ONT Re: Extension x Comprehension = Information
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Note 82
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TG = Tom Gollier
Tom,
Yes, I do see a number of problems here.
Usually when that happens it turns out to
be me and not Peirce that has the problem,
but right at the moment I am still trying
to stumble through several different ways
that I can see to deal with the issues.
When I was a kid, words like "denotative" and "connotative" were things that
we learned in English Literature classes and not among the bit of logic that
we got in Mathematics courses, so that is what comes first to mind when you
speak of the "traditional sense" of these words. What I got was something
like the "denotative meaning" being whatever a text refers to in the world
whereas the "connotative meaning" was just about any other notion that the
text brings to mind. So the first problem I see is with this "notion" --
that must be a "concept", which means a "mental symbol" to us Persians.
But maybe my English Lit teachers were not Persians, and they intended
us to understand the "intension" of the concept, all of which intents
and purposes are meant to be summoned together into a "comprehension".
And that appears to be just about where I'm stuck for the moment.
As a matter of fact, I was last night puzzling over this passage:
| It is important to distinguish between the two functions of a word:
| 1st to denote something -- to stand for something, and 2nd to mean
| something -- or as Mr. Mill phrases it -- to 'connote' something.
|
| What it denotes is called its 'Sphere'.
| What it connotes is called its 'Content'.
| Thus the 'sphere' of the word 'man' is for
| me every man I know; and for each of you it
| is every man you know. The 'content' of 'man'
| is all that we know of all men, as being two-
| legged, having souls, having language, &c., &c.
| It is plain that both the 'sphere' and the
| 'content' admit of more and less. ...
|
| Now the sphere considered as a quantity is called the Extension;
| and the content considered as quantity is called the Comprehension.
| Extension and Comprehension are also termed Breadth and Depth. So that
| a wider term is one which has a greater extension; a narrower one is
| one which has a less extension. A higher term is one which has a
| less Comprehension and a lower one has more.
|
| The narrower term is said to be contained under the wider one;
| and the higher term to be contained in the lower one.
|
| We have then:
|
| o-----------------------------o-----------------------------o
| | | |
| | What is 'denoted' | What is 'connoted' |
| | | |
| | Sphere | Content |
| | | |
| | Extension | Comprehension |
| | | |
| | ( wider | ( lower |
| | Breadth < | Depth < |
| | ( narrower | ( higher |
| | | |
| | What is contained 'under' | What is contained 'in' |
| | | |
| o-----------------------------o-----------------------------o
|
| CSP, CE 1, pages 459-460.
Is there anything here that forces us to read the "connotation"
as exclusively either the symbol, whether in the text or in the
mind, or as the objective intension, the hypostatic abstraction?
I just don't know yet.
Anyway, I notice that there is a lot more about this in the
previous year's lectures, so I will append a sample of that.
TG: There is certainly a distinction to be made between:
TG: 1. The traditional sense of "connotation" as involving
categories, predicates, some more general (higher),
some less general (lower), such that a several of
them might already take the connotation to a higher
level where adding more qualifiers would take it
no higher (that, say, "men, horses, kangaroos, and
whales have no attributes in common which are not
possessed by the entire class of mammals").
TG: 2. And another sense of "connotation" where conjoining terms further
restricts the extension of what is denoted regardless how many
are added (a 'spherical bright fragrant juicy tropical fruit').
Personally, I don't think the former, the idea of defining
connotation in terms of hierarchially organized systems,
is really tenable these days. Many such systems, each
of which is in and of itself, as "valid" as any other,
are possible while there are also more logical ways of
organizing them than just in an implicative hierarchy.
Further, as well versed as Peirce was in the traditional
view of categories, his own categories do not appear to be
of that nature.
TG: Of course, both kinds of examples involve connotation in the sense
there is no other way of denoting an extension other than by means
of the connotation (the rheme being in the form of "____ term").
That's why I like Peirce's definition of the proposition as
separately indicating its object (which did perhaps come later
in his career). The subject terms do nothing predicate terms
do not do -— each is a rheme -— but in doing what rhemes do
separately, in taking one as denotatively indicating what
might go in the blank and the other as connoatively saying
something about whatever does go in the blank, they give
rise to propositions. The distinction between denotation
and connotation -— while implict in terms in the sense
that each can be taken denotatively or connotatively -—
is made explicit only by combining two terms on the
level of propositions. And, dare we go even further,
and say the two are made explicit in one term, the
middle term, a concept, only on the level of an
argument?
