Information = Comprehension × Extension • Comment 2

Posted on October 12, 2024 by Jon Awbrey

Let’s examine Peirce’s second example of a disjunctive term — neat, swine, sheep, deer — within the style of lattice framework we used before.

Hence if we find out that neat are herbivorous, swine are herbivorous, sheep are herbivorous, and deer are herbivorous;  we may be sure that there is some class of animals which covers all these, all the members of which are herbivorous.  (468–469).

Accordingly, if we are engaged in symbolizing and we come to such a proposition as “Neat, swine, sheep, and deer are herbivorous”, we know firstly that the disjunctive term may be replaced by a true symbol.  But suppose we know of no symbol for neat, swine, sheep, and deer except cloven‑hoofed animals.  (469).

This is apparently a stock example of inductive reasoning Peirce is borrowing from traditional discussions, so let us pass over the circumstance that modern taxonomies may classify swine as omnivores.

In view of the analogical symmetries the disjunctive term shares with the conjunctive case, we can run through this example in fairly short order.  We have the following four terms.

\begin{array}{lll} s_1 & = & \mathrm{neat} \\ s_2 & = & \mathrm{swine} \\ s_3 & = & \mathrm{sheep} \\ s_4 & = & \mathrm{deer} \end{array}

Suppose u is the logical disjunction of the above four terms.

\begin{array}{lll} u & = & \texttt{((} s_1 \texttt{)(} s_2 \texttt{)(} s_3 \texttt{)(} s_4 \texttt{))} \end{array}

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 2. Disjunctive Term u, Taken as Subject

\text{Figure 2. Disjunctive Term}~ u, \text{Taken as Subject}

Here we have a situation which is dual to the structure of the conjunctive example.  There is a gap between the logical disjunction u, in lattice terminology, the least upper bound of the disjoined terms, u = \mathrm{lub} \{ s_1, s_2, s_3, s_4 \}, and what we might regard as the natural disjunction or natural lub of those terms, namely, v, cloven‑hoofed.

Once again, the sheer implausibility of imagining the disjunctive term u would ever be embedded exactly as such in a lattice of natural kinds leads to the evident naturalness of the induction to the implication v \Rightarrow w, namely, the rule that cloven‑hoofed animals are herbivorous.

Reference

  • 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.

Resources

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Mathstodon • Academia.edu
cc: Research Gate

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