ONT Re: Extension x Comprehension = Information

[Jon Awbrey <jawbrey@oakland.edu>]

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CH = Chris Hillman
CP = Charles Peirce
JA = Jon Awbrey
JD = Jean-Luc Delatre

CH: http://www.math.washington.edu/~hillman/papers.html

JA: i have only skimmed the "newconcept" paper and the "talk" abstract at this point --
    thanks again for the ghostscript tip, as my plugin finder did not know about it --
    as i was piqued by the idea of "galois information theory", but i did not yet
    find the equivalent of peirce's notion of information there, so clue me in
    if you think it's there somewhere.

JD: Hey! this is *your* job!!!

yes, sorry to be so lazy.  i will eventually work my way through the paper.
but i read my first paper on "galois theory of models" almost 20 years ago,
and so far, in spite of many tantalizing titles since then, i have not run
across anybody who rediscovered the equal of what i consider to be peirce's
crucial insights about information in this connection.  i think that there
are probably systemic reasons for this, having to do with the cognitive
styles that are engrained into most mathematical thinkers in our times.

JD: I will kindly try to translate for you from Peirce incredibly awkward and
    convoluted (yet, enlightened for the time) explanations, taken from your
    messages, into straight lattice theoretic statements matched to Hillman's
    so you will perhaps see that it's almost the same, just neater.
    I am also commenting Peirce statements & yours.

CP: | Yet there are combinations of words and combinations of conceptions
    | which are not strictly speaking symbols.  These are of two kinds
    | of which I will give you instances.  We have first cases like:
    |
    | 'man and horse and kangaroo and whale',
    |
    | and secondly, cases like:
    |
    | 'spherical bright fragrant juicy tropical fruit'.

JD: First thing first, here Peirce uses two different kinds
    of words in the two sentences.

JD: 1) 'man', 'horse', etc., ... stand for what is called 'intents' in
       Hillman's terminology and thus is a set of 'predicates' shared by all
       objects satisfying the concept ('man' -> 'Apterous', 'Biped', etc., ...)
       Translated to Hillman's 'man' is the name of the subset of his Y set
       which emcompass 'Apterous', 'Biped', etc... as *elements* of the Y set.
       These could be called essential attributes of 'man' and this is what
       Peirce call 'Content'.
 
JD: 2) 'spherical', 'bright, as well as 'Apterous', 'Biped', etc., ... are attributes,
       that is what Hillman calls 'predicates' (Page 2).  Although each one have an
       extension as well as 'man' and 'horse' have, the extension of a 'predicate' is
       just the counter-image of the singleton set:  <bright objects> = <| {'bright'},
       while <man objects> is the intersection of  all the counter-images of the
       "essential attributes" of 'man' (Peirce's 'Content', my wording on "essential")
       giving Hillman's 'extent', that is Peirce's 'Sphere'.

first off, peirce is using just one kind of words in the two cases.
they are all called "terms".  you may of course opt to use two kinds
of words to interpret him, if you wish, but then you will be in danger
of introducing distortions in the process of forming this interpretation.

indeed, one of the reasons that i prefer peirce's account is that he does not
begin by lading these ontological notions into his logical cart at the outset.
so his treatment is somewhat less theory-laden to start.  this is a good idea,
especially if we desire to use our logic to reason about the possibility of a
distinction between two kinds of terms, without prejudging the matterr in the
very form of the logic we use.

> As you can see, the X set of Hillman is the set (probably better named only
> a collection ...) of all 'things' in the universe.  While the Y set is the set
> of 'attributes', some subsets of which are distinguished as 'Content' (Peirce)
> or 'intents' (Hillman) and given a name which stands for the 'words' (Peirce)
> or 'concepts' (Hillman), although both are happily confusing the intentional
> and extensional meaning and you never know, unless you check carefully, if
> they are talking about subsets within X or Y, THAT's duality!!!

no, i think that peirce is quite clear about the distinction
between intensions (= attributes, properties, or qualities)
and extensions (= things in the universe of discourse).
moreover, he is admirably clear about the distinction
between the role of a sign and the role of an object,
which distinction almost everybody else confounds.

