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

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

Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign.

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 “spherical bright fragrant juicy tropical fruit” 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 and the Rule to the Case   The lack of any cases which are and not is expressed by the implication   Taking that in conjunction with the Rule gives the logical equivalence   But that reduces the Case to the Fact 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 that 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.

Remark

• Readers will notice I have slipped at this point from using symbol in the precise technical sense Peirce introduced at the beginning of this discussion to the more ordinary sense all of us, Peirce included, tend to use on other occasions.  Should it become a big problem we can always find a way to mark the distinction but so far it seems context has usually sufficed to resolve any likely confusion.

References

• 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.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

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