Difference between revisions of "Directory:Jon Awbrey/Papers/Information = Comprehension × Extension"
MyWikiBiz, Author Your Legacy — Monday June 05, 2023
Jump to navigationJump to searchJon Awbrey (talk | contribs) |
Jon Awbrey (talk | contribs) (update) |
||
| (46 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{DISPLAYTITLE:Information = Comprehension × Extension}} | {{DISPLAYTITLE:Information = Comprehension × Extension}} | ||
| − | + | '''Author: [[User:Jon Awbrey|Jon Awbrey]]''' | |
| − | + | Another angle from which to approach the incidence of signs and [[inquiry]] is by way of 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 extensions and intensions, 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: | ||
{| align="center" cellspacing="6" width="90%" | {| align="center" cellspacing="6" width="90%" | ||
| − | | align="center" | <math>\ | + | | align="center" | <math>\mathrm{Information} = \mathrm{Comprehension} \times \mathrm{Extension}\!</math> |
|} | |} | ||
But comprehending what in the world that might mean is a much longer story, the end of which your present teller has yet to reach. So, this time around, I will take up the story near the end of the beginning of the author's own telling of it, for no better reason than that's where I myself initially came in, or, at least, where it all started making any kind of sense to me. And from this point we will find it easy enough to flash both backward and forward, to and fro, as the occasions arise for doing so. | But comprehending what in the world that might mean is a much longer story, the end of which your present teller has yet to reach. So, this time around, I will take up the story near the end of the beginning of the author's own telling of it, for no better reason than that's where I myself initially came in, or, at least, where it all started making any kind of sense to me. And from this point we will find it easy enough to flash both backward and forward, to and fro, as the occasions arise for doing so. | ||
| − | ==Selections from Peirce's | + | ==Selections from Peirce's “Logic of Science” (1865–1866)== |
===Selection 1=== | ===Selection 1=== | ||
| Line 18: | Line 20: | ||
<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> | ||
| − | <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'' — ''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. | + | <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'' — ''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.</p> |
| + | |||
| + | <p>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> | ||
<p>(Peirce 1866, Lowell Lecture 7, CE 1, 467).</p> | <p>(Peirce 1866, Lowell Lecture 7, CE 1, 467).</p> | ||
| Line 65: | Line 69: | ||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | <p>Accordingly, if we are engaged in symbolizing and we come to such a proposition as | + | <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. 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.</p> |
<p>(Peirce 1866, Lowell Lecture 7, CE 1, 469).</p> | <p>(Peirce 1866, Lowell Lecture 7, CE 1, 469).</p> | ||
| Line 99: | Line 103: | ||
|} | |} | ||
| − | ===Discussion=== | + | ===Discussion 1=== |
At this point in his discussion, Peirce is relating the nature of inference, inquiry, and information to the character of the signs that are invoked in support of the overall process in question, a process that he is presently describing as ''symbolization''. | At this point in his discussion, Peirce is relating the nature of inference, inquiry, and information to the character of the signs that are invoked in support of the overall process in question, a process that he is presently describing as ''symbolization''. | ||
| Line 110: | Line 114: | ||
| | | | ||
<math>\begin{array}{lll} | <math>\begin{array}{lll} | ||
| − | t_1 & = & \ | + | t_1 & = & \mathrm{spherical} |
\\ | \\ | ||
| − | t_2 & = & \ | + | t_2 & = & \mathrm{bright} |
\\ | \\ | ||
| − | t_3 & = & \ | + | t_3 & = & \mathrm{fragrant} |
\\ | \\ | ||
| − | t_4 & = & \ | + | t_4 & = & \mathrm{juicy} |
\\ | \\ | ||
| − | t_5 & = & \ | + | t_5 & = & \mathrm{tropical} |
\\ | \\ | ||
| − | t_6 & = & \ | + | t_6 & = & \mathrm{fruit} |
\end{array}</math> | \end{array}</math> | ||
|} | |} | ||
| Line 133: | Line 137: | ||
|} | |} | ||
| − | What on earth could Peirce mean by saying that such a term is | + | 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”? |
In particular, let us consider the following statement: | In particular, let us consider the following statement: | ||
| Line 142: | Line 146: | ||
|} | |} | ||
| − | That is to say, if something <math>x\!</math> is said to be <math>z,\!</math> then we may guess fairly surely that <math>x\!</math> is really an orange, in other words, that <math>x\!</math> has all of the additional features that would be summed up quite succinctly in the much more constrained term <math>y,\!</math> where <math>y\!</math> means | + | That is to say, if something <math>x\!</math> is said to be <math>z,\!</math> then we may guess fairly surely that <math>x\!</math> is really an orange, in other words, that <math>x\!</math> has all of the additional features that would be summed up quite succinctly in the much more constrained term <math>y,\!</math> where <math>y\!</math> means “an orange”. |
| − | Figure 1 | + | Figure 1 shows the implication ordering of logical terms in the form of a ''lattice diagram''. |
