thus, every addition to our information about a term is an addition
| to the number of equivalents which that term has. Now, in whatever
| way a term gets to have a new equivalent, whether by an increase in
| our knowledge, or by a change in the things it denotes, this always
| results in an increase either of extension or comprehension without
| a corresponding decrease in the other quantity.
|
| For example we have here a number of circles
| dotted and undotted, crossed and uncrossed:
|
| (·X·) (···) (·X·) (···) ( X ) ( ) ( X ) ( )
|
| Here it is evident that the greater the extension the
| less the comprehension:
|
| o-------------------o-------------------o
| | | |
| | dotted | 4 circles |
| | | |
| o-------------------o-------------------o
| | | |
| | dotted & crossed | 2 circles |
| | | |
| o-------------------o-------------------o
|
| Now suppose we make these two terms 'dotted circle'
| and 'crossed and dotted circle' equivalent. This we can
| do by crossing our uncrossed dotted circles. In that way,
| we increase the comprehension of 'dotted circle' and at the
| same time increase the extension of 'crossed and dotted circle'
| since we now make it denote 'all dotted circles'.
|
| CSP, CE 1, pages 463-464.
Passage 3
| Thus every increase in the number of equivalents of any term increases either
| its extension or comprehension and 'conversely'. It may be said that there
| are no equivalent terms in logic, since the only difference between such
| terms would be merely external and grammatical, while in logic terms
| which have the same meaning are identical. I fully admit that.
| Indeed, the process of getting an equivalent for a term is
| an identification of two terms previously diverse. It is,
| in fact, the process of nutrition of terms by which they
| get all their life and vigor and by which they put forth
| an energy almost creative -- since it has the effect of
| reducing the chaos of ignorance to the cosmos of science.
| Each of these equivalents is the explication of what there is
| wrapt up in the primary -- they are the surrogates, the interpreters
| of the original term. They are new bodies, animated by that same soul.
| I call them the 'interpretants' of the term. And the quantity of these
| 'interpretants', I term the 'information' or 'implication' of the term.
|
| CSP, CE 1, pages 464-465.
Passage 4
| We must therefore modify the law of
| the inverse proportionality of
| extension and comprehension
| and instead of writing
|
| Extension x Comprehension = Constant
|
| which crudely expresses the fact
| that the greater the extension the
| less the comprehension, we must write
|
| Extension x Comprehension = Information
|
| which means that when the information
| is increased there is an increase of
| either extension or comprehension
| without any diminution of the
| other of these quantities.
|
| Now, ladies and gentlemen, as it is true that
| every increase of our knowledge is an increase
| in the information of a term -- that is, is an
| addition to the number of terms equivalent to
| that term -- so it is also true that the first
| step in the knowledge of a thing, the first
| framing of a term, is also the origin of the
| information of that term because it gives the
| first term equivalent to that term. I here
| announce the great and fundamental secret
| of the logic of science. There is no term,
| properly so called, which is entirely destitute
| of information, of equivalent terms. The moment
| an expression acquires sufficient comprehension
| to determine its extension, it already has more
| than enough to do so.
|
| CSP, CE 1, page 465.
Passage 5
| We are all, then, sufficiently familiar with the fact
| that many words have much implication; but I think we
| need to reflect upon the circumstance that every word
| implies some proposition or, what is the same thing,
| every word, concept, symbol has an equivalent term --
| or one which has become identified with it, --
| in short, has an 'interpretant'.
|
| Consider, what a word or symbol is; it is a sort of representation.
| Now a representation is something which stands for something. I will
| not undertake to analyze, this evening, this conception of 'standing
| for' something -- but, it is sufficiently plain that it involves the
| standing 'to' something 'for' something. A thing cannot stand for
| something without standing 'to' something 'for' that something.
| Now, what is this that a word stands 'to'? Is it a person? We
| usually say that the word 'homme' stands to a Frenchman for 'man'.
| It would be a little more precise to say that it stands 'to' the
| Frenchman's mind -- to his memory. It is still more accurate
| to say that it addresses a particular remembrance or image in
| that memory. And what 'image', what remembrance? Plainly,
| the one which is the mental equivalent of the word 'homme' --
| in short, its interpretant. Whatever a word addresses then
| or 'stands to', is its interpretant or identified symbol.
| Conversely, every interpretant is addressed by the word;
| for were it not so, did it not as it were overhear what
| the words says, how could it interpret what it says.
|
| There are doubtless some who cannot understand this metaphorical argument.
| I wish to show that the relation of a word to that which it addresses is
| the same as its relation to its equivalent or identified terms. For that
| purpose, I first show that whatever a word addresses is an equivalent term, --
| its mental equivalent. I next show that, since the intelligent reception
| of a term is the being addressed by that term, and since the explication
| of a term's implication is the intelligent reception of that term, that
| the interpretant or equivalent of a term which as we have already seen
| explicates the implication of a term is addressed by the term.
|
| The interpretant of a term, then, and that which it stands to are identical.
| Hence, since it is of the very essence of a symbol that it should stand 'to'
| something, every symbol -- every word and every 'conception' -- must have an
| interpretant -- or what is the same thing, must have information or implication.
|
| Let us now return to the information.
|
| CSP, CE 1, pages 466-467.
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