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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:
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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| align="center" | <math>\operatorname{Information} = \operatorname{Comprehension} \times \operatorname{Extension}</math>
<math>\operatorname{Information} = \operatorname{Comprehension} \times \operatorname{Extension}</math>
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===Selection 1===
===Selection 1===
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<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.</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.</p>
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===Selection 2===
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<p>For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.</p>
<p>For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.</p>
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===Selection 4===
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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).
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>A similar line of thought may be gone through in reference to hypothesis. In this case we must start with the consideration of the term:</p>
<p>A similar line of thought may be gone through in reference to hypothesis. In this case we must start with the consideration of the term:</p>
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<p>We have now seen how the mind is forced by the very nature of inference itself to make use of induction and hypothesis.</p>
<p>We have now seen how the mind is forced by the very nature of inference itself to make use of induction and hypothesis.</p>
In particular, let us consider the following statement:
In particular, let us consider the following statement:
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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.
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.
Let us now consider Peirce's alternate example of a disjunctive term, "neat, swine, sheep, deer".
Let us now consider Peirce's alternate example of a disjunctive term, "neat, swine, sheep, deer".
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<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.</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.</p>
I continue with the out lay of my incidental musings on the theme of ''approximate inference rules''.
I continue with the out lay of my incidental musings on the theme of ''approximate inference rules''.
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<p>For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.</p>
<p>For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.</p>
===Selection 7===
===Selection 7===
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<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>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>
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===Selection 8===
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<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>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>
===Selection 9===
===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>
<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>
===Selection 10===
===Selection 10===
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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).
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===
===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>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>
===Selection 12===
===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 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>
===Selection 13===
===Selection 13===
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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>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>
===Selection 14===
===Selection 14===
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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).
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).
===Selection 15===
===Selection 15===
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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>
<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>
<p>Thus suppose a blind man to be told that no red things are blue. He has previously known only that red is a color; and that certain things ''A'', ''B'', and ''C'' are red.</p>
<p>Thus suppose a blind man to be told that no red things are blue. He has previously known only that red is a color; and that certain things ''A'', ''B'', and ''C'' are red.</p>
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<p>But when he learns that no red thing is blue, ''non-blue'' is added to the comprehension of red, without the least diminution of its extension.</p>
<p>But when he learns that no red thing is blue, ''non-blue'' is added to the comprehension of red, without the least diminution of its extension.</p>
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<p>For example we have here a number of circles dotted and undotted, crossed and uncrossed:</p>
<p>For example we have here a number of circles dotted and undotted, crossed and uncrossed:</p>
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<font face="courier new"><pre>
<font face="courier new"><pre>
<p>Here it is evident that the greater the extension the less the comprehension:</p>
<p>Here it is evident that the greater the extension the less the comprehension:</p>
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<font face="courier new"><pre>
<font face="courier new"><pre>
===Selection 16===
===Selection 16===
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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).
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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===Selection 17===
===Selection 17===
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<p>We must therefore modify the law of the inverse proportionality of extension and comprehension and instead of writing:</p>
<p>We must therefore modify the law of the inverse proportionality of extension and comprehension and instead of writing:</p>
===Selection 18===
===Selection 18===
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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>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>
If you dreamed that this inquiry had come full circle then I inform you of what you already know, that there are always greater circles. I revert to Peirce's Harvard University Lectures of the year before, to pick up additional background material and a bit more motivation.
If you dreamed that this inquiry had come full circle then I inform you of what you already know, that there are always greater circles. I revert to Peirce's Harvard University Lectures of the year before, to pick up additional background material and a bit more motivation.
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<p>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.</p>
<p>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.</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''.
'''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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<p>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.</p>
<p>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.</p>
===Selection 20===
===Selection 20===
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<p>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.</p>
<p>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.</p>
===Selection 21===
===Selection 21===
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<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''.</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''.</p>
===Selection 22===
===Selection 22===
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<p>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?</p>
<p>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?</p>
===Selection 23===
===Selection 23===
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<p>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:</p>
<p>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:</p>
===Selection 24===
===Selection 24===
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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).
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).
===Selection 25===
===Selection 25===
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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).
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).