| − | {| align="center" | + | {| align="center" border="0" cellspacing="10" style="text-align:center; width:100%" |
| − | + | | [[File:ICE Figure 1.jpg|center]] | |
| − | + | |- | |
| − | + | | height="20px" | <math>\text{Figure 1.} ~~ \text{Conjunctive Term}~ z, ~\text{Taken as Predicate}\!</math> | |
| − | | | ||
| − | |||
| − | |||
| − | | | ||
| − | | | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | Figure 1. | ||
| − | </ | ||
|} | |} | ||
| − | What Peirce is saying about <math>z\!</math> not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction <math>z,\!</math> in lattice terms, the ''greatest lower bound'' (''glb'') of the conjoined terms, <math>z = \ | + | What Peirce is saying about <math>z\!</math> not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction <math>z,\!</math> in lattice terms, the ''greatest lower bound'' (''glb'') of the conjoined terms, <math>z = \mathrm{glb} \{ t_1, t_2, t_3, t_4, t_5, t_6 \},\!</math> and what we might regard as the ''natural conjunction'' or the ''natural glb'' of these terms, namely, <math>{y := \text{an orange}}.\!</math> 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 <math>z\!</math> and <math>y.\!</math> |
| − | ===Discussion=== | + | ===Discussion 2=== |
| − | Let | + | Let's examine Peirce's second example of a disjunctive term — ''neat, swine, sheep, deer'' — within the style of lattice framework that we used before. |
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| Line 188: | Line 166: | ||
<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> | ||
| − | <p>Accordingly, if we are engaged in symbolizing and we come to such a proposition as | + | <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 Peirce borrows from traditional discussions, so let us pass over the circumstance that modern taxonomies may classify swine as | + | This is apparently a stock example of inductive reasoning that Peirce borrows from traditional discussions, so let us pass over the circumstance that modern taxonomies may classify swine as omnivores. |
In view of the analogical symmetries that the disjunctive term shares with the conjunctive case, I think that we can run through this example in fairly short order. We have an aggregation over four terms: | In view of the analogical symmetries that the disjunctive term shares with the conjunctive case, I think that we can run through this example in fairly short order. We have an aggregation over four terms: | ||
| Line 198: | Line 176: | ||
| | | | ||
<math>\begin{array}{lll} | <math>\begin{array}{lll} | ||
| − | s_1 & = & \ | + | s_1 & = & \mathrm{neat} |
\\ | \\ | ||
| − | s_2 & = & \ | + | s_2 & = & \mathrm{swine} |
\\ | \\ | ||
| − | s_3 & = & \ | + | s_3 & = & \mathrm{sheep} |
\\ | \\ | ||
| − | s_4 & = & \ | + | s_4 & = & \mathrm{deer} |
\end{array}</math> | \end{array}</math> | ||
|} | |} | ||
| Line 213: | Line 191: | ||
| | | | ||
<math>\begin{array}{lll} | <math>\begin{array}{lll} | ||
| − | u & = & ((s_1)(s_2)(s_3)(s_4)) | + | u & = & \texttt{((} s_1 \texttt{)(} s_2 \texttt{)(} s_3 \texttt{)(} s_4 \texttt{))} |
\end{array}</math> | \end{array}</math> | ||
|} | |} | ||
| Line 219: | Line 197: | ||
Figure 2 depicts the situation that we have before us. | Figure 2 depicts the situation that we have before us. | ||
| − | {| align="center" | + | {| align="center" border="0" cellspacing="10" style="text-align:center; width:100%" |
| − | + | | [[File:ICE Figure 2.jpg|center]] | |
| − | + | |- | |
| − | + | | height="20px" | <math>\text{Figure 2.} ~~ \text{Disjunctive Term}~ u, ~\text{Taken as Subject}\!</math> | |
| − | | | ||
| − | | | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | | | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | Figure 2. | ||
| − | </ | ||
|} | |} | ||
| − | + | Here we have a situation that is dual to the structure of the conjunctive example. There is a gap between the logical disjunction <math>u,\!</math> in lattice terminology, the ''least upper bound'' (''lub'') of the disjoined terms, <math>u = \mathrm{lub} \{ s_1, s_2, s_3, s_4 \},\!</math> and what we might regard as the natural disjunction or natural lub of these terms, namely, <math>v,\!</math> ''cloven-hoofed''. | |
| − | Once again, the sheer implausibility of imagining that the disjunctive term <math>u\!</math> would ever be embedded exactly as such in a lattice of natural kinds | + | Once again, the sheer implausibility of imagining that the disjunctive term <math>u\!</math> would ever be embedded exactly as such in a lattice of natural kinds leads to the evident ''naturalness'' of the induction to <math>v \Rightarrow w,\!</math> namely, the rule that cloven-hoofed animals are herbivorous. |
| − | ===Discussion=== | + | ===Discussion 3=== |
| + | |||
| + | Peirce identifies inference with a process he describes as ''symbolization''. Let us consider what that might imply. | ||
| + | |||
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| + | | | ||
| + | <p>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''. [[Information_%3D_Comprehension_%C3%97_Extension#Selection_1|(467)]].</p> | ||
| + | |} | ||
