[Inquiry] Re: Extension x Comprehension = Information
Jon Awbrey
jawbrey at oakland.edu
Mon Mar 31 15:54:29 CST 2003
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ECI. Commentary Note 1
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Da capo
| Let us now return to the information.
| The information of a term is the measure of
| its superfluous comprehension. That is to say
| that the proper office of the comprehension is to
| determine the extension of the term. For instance,
| you and I are men because we possess those attributes --
| having two legs, being rational, &c. -- which make up the
| comprehension of 'man'. Every addition to the comprehension
| of a term lessens its extension up to a certain point, after
| that further additions increase the information instead.
|
| Thus, let us commence with the term 'colour'; add to the
| comprehension of this term, that of 'red'. 'Red colour'
| has considerably less extension than 'colour'; add to
| this the comprehension of 'dark'; 'dark red colour' has
| still less [extension]. Add to this the comprehension
| of 'non-blue' -- 'non-blue dark red colour' has the
| same extension as 'dark red colour', so that the
| 'non-blue' here performs a work of supererogation;
| it tells us that no 'dark red colour' is blue, but
| does none of the proper business of connotation,
| that of diminishing the extension at all.
|
| 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'.
|
| 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.
|
| CSP, CE 1, pages 467-468.
On running through this familiar yet ever strange refrain for another time,
I see that I have scarcely begun to trace the sinews of the linkages among
the three types of signs, "the differences there are in the manner in which
different representations stand for their objects", the matter of extension
and comprehension, and the whole life-cycle of inquiry that engages me most.
Al segno
Signs, inquiry, and information.
Let's focus on that for a while.
To put Peirce's examples more in line with the order of his
three categories, I will renumber them in the following way:
1. The conjunctive term "spherical bright fragrant juicy tropical fruit".
2.1. The disjunctive term "man or horse or kangaroo or whale".
2.2. The disjunctive term "neat or swine or sheep or deer".
Peirce suggests an analogy or a parallelism between
the corresponding elements of the following triples:
1. Conjunctive Term : Iconical Sign : Abductive Case
2. Disjunctive Term : Indicial Sign : Inductive Rule
Here is an overview of the two patterns of reasoning, along
with the first steps of an analysis in sign-theoretic terms:
1. Conjunctive term "spherical bright fragrant juicy tropical fruit".
| A similar line of thought may be gone through
| in reference to hypothesis. In this case we
| must start with the consideration of the term:
|
| 'spherical, bright, fragrant, juicy, tropical fruit'.
|
| Such a term, formed by the sum of the comprehensions of several terms,
| is called a conjunctive term. A conjunctive term has no extension
| adequate to its comprehension. Thus the only spherical bright
| fragrant juicy tropical fruit we know is the orange and that
| has many other characters besides these. Hence, such a term
| is of no use whatever. 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. On the other hand, if the conjunctive term is
| subject and we know that every spherical bright fragrant
| juicy tropical fruit necessarily has certain properties,
| it must be that we know more than that and can simplify the
| subject. Thus a conjunctive term may always be replaced by
| a simple one. So if we find that light is capable of producing
| certain phenomena which could only be enumerated by a long conjunction
| of terms, we may be sure that this compound predicate may be replaced
| by a simple one. And if only one simple one is known in which the
| conjunctive term is contained, this must be provisionally adopted.
|
| CSP, CE 1, page 470.
o-----------------------------o-----------------------------o
| Objective Framework | Interpretive Framework |
o-----------------------------o-----------------------------o
| |
| t_1 t_2 ... t_5 t_6 |
| o o ... o o |
| . . . . |
| . . . . |
| . . . . |
| .. .. |
| o z |
| * * Rule |
| * * y=>z |
| * * |
| * * |
| Fact * * |
| x=>z * o y |
| * * |
| * * |
| * * Case |
| * * x=>y |
| * * |
| o |
| x |
| |
o-----------------------------------------------------------o
| Conjunctive Predicate z, Abduction to the Case x => y |
o-----------------------------------------------------------o
| |
| S = I = {t_1, t_2, t_3, t_4, t_5, t_6, x, y, z} |
| |
| t_1 = "spherical" |
| t_2 = "bright" |
| t_3 = "fragrant" |
| t_4 = "juicy" |
| t_5 = "tropical" |
| t_6 = "fruit" |
| |
| x = "subject" |
| y = "orange" |
| z = "spherical bright fragrant juicy tropical fruit" |
| |
o-----------------------------------------------------------o
2. Disjunctive term "neat or swine or sheep or deer".
| 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. Now a disjunctive term -- such as 'neat swine
| sheep and deer', or 'man, horse, kangaroo, and whale' -- is not a true
| symbol. It does not denote what it does in consequence of its connotation,
| as a symbol does; on the contrary, no part of its connotation goes at all
| to determine what it denotes -- it is in that respect a mere accident if it
| denote anything. Its 'sphere' is determined by the concurrence of the four
| members, man, horse, kangaroo, and whale, or neat swine sheep and deer as
| the case may be.
|
| CSP, CE 1, 468-469.
o-----------------------------o-----------------------------o
| Objective Framework | Interpretive Framework |
o-----------------------------o-----------------------------o
| |
| w |
| o |
| * * Rule |
| * * v=>w |
| * * |
| * * |
| Fact * * |
| u=>w * o v |
| * * |
| * * |
| * * Case |
| * * u=>v |
| * * |
| o u |
| .. .. |
| . . . . |
| . . . . |
| . . . . |
| o o o o |
| s_1 s_2 s_3 s_4 |
| |
o-----------------------------------------------------------o
| Disjunctive Subject u, Induction to the Rule v => w |
o-----------------------------------------------------------o
| |
| S = I = {s_1, s_2, s_3, s_4, u, v, w} |
| |
| s_1 = "neat" |
| s_2 = "swine" |
| s_3 = "sheep" |
| s_4 = "deer" |
| |
| u = "neat or swine or sheep or deer" |
| v = "cloven-hoofed" |
| w = "herbivorous" |
| |
o-----------------------------------------------------------o
Jon Awbrey
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