Re: Category Theory • Henry Story

HS:

Could one not say that Frege also had a three part relation?  I guess:  for singular terms their Sense and Reference. […] His argument could be explained very simply.  Imagine you start with a theory of language where words only have referents.  Then since in point of fact Hesperus = Phosphorus, The Morning Star = The Evening Star, the simple theory of meaning would not allow one to explain how the discovery that they both were the planet Venus, came to be such a big event.  So sense cannot be reduced to reference.  Equalities can have informational content.

Peirce’s take on semiotics is often compared with Frege’s parsing of Sinn und Bedeutung.  There’s a long tradition concerned with the extension and intension of concepts and terms, also denotation and connotation, though the latter tends to be somewhat fuzzier from one commentator to the next.  The following paper by Peirce gives one of his characteristically thoroughgoing historical and technical surveys of the question.

• C.S. Peirce (1867) • Upon Logical Comprehension and Extension

The duality, inverse proportion, or reciprocal relation between extension and intension is the generic form of the more specialized galois correspondences we find in mathematics.  Peirce preferred the more exact term comprehension for a compound of many intensions.  In his Lectures on the Logic of Science (Harvard 1865, Lowell Institute 1866) he proposed his newfangled concept of information to integrate the dual aspects of comprehension and extension, saying the measures of comprehension and extension are inversely proportional only when the measure of information is constant.  The fundamental principle governing his “laws of information” could thus be expressed in the following formula.

The development of Peirce’s information formula is discussed in my ongoing study notes, consisting of selections from Peirce’s 1865–1866 Lectures on the Logic of Science and my commentary on them.

• Information = Comprehension × Extension

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
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