| − | + | Even if it were only a weaker analogy between inference and symbolization, a principle of logical continuity — what in physics is called a ''correspondence principle'' — would suggest parallels between steps of reasoning in the neighborhood of exact inferences and signs in the vicinity of genuine symbols. This would lead us to expect a correspondence between degrees of inference and degrees of symbolization that extends from exact to approximate or ''non-demonstrative'' inferences and from genuine to approximate or ''degenerate'' symbols. | |
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| Line 265: | Line 224: | ||
<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>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 | + | <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.</p> |
| − | <p> | + | <p>Likenesses denote nothing in particular; ''conventional signs'' connote nothing in particular.</p> |
| − | <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. [[Information_%3D_Comprehension_%C3%97_Extension#Selection_2|(467–468)]].</p> |
|} | |} | ||
| − | + | In addition to 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 also of the kinds of signs that lead up to genuine symbols, can all be recognized as pervasive Kantian themes. | |
| − | In order to draw out these themes | + | In order to draw out these themes and see how Peirce was led to develop their leading ideas, let us bring together our previous Figures, abstracting from their concrete details, and see if we can figure out what is going on here. |
| − | Figure 3 shows an abductive step of inquiry, as | + | Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign. |
| − | {| align="center" | + | {| align="center" border="0" cellspacing="10" style="text-align:center; width:100%" |
| − | + | | [[File:ICE Figure 3.jpg|center]] | |
| − | + | |- | |
| − | + | | height="20px" | <math>\text{Figure 3.} ~~ \text{Conjunctive Predicate}~ z, ~\text{Abduction of Case}~ \texttt{(} x \texttt{(} y \texttt{))}\!</math> | |
| − | | | ||
| − | | | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | | | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | Figure 3. | ||
| − | </ | ||
|} | |} | ||
| − | Figure 4 | + | Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign. |
| − | {| align="center" | + | {| align="center" border="0" cellspacing="10" style="text-align:center; width:100%" |
| − | + | | [[File:ICE Figure 4.jpg|center]] | |
| − | + | |- | |
| − | + | | height="20px" | <math>\text{Figure 4.} ~~ \text{Disjunctive Subject}~ u, ~\text{Induction of Rule}~ \texttt{(} v \texttt{(} w \texttt{))}\!</math> | |
| − | | | ||
| − | | | ||
| − | |||
| − | | | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | Figure 4. | ||
| − | </ | ||
|} | |} | ||
| − | + | ===Discussion 4=== | |
| − | + | There are still many things that puzzle me about Peirce's account at this point. I indicated a few of them by means of question marks at several places in the last two Figures. There is nothing for it but returning to the text and trying once more to follow the reasoning. | |
| + | |||
| + | Let's go back to Peirce's example of abductive inference and try to get a clearer picture of why he connects it with conjunctive terms and iconic signs. | ||
| − | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | + | Figure 1 shows the implication ordering of logical terms in the form of a ''lattice diagram''. |
| + | |||
| + | {| align="center" border="0" cellspacing="10" style="text-align:center; width:100%" | ||
| + | | [[File:ICE Figure 1.jpg|center]] | ||
| + | |- | ||
| + | | height="20px" | <math>\text{Figure 1.} ~~ \text{Conjunctive Term}~ z, ~\text{Taken as Predicate}\!</math> | ||
| + | |} | ||
| + | |||
| + | The relationship between conjunctive terms and iconic signs may be understood as follows. If there is anything that has all the properties described by the conjunctive term — ''spherical bright fragrant juicy tropical fruit'' — then sign users may use that thing as an icon of an orange, precisely by virtue of the fact that it shares those properties with an orange. But the only natural examples of things that have all those properties are oranges themselves, so the only thing that can serve as a natural icon of an orange by virtue of those very properties is that orange itself or another orange. | ||
| + | |||
| + | ===Discussion 5=== | ||
| + | |||
| + | Let's stay with Peirce's example of abductive inference a little longer and try to clear up the more troublesome confusions that tend to arise. | ||
| + | |||
| + | Figure 1 shows the implication ordering of logical terms in the form of a ''lattice diagram''. | ||
| + | |||
| + | {| align="center" border="0" cellspacing="10" style="text-align:center; width:100%" | ||
| + | | [[File:ICE Figure 1.jpg|center]] | ||
| + | |- | ||
| + | | height="20px" | <math>\text{Figure 1.} ~~ \text{Conjunctive Term}~ z, ~\text{Taken as Predicate}\!</math> | ||
| + | |} | ||
| + | |||
| + | One thing needs to be stressed at this point. It is important to recognize that the conjunctive term itself — namely, the syntactic string “spherical bright fragrant juicy tropical fruit” — is not an icon but a symbol. It has its place in a formal system of symbols, for example, a propositional calculus, where it would normally be interpreted as a logical conjunction of six elementary propositions, denoting anything in the universe of discourse that has all six of the corresponding properties. The symbol denotes objects that may be taken as icons of oranges by virtue of bearing those six properties. But there are no objects denoted by the symbol that aren't already oranges themselves. Thus we observe a natural reduction in the denotation of the symbol, consisting in the absence of cases outside of oranges that have all the properties indicated. | ||