===Selection 26===
===Selection 26===
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<p>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.</p>
<p>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.</p>
===Selection 27===
===Selection 27===
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<p>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.</p>
<p>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.</p>
===Selection 28===
===Selection 28===
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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>
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>
===Selection 29===
===Selection 29===
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<p>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.</p>
<p>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.</p>
===Selection 30===
===Selection 30===
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<p>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.</p>
<p>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.</p>
===Selection 31===
===Selection 31===
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<p>All these principles must as principles be universal. Hence they are as follows:—</p>
<p>All these principles must as principles be universal. Hence they are as follows:—</p>
===Selection 32===
===Selection 32===
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<p>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.</p>
<p>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.</p>
===Selection 33===
===Selection 33===
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<p>In every induction we have given some remarkable fact or piece of information:</p>
<p>In every induction we have given some remarkable fact or piece of information:</p>
===Selection 34===
===Selection 34===
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<p>In the case of hypothesis we have given some remarkable state of things:</p>
<p>In the case of hypothesis we have given some remarkable state of things:</p>
===Selection 35===
===Selection 35===
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<p>The last lecture was devoted to the fundamental inquiry of the whole course, that of the grounds of inference.</p>
<p>The last lecture was devoted to the fundamental inquiry of the whole course, that of the grounds of inference.</p>
===Selection 36===
===Selection 36===
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<p>There are certain pseudo-symbols which are formed by combinations of symbols, and which must therefore be considered in logic, which lack either denotation or connotation. Thus, ''cats and stoves'' is a symbol wanting in connotation because it does not purport to relate to any definite quality. ''Tailed men'' wants denotation; for though it implies that there are men and that there are tailed things, it does not deny that these classes are mutually exclusive. All such terms are totally wanting in ''information''.</p>
<p>There are certain pseudo-symbols which are formed by combinations of symbols, and which must therefore be considered in logic, which lack either denotation or connotation. Thus, ''cats and stoves'' is a symbol wanting in connotation because it does not purport to relate to any definite quality. ''Tailed men'' wants denotation; for though it implies that there are men and that there are tailed things, it does not deny that these classes are mutually exclusive. All such terms are totally wanting in ''information''.</p>
===Selection 37===
===Selection 37===
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<p>The difference between subject and predicate was also considered in the last lecture. The subject is usually defined as the term determined by the proposition, but as the predicates of ''A'', ''E'', and ''I'' are also determined, this definition is inadequate. We were led to substitute for it the following:—</p>
<p>The difference between subject and predicate was also considered in the last lecture. The subject is usually defined as the term determined by the proposition, but as the predicates of ''A'', ''E'', and ''I'' are also determined, this definition is inadequate. We were led to substitute for it the following:—</p>
===Selection 38===
===Selection 38===
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<p>Having thus far established:</p>
<p>Having thus far established:</p>
Peirce continues his remarks on the problem of the grounds of inference:
Peirce continues his remarks on the problem of the grounds of inference:
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<p>In the first place with reference to the nature of the problem itself. It is not required to prove that deduction, induction, or hypothesis are valid. On the contrary, they are to be accepted as conditions of thought. It had been shown in previous lectures that they are so. Nor was a mode of calculating the probability of an induction or hypothesis now demanded; this being a merely subsidiary problem at best and one which may for ought we could yet see, be absurd. What we now wanted was an articulate statement and a satisfactory demonstration of those transcendental laws which give rise to the possibility of each kind of inference.</p>
<p>In the first place with reference to the nature of the problem itself. It is not required to prove that deduction, induction, or hypothesis are valid. On the contrary, they are to be accepted as conditions of thought. It had been shown in previous lectures that they are so. Nor was a mode of calculating the probability of an induction or hypothesis now demanded; this being a merely subsidiary problem at best and one which may for ought we could yet see, be absurd. What we now wanted was an articulate statement and a satisfactory demonstration of those transcendental laws which give rise to the possibility of each kind of inference.</p>
===Selection 40===
===Selection 40===
