ONT Re: Extension x Comprehension = Information
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Let us now consider Peirce's alternate example of a disjunctive term,
"neat, swine, sheep, deer", which he commonly borrows from classical
and scholastic discussions as a stock example of inductive reasoning.
| 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.
| 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.
In view of the analogical symmetries that it shares with the
conjunctive case, I think that we can run through this example
in fairly short order. We have the aggregation over four terms:
| s_1 = neat
| s_2 = swine
| s_3 = sheep
| s_4 = deer
Suppose that u is the logical disjunction of these terms:
| u = ((s_1)(s_2)(s_3)(s_4)).
Figure 2 illustrates the situation that we have before us.
| w = herbivorous
| o
| * * Rule
| * * v=>w
| * *
| * *
| * *
| Fact * o v = cloven-hoofed
| * *
| * *
| * * Case
| * *
| * *
| o u = ((neat(swine)(sheep)(deer))
| .. ..
| . . . .
| . . . .
| . . . .
| . . . .
| o o o o
| s_1 s_2 s_3 s_4
|
| Figure 2. Disjunctive Term u, Taken As Subject
In a similar but dual fashion to what we observed before, there is a gap
between the the logical disjunction u, expressed in lattice terminology,
the "least upper bound" (lub) of the disjoined terms, u = lub(s_j), and
what we might well call their "natural disjunction" v = cloven-hoofed.
Once again, the sheer implausibility of imagining that
the disjunctive term u would ever be embedded exactly
per se in a lattice of natural kinds, leads to the
evident "naturalness" of the induction to v => w,
namely, the rule that cloven-hoofed animals are
herbivorous. Yes, that means unicorns, too.
Stock example, indeed!
Jon Awbrey
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