| + | |||
| + | The above analysis provides another way to understand the abductive inference that reasons from the Fact <math>x \Rightarrow z\!</math> and the Rule <math>y \Rightarrow z\!</math> to the Case <math>x \Rightarrow y.\!</math> The lack of any cases that are <math>z\!</math> and not <math>y\!</math> is expressed by the implication <math>z \Rightarrow y.\!</math> Taking this together with the Rule <math>y \Rightarrow z\!</math> gives the logical equivalence <math>y = z.\!</math> But this reduces the Case <math>x \Rightarrow y\!</math> to the Fact <math>x \Rightarrow z\!</math> and so the Case is justified. | ||
| + | |||
| + | Viewed in the light of the above analysis, Peirce's example of abductive reasoning exhibits an especially strong form of inference, almost deductive in character. Do all abductive arguments take this form, or may there be weaker styles of abductive reasoning that enjoy their own levels of plausibility? That must remain an open question at this point. | ||
| + | |||
| + | ===Discussion 6=== | ||
| + | |||
| + | Figure 2 shows the implication ordering of logical terms in the form of a ''lattice diagram''. | ||
| + | |||
| + | {| align="center" border="0" cellspacing="10" style="text-align:center; width:100%" | ||
| + | | [[File:ICE Figure 2.jpg|center]] | ||
| + | |- | ||
| + | | height="20px" | <math>\text{Figure 2.} ~~ \text{Disjunctive Term}~ u, ~\text{Taken as Subject}\!</math> | ||
| + | |} | ||
| + | |||
| + | ===Selection 7=== | ||
| + | |||
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
<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> | ||
| Line 576: | Line 525: | ||
<p>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.</p> | <p>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.</p> | ||
| − | <p>(Peirce 1866, Lowell Lecture 7, CE 1, 464–465). | + | <p>(Peirce 1866, Lowell Lecture 7, CE 1, 464–465).</p> |
|} | |} | ||
| Line 585: | Line 534: | ||
<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> | ||
|- | |- | ||
| − | | align="center" | <p><math>\ | + | | align="center" | <p><math>\mathrm{Extension} \times \mathrm{Comprehension} = \mathrm{Constant}~\!</math></p> |
|- | |- | ||
| | | | ||
<p>which crudely expresses the fact that the greater the extension the less the comprehension, we must write</p> | <p>which crudely expresses the fact that the greater the extension the less the comprehension, we must write</p> | ||
|- | |- | ||
| − | | align="center" | <p><math>\ | + | | align="center" | <p><math>\mathrm{Extension} \times \mathrm{Comprehension} = \mathrm{Information}~\!</math></p> |
|- | |- | ||
| | | | ||
| Line 617: | Line 566: | ||
|} | |} | ||
| − | ===Discussion=== | + | ===Discussion 7=== |
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. | ||
| Line 751: | Line 700: | ||
|- | |- | ||
| align="center" | | | align="center" | | ||
| − | <math>{}_\text{man}\overbrace{{}_\text{risible} \qquad\qquad\qquad\qquad {}_\text{man non-}}^{\text{man}}{}_\text{risible}</math> | + | <math>{}_\text{man}\overbrace{{}_\text{risible} \qquad\qquad\qquad\qquad {}_\text{man non-}}^{\text{man}}{}_\text{risible}\!</math> |
|- | |- | ||
| | | | ||
| Line 767: | Line 716: | ||
<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> | ||
|- | |- | ||
| − | | align="center" | <math>\ | + | | align="center" | <math>\mathrm{comprehension} \times \mathrm{extension} = \mathrm{constant},\!</math> |
|- | |- | ||
| <p>we may say:</p> | | <p>we may say:</p> | ||
|- | |- | ||
| − | | align="center" | <math>\ | + | | align="center" | <math>\mathrm{comprehension} \times \mathrm{extension} = \mathrm{information}.\!</math> |
|- | |- | ||
| | | | ||
| Line 822: | Line 771: | ||
|- | |- | ||
| align="center" | | | align="center" | | ||
| − | <math>\text{Conjunctive} \quad \text{Simple} \quad \text{Enumerative}</math> | + | <math>\text{Conjunctive} \quad \text{Simple} \quad \text{Enumerative}\!</math> |
|- | |- | ||
| <p>propositions so related to:</p> | | <p>propositions so related to:</p> | ||
|- | |- | ||
| align="center" | | | align="center" | | ||
| − | <math>\text{Denotative} \quad \text{Informative} \quad \text{Connotative}</math> | + | <math>\text{Denotative} \quad \text{Informative} \quad \text{Connotative}\!</math> |
|- | |- | ||
| | | | ||
| Line 876: | Line 825: | ||
<p>The ground of deductive inference then must be established deductively; that is by reasoning from determinant to determinate, or in other words by reasoning from definition. But this kind of reasoning can only be applied to an object whose character depends upon its definition. Now of most objects it is the definition which depends upon the character; and so the definition must therefore itself rest on induction or hypothesis. But the principle of deduction must rest on nothing but deduction, and therefore it must relate to something whose character depends upon its definition. Now the only objects of which this is true are symbols; they indeed are created by their definition; while neither forms nor things are. Hence, the principle of deduction must relate to the symbolizability of of symbols.</p> | <p>The ground of deductive inference then must be established deductively; that is by reasoning from determinant to determinate, or in other words by reasoning from definition. But this kind of reasoning can only be applied to an object whose character depends upon its definition. Now of most objects it is the definition which depends upon the character; and so the definition must therefore itself rest on induction or hypothesis. But the principle of deduction must rest on nothing but deduction, and therefore it must relate to something whose character depends upon its definition. Now the only objects of which this is true are symbols; they indeed are created by their definition; while neither forms nor things are. Hence, the principle of deduction must relate to the symbolizability of of symbols.</p> | ||
