Logic As Semiotic

Peircers,

Here's a collection of notes on Peirce's concept of "Logic As Semiotic"
that I gathered together the last couple of times the subject came up
in various discussions on the web.

Jon

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

LAS.  Logic As Semiotic

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

LAS.  Note 1

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

| Logic, in its general sense, is, as I believe I have shown, only another
| name for 'semiotic' (Greek 'semeiotike'), the quasi-necessary, or formal,
| doctrine of signs.  By describing the doctrine as "quasi-necessary", or
| formal, I mean that we observe the characters of such signs as we know,
| and from such an observation, by a process which I will not object to
| naming Abstraction, we are led to statements, eminently fallible, and
| therefore in one sense by no means necessary, as to what 'must be' the
| characters of all signs used by a "scientific" intelligence, that is to
| say, by an intelligence capable of learning by experience.  As to that
| process of abstraction, it is itself a sort of observation.
|
| The faculty which I call abstractive observation is one which
| ordinary people perfectly recognize, but for which the theories
| of philosophers sometimes hardly leave room.  It is a familiar
| experience to every human being to wish for something quite
| beyond his present means, and to follow that wish by the
| question, "Should I wish for that thing just the same, if
| I had ample means to gratify it?"  To answer that question,
| he searches his heart, and in doing so makes what I term an
| abstractive observation.  He makes in his imagination a sort
| of skeleton diagram, or outline sketch, of himself, considers
| what modifications the hypothetical state of things would
| require to be made in that picture, and then examines it,
| that is, 'observes' what he has imagined, to see whether
| the same ardent desire is there to be discerned.  By such
| a process, which is at bottom very much like mathematical
| reasoning, we can reach conclusions as to what 'would be'
| true of signs in all cases, so long as the intelligence
| using them was scientific.
|
| Charles Sanders Peirce, 'Collected Papers', CP 2.227.
| Eds. Note.  "From an unidentified fragment, c. 1897".

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

LAS.  Note 2

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

| Logic is an analysis of forms not a study of the mind.
| It tells 'why' an inference follows not 'how' it arises
| in the mind.  It is the business therefore of the logician
| to break up complicated inferences from numerous premisses
| into the simplest possible parts and not to leave them
| as they are.
|
| C.S. Peirce, 'Chronological Edition', CE 1, p. 217.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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

LAS.  Note 3

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

| Some reasons having now been given for adopting the
| unpsychological conception of the science, let us now
| seek to make this conception sufficiently distinct to
| serve for a definition of logic.  For this purpose we
| must bring our 'logos' from the abstract to the concrete,
| from the absolute to the dependent.  There is no science
| of absolutes.  The metaphysical logos is no more to us
| than the metaphysical soul or the metaphysical matter.
| To the absolute Idea or Logos, the dependent or relative
| 'word' corresponds.  The word 'horse', is thought of as
| being a word though it be unwritten, unsaid, and unthought.
| It is true, it must be considered as having been thought;
| but it need not have been thought by the same mind which
| regards it as being a word.  I can think of a word in
| Feejee, though I can attach no definite articulation to
| it, and do not guess what it would be like.  Such a word,
| abstract but not absolute, is no more than the genus of
| all symbols having the same meaning.  We can also think
| of the higher genus which contains words of all meanings.
| A first approximation to a definition, then, will be that
| logic is the science of representations in general, whether
| mental or material.  This definition coincides with Locke's.
| It is however too wide for logic does not treat of all kinds
| of representations.  The resemblance of a portrait to its
| object, for example, is not logical truth.  It is necessary,
| therefore, to divide the genus representation according to
| the different ways in which it may accord with its object.
|
| The first and simplest kind of truth is the resemblance of a copy.
| It may be roughly stated to consist in a sameness of predicates.
| Leibniz would say that carried to its highest point, it would
| destroy itself by becoming identity.  Whether that is true or
| not, all known resemblance has a limit.  Hence, resemblance
| is always partial truth.  On the other hand, no two things
| are so different as to resemble each other in no particular.
| Such a case is supposed in the proverb that Dreams go by
| contraries, -- an absurd notion, since concretes have no
| contraries.  A false copy is one which claims to resemble
| an object which it does not resemble.  But this never fully
| occurs, for two reasons;  in the first place, the falsehood
| does not lie in the copy itself but in the 'claim' which is
| made for it, in the 'superscription' for instance;  in the
| second place, as there must be 'some' resemblance between
| the copy and its object, this falsehood cannot be entire.
| Hence, there is no absolute truth or falsehood of copies.
| Now logical representations have absolute truth and
| falsehood as we know 'à posteriori' from the law
| of excluded middle.  Hence, logic does not treat
| of copies.
|
| The second kind of truth, is the denotation of a sign,
| according to a previous convention.  A child's name, for
| example, by a convention made at baptism, denotes that person.
| Signs may be plural but they cannot have genuine generality because
| each of the objects to which they refer must have been fixed upon
| by convention.  It is true that we may agree that a certain sign
| shall denote a certain individual conception, an individual act
| of an individual mind, and that conception may stand for all
| conceptions resembling it;  but in this case, the generality
| belongs to the 'conception' and not to the sign.  Signs,
| therefore, in this narrow sense are not treated of in
| logic, because logic deals only with general terms.
|
| The third kind of truth or accordance of a representation
| with its object, is that which inheres in the very nature
| of the representation whether that nature be original or
| acquired.  Such a representation I name a 'symbol'.
|
| C.S. Peirce, 'Chronological Edition', CE 1, pp. 169-170.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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