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Note 34
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If you dreamed that this inquiry had come full circle then I inform
you of what you already know, that there are always broader circles.
I revert to Peirce's Harvard University Lectures of the year before,
to pick up additional background material and a bit more motivation.
| We are already familiar with the distinction between the extension and
| comprehension of terms. A term has comprehension in virtue of having
| a meaning and has extension in virtue of being applicable to objects.
| The meaning of a term is called its 'connotation'; its applicability
| to things its 'denotation'. Every symbol 'denotes' by 'connoting'.
| A representation which 'denotes' without connoting is a mere 'sign'.
| If it 'connotes' without thereby 'denoting', it is a mere copy.
|
| It is universally held that extension and comprehension
| are in reciprocal relation; thus if 'horse' be divided
| into 'black horse' and 'non-black horse', 'black horse'
| has more intension and therefore less extension than
| 'horse'.
|
| It behooves me to say what the distinction between extension and
| comprehension is upon my view of logic. Before doing so, however,
| I must remark that the distinction extends to propositions; there
| are extensive and intensive propositions.
|
| An extensive proposition is defined to be one which
| states the relation between the extension of two terms.
|
| An intensive proposition is one which states the relation
| between the intension or comprehension of two terms.
|
| Subordination in extension is expressed by the term 'contained under'.
|
| Subordination in intension is expressed by the term 'contained in'.
|
| Hence in the case of affirmatives;
| an extensive judgment is expressed
| by the formula
|
| 'A' is contained under 'B'
|
| an equivalent intensive proposition
| by the formula
|
| 'B' is contained in 'A'.
|
| Thus 'black horse' is contained under 'horse',
| and 'horse' [is contained in 'black horse'].
|
| CSP, CE 1, page 272.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 35
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| What we have to distinguish, therefore, is not so much the
| quantity of extension from the quantity of intension as it
| is the object of connotation from the object of denotation.
| In analytical judgments there is no denotation at all. In
| a synthetical judgment the subject is an object of denotation.
Nota Bene. In the Table below the form "XY" indicates a premiss of
a classical syllogism where X is the subject and Y is the predicate.
Also, I suspect that the Third Figure syllogism ought to be XY & XZ.
| o----------------------o-------------------------o-------------------o
| | | | |
| | | ( Subject: O of C | ( XY |
| | Analytic | < | 2nd Fig. < |
| | | ( Predicate: O of C | ( ZY |
| | | | |
| o----------------------o-------------------------o-------------------o
| | | | |
| | | ( Subject: O of D | ( YX |
| | Synthetic Intensive | < | 1st Fig. < |
| | | ( Predicate: O of C | ( ZY |
| | | | |
| o----------------------o-------------------------o-------------------o
| | | | |
| | | ( Subject: O of D | ( YX |
| | Extensive | < | 3rd Fig. < |
| | | ( Predicate: O of D | ( ZX |
| | | | |
| o----------------------o-------------------------o-------------------o
|
| There cannot be a judgment whose subject is an object of connotation and
| whose predicate is an object of denotation. For a symbol 'denotes' by virtue
| of 'connoting' and not 'vice versa', hence the object of connotation determines
| the object of denotation and not 'vice versa', in the sense in which the subject
| of a proposition is the term determined and the predicate is the determining term.
| Whence if one of the terms is an object of connotation and the other is an object
| of denotation, the latter is the subject and not the former.
|
| In the other two cases, there is no difference between subject and predicate;
| except that one may be regarded as taken first.
|
| Thus these cases in which both terms are of the same kind are two kinds of
| twists of the first kind, just as the 2nd and 3rd Figures of Syllogism are
| right-handed and left-handed twists of the 1st. This is expressed in the
| above Table.
|
| A proposition would usually be called intensive if its
| predicate were an object of connotation; hence we have
| three kinds of propositions given by these two; namely,
|
| Analytic.
|
| Synthetic Intensive.
|
| Extensive.
|
| There is no such thing as an analytic extensive proposition.
| For an analytic proposition containing no object of denotation
| is merely the expression of a relation of comprehension. Of course
| from an analytic proposition a synthetic one may be immediately inferred.
| From
|
| Man is mortal
|
| we may infer
|
| All men are mortals
|
| but the predicate 'mortals' is not a mere result of the analysis of 'men'.
| I have here slightly narrowed Kant's definition of the analytic judgment so
| as to make it not merely needless but impossible to test one by experience.
