Difference between revisions of "Directory:Jon Awbrey/Papers/Information = Comprehension × Extension"

Another angle from which to approach the incidence of [[sign]]s and [[inquiry]] is by way of [[Charles Sanders Peirce|Peirce]]'s "[[laws of information]]" and the corresponding theory of information that he developed from the time of his lectures on the "Logic of Science" at Harvard University (1865) and the Lowell Institute (1866).

When it comes to the supposed reciprocity between [[extension (logic)|extension]]s and [[intension (logic)|intension]]s, Peirce, of course, has another idea, and I would say a better idea, in part, because it forms the occasion for him to bring in his new-fangled notion of "[[information]]" to mediate the otherwise static dualism between the other two. The development of this novel idea brings Peirce to enunciate this formula:

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<math>\operatorname{Information} = \operatorname{Comprehension} \times \operatorname{Extension}</math>

==Selections from Peirce's "Logic of Science" (1865&ndash;1866)==

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<p>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'' &mdash; ''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''.</p>

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<p>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.</p>

<p>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. (Peirce 1866, "Lowell Lecture 7", CE 1, 467–468).</p>

{| align="center" cellpadding="8" width="90%" <!--QUOTE-->

 

<center>man and horse and kangaroo and whale,</center>

 

<center>spherical bright fragrant juicy tropical fruit.</center>

<p>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.</p>

<p>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.</p>

<p>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.  (Peirce 1866, "Lowell Lecture 7", CE 1, 468–469).</p>

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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. (Peirce 1866, "Lowell Lecture 7", CE 1, 469).

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<p>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.  (Peirce 1866, "Lowell Lecture 7", CE 1, 470).</p>

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<p>The answer is that which I gave a week ago.  Namely, that there is a certain vague tendency for the whole to be like any of its parts taken at random because it is composed of its parts.  And, therefore, there must be some slight preponderance of true over false scientific inferences.  Now the falsity in conclusions is eliminated and neutralized by opposing falsity while the slight tendency to the truth is always one way and is accumulated by experience.  The same principle of balancing of errors holds alike in observation and in reasoning. (Peirce 1866, "Lowell Lecture 7", CE 1, 470–471.</p>

===Discussion===

: ''t''₁ = ''spherical''

: ''t''₂ = ''bright''

: ''t''₃ = ''fragrant''

: ''t''₄ = ''juicy''

: ''t''₅ = ''tropical''

: ''t''₆ = ''fruit''

Suppose that ''z'' is the logical conjunction of these six terms:

: ''z'' = ''t''₁ ''t''₂ ''t''₃ ''t''₄ ''t''₅ ''t''₆

What on earth could Peirce mean by saying that such a term is "not a true symbol", or that it is of "no use whatever"?

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That is to say, if something ''x'' is said to be ''z'', then we may guess fairly surely that ''x'' is really an orange, in other words, that ''x'' has all of the additional features that would be summed up quite succinctly in the much more constrained term ''y''&nbsp;=&nbsp;''an&nbsp;orange''.

Figure 1 depicts the situation that is being contemplated here.

<font face="courier new"><pre>

o---------------------------------------------------------------------o

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` o ` ` o ` `...` ` o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` * ` `*` ` ` ` `*` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` o z = spherical bright fragrant juicy tropical fruit` |

| ` ` ` ` ` ` ` * ` ` * y=>z` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

Figure 1. Conjunctive Term z, Taken as Predicate

</pre></font>

What Peirce is saying about ''z'' not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction ''z'', in lattice terms, the ''greatest lower bound'' (''glb'') of the conjoined terms, ''z'' = ''glb''{''t''_''j''&nbsp;:&nbsp;''j''&nbsp;=&nbsp;1&nbsp;to&nbsp;6}, and what we might regard as the "natural conjunction" or the "natural glb" of these terms, namely, ''y''&nbsp;=&nbsp;''an&nbsp;orange''.  That is to say, there is an extra measure of constraint that goes into forming the natural kinds lattice from the free lattice that logic and set theory would otherwise impose.  The local manifestations of this global information are meted out over the structure of the natural lattice by just such abductive gaps as the one between ''z'' and ''y''.

===Discussion===

Let us now consider Peirce's alternate example of a disjunctive term, "neat, swine, sheep, deer".

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<p>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.</p>

This is apparently a stock example of inductive reasoning that Peiece borrows from traditional discussions, so let us pass over the circumstance that modern taxonomies may classify swine as omniverous.