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<p>Each ground-principle must be proved entirely by that same kind of inference which it supports. But we cannot arrive at any conclusion by mere deduction except about symbols. We cannot arrive at any conclusion by mere induction except about things. And we cannot arrive at any conclusion by mere hypothesis except about forms.</p>
<p>Each ground-principle must be proved entirely by that same kind of inference which it supports. But we cannot arrive at any conclusion by mere deduction except about symbols. We cannot arrive at any conclusion by mere induction except about things. And we cannot arrive at any conclusion by mere hypothesis except about forms.</p>
===Selection 41===
===Selection 41===
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<p>Every induction, then, and every hypothesis yields a certain amount of truth.</p>
<p>Every induction, then, and every hypothesis yields a certain amount of truth.</p>
===Selection 42===
===Selection 42===
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<p>In this way truth is measured upon a scale of numbers from ''one'' to ''infinity''. And thus we cannot measure the ratio of the truth to the falsehood but only the ratio between the pregnancy of two sets of facts. Of any particular conclusion therefore we can only judge by ascertaining by further experience whether it can be improved. But the comparative usefulness of the facts upon which it proceeds may be estimated with an approach to precision.</p>
<p>In this way truth is measured upon a scale of numbers from ''one'' to ''infinity''. And thus we cannot measure the ratio of the truth to the falsehood but only the ratio between the pregnancy of two sets of facts. Of any particular conclusion therefore we can only judge by ascertaining by further experience whether it can be improved. But the comparative usefulness of the facts upon which it proceeds may be estimated with an approach to precision.</p>
===Selection 43===
===Selection 43===
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<p>I fear I have wearied you in these lectures by dwelling so much upon merely logical forms. But to the pupil of Kant as to the pupil of Aristotle the Analytic of Logic is the foundation of Metaphysics. We find ourselves in all our discourse taking certain points for granted which we cannot have observed. The question therefore is what may we take for granted independent of all experience. The answer to this is metaphysics. But it is plain that we can thus take for granted only what is involved in logical forms. Hence, the necessity of studying these forms. In these lectures, one set of Logical forms has been pretty thoroughly studied; that of Hypothesis, Deduction, Induction. Another set has been partly studied, that of Denotation, Information, Connotation.</p>
<p>I fear I have wearied you in these lectures by dwelling so much upon merely logical forms. But to the pupil of Kant as to the pupil of Aristotle the Analytic of Logic is the foundation of Metaphysics. We find ourselves in all our discourse taking certain points for granted which we cannot have observed. The question therefore is what may we take for granted independent of all experience. The answer to this is metaphysics. But it is plain that we can thus take for granted only what is involved in logical forms. Hence, the necessity of studying these forms. In these lectures, one set of Logical forms has been pretty thoroughly studied; that of Hypothesis, Deduction, Induction. Another set has been partly studied, that of Denotation, Information, Connotation.</p>
===Anthematic Note 1===
===Anthematic Note 1===
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Each man has his own peculiar character. It enters into all he does. It is in his consciousness and not a mere mechanical trick, and therefore it is by the principles of the last lecture a cognition; but as it enters into all his cognition, it is a cognition of ''things in general''. It is therefore the man's philosophy, his way of regarding things; not a philosophy of the head alone — but one which pervades the whole man. This idiosyncrasy is the idea of the man, and if this idea is true he lives forever; if false, his individual soul has but a contingent existence. (Peirce 1866, CE 1, 501).
Each man has his own peculiar character. It enters into all he does. It is in his consciousness and not a mere mechanical trick, and therefore it is by the principles of the last lecture a cognition; but as it enters into all his cognition, it is a cognition of ''things in general''. It is therefore the man's philosophy, his way of regarding things; not a philosophy of the head alone — but one which pervades the whole man. This idiosyncrasy is the idea of the man, and if this idea is true he lives forever; if false, his individual soul has but a contingent existence. (Peirce 1866, CE 1, 501).
===Anthematic Note 2===
===Anthematic Note 2===
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That the idiosyncrasy of a man — his peculiar character — is his peculiar philosophy, is best seen in the earliest stages of its formation before those complications have been developed which render it difficult to seize upon it. The cunning speeches of children just as they begin to talk often startle one by their philosophical nature. The drawer of ''Harper's Magazine'' has been filled for years with the sayings of "our three year old" — who seems blessed with perennial three-year-old-ness — but if all these stories are true, they are very valuable as showing the character of the childish mind in general, and particularly the philosophical tendencies of children. I shall not trouble you with the recitation of any of these funny stories — they are stale and therefore flat; but I will mention a case, which has nothing laughable in it — but which illustrates remarkably well how the peculiar differences of men are differences of philosophian method. (Peirce 1866, CE 1, 501).