|
| CSP, CE 1, pages 272-274.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 36
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| We come now to an objection to the division of propositions
| which I have just given which will require us to examine the
| matter somewhat more deeply. It may be said: the copula in
| all cases establishes an identity between two terms. Hence
| as in one of the propositions the object of denotation is
| the subject and the object of connotation the predicate,
| these two objects are identical and hence the division
| into three kinds is a distinction without a difference.
|
| In order to answer this objection we must revert to that distinction
| between 'thing', 'image', and 'form' established in the lecture upon
| the definition of logic. A representation is anything which may be
| regarded as standing for something else. Matter or thing is that
| for which a representation might stand prescinded from all that
| could constitute a relation with any representation. A form is
| the relation between a representation and thing prescinded from
| both representation and thing. An image is a representation
| prescinded from thing and form.
|
| Derived directly from this abstractest triad was another less abstract.
| This is Object--Equivalent-Representation--Logos. The 'object' is
| a thing corresponding to a representation regarded as actual.
| The equivalent representation is a representation in any
| language equivalent to a representation regarded
| as actual. A Logos is a form constituting
| the relation between an object and a
| representation regarded as actual.
|
| Every symbol may be said in three different senses to be determined by its
| 'object', its 'equivalent representation', and its 'logos'. It stands for
| its 'object', it translates its 'equivalent representation', it realizes
| its 'logos'.
|
| CSP, CE 1, page 274.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 37
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| Every symbol may be said in three different senses
| to be determined by its 'object', its 'equivalent
| representation', and its 'logos'. It stands for
| its 'object', it translates its 'equivalent
| representation', it realizes its 'logos'.
|
| As every symbol is determined in these three ways, Symbols, as such,
| are subject to three laws one of which is the 'conditio sine qua non'
| of its standing for anything, the second of its translating anything,
| and the third of its realizing anything. The first law is Logic, the
| second Universal Rhetoric, the third Universal Grammar.
|
| But an object is a thing informed and represented.
| An equivalent representation is an image which is
| itself represented and realized, and a logos is
| a form, embodied in an object and representation.
|
| Hence the object of a symbol implies in itself both thing, form, and image.
| And hence regarded as containing one or other of these three elements it may be
| distinguished as 'material object', 'formal object', and 'representative object'.
| Now so far as the object of a symbol contains the 'thing', so far the symbol
| stands for something and so far it denores. So far as its object embodies
| a form, so far the symbol has a meaning and so far it connotes. Thus we see
| that the 'denotative object' and the 'connotative object' are in fact identical;
| and therefore an analytic, an intensive synthetic, and an extensive proposition
| may all represent the same fact and yet the mode in which they are obtained and
| the relation of the proposition to that fact are necessarily very different.
|
| CSP, CE 1, pages 274-275.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 38
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| But since the object contains three elements,
| thing, image, form, we ought to have another kind
| of object besides the denotative and connotative.
| What is this?
|
| If we suppose ourselves to know no more of man than what is
| contained in the definition Man is the rational animal, then
| we might divide man into 'man risible' and 'man non-risible'.
|
| man
| ___________________|___________________
| / \
| man risible man non-risible
|
| And then the connotation of 'man' would be less than that
| of either 'man risible' or 'man non-risible'. And conversely
| 'man risible' and 'man non-risible' would have a less extension
| than 'man'. But we afterwards find that the class 'man non-risible'
| does not exist and is impossible. Henceforward the idea of man and
| that of risible man are changed. The 'extension' of risible man has
| become equal to that of 'men' and the comprehension of 'man' has become
| equal to that of 'risible man'. And how has this change in the relations
| of the terms been effected?
|
| Before the information we knew (let us say) that there were certain risible
| men whom we may denote by 'A' and there were other men who might or might not
| be risible whom we will denote by 'BB’'. We have now found that 'BB’' are also
| risible. When we said all men before we meant 'A + B + B’'; when we say all men
| now we mean the same. The extension of 'man' then has not changed. When we said
| risible men before we denoted 'A + B?' that is to say the whole of 'A' but none
| of 'B' for certain; but now when we say risible men we denote 'A + B + B’'.
| Hence the extension of risible men has 'increased', so as to become equal to
| that of 'men'. On the other hand the intension of 'risible man' is now as
| it was before, composed of 'risible', 'rational', and 'animal'; while the
| comprehension of 'man' which before contained only 'rational' and 'animal',
| now contains 'risible' also.