LAS.  Note 4

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

| How often do we think of the thing in algebra?
| When we use the symbol of multiplication we do not
| even think out the conception of multiplication, we think
| merely of the laws of that symbol, which coincide with the
| laws of the conception, and what is more to the purpose,
| coincide with the laws of multiplication in the object.
| Now, I ask, how is it that anything can be done with
| a symbol, without reflecting upon the conception,
| much less imagining the object that belongs to it?
| It is simply because the symbol has acquired a nature,
| which may be described thus, that when it is brought before
| the mind certain principles of its use -- whether reflected on
| or not -- by association immediately regulate the action of the
| mind;  and these may be regarded as laws of the symbol itself
| which it cannot 'as a symbol' transgress.
|
| C.S. Peirce, 'Chronological Edition', CE 1, p. 173.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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

LAS.  Note 5

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

| Finally, these principles as principles applying not to this or that
| symbol, form, thing, but to all equally, must be universal.  And as
| grounds of possibility they must state what is possible.  Now what
| is the universal principle of the possible symbolization of symbols?
| It is that all symbols are symbolizable.  And the other principles
| must predicate the same thing of forms and things.
|
| These, then, are the three principles of inference.  Our next business is
| to demonstrate their truth.  But before doing so, let me repeat that these
| principles do not serve to prove that the kinds of inference are valid, since
| their own proof, on the contrary, must rest on the assumption of that validity.
| Their use is only to show what the condition of that validity is.  Hence, the
| only proof of the truth of these principles is this;  to show, that if these
| principles be admitted as sufficient, and if the validity of the several kinds
| of inference be also admitted, that then the truth of these principles follows
| by the respective kinds of inference which each establishes.
|
| C.S. Peirce, 'Chronological Edition', CE 1, pp. 184-185.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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

LAS.  Note 6

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

| To prove then, first, that all symbols are symbolizable.
| Every syllogism consists of three propositions with two terms
| each, a subject and a predicate, and three terms in all each term
| being used twice.  It is obvious that one term must occur both as
| subject and predicate.  Now a predicate is a symbol of its subject.
| Hence in all reasoning 'à priori' a symbol must be symbolized.
| But as reasoning 'à priori' is possible about a statement
| without reference to its predicate, all symbols must be
| symbolizable.
|
| 2nd To prove that all forms are symbolizable.
| Since this proposition relates to pure form it is
| sufficient to show that its consequences are true.
| Now the consequence will be that if a symbol of any
| object be given, but if this symbol does not adequately
| represent any form then another symbol more formal may
| always be substituted for it, or in other words as soon
| as we know what form it ought to symbolize the symbol may
| be so changed as to symbolize that form.  But this process
| is a description of inference 'à posteriori'.  Thus in the
| example relating to light;  the symbol of "giving such and
| such phenomena" which is altogether inadequate to express a
| form is replaced by "ether-waves" which is much more formal.
| The consequence then of the universal symbolization of forms
| is the inference 'à posteriori', and there is no truth or
| falsehood in the principle except what appears in the
| consequence.  Hence, the consequence being valid,
| the principle may be accepted.
|
| 3rd To prove that all things may be symbolized.
| If we have a proposition, the subject of which is not
| properly a symbol of the thing it signifies;  then in case
| everything may be symbolized, it is possible to replace this
| subject by another which is true of it and which does symbolize
| the subject.  But this process is inductive inference.  Thus having
| observed of a great variety of animals that they all eat herbs, if I
| substitute for this subject which is not a true symbol, the symbol
| "cloven-footed animals" which is true of these animals, I make an
| induction.  Accordingly I must acknowledge that this principle
| leads to induction;  and as it is a principle of objects,
| what is true of its subalterns is true of it;  and since
| induction is always possible and valid, this principle
| is true.
|
| C.S. Peirce, 'Chronological Edition', CE 1, pp. 185-186.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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