That the idiosyncrasy of a man — his peculiar character — is his peculiar philosophy, is best seen in the earliest stages of its formation before those complications have been developed which render it difficult to seize upon it. The cunning speeches of children just as they begin to talk often startle one by their philosophical nature. The drawer of ''Harper's Magazine'' has been filled for years with the sayings of "our three year old" — who seems blessed with perennial three-year-old-ness — but if all these stories are true, they are very valuable as showing the character of the childish mind in general, and particularly the philosophical tendencies of children. I shall not trouble you with the recitation of any of these funny stories — they are stale and therefore flat; but I will mention a case, which has nothing laughable in it — but which illustrates remarkably well how the peculiar differences of men are differences of philosophian method. (Peirce 1866, CE 1, 501).
===Anthematic Note 3===
===Anthematic Note 3===
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A certain child who is rather backward in learning to speak, — not from dullness, but from a want of aptitude in imitating the words which it hears, — has got to use three words only; and what are these? ''Name'', ''story'', and ''matter''. He says ''name'' when he wishes to know the name of a person or thing; ''story'' when he wishes to hear a narration or description; and ''matter'' — a highly abstract and philosophical term — when he wishes to be acquainted with the cause of anything. ''Name'', ''story'', and ''matter'', therefore, make the foundation of this child's philosophy. What a wonderful thing that his individuality should have been shown so strongly, at that age, in selecting those three words out of all the equally common ones which he heard about him. Already he has made his list of categories, which is the principal part of any philosophy. (Peirce 1866, CE 1, 501).
A certain child who is rather backward in learning to speak, — not from dullness, but from a want of aptitude in imitating the words which it hears, — has got to use three words only; and what are these? ''Name'', ''story'', and ''matter''. He says ''name'' when he wishes to know the name of a person or thing; ''story'' when he wishes to hear a narration or description; and ''matter'' — a highly abstract and philosophical term — when he wishes to be acquainted with the cause of anything. ''Name'', ''story'', and ''matter'', therefore, make the foundation of this child's philosophy. What a wonderful thing that his individuality should have been shown so strongly, at that age, in selecting those three words out of all the equally common ones which he heard about him. Already he has made his list of categories, which is the principal part of any philosophy. (Peirce 1866, CE 1, 501).
===Anthematic Note 4===
===Anthematic Note 4===
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Constantly, in using these words, this philosophy becomes more and more impressed upon him until, when he arrives at maturity of intellect, he may be able to show that it is a profound and legitimate classification. Tell me a man's ''name'', his ''story'', and his ''matter'' or character; and I know about all there is to know of him. Aristotle says there are two questions to be asked concerning anything: the ''oti'' and the ''dioti'', the ''what'' and the ''why'' — the account of premisses and the rational account or explanation; or as this child would say the 'story' and the ''matter''; but Aristotle has not noticed that previous to either of these questions must come the fixing of the attention upon the object — the determination of the mind to it as an object — and the demand for this determination is asking for its ''name''. Here we have therefore in this child, a philosophy which furnishes an emendation upon the mighty Aristotle — the leader of the thought of ages, the prince of philosophers. (Peirce 1866, CE 1, 501–502).
Constantly, in using these words, this philosophy becomes more and more impressed upon him until, when he arrives at maturity of intellect, he may be able to show that it is a profound and legitimate classification. Tell me a man's ''name'', his ''story'', and his ''matter'' or character; and I know about all there is to know of him. Aristotle says there are two questions to be asked concerning anything: the ''oti'' and the ''dioti'', the ''what'' and the ''why'' — the account of premisses and the rational account or explanation; or as this child would say the 'story' and the ''matter''; but Aristotle has not noticed that previous to either of these questions must come the fixing of the attention upon the object — the determination of the mind to it as an object — and the demand for this determination is asking for its ''name''. Here we have therefore in this child, a philosophy which furnishes an emendation upon the mighty Aristotle — the leader of the thought of ages, the prince of philosophers. (Peirce 1866, CE 1, 501–502).
===Anthematic Note 5===
===Anthematic Note 5===
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But why should I presume to expound that soul's philosophy; could I enter fully into it he would have no private personality — he would not be the mysterious Island that every soul is to every other. No, I dare not attempt to fathom the awful depths of that child's possibilities; when he grows up, in some way and to some degree he will manifest his character, his philosophy; then we can judge as much of it as we can see, but its intrinsic worth we never can judge; it is hid forever in the bosom of its God. (Peirce 1866, CE 1, 502).