|
| CSP, CE 1, pages 275-276.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 39
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| Thus the process of information disturbs the relations
| of extension and comprehension for a moment and the class
| which results from the equivalence of two others has a greater
| intension than one and a greater extension than the other. Hence,
| we may conveniently alter the formula for the relations of extension
| and comprehension; thus, instead of saying that one is the reciprocal
| of the other or
|
| comprehension x extension = constant
|
| we may say
|
| comprehension x extension = information.
|
| We see then that all symbols besides their denotative and
| connotative objects have another; their informative object.
| The denotative object is the total of possible things denoted.
| The connotative object is the total of symbols translated or implied.
| The informative object is the total of forms manifested and is measured
| by the amount of intension the term has, over and above what is necessary
| for limiting its extension. For example the denotative object of 'man' is
| such collections of matter the word knows while it knows them i.e. while they
| are organized. The connotative object of 'man' is the total form which the word
| expresses. The informative object of 'man' is the total fact which it embodies;
| or the value of the conception which is its equivalent symbol.
|
| CSP, CE 1, page 276.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 40
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| Abstract words such as 'truth', 'honor', by the way, are somewhat difficult
| to understand. It seems to me that they are simply fictions. Every word
| must denote some 'thing'; these are names for certain fictitious things
| which are supposed for the purpose of indicating that the object of
| a concrete term is meant as it would be did it contain either no
| information or a certain amount of information. Thus "charity
| is a virtue" means "What is charitable is virtuous -- by the
| definition of charity and not by reason of what is known
| about it". Hence, only analytical propositions are
| possible of abstract terms; and on this account
| they are peculiarly useful in metaphysics
| where the question is what can we know
| without any information.
|
| CSP, CE 1, pages 276-277.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 41
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| Coming back now to propositions, we should first remark that just as the
| framing of a term is a process of symbolization so also is the framing
| of a proposition. No proposition is supposed to leave its terms as it
| finds them. Some symbol is determined by every proposition. Hence,
| since symbols are determined by their objects; and there are three
| objects of symbols, the connotative, denotative, informative; it
| follows that there will be three kinds of propositions, such as
| alter the denotation, the information, and the connotation of
| their terms respectively. But when information is determined
| both connotation and information [?] are determined; hence
| the three kinds will be 1st Such as determine connotation,
| 2nd Such as determine denotation, 3rd Such as determine
| both denotation and connotation.
|
| CSP, CE 1, page 277.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 42
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| The difference between connotation, denotation, and information
| supplies the basis for another division of terms and propositions;
| a division which is related to the one we have just considered in
| precisely the same way as the division of syllogism into 3 figures
| is related to the division into Deduction, Induction, and Hypothesis.
| Every symbol which has connotation and denotation has also information.
| For by the denotative character of a symbol, I understand application
| to objects implied in the symbol itself. The existence therefore of
| objects of a certain kind is implied in every connotative denotative
| symbol; and this is information.
|
| Now there are certain imperfect or false symbols produced
| by the combination of true symbols which have lost either
| their denotation or their connotation.
|
| When symbols are combined together in extension,
| as for example in the compound term "cats and dogs",
| their sum possesses denotation but no connotation
| or at least no connotation which determines their
| denotation. Hence, such terms, which I prefer to
| call 'enumerative' terms, have no information, and
| it remains unknown whether there be any real kind
| corresponding to cats and dogs taken together.
|
| On the other hand, when symbols are combined together in
| comprehension, as for example in the compound "tailed men",
| the product possesses connotation but no denotation, it not
| being therein implied that there may be any 'tailed men'.
| Such conjunctive terms have therefore no information.
|
| Thirdly, there are names purporting to be of real kinds,
| as 'men'; and these are perfect symbols.
|
| Enumerative terms are not truly symbols but only signs;
| and Conjunctive terms are copies; but these copies and
| signs must be considered in symbolistic because they are
| composed of symbols.
|
| When an enumerative term forms the subject of a grammatical proposition,
| as when we say "cats and dogs have tails", there is no logical unity in the
| proposition at all. Logically, therefore, it is two propositions and not one.
| The same is the case when a conjunctive proposition forms the predicate of a
| sentence; for to say "hens are feathered bipeds" is simply to predicate two
| unconnected marks of them.
|
| When an enumerative term as such is the predicate of a proposition, that
| proposition cannot be a denotative one, for a denotative proposition is one
| which merely analyzes the denotation of its predicate, but the denotation of
| an enumerative term is analyzed in the term itself; hence if an enumerative
| term as such were the predicate of a proposition, that proposition would be
| equivalent in meaning to its own predicate.