JD: Once you grasp the image/counter-image idea as explained on
    Page 3 of Hillman's "What is a Concept?" you can dispense with
    all the arduous explanations about "wider terms", "narrower terms",
    "higher terms", "lower terms", "contained under", "contained in",
    "denote", "connote" and whatnot ... We are just talking about subsets
    from X and Y and their mapping thru the R relation (just a subset of XxY
    corresponding to the current ascertained knowledge) and closures <||> & |><|
    using De Morgan's laws.

i did get a heap of this stuff once upon a time.
but so far i think the fca folks are guilty of
confusing concepts (= symbols in the mind)
with the objects of these concepts, and
this would not be a good thing to do.

cf.  http://suo.ieee.org/ontology/msg03799.html

JD: So now, we are ready to restate and comment some
    Peirce excerpts you mentioned in your mails:

CP: | The first of these terms has no comprehension which is adequate to the
    | limitation of the extension.  In fact, men, horses, kangaroos, and whales
    | have no attributes in common which are not possessed by the entire class of
    | mammals.  For this reason, this disjunctive term, man and horse and kangaroo
    | and whale, is of no use whatever.  For suppose it is the subject of a sentence;
    | suppose we know that men and horses and kangaroos and whales have some common
    | character.  Since they have no common character which does not belong to the
    | whole class of mammals, it is plain that 'mammals' may be substituted for
    | this term.  Suppose it is the predicate of a sentence, and that we know
    | that something is either a man or a horse or a kangaroo or a whale;  then,
    | the person who has found out this, knows more about this thing than that it
    | is a mammal;  he therefore knows which of these four it is for these four have
    | nothing in common except what belongs to all other mammals.  Hence in this case
    | the particular one may be substituted for the disjunctive term.  A disjunctive
    | term, then, -- one which aggregates the extension of several symbols, -- may
    | always be replaced by a simple term.

JD: This is only true if the "common character" which has
    been spotted is in fact common to all mammals and not
    a newly discovered characteristic shared only by men,
    horses, kangaroos, and whales among mammals!  In this case
    there *is* in fact a possible new "comprehension which is
    adequate to the limitation of the extension", just add this
    new 'attribute' to the 'Content' mammals and give a name to
    this new 'Content', say "weirdo-mammals", if you don't do that
    you don't have a name to use to replace "men, horses, kangaroos,
    and whales" because you would abusively make statements about
    mammals which should apply only to "weirdo-mammals".

it is necessary to understand the rules of the natural kinds game.
assume that X is the universe of discourse.  everybody knows what
it would be like to work within the power set of X, the lattice
Set(X) = (Pow(X), =>).  but the natural kinds lattice Nat(X)
that we are given to work with is more restricted than this.
today we would talk of "accessible predicates" and such.
yes, it sounds silly to speak of accessible predicates
in a finite universe, but it is merely an exposition,
not a competition.  or maybe it is.  try to imagine
that you are on a tv game show where there's a lot
of money riding on it if you can just manage to
simulate the thinking of an "ordinary thinker"
and not that of a nit-picking wiseacre.  yes,
it's easier said than done, but we must try.

if the game-show host says "men, horses, kangaroos, whales",
the ordinary thinker is already punching the buzzer and
screaming "mammals!"

JD: Back to Hillman, what makes a proper concept?
    By Lemmas 3 and 4, Pages 3/4, the fact that the comprehension and
    extension of the concept are closed under <||> and |><|, that is,
    the image of the extension under |> is exactly the comprehension
    and the counter-image of the comprehension under <| is exactly the
    extension.  So in the case above, either we cannot find an attribute
    specific to only men, horses, kangaroos, and whales and Peirce is
    right because the closure of their extension thru |><| will map
    back to all mammals.  Or we *can* find a specific attribute and
    the closure of |><| will map back exactly as well as the closure
    of <||> from the 'Content' mammals supplemented with this new
    attribute will map back to itself.  In this case we can
    legitimately introduce a name for this new concept.