| − | <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 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 | + | <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 these principles must as principles be universal. Hence they are as follows:—</p> | ||
|- | |- | ||
| − | | align="center" | < | + | | align="center" | <math>\text{All things, forms, symbols are symbolizable.}~\!</math> |
|- | |- | ||
| | | | ||
| Line 894: | Line 843: | ||
<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> | ||
|- | |- | ||
| − | | align="center" | <math>\text{All things, forms, symbols are symbolizable.}\!</math> | + | | align="center" | <math>\text{All things, forms, symbols are symbolizable.}~\!</math> |
|- | |- | ||
| | | | ||
| Line 920: | Line 869: | ||
<p>Thus we have in order of strength Deduction, Induction, Hypothesis. Deduction, in fact, is the only demonstration; yet no one thinks of questioning a good induction, while hypothesis is proverbially dangerous. ''Hypotheses non fingo'', said Newton, striving to place his theory on a basis of strict induction. Yet it is hypotheses with which we must start; the baby when he lies turning his fingers before his eyes is making a hypothesis as to the connection of what he sees and what he feels. Hypotheses give us our facts. Induction extends our knowledge. Deduction makes it distinct.</p> | <p>Thus we have in order of strength Deduction, Induction, Hypothesis. Deduction, in fact, is the only demonstration; yet no one thinks of questioning a good induction, while hypothesis is proverbially dangerous. ''Hypotheses non fingo'', said Newton, striving to place his theory on a basis of strict induction. Yet it is hypotheses with which we must start; the baby when he lies turning his fingers before his eyes is making a hypothesis as to the connection of what he sees and what he feels. Hypotheses give us our facts. Induction extends our knowledge. Deduction makes it distinct.</p> | ||
| − | <p>(Peirce 1865, | + | <p>(Peirce 1865, Harvard Lecture 10 : Grounds of Induction, CE 1, 283).</p> |
|} | |} | ||
| Line 928: | Line 877: | ||
| | | | ||
<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> | ||
| − | + | |- | |
| − | + | | align="center" | <math>S ~\mathrm{is}~ B\!</math> | |
| − | + | |- | |
| − | <p>where B is an object of connotation. We infer that something else:</p> | + | | |
| − | + | <p>where <math>B\!</math> is an object of connotation. We infer that something else:</p> | |
| − | + | |- | |
| − | + | | align="center" | <math>\Sigma ~\mathrm{is}~ B\!</math> | |
| − | <p>Let us suppose that | + | |- |
| + | | | ||
| + | <p>Let us suppose that <math>\Sigma\!</math> contains more ''information'' than <math>S.\!</math> Then, if <math>\Sigma\!</math> is no more extensive than <math>S,\!</math> <math>\Sigma ~\mathrm{is}~ B\!</math> is a better judgment than <math>S ~\mathrm{is}~ B\!</math> because it contains more information without predicating <math>B\!</math> of anything doubtful.</p> | ||
<p>Thus, it is better to say "All men are mortal" than "all rational animals are mortal" for the former implies the latter and contains no more possibility of error and is more ''distinct''.</p> | <p>Thus, it is better to say "All men are mortal" than "all rational animals are mortal" for the former implies the latter and contains no more possibility of error and is more ''distinct''.</p> | ||
| − | <p>But in every case of induction | + | <p>But in every case of induction <math>\Sigma\!</math> is also more extensive than <math>S.\!</math> Then in case <math>S\!</math> is a true symbol and <math>S ~\mathrm{is}~ B\!</math> is a single true judgment, this judgment or proposition must be the result of induction, as we saw in the last lecture that all propositions are. The question is, therefore, which is the preferable theory, <math>S ~\mathrm{is}~ B\!</math> or <math>\Sigma ~\mathrm{is}~ B.\!</math> The greater information of <math>\Sigma\!</math> causes the latter theory to contain more truth but its greater extension renders it liable to more error. If in <math>\Sigma\!</math> the extension of <math>S\!</math> is increased more than the information is, the connotation will be diminished and ''vice versa''. Accordingly the greater the connotation of <math>\Sigma\!</math> relatively to that of <math>S,\!</math> the better is the theory proposed, <math>\Sigma ~\mathrm{is}~ B.\!</math></p> |