LAS.  Note 7

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

| Having discovered and demonstrated the grounds of the possibility of
| the three inferences, let us take a preliminary glance at the manner in
| which additions to these principles may make them grounds of proceedure.
|
| The principle of inference 'à priori' has been apodictically demonstrated;
| the principle of inductive inference has been shown upon sufficient evidence
| to be true;  the principle of inference 'à posteriori' has been shown to be one
| which nothing can contradict.  These three degrees of modality in the principles of
| the three inferences show the amount of certainty which each is capable of affording.
| Inference 'à priori' is as we all know the only apodictic proceedure;  yet no one
| thinks of questioning a good induction;  while inference 'à posteriori' is
| proverbially uncertain.  'Hypotheses non fingo', said Newton;  striving
| to place his theory on a firm inductive basis.  Yet provisionally we
| must make hypotheses;  we start with them;  the baby when he lies
| turning his fingers before his eyes is testing a hypothesis he has
| already formed, as to the connection of touch and sight.  Apodictic
| reasoning can only be applied to the manipulation of our knowledge;
| it never can extend it.  So that it is an induction which eventually
| settles every question of science;  and nine-tenths of the inferences
| we draw in any hour not of study are of this kind.
|
| C.S. Peirce, 'Chronological Edition', CE 1, p. 186.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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

LAS.  Note 8

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

| The first distinction we found it necessary to draw --
| the first set of conceptions we have to signalize --
| forms a triad
|
|    Thing   Representation   Form.
|
| Kant you remember distinguishes in all mental representations the
| matter and the form.  The distinction here is slightly different.
| In the first place, I do not use the word 'Representation' as
| a translation of the German 'Vorstellung' which is the general
| term for any product of the cognitive power.  Representation,
| indeed, is not a perfect translation of that term, because it
| seems necessarily to imply a mediate reference to its object,
| which 'Vorstellung' does not.  I however would limit the term
| neither to that which is mediate nor to that which is mental,
| but would use it in its broad, usual, and etymological sense
| for anything which is supposed to stand for another and which
| might express that other to a mind which truly could understand
| it.  Thus our whole world -- that which we can comprehend -- is
| a world of representations.
|
| No one can deny that there are representations, for every thought is one.
| But with 'things' and 'forms' scepticism, though still unfounded, is at first
| possible.  The 'thing' is that for which a representation might stand prescinded
| from all that would constitute a relation with any representation.  The 'form' is
| the respect in which a representation might stand for a thing, prescinded from both
| thing and representation.  We thus see that 'things' and 'forms' stand very differently
| with us from 'representations'.  Not in being prescinded elements, for representations
| also are prescinded from other representations.  But because we know representations
| absolutely, while we only know 'forms' and 'things' through representations.  Thus
| scepticism is possible concerning 'them'.  But for the very reason that they are
| known only relatively and therefore do not belong to our world, the hypothesis
| of 'things' and 'forms' introduces nothing false.  For truth and falsity only
| apply to an object as far as it can be known.  If indeed we could know things
| and forms in themselves, then perhaps our representations of them might
| contradict this knowledge.  But since all that we know of them we know
| through representations, if our representations be consistent they
| have all the truth that the case admits of.
|
| C.S. Peirce, 'Chronological Edition', CE 1, pp. 256-257.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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

LAS.  Note 9

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

| We found representations to be of three kinds
|
|    Signs   Copies   Symbols.
|
| By a 'copy', I mean a representation whose agreement with
| its object depends merely upon a sameness of predicates.
|
| By a 'sign', I mean a representation whose reference to
| its object is fixed by convention.
|
| By a 'symbol', I mean one which upon being presented to the mind --
| without any resemblance to its object and without any reference to
| a previous convention -- calls up a concept.  I consider concepts,
| themselves, as a species of symbols.
|
| A symbol is subject to three conditions.  First it must represent an object,
| or informed and representable thing.  Second it must be a manifestation of
| a 'logos', or represented and realizable form.  Third it must be translatable
| into another language or system of symbols.
|
| The science of the general laws of relations of symbols to logoi is general grammar.
| The science of the general laws of their relations to objects is logic.  And the
| science of the general laws of their relations to other systems of symbols is
| general rhetoric.
|
| C.S. Peirce, 'Chronological Edition', CE 1, pp. 257-258.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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