CP: | Now those who are not accustomed to the homologies of the conceptions of
    | men and words, will think it very fanciful if I say that this concurrence
    | of four terms to determine the sphere of a disjunctive term resembles the
    | arbitrary convention by which men agree that a certain sign shall stand
    | for a certain thing.  And yet how is such a convention made?  The men
    | all look upon or think of the thing and each gets a certain conception
    | and then they agree that whatever calls up or becomes an object of that
    | conception in either of them shall be denoted by the sign.  In the one
    | case, then, we have several different words and the disjunctive term
    | denotes whatever is the object of either of them.  In the other case,
    | we have several different conceptions -- the conceptions of different
    | men -- and the conventional sign stands for whatever is an object of
    | either of them.  It is plain the two cases are essentially the same,
    | and that a disjunctive term is to be regarded as a conventional sign
    | or index.  And we find both agree in having a determinate extension
    | but an inadequate comprehension.

JD: Does he really says that the extension of a conventional sign is the
    union of the extensions intended by each of the men using this sign?
    Very likely of course that no proper comprehension can be found, but
    that means that all efforts to building any shared ontology are doomed!

It's more like an analogy, "homology", or isomorphism.
I will try to draw a diagram of this later.

CP: | 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.  There is but one
    | objection to substituting this for the disjunctive term;  it is
    | that we should, then, say more than we have observed.  In short,
    | it has a superfluous information.  But we have already seen that
    | this is an objection which must always stand in the way of taking
    | symbols.  If therefore we are to use symbols at all we must use
    | them notwithstanding that.  Now all thinking is a process of
    | symbolization, for the conceptions of the understanding are
    | symbols in the strict sense.  Unless, therefore, we are to
    | give up thinking altogeher we must admit the validity of
    | induction.  But even to doubt is to think.  So we cannot
    | give up thinking and the validity of induction must be
    | admitted.

JD: This is off topic!
    Mixing concept building and induction is just introducing more confusion.
    The union of extensions for "Neat, swine, sheep, and deer" will map thru |>
    to "cloven-hoofed herbivorous animals" and it is uncontroversial that this
    is a proper concept (closed under <||>) and that it includes Neat, swine,
    sheep, and deer but this is all that we can say.

no, for peirce, as for kant before him, "concept building" has everything
to do with the whole process of inquiry, whether everyday problem-solving
or scientific reasoning, in which the concept is first conceived, and if
it is lucky, continues to find use.  if the fca version does not deal with
this aspect of concepts, then it has missed a crucial aspect of the concept
of a concept.

JD: Whether there exist "cloven-hoofed herbivorous animals" other
    than the ones mentioned or whether we want to assume that all
    "cloven-hoofed animals" are herbivorous are entirely separate
    questions and should not interfere with concept building
    considerations.

JA: 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.

JD: Irrelevant!  (Sorry ...)

CP: | 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.

JD: I bet that I can find a 'spherical, bright, fragrant,
    juicy, tropical fruit' which is not an orange!

the fact that you would have to spend time
thinking of it proves the rule in question,
and this is the whole point of the example.

JD: Assuming for a moment Peirce is true with this, this means
    that the closure under <||> of 'spherical, bright, fragrant,
    juicy, tropical fruit' brings us back to a more specific
    'Content' which happens to be the set of all "essential"
    attributes of an orange.  But this is only true until I
    find my other strange fruit, at which point all previous
    replacements of 'spherical, bright, fragrant, juicy,
    tropical fruit' by 'an orange' will be screwed up.
    I made a similar point a while ago in the CG list
    about unicity of individual markers, in that case
    the screwed info was on the "things" side.

keep looking for strange fruits.  in the mean time,
your naturally naive opponent is walking off with
all the loot.

JA: The way the joke goes, the straight man "defines" a human being H
    as an "apterous biped" A·B, a two-legged critter without feathers,
    and then the wiseguy hits him over the head with a plucked chicken,
    and by dint of this koan, he achieves enlightenment about the marks
    that distinguish kindness of the artless kind from the crasser kinds
    of artificial kindness.  Leastwise, anyways -- so I've heard it told.

JD: Of course choosing a poor subset of essential attributes of a concept
    brings the risk that the <||> closure will not map back to what you
    expected, so what?