<p>Which of the two theories to select in any case will depend upon the motives which influence us. In a desperate practical case, if one's life depends upon taking the right one, he ought to select the one whose subject has the greatest connotation. In a cool speculation where safety is the essential; the least extensive should be taken.</p> | <p>Which of the two theories to select in any case will depend upon the motives which influence us. In a desperate practical case, if one's life depends upon taking the right one, he ought to select the one whose subject has the greatest connotation. In a cool speculation where safety is the essential; the least extensive should be taken.</p> | ||
| − | <p>So much for the preference between two theories. But in proceeding from fact to theory | + | <p>So much for the preference between two theories. But in proceeding from fact to theory — in such a case as that about ''neat'', ''swine'', ''sheep'', and ''deer'' — <math>S\!</math> is a mere enumerative term and has no connotation at all. In this case therefore <math>\Sigma\!</math> increases the connotation of <math>S\!</math> absolutely and <math>\Sigma ~\mathrm{is}~ B\!</math> ought therefore to be absolutely preferred to <math>S ~\mathrm{is}~ B\!</math> and be accepted assertorically; as long as there is no question between this theory and some other and as long as it is not opposed by some other induction.</p> |
| + | |||
| + | <p>(Peirce 1865, Harvard Lecture 10 : Grounds of Induction, CE 1, 285).</p> | ||
|} | |} | ||
| − | |||
| − | |||
===Selection 34=== | ===Selection 34=== | ||
| Line 953: | Line 904: | ||
| | | | ||
<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> | ||
| + | |- | ||
| + | | align="center" | <math>X ~\mathrm{is}~ P\!</math> | ||
| + | |- | ||
| + | | | ||
| + | <p>where <math>X\!</math> is an object of denotation; we explain this by supposing that:</p> | ||
| + | |- | ||
| + | | align="center" | <math>X ~\mathrm{is}~ \Pi\!</math> | ||
| + | |- | ||
| + | | | ||
| + | <p>and <math>\Pi\!</math> always contains more information than <math>P.\!</math> If <math>\Pi,\!</math> therefore, has no more comprehension than <math>P,\!</math> it is better to say <math>X ~\mathrm{is}~ \Pi\!</math> than <math>X ~\mathrm{is}~ P.\!</math></p> | ||
| − | + | <p>It is ''clearer'' to say that Every man is mortal than to say that Every man is either a good mortal or a bad mortal.</p> | |
| − | <p> | + | <p>But in the case of hypothesis, <math>\Pi\!</math> always comprehends more than <math>P.\!</math> To decide then between the two; we have to consider whether <math>\Pi\!</math> has more denotation than <math>P\!</math> for if it has, the information of <math>P\!</math> is increased more in <math>\Pi\!</math> than its comprehension is and ''vice versa''; and we must be decided which to take by our motives.</p> |
| − | + | <p>This is the case of a preference between hypotheses. But in the first proceedure from facts, <math>P\!</math> is a mere conjunctive term, destitute of any denotation before this proposition. Hence in this case the information is increased absolutely, the connotation only relatively, and the hypothesis is absolutely needed and must be taken as a ''pis aller'' unless opposed by some other argument and until a better one presents itself.</p> | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | <p>This is the case of a preference between hypotheses. But in the first proceedure from facts, P is a mere conjunctive term, destitute of any denotation before this proposition. Hence in this case the information is increased absolutely, the connotation only relatively, and the hypothesis is absolutely needed and must be taken as a ''pis aller'' unless opposed by some other argument and until a better one presents itself.</p> | ||
<p>Polarization for instance is a series of phenomena which it is impossible to name or define without the use of a hypothesis.</p> | <p>Polarization for instance is a series of phenomena which it is impossible to name or define without the use of a hypothesis.</p> | ||
| − | <p>(Peirce 1865, | + | <p>(Peirce 1865, Harvard Lecture 10 : Grounds of Induction, CE 1, 285–286).</p> |
|} | |} | ||
| − | |||
| − | |||
===Selection 35=== | ===Selection 35=== | ||
| Line 981: | Line 932: | ||
<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> | ||
| − | <p>We first | + | <p>We first distinguished three kinds of reference which every true symbol has to its object.</p> |
<p>In the first place, every true symbol is applicable to some real thing. Hence, every symbol whether true or not asserts itself to be applicable to some real thing. This is the ''denotation'' of the symbol. All that we know of things is as denotative objects of symbols. And thus all denotation is comparative, merely. One symbol has more denotation than another or is more extensive when it asserts itself to be applicable to all the things of which the first asserts itself to be applicable and also to others.</p> | <p>In the first place, every true symbol is applicable to some real thing. Hence, every symbol whether true or not asserts itself to be applicable to some real thing. This is the ''denotation'' of the symbol. All that we know of things is as denotative objects of symbols. And thus all denotation is comparative, merely. One symbol has more denotation than another or is more extensive when it asserts itself to be applicable to all the things of which the first asserts itself to be applicable and also to others.</p> | ||
| Line 993: | Line 944: | ||
