[Inquiry] Re: Extension x Comprehension = Information

[Jon Awbrey]

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ECI.  Note 8

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At this point in his discussion, Peirce is relating the nature of
inference, inquiry, and information to the character of the signs
that are invoked in support of the overall process in question,
a process that he is presently describing as "symbolization".

In the interests of the maximum possible clarity I would like
to pause for a while and try to extract from Peirce's account
a couple of quick sketches, designed to show how the examples
that he gives of a "conjunctive term" and a "disjunctive term"
might look if they were cast within a lattice-theoretic frame.

Let's examine Peirce's example of a conjunctive term,
"spherical, bright, fragrant, juicy, tropical fruit",
within a lattice framework.  We have these six terms:

   t_1  =  spherical
   t_2  =  bright
   t_3  =  fragrant
   t_4  =  juicy
   t_5  =  tropical
   t_6  =  fruit

Suppose that z is the logical conjunction of these terms:

   z  =  t_1  t_2  t_3  t_4  t_5  t_6.

What on earth could Peirce mean by saying that such a term
is "not a true symbol", or that it is of "no use whatever"?

In particular, let us consider the following statement:

| If it occurs in the predicate and something is said to be a
| spherical bright fragrant juicy tropical fruit, since there
| is nothing which is all this which is not an orange, we may
| say that this is an orange at once.

That is to say, if something x is said to be z, then we may guess
fairly surely that x is really an orange, in other words, that x
has all of the additional features that would be summed up quite
succinctly in the much more constrained term "y" = "an orange".

Figure 1 depicts the situation that is being contemplated here.

o---------------------------------------------------------------------o
|                                                                     |
|  t_1   t_2         t_5   t_6                                        |
|   o     o    ...    o     o                                         |
|     .    .         .    .                                           |
|       .   .       .   .                                             |
|         .  .     .  .                                               |
|           . .   . .                                                 |
|             .. ..                                                   |
|               o z = spherical bright fragrant juicy tropical fruit  |
|               * *                                                   |
|               *   *   Rule                                          |
|               *     * y=>z                                          |
|               *       *                                             |
|               *         *                                           |
|          Fact *           o y = orange                              |
|          x=>z *         *                                           |
|               *       *                                             |
|               *     * Case                                          |
|               *   *   x=>y                                          |
|               * *                                                   |
|               o                                                     |
|               x = subject                                           |
|                                                                     |
o---------------------------------------------------------------------o
Figure 1.  Conjunctive Term z, Taken as Predicate

As far as I am presently able to understand it, what Peirce is saying
about z not being a genuinely useful symbol can be explained in terms
of the gap between the logical conjunction z, in lattice terms, the
"greatest lower bound" (glb) of the conjoined terms, z = glb(t_j),
and what we might call the "natural conjunction" y = an orange.
That is to say, there is an extra measure of constraint that
goes into forming the natural kinds lattice from the free
lattice that logic and set theory would otherwise impose.
The local manifestations of this global information are
meted out over the structure of the natural lattice by
just such abductive gaps as the one between z and y.

Jon Awbrey

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