This is just the nature of the natural situation,
of which it's our task to find a natural account.

have to break here, will get to the rest later.

jon awbrey

JD: Indeed, assuming that by "essential' attributes or 'Content' as
    Peirce names it we mean all attributes shared by all instances
    of the concept, it is likely that many subsets of the 'Content'
    will be selective enough such as to map back to the full 'Content'
    under <||> but not *all* subsets will do.

JD: This is probably the way we use ourselves when we recognise
    something or someone from just a few hints ('pars pro toto').

CP: | You never can narrow down to an individual.
    | Do you say Daniel Webster is an individual?
    | He is so in common parlance,
    | but in logical strictness he is not.
    | We think of certain images in our memory --
    | a platform and a noble form uttering convincing and patriotic words --
    | a statue --
    | certain printed matter --
    | and we say that which
    | that speaker and the
    | man whom that statue
    | was taken for and the
    | writer of this speech --
    | that which these are in
    | common is Daniel Webster.
    | Thus, even the proper name
    | of a man is a general term or
    | the name of a class, for it names
    | a class of sensations and thoughts.
    | The true individual term the absolutely
    | singular 'this' & 'that' cannot be reached.
    | Whatever has comprehension must be general.

CP: | In like manner, it is impossible to find any simple term.
    | This is obvious from this consideration.  If there is
    | any simple term, simple terms are innumerable for in
    | that case all attributes which are not simple are
    | made up of simple attributes.  Now none of these
    | attributes can be affirmed or denied universally
    | of whatever has any one.  For let 'A' be one
    | simple term and 'B' be another.  Now suppose
    | we can say All 'A' is 'B';  then 'B' is
    | contained in 'A'.  If, therefore, 'A'
    | contains anything but 'B' it is
    | a compound term, but 'A' is
    | different from 'B', and is
    | simple;  hence it cannot
    | be that All 'A' is 'B'.
    | Suppose No 'A' is 'B', then
    | not-'B' is contained in 'A';
    | if therefore 'A' contains anything
    | besides not-'B' it is not a simple term;
    | but if it is the same as not-'B', it is not a
    | simple term but is a term relative to 'B'.  Now it is a
    | simple term and therefore Some 'A' is 'B'.  Hence if we take
    | any two simple terms and call one 'A' and the other 'B' we have
    |
    |       Some 'A' is 'B'
    |
    | and   Some 'A' is not 'B'
    |
    | or in other words the universe will contain every possible kind of thing
    | afforded by the permutation of simple qualities.  Now the universe does not
    | contain all these things;  it contains no 'well-known green horse'.  Hence the
    | consequence of supposing a simple term to exist is an error of fact.  There
    | are several other ways of showing this besides the one that I have adopted.
    | They all concur to show that whatever has extension must be composite.

JD: Give up ontology building right now!
    This is a common result of desperates attempts to attain
    absolute truth.  It's absolutely of no interest outside
    of philosophy.  When I use a road map I perfectly know
    that it is not accurate and may even contains factual
    errors and I don't give a shit! It usually works for
    my purpose.  The question to ask is how to build an
    usable ontology with respect to the goal at hand and
    this obviously imply having absolute referents for
    individuals as well as primitive terms for attributes.
    What could be the 'attributes' of the green color?
    Nonsense question!