<p>Thus, no matter how general a symbol may be, it must have some connotation limiting its denotation; it must refer to some determinate form; but it must also connote ''reality'' in order to denote at all; but ''all'' that has any determinate form has reality and thus this reality is a part of the connotation which does not limit the extension of the symbol.</p> | <p>Thus, no matter how general a symbol may be, it must have some connotation limiting its denotation; it must refer to some determinate form; but it must also connote ''reality'' in order to denote at all; but ''all'' that has any determinate form has reality and thus this reality is a part of the connotation which does not limit the extension of the symbol.</p> | ||
| − | <p>And so every symbol has information. To say that a symbol has information is as much as to say that it implies that it is equivalent to another symbol different in connotation. | + | <p>And so every symbol has information. To say that a symbol has information is as much as to say that it implies that it is equivalent to another symbol different in connotation.</p> |
| + | |||
| + | <p>(Peirce 1865, Harvard Lecture 11, CE 1, 286–288).</p> | ||
|} | |} | ||
| Line 1,003: | Line 956: | ||
<p>In short the formula:</p> | <p>In short the formula:</p> | ||
| − | + | |- | |
| − | + | | align="center" | <math>\mathrm{Connotation} \times \mathrm{Denotation} = \mathrm{Information}\!</math> | |
| − | + | |- | |
| + | | | ||
<p>holds good thoroughly.</p> | <p>holds good thoroughly.</p> | ||
| − | <p>(Peirce 1865, | + | <p>(Peirce 1865, Harvard Lecture 11, CE 1, 288).</p> |
|} | |} | ||
| Line 1,015: | Line 969: | ||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | <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 | + | <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 <math>\mathrm{A},\!</math> <math>\mathrm{E},\!</math> and <math>\mathrm{I}~\!</math> are also determined, this definition is inadequate. We were led to substitute for it the following:—</p> |
<p>The subject is the term determined in connotation and determining denotation; the predicate is the term determined in denotation and determining in connotation.</p> | <p>The subject is the term determined in connotation and determining denotation; the predicate is the term determined in denotation and determining in connotation.</p> | ||
| Line 1,022: | Line 976: | ||
<p>Thus we have three kinds of judgments:</p> | <p>Thus we have three kinds of judgments:</p> | ||
| − | + | |- | |
| − | + | | align="center" | | |
| − | + | <math>\begin{matrix} | |
| − | + | \mathrm{IC} | |
| − | + | \\ | |
| + | \mathrm{DC} | ||
| + | \\ | ||
| + | \mathrm{DI} | ||
| + | \end{matrix}</math> | ||
| + | |- | ||
| + | | | ||
<p>In the first case the subject is informative, the predicate connotative; that is to say, the connotation of the symbol which forms the subject is explicated in the predicate. Such judgments, usually called explicatory or analytic, I call connotative.</p> | <p>In the first case the subject is informative, the predicate connotative; that is to say, the connotation of the symbol which forms the subject is explicated in the predicate. Such judgments, usually called explicatory or analytic, I call connotative.</p> | ||
| Line 1,033: | Line 993: | ||
<p>In the third case the subject is denotative, the predicate is informative. That is, the thing which the subject denotes is offered as an example of the application of the symbol which forms the predicate. I call such judgments denotative.</p> | <p>In the third case the subject is denotative, the predicate is informative. That is, the thing which the subject denotes is offered as an example of the application of the symbol which forms the predicate. I call such judgments denotative.</p> | ||
| − | <p>(Peirce 1865, | + | <p>(Peirce 1865, Harvard Lecture 11, CE 1, 288–289).</p> |
|} | |} | ||
| Line 1,053: | Line 1,013: | ||
| valign=top | 5th. || The fact that every mental representation is a symbol in a loose sense, and that every conception is so strictly; | | valign=top | 5th. || The fact that every mental representation is a symbol in a loose sense, and that every conception is so strictly; | ||
|- | |- | ||
| − | | valign=top | 6th. || The fact that hypothesis gives terms or problematic propositions; inductions propositions strictly speaking | + | | valign=top | 6th. || The fact that hypothesis gives terms or problematic propositions; inductions propositions strictly speaking — assertory propositions; and deduction apodictic propositions or syllogisms proper. That thus every elementary conception implies hypothesis and every judgment induction; |
|- | |- | ||
| valign=top | 7th. || The relations of denotation, connotation, and information; and | | valign=top | 7th. || The relations of denotation, connotation, and information; and | ||
| Line 1,060: | Line 1,020: | ||
|} | |} | ||
| − | <p>we found ourselves in a condition to solve the question of the grounds of inference by putting together these materials. | + | <p>we found ourselves in a condition to solve the question of the grounds of inference by putting together these materials.</p> |
| + | |||
| + | <p>(Peirce 1865, CE 1, 289).</p> | ||
|} | |} | ||
| Line 1,071: | Line 1,033: | ||
<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> | ||