: ''s''₁ = ''neat''

: ''s''₂ = ''swine''

: ''s''₃ = ''sheep''

: ''s''₄ = ''deer''

Suppose that ''u'' is the logical disjunction of these four terms:

: ''u'' = ((''s''₁)(''s''₂)(''s''₃)(''s''₄)).

<font face="courier new"><pre>

| ` ` ` ` ` ` ` w = herbivorous ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` * ` * ` v=>w` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` * ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` * ` ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` * ` ` ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` `Fact * ` ` ` ` ` o v = cloven-hoofed ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` `u=>w * ` ` ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` * ` ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` * ` * ` u=>v` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` o u = ((neat)(swine)(sheep)(deer))` ` ` ` ` ` ` ` ` ` |

| `s_1` `s_2` ` ` ` `s_3` `s_4` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

Figure 2.  Disjunctive Term u, Taken as Subject

</pre></font>

In a similar but dual fashion to the preceding consideration, there is a gap between the the logical disjunction ''u'', in lattice terminology, the ''least upper bound'' (''lub'') of the disjoined terms, ''u'' = ''lub''{''s''_''j''&nbsp;:&nbsp;''j''&nbsp;=&nbsp;1&nbsp;to&nbsp;4}, and what we might regard as the "natural disjunction" or the "natural lub", namely, ''v''&nbsp;=&nbsp;''cloven-hoofed''.

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'' &rArr; ''w'', namely, the rule that cloven-hoofed animals are herbivorous.

===Discussion===

I continue with the out lay of my incidental musings on the theme of ''approximate inference rules''.

{| align="center" cellpadding="8" width="90%" <!--QUOTE-->

<p>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.</p>

<p>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.</p>

<p>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.  (Peirce 1866, "Lowell Lecture 7", CE 1, 467–468).</p>

Aside from Aristotle, the influence of Kant on Peirce is very strongly marked in these earliest expositions.  The invocations of "conceptions of the understanding", the "use" of concepts and thus of symbols in reducing the manifold of extension, and the not so subtle hint of the synthetic à priori in Peirce's discussion, not only of natural kinds, but of the kinds of signs that lead up to genuine symbols, can all be recognized as being reprises of dominant, pervasive Kantian themes.

In order to draw out these themes, and to see how Peirce was led and often inspired to develop their main motives, let us bring together our previous Figures, abstracting away from all of those distractingly ephemeral details about defunct stockyards full of imaginary beasts, and see if we can see what is really going to go on here.

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

<font face="courier new"><pre>

| ` ` ` ` ` ` ` o z = icon? ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` * ` ` * y=>z` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` `Fact * ` ` ` ` ` o y = object? ` ` ` ` ` ` ` ` ` |

| ` ` ` ` `x=>z * ` ` ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` * ` * ` x=>y` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

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

Figure 4. Disjunctive Subject u, Induction of Rule (v (w))

I have up to this point followed Peirce's suggestions somewhat unthinkingly, but I can tell you now that previous unfortunate experience has led me concurrently to remain suspicious of all attempts to conflate the types of signs and the roles of terms in arguments quite so facilely, so I will keep that as a topic for future inquiry.

===Selection 7===

<p>It is obvious that all deductive reasoning has a common property unshared by the other kinds — in being purely ''explicatory''.  Buffier mentions a definition of logic as the art of confessing in the conclusion what we have avowed in the premisses.  This bit of satire translated into the language of sobriety — amounts to charging that the logicians confine their attention exclusively to deductive reasoning.  A charge which against the logicians of other days, was quite just.</p>

<p>Explication direct consists in simply substituting for a word what is implied in that word.  A statement therefore in order to imply something not expressed must either say that a word denotes something or else that something is meant by a word.  Then the direct explication consists in saying that that what a word denotes is what is meant by the word.</p>

<p>Indirect explication consists in saying that what is not what is meant by the word is not denoted by the word or else in saying that which what a word denotes is not is not meant by the word.</p>

<p>Explication in general, then, may be said to be the application of the maxim that what a word denotes is what is meant by the word. (Peirce 1866, "Lowell Lecture 7", CE 1, 458–459).</p>

{| align="center" cellpadding="8" width="90%" <!--QUOTE-->

<p>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.</p>

<p>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.  &hellip;</p>

<p>The narrower term is said to be contained under the wider one; and the higher term to be contained in the lower one.</p>

