Information = Comprehension × Extension • Comment

Information = Comprehension × Extension • Comment 1
• https://inquiryintoinquiry.com/2024/10/11/information-comprehension-x-extension-comment-1-a/

Re: Information = Comprehension × Extension • Selection 1
• https://inquiryintoinquiry.com/2024/10/05/information-comprehension-x-extension-selection-1-a/

All,

Selection 1 ends with Peirce drawing the following conclusion about the
links between information, comprehension, inference, and symbolization.

❝Thus information measures the superfluous comprehension.
And, hence, whenever we make a symbol to express any thing
or any attribute we cannot make it so empty that it shall
have no superfluous comprehension.

❝I am going, next, to show that inference is symbolization
and that the puzzle of the validity of scientific inference
lies merely in this superfluous comprehension and is therefore
entirely removed by a consideration of the laws of information.❞

(Peirce 1866, p. 467)

At this point in his inventory of scientific reasoning, Peirce is
relating the nature of inference, information, and inquiry to the
character of the signs mediating the process in question, a process
he describes as “symbolization”.

In the interest of clarity let's draw from Peirce's account
a couple of quick sketches, designed to show how the examples
he gives of conjunctive terms and disjunctive terms might look
if they were cast within a lattice‑theoretic framework.

Re: Information = Comprehension × Extension • Selection 5
• https://inquiryintoinquiry.com/2024/10/09/information-comprehension-x-extension-selection-5-a/

Looking back on Selection 5, let's first examine Peirce's example of a
conjunctive term — “spherical, bright, fragrant, juicy, tropical fruit” —
within a lattice framework. We have the following six terms.

t₁ = spherical
t₂ = bright
t₃ = fragrant
t₄ = juicy
t₅ = tropical
t₆ = fruit

Suppose z is the logical conjunction of the above six terms.

z = t₁ ∙ t₂ ∙ t₃ ∙ t₄ ∙ t₅ ∙ t₆

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, 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.❞
(Peirce 1866, p. 470).

In other words, if something x is said to be z then we may guess fairly
surely x is really an orange, in short, x has all the additional features
otherwise summed up quite succinctly in the much more constrained term y,
where y means “an orange”.

Figure 1 shows the implication ordering of logical terms
in the form of a “lattice diagram”.

Figure 1. Conjunctive Term z, Taken as Predicate
• https://inquiryintoinquiry.files.wordpress.com/2016/10/ice-figure-1.jpg

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 of the conjoined terms,
z = glb{t₁, t₂, t₃, t₄, t₅, t₆}, and what we might regard as the
natural conjunction or natural glb of those terms, namely, y,
“an orange”.

In sum there is an extra measure of constraint which goes into forming the
natural kinds lattice from the free lattice which logic and set theory would
otherwise impose as a default background. The local manifestations of that
global information are meted out over the structure of the natural lattice
by just such abductive gaps as the one we observe between z and y.

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 —

Inquiry Blog • Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/

OEIS Wiki • Information = Comprehension × Extension
• https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension

C.S. Peirce • Upon Logical Comprehension and Extension
• https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm

Regards,

Jon

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Information = Comprehension × Extension • Comment 2
• https://inquiryintoinquiry.com/2024/10/12/information-comprehension-x-extension-comment-2-a/

All,

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.

s₁ = neat
s₂ = swine
s₃ = sheep
s₄ = deer

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

u = ((s₁)(s₂)(s₃)(s₄))

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

Figure 2. Disjunctive Term u, Taken as Subject
• https://inquiryintoinquiry.files.wordpress.com/2016/10/ice-figure-2.jpg

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 = lub{s₁, s₂, s₃, s₄}, 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 ⇒ w, namely,
the rule that cloven‑hoofed animals are herbivorous.
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Information = Comprehension × Extension • Comment 3
• https://inquiryintoinquiry.com/2024/10/13/information-comprehension-x-extension-comment-3-a/

All,

Peirce identifies inference with a process he describes
as “symbolization”. Let us consider what that might imply.
entirely removed by a consideration of the laws of “information”.❞
(467)

Even if it were only a rough analogy between inference and symbolization,
a principle of logical continuity, what is known in physics as a “correspondence
principle”, would suggest parallels between steps of reasoning in the neighborhood
of exact inferences and signs in the vicinity of genuine symbols. This would lead us
to expect a correspondence between degrees of inference and degrees of symbolization
extending from exact to approximate (“non‑demonstrative”) inferences and from genuine
to approximate (“degenerate”) symbols.