|
| On the other hand, if a conjunctive term as such is the subject of a proposition,
| that proposition cannot be connotative, for the connotation of a conjunctive term
| is already analyzed in the term itself, and a connotative proposition does no more
| than analyze the connotation of its subject.
|
| Thus, we have
|
| Conjunctive, Simple, Enumerative
|
| propositions so related to
|
| Denotative, Informative, Connotative
|
| propositions that what is on the left hand of
| one line cannot be on the right hand of the other.
|
| CSP, CE 1, pages 278-279.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 43
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| We are now in a condition to discuss the question of the grounds
| of scientific inference. The problem naturally divides itself
| into parts: 1st To state and prove the principles upon which
| the possibility in general of each kind of inference depends,
| 2nd To state and prove the rules for making inferences in
| particular cases.
|
| The first point I shall discuss in the remainder of this lecture;
| the second I shall scarcely be able to touch upon in these lectures.
|
| Inference in general obviously supposes symbolization; and all
| symbolization is inference. For every symbol as we have seen
| contains information. And in the last lecture we saw that
| all kinds of information involve inference. Inference,
| then, is symbolization. They are the same notions.
| Now we have already analyzed the notion of a 'symbol',
| and we have found that it depends upon the possibility
| of representations acquiring a nature, that is to say
| an immediate represenative power. This principle is
| therefore the ground of inference in general.
|
| CSP, CE 1, pages 279-280.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 44
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| But there are three distinct kinds of inference;
| inconvertible and different in their conception.
| There must, therefore, be three different principles
| to serve for their grounds. These three principles
| must also be indemonstrable; that is to say, each
| of them so far as it can be proved must be proved
| by means of that kind of inference of which it is
| the ground. For if the principle of either kind of
| inference were proved by another kind of inference,
| the former kind of inference would be reduced to the
| latter; and since the different kinds of inference are
| in all respects different this cannot be. You will say
| that it is no proof of these principles at all to support
| them by that which they themselves support. But I take it
| for granted at the outset, as I said at the beginning of my
| first lecture, that induction and hypothesis have their own
| validity. The question before us is 'why' they are valid.
| The principles, therefore, of which we are in search,
| are not to be used to prove that the three kinds
| of inference are valid, but only to show how
| they come to be valid, and the proof of
| them consists in showing that they
| determine the validity of the
| three kinds of inference.
|
| CSP, CE 1, page 280.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 45
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| But these three principles must have this in common that they refer to
| 'symbolization' for they are principles of inference which is symbolization.
| As grounds of the possibility of inference they must refer to the possibility
| of symbolization or symbolizability. And as logical principles they must relate
| to the reference of symbols to objects; for logic has been defined as the science
| of the general conditions of the relations of symbols to objects. But as three
| different principles they must state three different relations of symbols
| to objects. Now we have already found that a symbol has three different
| relations of objects; namely connotation, denotation, and information
| which are its relations to the object considered as a thing, a form,
| and an equivalent representation. Hence, it is obvious that these
| three principles must relate to the symbolizability of things, of
| forms, and of symbols.
|
| Our next business is to find which is which.
|
| CSP, CE 1, pages 280-281.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 46
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| Our next business is to find which is which.
| For this purpose we must consider that each
| principle is to be proved by the kind of
| inference which it supports.
|
| The ground of deductive inference then must be established deductively;
| that is by reasoning from determinant to determinate, or in other words
| by reasoning from definition. But this kind of reasoning can only be
| applied to an object whose character depends upon its definition.
| Now of most objects it is the definition which depends upon the
| character; and so the definition must therefore itself rest on
| induction or hypothesis. But the principle of deduction must
| rest on nothing but deduction, and therefore it must relate
| to something whose character depends upon its definition.
| Now the only objects of which this is true are symbols;
| they indeed are created by their definition; while
| neither forms nor things are. Hence, the principle
| of deduction must relate to the symbolizability of
| of symbols.
|
| The principle of hypothetic inference must be established hypothetically,
| that is by reasoning from determinate to determinant. Now it is clear that
| this kind of reasoning is applicable only to that which is determined by what
| it determines; or that which is only subject to truth and falsehood so far as
| its determinate is, and is thus of itself pure 'zero'. Now this is the case
| with nothing whatever except the pure forms; they indeed are what they
| are only in so far as they determine some symbol or object. Hence the
| principle of hypothetic inference must relate to the symbolizability
| of forms.