<p>We have then:</p>

<font face="courier new"><pre>

<p>The principle of explicatory or deductive reasoning then is that:</p>

<p>Every part of a word's Content belongs to every part of its Sphere,</p>

<p>or:</p>

<p>Whatever is contained ''in'' a word belongs to whatever is contained under it.</p>

<p>Now this maxim would not be true if the Extension and Comprehension were directly proportional to one another;  this is to say if the Greater the one the greater the other.  For in that case, though the whole Content would belong to the whole Sphere;  yet only a particular part of it would belong to a part of that Sphere and not every part to every part.  On the other hand if the Comprehension and Extension were not in some way proportional to one another, that is if terms of different spheres could have the same content or terms of the same content different spheres;  then there would be no such fact as a content's ''belonging'' to a sphere and hence again the maxim would fail.  For the maxim to be true, then, it is absolutely necessary that the comprehension and extension should be inversely proportional to one another.  That is that the greater the sphere, the less the content.</p>

<p>Now this evidently true.  If we take the term ''man'' and increase its ''comprehension'' by the addition of ''black'', we have ''black man'' and this has less ''extension'' than ''man''.  So if we take ''black man'' and add ''non-black man'' to its sphere, we have ''man'' again, and so have decreased the comprehension.  So that whenever the extension is increased the comprehension is diminished and ''vice versa''.  (Peirce 1866, "Lowell Lecture 7", CE 1, 459–460).</p>

===Selection 9===

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<p>The highest terms are therefore broadest and the lowest terms the narrowest.  We can take a term so broad that it contains all other spheres under it. Then it will have no content whatever.  There is but one such term — with its synonyms — it is ''Being''. We can also take a term so low that it contains all other content within it.  Then it will have no sphere whatever.  There is but one such term — it is ''Nothing''.</p>

<font face="courier new"><pre>

o------------------------o------------------------o

</pre></font>

<p>We can conceive of terms so narrow that they are next to nothing, that is have an absolutely individual sphere.  Such terms would be innumerable in number.  We can also conceive of terms so high that they are next to ''being'', that is have an entirely simple content.  Such terms would also be innumerable.</p>

<font face="courier new"><pre>

</pre></font>

<p>(Peirce 1866, "Lowell Lecture 7", CE 1, 460).</p>

|}

 

 

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|

But such terms though conceivable in one sense — that is intelligible in their conditions — are yet impossible. 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.  (Peirce 1866, "Lowell Lecture 7", CE 1, 461).

===Selection 11===

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<p>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:</p>

<p><center>Some ''A'' is ''B''</center>

<center>Some ''A'' is not ''B''</center></p>

<p>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.  (Peirce 1866, "Lowell Lecture 7", CE 1, 461).</p>

===Selection 12===

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<p>The moment, then, that we pass from nothing and the vacuity of being to any content or sphere, we come at once to a composite content and sphere.  In fact, extension and comprehension — like space and time — are quantities which are not composed of ultimate elements;  but every part however small is divisible.</p>

<p>The consequence of this fact is that when we wish to enumerate the sphere of a term — a process termed ''division'' — or when we wish to run over the content of a term — a process called ''definition'' — since we cannot take the elements of our enumeration singly but must take them in groups, there is danger that we shall take some element twice over, or that we shall omit some.  Hence the extension and comprehension which we know will be somewhat indeterminate.  But we must distinguish two kinds of these quantities.  If we were to subtilize we might make other distinctions but I shall be content with two.  They are the extension and comprehension relatively to our actual knowledge, and what these would be were our knowledge perfect.</p>

<p>Logicians have hitherto left the doctrine of extension and comprehension in a very imperfect state owing to the blinding influence of a psychological treatment of the matter.  They have, therefore, not made this distinction and have reduced the comprehension of a term to what it would be if we had no knowledge of fact at all.  I mention this because if you should come across the matter I am now discussing in any book, you would find the matter left in quite a different state.  (Peirce 1866, "Lowell Lecture 7", CE 1, 462).</p>

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<p>With me — the ''Sphere'' of a term is all the things we know that it applies to, or the disjunctive sum of the subjects to which it can be predicate in an affirmative subsumptive proposition.  The ''content'' of a term is all the attributes it tells us, or the conjunctive sum of the predicates to which it can be made subject in a universal necessary proposition.</p>

<p>The maxim then which rules explicatory reasoning is that any part of the content of a term can be predicated of any part of its sphere.  (Peirce 1866, "Lowell Lecture 7", CE 1, 462).</p>