❝For this purpose, I must call your attention to the differences there are
in the manner in which different representations stand for their objects.

❝In the first place there are likenesses or copies — such as “statues”, “pictures”,
“emblems”, “hieroglyphics”, and the like. Such representations stand for their
objects only so far as they have an actual resemblance to them — that is agree
with them in some characters. The peculiarity of such representations is that
they do not determine their objects — they stand for anything more or less; for
they stand for whatever they resemble and they resemble everything more or less.

❝The second kind of representations are such as are set up by a convention of men
or a decree of God. Such are “tallies”, “proper names”, &c. The peculiarity of
these “conventional signs” is that they represent no character of their objects.

❝Likenesses denote nothing in particular; “conventional signs” connote nothing
in particular.

❝The third and last kind of representations are “symbols” or general representations.
They connote attributes and so connote them as to determine what they denote. To this
class belong all “words” and all “conceptions”. Most combinations of words are also
symbols. A proposition, an argument, even a whole book may be, and should be, a single
symbol.❞ (467–468)

In addition to Aristotle, the influence of Kant on Peirce is very strongly marked in
these earliest expositions. The invocations of “conceptions of the understanding”,
the “use of concepts” and thus of symbols in reducing the manifold of extension,
and the not so subtle hint of the synthetic à priori in Peirce's discussion, not
only of natural kinds but also of the kinds of signs leading up to genuine symbols,
can all be recognized as pervasive Kantian themes.

In order to draw out these themes and see how Peirce was led to develop their
leading ideas, let us bring together our previous Figures, abstracting from
their concrete details, and see if we can figure out what is going on.

Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign.

Figure 3. Conjunctive Predicate z, Abduction of Case x ⇒ y
https://inquiryintoinquiry.files.wordpress.com/2016/10/ice-figure-3.jpg

Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign.

Figure 4. Disjunctive Subject u, Induction of Rule v ⇒ w
• https://inquiryintoinquiry.files.wordpress.com/2016/10/ice-figure-4.jpg
Regards,

Jon

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Information = Comprehension × Extension • Comment 4
• https://inquiryintoinquiry.com/2024/10/14/information-comprehension-x-extension-comment-4-a/

Re: Information = Comprehension × Extension • Comment 3
• https://inquiryintoinquiry.com/2024/10/13/information-comprehension-x-extension-comment-3-a/

All,

Many things still puzzle me about Peirce's account at this point. The question
marks in the Figures of the previous post indicate a number of places I have
remaining questions about. There is nothing for it but returning to Peirce's
text and trying again to follow his reasoning.

Let's go back to Peirce's example of abductive inference and try to get a
clearer picture of why he connects it with conjunctive terms and iconic signs.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.
The relationship between conjunctive terms and iconic signs may be understood along
the following lines. If there is anything with all the properties described by the
conjunctive term “spherical bright fragrant juicy tropical fruit” then sign users
may use that thing as an icon of an orange, precisely by virtue of the fact it
shares those properties with an orange. But the only natural examples of things
with all those properties are oranges themselves, so the only thing qualified to
serve as a natural icon of an orange by virtue of those very properties is that
orange itself or another orange.
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Information = Comprehension × Extension • Comment 5
• https://inquiryintoinquiry.com/2024/10/16/information-comprehension-x-extension-comment-5-a/

All,

Let's stay with Peirce's example of abductive inference a little longer
and try to clear up the more troublesome confusions tending to arise.
One thing needs to be stressed at this point. It is important to
recognize the conjunctive term itself — namely, the syntactic string
“spherical bright fragrant juicy tropical fruit” — is not an icon but
a symbol. It has its place in a formal system of symbols, for example,
a propositional calculus, where it would normally be interpreted as a
logical conjunction of six elementary propositions, denoting anything
in the universe of discourse with all six of the corresponding properties.

