[Inquiry] Re: Extension x Comprehension = Information

[Jon Awbrey]

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ECI.  Note 19

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| 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.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce:  A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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