But why should I presume to expound that soul's philosophy; could I enter fully into it he would have no private personality — he would not be the mysterious Island that every soul is to every other. No, I dare not attempt to fathom the awful depths of that child's possibilities; when he grows up, in some way and to some degree he will manifest his character, his philosophy; then we can judge as much of it as we can see, but its intrinsic worth we never can judge; it is hid forever in the bosom of its God. (Peirce 1866, CE 1, 502).
===Anthematic Note 6===
===Anthematic Note 6===
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<p>In dialectica autem vestra nullam existimavit esse nec ad melius vivendum nec ad commodius disserendum viam.</p>
<p>In dialectica autem vestra nullam existimavit esse nec ad melius vivendum nec ad commodius disserendum viam.</p>
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<p>Who hath learnt any wit or understanding in Logique? Where are her faire promises? Nec ad melius vivendum, nec ad commodius disserendum: Neither to live better nor to dispute fitter.</p>
<p>Who hath learnt any wit or understanding in Logique? Where are her faire promises? Nec ad melius vivendum, nec ad commodius disserendum: Neither to live better nor to dispute fitter.</p>
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Gentlemen and ladies, I announce to you this theory of immortality for the first time. It is poorly said, poorly thought; but its foundation is the rock of truth. And at least it will serve to illustrate what use might be made by mightier hands of this reviled science, logic, ''nec ad melius vivendum, nec ad commodius disserendum''. (Peirce 1866, CE 1, page 502).
Gentlemen and ladies, I announce to you this theory of immortality for the first time. It is poorly said, poorly thought; but its foundation is the rock of truth. And at least it will serve to illustrate what use might be made by mightier hands of this reviled science, logic, ''nec ad melius vivendum, nec ad commodius disserendum''. (Peirce 1866, CE 1, page 502).
===Incidental Note 4===
===Incidental Note 4===
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<p>The Critique Of Pure Reason</p>
<p>The Critique Of Pure Reason</p>
No, I mean *really, really* irritating doubts ...
No, I mean *really, really* irritating doubts ...
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<p>The precursors of hatred arise from the infant's response to what William James (1890) called the "booming buzzing confusion" that assaults the infant's sensorium at perception's birth. The "stranger anxiety" evident as early as eight months indicates that the mental capacity to perceive differences in objects and to organize subjective psychic forces has already begun. Freud (1915) tells us that "hate, as a relation to objects, is older than love" (p. 139). Freud continues, "As an expression of the reaction of unpleasure evoked by objects, it always remains in an intimate relation with the self-preservative instincts; so that sexual and ego-instincts can readily develop an antithesis which repeats that of love and hate".</p>
<p>The precursors of hatred arise from the infant's response to what William James (1890) called the "booming buzzing confusion" that assaults the infant's sensorium at perception's birth. The "stranger anxiety" evident as early as eight months indicates that the mental capacity to perceive differences in objects and to organize subjective psychic forces has already begun. Freud (1915) tells us that "hate, as a relation to objects, is older than love" (p. 139). Freud continues, "As an expression of the reaction of unpleasure evoked by objects, it always remains in an intimate relation with the self-preservative instincts; so that sexual and ego-instincts can readily develop an antithesis which repeats that of love and hate".</p>
Ah Bartleby! Ah Humanity!
Ah Bartleby! Ah Humanity!
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<p>At first Bartleby did an extraordinary quantity of writing. As if long famishing for something to copy, he seemed to gorge himself on my documents. There was no pause for digestion. He ran a day and night line, copying by sun-light and by candle-light. I should have been quite delighted with his application, had he been cheerfully industrious. But he wrote on silently, palely, mechanically.</p>
<p>At first Bartleby did an extraordinary quantity of writing. As if long famishing for something to copy, he seemed to gorge himself on my documents. There was no pause for digestion. He ran a day and night line, copying by sun-light and by candle-light. I should have been quite delighted with his application, had he been cheerfully industrious. But he wrote on silently, palely, mechanically.</p>
Very often the reason that one is interested in these varieties of fibers under a given function is so that one can follow them "upstream" or "backward", functionally speaking, in other words, toward the "source" of the functional value under investigation. That leads rather naturally to the other mathematical usage for the word "fiber" that I have in mind. Here are the definitions as I formulated them in my dissertation proposal:
Very often the reason that one is interested in these varieties of fibers under a given function is so that one can follow them "upstream" or "backward", functionally speaking, in other words, toward the "source" of the functional value under investigation. That leads rather naturally to the other mathematical usage for the word "fiber" that I have in mind. Here are the definitions as I formulated them in my dissertation proposal:
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<p>The "fiber" of a codomain element y in Y under a function f : X -> Y is the subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X.</p>
<p>The "fiber" of a codomain element y in Y under a function f : X -> Y is the subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X.</p>
Just by way of introducing a few bits of useful terminology, I take the liberty of expressing the following observations:
Just by way of introducing a few bits of useful terminology, I take the liberty of expressing the following observations:
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===Incidental Note 13===
===Incidental Note 13===
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'''Comprehension.''' The sum of characteristics which connote a class notion symbolized by a general term. Also, the features common to a number of instances or objects. Thus, the ''connotation'' (''qv'') or ''intension'' (''qv'') of a concept. (Otto F. Kraushaar, in D.D. Runes (ed.) ''Dictionary of Philosophy'', 1962).