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We come next to consider inductions.  In inferences of this kind we proceed as if upon the principle that as is a sample of a class so is the whole class.  The word ''class'' in this connection means nothing more than what is denoted by one term, — or in other words the sphere of a term.  Whatever characters belong to the whole sphere of a term constitute the content of that term.  Hence the principle of induction is that whatever can be predicated of a specimen of the sphere of a term is part of the content of that term.  And what is a specimen?  It is something taken from a class or the sphere of a term, at random — that is, not upon any further principle, not selected from a part of that sphere;  in other words it is something taken from the sphere of a term and not taken as belonging to a narrower sphere.  Hence the principle of induction is that whatever can be predicated of something taken as belonging to the sphere of a term is part of the content of that term.  But this principle is not axiomatic by any means.  Why then do we adopt it? (Peirce 1866, "Lowell Lecture 7", CE 1, 462–463).

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<p>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.</p>

{| align="center" cellpadding="8" width="90%" <!--QUOTE-->

| The comprehension of red then has been for him ||&nbsp;|| ''color''.

| Its extension has been                        ||&nbsp;|| ''A'', ''B'', ''C''.

{| align="center" cellpadding="8" width="90%" <!--QUOTE-->

 

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|

<font face="courier new"><pre>

| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |

| dotted` ` ` ` ` ` | 4 circles ` ` ` ` |

| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |

| dotted & crossed` | 2 circles ` ` ` ` |

| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |

</pre></font>

<p>Now suppose we make these two terms ''dotted circle'' and ''crossed and dotted circle'' equivalent.  This we can do by crossing our uncrossed dotted circles.  In that way, we increase the comprehension of ''dotted circle'' and at the same time increase the extension of ''crossed and dotted circle'' since we now make it denote ''all dotted circles''.  (Peirce 1866, "Lowell Lecture 7", CE 1, 463–464).</p>

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Thus every increase in the number of equivalents of any term increases either its extension or comprehension and ''conversely''.  It may be said that there are no equivalent terms in logic, since the only difference between such terms would be merely external and grammatical, while in logic terms which have the same meaning are identical.  I fully admit that.  Indeed, the process of getting an equivalent for a term is an identification of two terms previously diverse.  It is, in fact, the process of nutrition of terms by which they get all their life and vigor and by which they put forth an energy almost creative — since it has the effect of reducing the chaos of ignorance to the cosmos of science.  Each of these equivalents is the explication of what there is wrapt up in the primary — they are the surrogates, the interpreters of the original term.  They are new bodies, animated by that same soul.  I call them the ''interpretants'' of the term.  And the quantity of these ''interpretants'', I term the ''information'' or ''implication'' of the term. (Peirce 1866, "Lowell Lecture 7", CE 1, 464–465).

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<p>We must therefore modify the law of the inverse proportionality of extension and comprehension and instead of writing:</p>

<center>Extension × Comprehension = Constant,</center>

 

<p>which crudely expresses the fact that the greater the extension the less the comprehension, we must write:</p>

 

 

<p>which means that when the information is increased there is an increase of either extension or comprehension without any diminution of the other of these quantities.</p>

<p>Now, ladies and gentlemen, as it is true that every increase of our knowledge is an increase in the information of a term — that is, is an addition to the number of terms equivalent to that term — so it is also true that the first step in the knowledge of a thing, the first framing of a term, is also the origin of the information of that term because it gives the first term equivalent to that term.  I here announce the great and fundamental secret of the logic of science.  There is no term, properly so called, which is entirely destitute of information, of equivalent terms.  The moment an expression acquires sufficient comprehension to determine its extension, it already has more than enough to do so.  (Peirce 1866, "Lowell Lecture 7", CE 1, 465).</p>

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<p>We are all, then, sufficiently familiar with the fact that many words have much implication;  but I think we need to reflect upon the circumstance that every word implies some proposition or, what is the same thing, every word, concept, symbol has an equivalent term — or one which has become identified with it, — in short, has an ''interpretant''.</p>

<p>Consider, what a word or symbol is;  it is a sort of representation.  Now a representation is something which stands for something.  I will not undertake to analyze, this evening, this conception of ''standing for'' something — but, it is sufficiently plain that it involves the standing ''to'' something ''for'' something.  A thing cannot stand for something without standing ''to'' something ''for'' that something.  Now, what is this that a word stands ''to''? Is it a person?  We usually say that the word ''homme'' stands to a Frenchman for ''man''.  It would be a little more precise to say that it stands ''to'' the Frenchman's mind — to his memory.  It is still more accurate to say that it addresses a particular remembrance or image in that memory.  And what ''image'', what remembrance?  Plainly, the one which is the mental equivalent of the word ''homme'' — in short, its interpretant.  Whatever a word addresses then or ''stands to'', is its interpretant or identified symbol.  Conversely, every interpretant is addressed by the word;  for were it not so, did it not as it were overhear what the words says, how could it interpret what it says.</p>