LAS.  Note 10

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

| When have then three different kinds of inference.
|
|    Deduction or inference 'à priori',
|
|    Induction or inference 'à particularis', and
|
|    Hypothesis or inference 'à posteriori'.
|
| It is necessary now to examine this classification critically.
|
| And first let me specify what I claim for my invention.  I do not claim that it is
| a natural classification, in the sense of being right while all others are wrong.
| I do not know that such a thing as a natural classification is possible in the
| nature of the case.  The science which most resembles logic is mathematics.
| Now among mathematical forms there does not seem to be any natural classification.
| It is true that in the solutions of quadratic equations, there are generally two
| solutions from the positive and negative values of the root with an impossible
| gulf between them.  But this classing is owing to the forms being restricted
| by the conditions of the problem;  and I believe that all natural classes arise
| from some problem -- something which was to be accomplished and which could be
| accomplished only in certain ways.  Required to make a musical instrument;
| you must set either a plate or a string in vibration.  Required to make
| an animal;  it must be either a vertebrate, an articulate, a mollusk, or
| a radiate.  However this may be, in Geometry we find ourselves free to make
| several different classifications of curves, either of which shall be equally
| good.  In fact, in order to make any classification of them whatever we must
| introduce the purely arbitrary element of a system of coördinates or something
| of the kind which constitutes the point of view from which we regard the curves
| and which determines their classification completely.  Now it may be said that
| one system of coördinates is more 'natural' than another;  and it is obvious
| that the conditions of binocular vision limit us in our use of our eyes to
| the use of particular coördinates.  But this fact that one such system
| is more natural to us has clearly nothing to do with pure mathematics
| but is merely introducing a problem;  given two eyes, required to form
| geometrical judgements, how can we do it?  In the same way, I conceive
| that the syllogism is nothing but the system of coördinates or method of
| analysis which we adopt in logic.  There is no reason why arguments should
| not be analyzed just as correctly in some other way.  It is a great mistake to
| suppose that arguments as they are thought are often syllogisms, but even if this
| were the case it would have no bearing upon pure logic as a formal science.  It is
| the principal business of the logician to analyze arguments into their elements just
| as it is part of the business of the geometer to analyze curves;  but the one is no
| more bound to follow the natural process of the intellect in his analysis, than the
| other is bound to follow the natural process of perception.
|
| C.S. Peirce, 'Chronological Edition', CE 1, pp. 267-268.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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

LAS.  Logic As Semiotic

Ontology List -- Aug 2001

00.  http://suo.ieee.org/ontology/thrd48.html#03070
01.  http://suo.ieee.org/ontology/msg03070.html
02.  http://suo.ieee.org/ontology/msg03171.html
03.  http://suo.ieee.org/ontology/msg03178.html
04.  http://suo.ieee.org/ontology/msg03179.html
05.  http://suo.ieee.org/ontology/msg03184.html
06.  http://suo.ieee.org/ontology/msg03187.html
07.  http://suo.ieee.org/ontology/msg03189.html
08.  http://suo.ieee.org/ontology/msg03190.html
09.  http://suo.ieee.org/ontology/msg03192.html
10.  http://suo.ieee.org/ontology/msg03193.html

Inquiry List -- Dec 2004

00.  http://stderr.org/pipermail/inquiry/2004-December/thread.html#2219
01.  http://stderr.org/pipermail/inquiry/2004-December/002219.html
02.  http://stderr.org/pipermail/inquiry/2004-December/002220.html
03.  http://stderr.org/pipermail/inquiry/2004-December/002221.html
04.  http://stderr.org/pipermail/inquiry/2004-December/002222.html
05.  http://stderr.org/pipermail/inquiry/2004-December/002223.html
06.  http://stderr.org/pipermail/inquiry/2004-December/002224.html
07.  http://stderr.org/pipermail/inquiry/2004-December/002225.html
08.  http://stderr.org/pipermail/inquiry/2004-December/002226.html
09.  http://stderr.org/pipermail/inquiry/2004-December/002227.html
10.  http://stderr.org/pipermail/inquiry/2004-December/002228.html

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

facebook page: https://www.facebook.com/JonnyCache
inquiry list: http://stderr.org/pipermail/inquiry/
mwb: http://www.mywikibiz.com/Directory:Jon_Awbrey
knol profile: http://knol.google.com/k/Jon-Awbrey#
oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
polmic: www.policymic.com/profiles/1110/Jon-Awbrey