{ Information = Comprehension × Extension } Revisited • Comment 5

Let’s stay with Peirce’s example of abductive inference a little longer and try to clear up the more troublesome confusions that tend to arise.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 1. Conjunctive Term z, Taken as Predicate

Figure 1. Conjunctive Term z, Taken as Predicate

One thing needs to be stressed at this point.  It is important to recognize the conjunctive term itself — namely, the syntactic string “spherical bright fragrant juicy tropical fruit” — is not an icon but a symbol.  It has its place in a formal system of symbols, for example, a propositional calculus, where it would normally be interpreted as a logical conjunction of six elementary propositions, denoting anything in the universe of discourse with all six of the corresponding properties.  The symbol denotes objects which may be taken as icons of oranges by virtue of their bearing those six properties in common with oranges.  But there are no objects denoted by the symbol which aren’t already oranges themselves.  Thus we observe a natural reduction in the denotation of the symbol, consisting in the absence of cases outside of oranges which have all the properties indicated.

The above analysis provides another way to understand the abductive inference from the Fact x \Rightarrow z and the Rule y \Rightarrow z to the Case x \Rightarrow y.  The lack of any cases which are z and not y is expressed by the implication z \Rightarrow y.  Taking this together with the Rule y \Rightarrow z gives the logical equivalence y = z.  But this reduces the Case x \Rightarrow y to the Fact x \Rightarrow z and so the Case is justified.

Viewed in the light of the above analysis, Peirce’s example of abductive reasoning exhibits an especially strong form of inference, almost deductive in character.  Do all abductive arguments take this form, or may there be weaker styles of abductive reasoning which enjoy their own levels of plausibility?  That must remain an open question at this point.


  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.


cc: CyberneticsOntolog ForumStructural ModelingSystems Science

This entry was posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information = Comprehension × Extension, Information Theory, Inquiry, Intension, Logic, Logic of Science, Peirce, Peirce's Categories, Pragmatic Semiotic Information, Pragmatism, Scientific Method, Semiotics, Sign Relations and tagged , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

3 Responses to { Information = Comprehension × Extension } Revisited • Comment 5

  1. Pingback: Survey of Pragmatic Semiotic Information • 4 | Inquiry Into Inquiry

  2. Pingback: Survey of Pragmatic Semiotic Information • 5 | Inquiry Into Inquiry

  3. Pingback: C.S. Peirce and Category Theory • 4 | Inquiry Into Inquiry

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.