<p>There are doubtless some who cannot understand this metaphorical argument.  I wish to show that the relation of a word to that which it addresses is the same as its relation to its equivalent or identified terms.  For that purpose, I first show that whatever a word addresses is an equivalent term, — its mental equivalent.  I next show that, since the intelligent reception of a term is the being addressed by that term, and since the explication of a term's implication is the intelligent reception of that term, that the interpretant or equivalent of a term which as we have already seen explicates the implication of a term is addressed by the term.</p>

<p>The interpretant of a term, then, and that which it stands to are identical.  Hence, since it is of the very essence of a symbol that it should stand ''to'' something, every symbol — every word and every ''conception'' — must have an interpretant — or what is the same thing, must have information or implication.</p>

<p>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.  (Peirce 1866, "Lowell Lecture 7", CE 1, 466–467).</p>

===Discussion===

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<center>''A'' is contained under ''B'',</center>

 

 

<p>Thus ''black horse'' is contained under ''horse'', and ''horse'' [is contained in ''black horse''].  (Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 272).</p>

'''Nota Bene.'''  In the Table below a label of the form ''XY'' indicates a premiss of a classical syllogism in which ''X'' is the subject and ''Y'' is the predicate.  Also, I suspect that the Third Figure syllogism ought to be ''XY'' & ''XZ''.

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<font face="courier new"><pre>

| ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` ` | ( ` `Subject: `O of C | ` ` ` ` ` ( `XY |

| Analytic` ` ` ` ` ` | < ` ` ` ` ` ` ` ` ` ` | 2nd Fig.` < ` ` |

| ` ` ` ` ` ` ` ` ` ` | ( `Predicate: `O of C | ` ` ` ` ` ( `ZY |

| ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` ` | ( ` `Subject: `O of D | ` ` ` ` ` ( `YX |

| Synthetic Intensive | < ` ` ` ` ` ` ` ` ` ` | 1st Fig.` < ` ` |

| ` ` ` ` ` ` ` ` ` ` | ( `Predicate: `O of C | ` ` ` ` ` ( `ZY |

| ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` ` | ( ` `Subject: `O of D | ` ` ` ` ` ( `YX |

| Extensive ` ` ` ` ` | < ` ` ` ` ` ` ` ` ` ` | 3rd Fig.` < ` ` |

| ` ` ` ` ` ` ` ` ` ` | ( `Predicate: `O of D | ` ` ` ` ` ( `ZX |

| ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` |

</pre></font>

 

<p>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.</p>

: <p>Analytic.</p>

: <p>Synthetic Intensive.</p>

: <p>Extensive.</p>

 

<p>we may infer:</p>

 

 

<p>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.  (Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 272–274).</p>

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<p>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.</p>

<p>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''.  (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 274).</p>

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<p>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. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 274–275).</p>

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<p>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?</p>

<p>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''’ [— perhaps ''B'' + ''B''’ was intended?].  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''&nbsp;?, 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.  (Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 275–276).</p>

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<p><center>comprehension × extension = constant,</center>

 

<p>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.  (Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 276).</p>

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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. (Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 276–277).

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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 [— perhaps "denotation" ?] 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. (Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 277).

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<p>propositions that what is on the left hand of one line cannot be on the right hand of the other.  (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 278–279).</p>

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<p>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 representative power.  This principle is therefore the ground of inference in general. (Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 279–280).</p>

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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. (Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 280).</p>

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<p>(Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 280–281).</p>

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<p>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.</p>

<p>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.</p>

<p>All these principles must as principles be universal.  Hence they are as follows:—</p>

 

<p>All things, forms, symbols are symbolizable.</p>

 

<p>(Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 281–282).</p>

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<p>All these principles must as principles be universal.  Hence they are as follows:—</p>

<p>All things, forms, symbols are symbolizable.</p>

<p>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.</p>

<p>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.</p>

 

<p>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.  (Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 282–283).</p>

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<p>(Peirce 1865, "Harvard Lecture 10.  Grounds of Induction", CE 1, 283).</p>

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