The symbol denotes objects which may be taken as icons of oranges by
virtue of their bearing those six properties in common with oranges.
But there are no objects denoted by the symbol which aren't already
oranges themselves. Thus we observe a natural reduction in the
denotation of the symbol, consisting in the absence of cases
outside of oranges which have all the properties indicated.

The above analysis provides another way to understand the abductive inference
from the Fact x ⇒ z and the Rule y ⇒ z to the Case x ⇒ y. The lack of any
cases which are z and not y is expressed by the implication z ⇒ y. Taking
this together with the Rule y ⇒ z gives the logical equivalence y = z. But
this reduces the Case x ⇒ y to the Fact x ⇒ z and so the Case is justified.

Viewed in the light of the above analysis, Peirce's example of abductive
reasoning exhibits an especially strong form of inference, almost deductive
in character. Do all abductive arguments take this form, or may there be
weaker styles of abductive reasoning which enjoy their own levels of
plausibility? That must remain an open question at this point.
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Information = Comprehension × Extension • Comment 6
• https://inquiryintoinquiry.com/2024/10/23/information-comprehension-x-extension-comment-6-a/

Re: Information = Comprehension × Extension • Comment 2
• https://inquiryintoinquiry.com/2024/10/12/information-comprehension-x-extension-comment-2-a/

Returning to Peirce's example of inductive inference, let's try to
get a clearer picture of why he connects it with disjunctive terms
and indicial signs. At this point in time I can't say I'm entirely
satisfied with my understanding of the relationship between disjunctive
terms, indicial signs, and inductive inferences as presented by Peirce
in his early accounts. What follows is just one of the simplest and
least question‑begging attempts at rational reconstruction I've been
able to devise.
If there is any distinguishing feature shared by all the instances under the
disjunctive description “neat, swine, sheep, deer” then sign users may take
that feature as a predictor of being herbivorous, precisely because all the
things under the disjunctive description are herbivorous. But everything
under the disjunctive description is cloven‑hoofed, so the cases under the
disjunctive description serve to indicate, support, or witness the utility
of the induction from cloven‑hoofed to herbivorous.
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Information = Comprehension × Extension • Comment 7
• https://inquiryintoinquiry.com/2024/10/28/information-comprehension-x-extension-comment-7-a/

Let's stay with Peirce's example of inductive inference a little longer
One final point needs to be stressed. It is important to recognize the
disjunctive term itself — the syntactic formula “neat, swine, sheep, deer”
or any logically equivalent formula — is not an index but a symbol. It has
the character of an artificial symbol which is constructed to fill a place in
a formal system of symbols, for example, a propositional calculus. In that
setting it would normally be interpreted as a logical disjunction of four
which has any of the four corresponding properties.

The artificial symbol “neat, swine, sheep, deer” denotes objects which serve
as indices of the genus herbivore by virtue of their belonging to one of the
four named species of herbivore. But there is in addition a natural symbol
which serves to unify the manifold of given species, namely, the concept of
a cloven‑hoofed animal.

As a symbol or general representation, the concept of a cloven‑hoofed animal
must connote an attribute and connote so as to determine what it denotes.
Thus we observe a natural expansion in the connotation of the symbol,
amounting to what Peirce calls the “superfluous comprehension” or
information added by an “ampliative” or synthetic inference.

In sum we have sufficient information to motivate an inductive inference,
from the Fact u ⇒ w and the Case u ⇒ v to the Rule v ⇒ w.
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