{ Information = Comprehension × Extension } • Comment 2

Let’s examine Peirce’s second example of a disjunctive term — neat, swine, sheep, deer — within the style of lattice framework that 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 that Peirce borrows from traditional discussions, so let us pass over the circumstance that modern taxonomies may classify swine as omnivores.

In view of the analogical symmetries that the disjunctive term shares with the conjunctive case, I think that we can run through this example in fairly short order.  We have an aggregation over four terms:

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

Suppose that 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 depicts the situation that we have before us.

Figure 2. Disjunctive Term u, Taken as Subject

Figure 2. Disjunctive Term u, Taken as Subject

Here we have a situation that 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 (lub) 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 these terms, namely, v, cloven-hoofed.

Once again, the sheer implausibility of imagining that 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 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.

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