CP: | To explain this, we must remember that the process of induction is a
    | process of adding to our knowledge;  it differs therein from deduction --
    | which merely explicates what we know -- and is on this very account called
    | scientific inference.  Now deduction rests as we have seen upon the inverse
    | proportionality of the extension and comprehension of every term;  and this
    | principle makes it impossible apparently to proceed in the direction of
    | ascent to universals.  But a little reflection will show that when our
    | knowledge receives an addition this principle does not hold.
    |
    | Thus suppose a blind man to be told that no red things are
    | blue.  He has previously known only that red is a color;
    | and that certain things 'A', 'B', and 'C' are red.
    |
    |    The comprehension of red then has been for him   'color'.
    |    Its extension has been                           'A', 'B', 'C'.
    |
    | But when he learns that no red thing is blue, 'non-blue'
    | is added to the comprehension of red, without the least
    | diminution of its extension.
    |
    |    Its comprehension becomes   'non-blue color'.
    |    Its extension remains       'A', 'B', 'C'.
    |
    | Suppose afterwards he learns that a fourth thing 'D' is red.
    | Then, the comprehension of 'red' remains unchanged, 'non-blue color';
    | while its extension becomes 'A', 'B', 'C', and 'D'.  Thus, the rule
    | that the greater the extension of a term the less its comprehension
    | and 'vice versa', holds good only so long as our knowledge is not
    | added to;  but as soon as our knowledge is increased, either the
    | comprehension or extension of that term which the new information
    | concerns is increased without a corresponding decrease of the other
    | quantity.
    |
    | The reason why this takes place is worthy of notice.  Every addition to
    | the information which is incased in a term, results in making some term
    | equivalent to that term.  Thus when the blind man learns that 'red' is
    | not-blue, 'red not-blue' becomes for him equivalent to 'red'.  Before
    | that, he might have thought that 'red not-blue' was a little more
    | restricted term than 'red', and therefore it was so to him, but
    | the new information makes it the exact equivalent of red.
    | In the same way, when he learns that 'D' is red, the
    | term 'D-like red' becomes equivalent to 'red'.
    |
    | Thus, every addition to our information about a term is an addition
    | to the number of equivalents which that term has.  Now, in whatever
    | way a term gets to have a new equivalent, whether by an increase in
    | our knowledge, or by a change in the things it denotes, this always
    | results in an increase either of extension or comprehension without
    | a corresponding decrease in the other quantity.

JD: So, here we are, to YOUR POINT ABOUT INFORMATION!

JD: First I will say that the way Peirce use the word "information"
    is  quite loosy an only qualitative.  We do feel that we gained
    some knowledge but what is it?

> Let me rephrase Peirce example in Hillman's terms.
> 
> Telling the blind man that *all* 'red' things are also 'non-blue' will
> make him to change *his* knowledge database R = subset of XxY
> (if he is using Hillman's data representation for such purpose)
> in such a way that all instances of 'red' things will have, not only
> a pair (x , 'red') in R but also a pair (x, 'non-blue').
> Which means that doing the <||> closure on the {'red'} set will now
> map back to {'red', 'non-blue'}, an "enriched" content.
> The ambiguous point is what do we now mean by "red" when we speak of
> 'Content' along Peirce's teminlogy? He gives no hint about how his
> new "information" is stored.
> Does "red" still means only {'red'}, in which case I cannot fathom
> where we got any new information at all.
> Or, Does "red" now means the <||> closure of {'red'},
> that is {'red', 'non-blue'}, in which case I, for myself, can clearly
> see where some new information has been introduced and can even try
> to measure it according to the cardinality of the sets (or whatever,
> I did not really investigate this question).
> Of course, Hillman did not explicitely told us that this happens, but
> he does not need to, he provided a clean structure we can work with
> instead of murky "philosophic" considerations about "information".
> 
> > the crucial thing about peirce's idea is that it breaks the symmetry
> > of extension and intension and integrates them within the notion of
> > information.  apart from all the details of how right he got it,
> 
> Right, may be, useless for sure.
> If we get no mathematical model nor data structure what do we do?
> 
> > the big picture for us is this:  we want to know how theories
> > change and grow in contact with reality, more than just their
> > anatomy and physiology, their development and evolution over
> > the short and the long haul.  and we need a framework for
> > talking about theories, ontological or otherwise, that
> > allows us to think about the dynamics of inquiry.
> 
> Use R = subset of XxY and the closures <||> and |><|, you are in
> for a few surprises about how nastily the 'Content' slips under
> your feet when you add new instances of a concept with some
> missing or extraneous attributes (penguins not flying, black
> swans, etc...) but this will bring you an USEFULL insight into
> those questions not just interesting riddles.
> 
> Beyond page 5 Hillman paper is just developments about his favorite
> topics related to physic. We don't even need to care about Galois
> to make use of the closures in ontology building.

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