[Inquiry] Re: Extension x Comprehension = Information

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

ECI.  Note 38

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

| In every induction we have given some
| remarkable fact or piece of information:
|
| S is B,
|
| where B is an object of connotation.
| We infer that something else:
|
| T is B.
|
| Let us suppose that T contains more 'information'
| than S.  Then, if T is no more extensive than S,
| "T is B" is a better judgment than "S is B"
| because it contains more information without
| predicating B of anything doubtful.
|
| Thus, it is better to say "All men are mortal"
| than "all rational animals are mortal" for the
| former implies the latter and contains no more
| possibility of error and is more 'distinct'.
|
| But in every case of induction T is also more extensive
| than S.  Then in case S is a true symbol and "S is B" is
| a single true judgment, this judgment or proposition must
| be the result of induction, as we saw in the last lecture
| that all propositions are.  The question is, therefore,
| which is the preferable theory, "S is B" or "T is B".
| The greater information of T causes the latter theory
| to contain more truth but its greater extension renders
| it liable to more error.  If in T the extension of S is
| increased more than the information is, the connotation
| will be diminished and 'vice versa'.  Accordingly the
| greater the connotation of T relatively to that of S,
| the better is the theory proposed, "T is B".
|
| Which of the two theories to select in any case will depend upon
| the motives which influence us.  In a desperate practical case,
| if one's life depends upon taking the right one, he ought to
| select the one whose subject has the greatest connotation.
| In a cool speculation where safety is the essential;
| the least extensive should be taken.
|
| So much for the preference between two theories.  But in proceeding from
| fact to theory -- in such a case as that about 'neat', 'swine', 'sheep',
| and 'deer' -- S is a mere enumerative term and has no connotation at all.
| In this case therefore T increases the connotation of S absolutely and
| "T is B" ought therefore to be absolutely preferred to "S is B" and be
| accepted assertorically;  as long as there is no question between this
| theory and some other and as long as it is not opposed by some other
| induction.
|
| CSP, CE 1, page 285.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce:  A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

Nota Bene.  For the sake of readability in this transcription,
I supply quotation marks around formulas and change a couple of
Greek letters to Roman characters, using T for Sigma and Q for Pi.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o