|
| The principle of inductive inference must be established inductively,
| that is by reasoning from parts to whole. This kind of reasoning can
| apply only to those objects whose parts collectively are their whole.
| Now of symbols this is not true. If I write 'man' here and 'dog' here
| that does not constitute the symbol of 'man and dog', for symbols have
| to be reduced to the unity of symbolization which Kant calls the unity
| of apperception and unless this be indicated by some special mark they
| do not constitute a whole. In the same way forms have to determine the
| same matter before they are added; if the curtains are green and the
| wainscot yellow that does not make a 'yellow-green'. But with things
| it is altogether different; wrench the blade and handle of a knife
| apart and the form of the knife has disappeared but they are the
| same thing -- the same matter -- that they were before. Hence,
| the principle of induction must relate to the symbolizability
| of things.
|
| All these principles must as principles be universal.
| Hence they are as follows:--
|
| All things, forms, symbols are symbolizable.
|
| CSP, CE 1, pages 281-282.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 47
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| All these principles must as principles be universal.
| Hence they are as follows:--
|
| All things, forms, symbols are symbolizable.
|
| The next step is to prove each of these principles.
| First then, to prove deductively that all symbols are
| symbolizable. In every syllogism there is a term which
| is predicate and subject. But a predicate is a symbol
| of its subject. Hence, in every deduction a symbol is
| symbolized. Now deduction is valid independently of
| the matter of the judgment. Hence all symbols are
| symbolizable.
|
| Next; to prove inductively that all things are symbolizable.
| For this purpose we must take all the collocations of things we
| can and judge by them. Now all these collocations of things have
| been selected upon some principle; this principle of selection is
| a predicate of them and a 'concept'. Being a concept it is a symbol.
| And it partakes of that peculiarity of symbols that it must have
| information. We have no concepts which do not denote some things
| as well as connoting; because all our thought begins with experience.
| But a symbol which has connotation and denotation contains information.
| Whatever symbol contains information contains more connotation than is
| necessary to limit its possible denotation to those things which it
| may denote. That is every symbol contains more than is sufficient
| for a principle of selection. Hence every selected collocation of
| things must have something more than a mere principle of selection,
| it must have another common quality. Now by induction this common
| quality may be predicated of the whole possible denotation of the
| concept which serves as principle of selection. And thus every
| collocation of things we can select is symbolized by its principle
| of selection. Now by induction we pass from this statement that all
| things we can take are symbolizable to the principle that all things
| are symbolzable. Q.E.D. This argument though inductive in form is
| of the highest possible validity, for no case can possibly arise to
| contradict it.
|
| Thirdly, we have to prove hypothetically that all forms are symbolizable.
| For this purpose we must consider that 'forms' are nothing unless they
| are embodied, and then they constitute the synthesis of the matter.
| Hence the knowledge of them cannot be directly given but must be
| obtained by hypothesis. Now we have to explain this fact, that
| all forms are to be regarded as subjects for hypothesis, by a
| hypothesis. For this purpose, we should reflect that whatever
| is symbolizable is symbolized by terms and their combinations.
| Now we saw at the last lecture that the process of obtaining
| a new term is a hypothetic inference. So that everything
| which is symbolizable is to be regarded as a subject for
| hypothesis. This accounts for the same thing being true
| of forms, if we make the hypothesis that all forms are
| symbolizable. Q.E.D. This argument though only an
| hypothesis could not have been stronger for the
| conclusion involves no matter of fact at all.
|
| CSP, CE 1, pages 282-283.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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Note 48
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| Thus the three grounds of inference are proved.
| All have been made certain. But the manner in
| which they have attained to certainty indicates
| a very different general strength of the three
| kinds of inference.
|
| The hypothetic argument became certain only by speaking of
| that which has no sense except when this principle is true.
|
| The inductive argument became certain only by taking into
| account all that could possibly be known.
|
| The deductive argument alone was strictly demonstrative.
|
| Thus we have in order of strength Deduction, Induction, Hypothesis.
| Deduction, in fact, is the only demonstration; yet no one thinks of
| questioning a good induction, while hypothesis is proverbially dangerous.
| 'Hypotheses non fingo', said Newton, striving to place his theory on a basis
| of strict induction. Yet it is hypotheses with which we must start; the baby
| when he lies turning his fingers before his eyes is making a hypothesis as to
| the connection of what he sees and what he feels. Hypotheses give us our facts.
| Induction extends our knowledge. Deduction makes it distinct.
|
| CSP, CE 1, page 283.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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