'''Comprehension.''' The sum of characteristics which connote a class notion symbolized by a general term. Also, the features common to a number of instances or objects. Thus, the ''connotation'' (''qv'') or ''intension'' (''qv'') of a concept. (Otto F. Kraushaar, in D.D. Runes (ed.) ''Dictionary of Philosophy'', 1962).
Peirce's incipient theory of information, that he appears to have developed by sheer force of logical insight from his early understanding of signs and scientific inquiry, is not an easy subject to grasp in its developing state. An attempt to follow his reasoning step by step might well begin with this:
Peirce's incipient theory of information, that he appears to have developed by sheer force of logical insight from his early understanding of signs and scientific inquiry, is not an easy subject to grasp in its developing state. An attempt to follow his reasoning step by step might well begin with this:
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<p>Let us now return to the information.</p>
<p>Let us now return to the information.</p>
===Commentary Note 2===
===Commentary Note 2===
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<p>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.</p>
<p>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.</p>
===Commentary Note 3===
===Commentary Note 3===
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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''.
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''.
===Commentary Note 4===
===Commentary Note 4===
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<p>For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.</p>
<p>For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.</p>
</pre></font>
</pre></font>
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<p>A similar line of thought may be gone through in reference to hypothesis. In this case we must start with the consideration of the term:</p>
<p>A similar line of thought may be gone through in reference to hypothesis. In this case we must start with the consideration of the term:</p>
</pre></font>
</pre></font>
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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. Peirce, CE 1, 468-469).
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. Peirce, CE 1, 468-469).
'''Passage 1'''
'''Passage 1'''
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<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>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>
'''Passage 2'''
'''Passage 2'''
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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>
<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>
'''Passage 3'''
'''Passage 3'''
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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 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>
'''Passage 4'''
'''Passage 4'''
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<p>Yet there are combinations of words and combinations of conceptions which are not strictly speaking symbols. These are of two kinds of which I will give you instances. We have first cases like:</p>
<p>Yet there are combinations of words and combinations of conceptions which are not strictly speaking symbols. These are of two kinds of which I will give you instances. We have first cases like:</p>
'''Passage 1'''
'''Passage 1'''
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<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>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>
I am going to treat Peirce's use of the ''quantity consideration'' as a significant operator that transforms its argument from the syntactic domain ''S'' ∪ ''I'' to the objective domain ''O''.
I am going to treat Peirce's use of the ''quantity consideration'' as a significant operator that transforms its argument from the syntactic domain ''S'' ∪ ''I'' to the objective domain ''O''.
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<p>Now the sphere considered as a quantity is called the Extension;</br>
<p>Now the sphere considered as a quantity is called the Extension;</br>
2.1. "men or horses or kangaroos or whales" (extensional disjunction).
2.1. "men or horses or kangaroos or whales" (extensional disjunction).
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<p>Yet there are combinations of words and combinations of conceptions which are not strictly speaking symbols.</p>
<p>Yet there are combinations of words and combinations of conceptions which are not strictly speaking symbols.</p>
2. Conventions, Disjunctive Terms, Indexical Signs, Inductive Rules (cont.)
2. Conventions, Disjunctive Terms, Indexical Signs, Inductive Rules (cont.)
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<p>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?</p>
<p>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?</p>
At this point it will help to jump ahead a bit in time, and to take in the more systematic account of the same material from Peirce's "New List of Categories" (1867).
