[Inquiry] Re: Extension x Comprehension = Information

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

ECI.  Note 9

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

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 depicts the situation that we have before us.

o---------------------------------------------------------------------o
|                                                                     |
|               w = herbivorous                                       |
|               o                                                     |
|               * *     Rule                                          |
|               *   *   v=>w                                          |
|               *     *                                               |
|               *       *                                             |
|               *         *                                           |
|          Fact *           o v = cloven-hoofed                       |
|          u=>w *         *                                           |
|               *       *                                             |
|               *     * Case                                          |
|               *   *   u=>v                                          |
|               * *                                                   |
|               o u = ((neat)(swine)(sheep)(deer))                    |
|             .. ..                                                   |
|           . .   . .                                                 |
|         .  .     .  .                                               |
|       .   .       .   .                                             |
|     .    .         .    .                                           |
|   o     o           o     o                                         |
|  s_1   s_2         s_3   s_4                                        |
|                                                                     |
o---------------------------------------------------------------------o
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.

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o