At this point it will help to jump ahead a bit in time, and to take in the more systematic account of the same material from Peirce's "New List of Categories" (1867).
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I shall now show how the three conceptions of reference to a ground, reference to an object, and reference to an interpretant are the fundamental ones of at least one universal science, that of logic. (Peirce 1867, CP 1.559).
I shall now show how the three conceptions of reference to a ground, reference to an object, and reference to an interpretant are the fundamental ones of at least one universal science, that of logic. (Peirce 1867, CP 1.559).
We will have occasion to consider this paragraph in detail later, but for the present purpose let's hurry on down to the end of it.
We will have occasion to consider this paragraph in detail later, but for the present purpose let's hurry on down to the end of it.
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<p>In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion. In deductive argument, the conclusion is represented by the premisses as by a general sign under which it is contained. In hypotheses, something ''like'' the conclusion is proved, that is, the premisses form a likeness of the conclusion. Take, for example, the following argument:</p>
<p>In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion. In deductive argument, the conclusion is represented by the premisses as by a general sign under which it is contained. In hypotheses, something ''like'' the conclusion is proved, that is, the premisses form a likeness of the conclusion. Take, for example, the following argument:</p>
Let us look to Peirce's ''New List'' of the next year for guidance:
Let us look to Peirce's ''New List'' of the next year for guidance:
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<p>In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion. …</p>
<p>In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion. …</p>
I was trying to understand the things that Peirce said and wrote about the conventional, disjunctive, indexical, inductive complex of notions in the period 1865–1867. And I was focused for the moment on this bit:
I was trying to understand the things that Peirce said and wrote about the conventional, disjunctive, indexical, inductive complex of notions in the period 1865–1867. And I was focused for the moment on this bit:
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<p>In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion. …</p>
<p>In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion. …</p>
The ''New List'' (1867) account of the relationship between the kinds of signs and the kinds of arguments says this:
The ''New List'' (1867) account of the relationship between the kinds of signs and the kinds of arguments says this:
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In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object.
In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object.
This generality may be broken down according to the role of the premisses:
This generality may be broken down according to the role of the premisses:
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The premisses may afford a likeness, index, or symbol of the conclusion.
The premisses may afford a likeness, index, or symbol of the conclusion.
Premisses (Index):
Premisses (Index):
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<p>''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub> are taken as samples of the collection ''M''.</p>
<p>''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub> are taken as samples of the collection ''M''.</p>
Conclusion (Object):
Conclusion (Object):
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All ''M'' is ''P''.
All ''M'' is ''P''.
Remark:
Remark:
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Hence the first premiss amounts to saying that "''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub>" is an index of ''M''. Hence the premisses are an index of the conclusion.
Hence the first premiss amounts to saying that "''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub>" is an index of ''M''. Hence the premisses are an index of the conclusion.
I got as far as sketching a few readings of the penultimate sentence:
I got as far as sketching a few readings of the penultimate sentence:
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Hence the first premiss amounts to saying that "''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub>" is an index of ''M''.
Hence the first premiss amounts to saying that "''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub>" is an index of ''M''.
Uncertain as my comprehension remains at this point, I will have to leave it in suspension for the time being. But let me make an initial pass at the final sentence, so as not to leave an utterly incomplete impression of the whole excerpt.
Uncertain as my comprehension remains at this point, I will have to leave it in suspension for the time being. But let me make an initial pass at the final sentence, so as not to leave an utterly incomplete impression of the whole excerpt.
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Hence the premisses are an index of the conclusion.
Hence the premisses are an index of the conclusion.
The first premiss is this:
The first premiss is this:
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''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub> are taken as samples of the collection ''M''.
''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub> are taken as samples of the collection ''M''.
The second premiss is this:
The second premiss is this:
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''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub> are ''P''.
''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub> are ''P''.
Together these premisses form an index of the conclusion, namely:
Together these premisses form an index of the conclusion, namely:
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All ''M'' is ''P''.
All ''M'' is ''P''.
And all of this is said to be so because:
And all of this is said to be so because:
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In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object.
In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object.
Here's the "New List" text about the relations between the types of signs and the types of inference, that is, the morphological and temporal constituents of inquiry:
Here's the "New List" text about the relations between the types of signs and the types of inference, that is, the morphological and temporal constituents of inquiry:
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<p>In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion.</p>
<p>In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion.</p>