| − | <p>Those grounds of possibility we found to be that All things, forms, symbols are symbolizable. For these laws must refer to symbolization because symbolization and inference are the same. As grounds of possibility they must refer to the possibility of symbolization. As logical laws they must consider the reference of symbols in general to objects. Now symbols in general have three relations to objects; namely so far as the latter contain things, forms, symbols. Finally as general principles they must be universal. | + | <p>Those grounds of possibility we found to be that All things, forms, symbols are symbolizable. For these laws must refer to symbolization because symbolization and inference are the same. As grounds of possibility they must refer to the possibility of symbolization. As logical laws they must consider the reference of symbols in general to objects. Now symbols in general have three relations to objects; namely so far as the latter contain things, forms, symbols. Finally as general principles they must be universal.</p> |
| + | |||
| + | <p>(Peirce 1865, CE 1, 289–290).</p> | ||
|} | |} | ||
| Line 1,084: | Line 1,048: | ||
<p>The three principles were proved by the several kinds of inference with certainty. The inductive proof attained certainty by considering all the instances that could be taken. And the hypothetic inference attained certainty by having only a subjective character.</p> | <p>The three principles were proved by the several kinds of inference with certainty. The inductive proof attained certainty by considering all the instances that could be taken. And the hypothetic inference attained certainty by having only a subjective character.</p> | ||
| − | <p>The influence of the three principles was shown in the case of deduction by the rule of ''Nota notae'' without which there could be no deduction. In the case of Induction by the affirmative denotative proposition which must always be the first premiss. And in the case of Hypothesis by the Universal connotative proposition which must always be the second premiss. | + | <p>The influence of the three principles was shown in the case of deduction by the rule of ''Nota notae'' without which there could be no deduction. In the case of Induction by the affirmative denotative proposition which must always be the first premiss. And in the case of Hypothesis by the Universal connotative proposition which must always be the second premiss.</p> |
| + | |||
| + | <p>(Peirce 1865, CE 1, 290).</p> | ||
|} | |} | ||
| Line 1,095: | Line 1,061: | ||
<p>I might also show that no induction or hypothesis is completely true except such as we call cognitions ''a priori''. For the chance against it is infinite. Hence, the question what is the 'probability' of an induction or hypothesis is senseless and the true question is how much truth does an induction contain. For the same reasons by how much truth should not be meant what proportion of inferences therefrom are true but simply of how much value are certain premisses in giving us truth by induction or hypothesis.</p> | <p>I might also show that no induction or hypothesis is completely true except such as we call cognitions ''a priori''. For the chance against it is infinite. Hence, the question what is the 'probability' of an induction or hypothesis is senseless and the true question is how much truth does an induction contain. For the same reasons by how much truth should not be meant what proportion of inferences therefrom are true but simply of how much value are certain premisses in giving us truth by induction or hypothesis.</p> | ||
| − | <p>We must distinguish therefore the truth which an inductive or hypothetic conclusion may have by accident from that which it must have from the nature of the facts explained. The former cannot properly be estimated. The latter can. For to consider first induction; if the same conclusion result inductively as the least truthful explanation possible of two different sets of facts, it is plain that a certain amount of truth it is obliged to have on account of each instance, that is on account of the extension of the subject of the fact. And each instance determines a certain amount of truth independently of the others. So that the number of different kinds of instances measures the least amount of truth the induction can have. In the same way with hypothesis the number of different properties explained measures the least possible truth of the hypothesis. | + | <p>We must distinguish therefore the truth which an inductive or hypothetic conclusion may have by accident from that which it must have from the nature of the facts explained. The former cannot properly be estimated. The latter can. For to consider first induction; if the same conclusion result inductively as the least truthful explanation possible of two different sets of facts, it is plain that a certain amount of truth it is obliged to have on account of each instance, that is on account of the extension of the subject of the fact. And each instance determines a certain amount of truth independently of the others. So that the number of different kinds of instances measures the least amount of truth the induction can have. In the same way with hypothesis the number of different properties explained measures the least possible truth of the hypothesis.</p> |
| + | |||
| + | <p>(Peirce 1865, CE 1, 293–294).</p> | ||
|} | |} | ||
| Line 1,106: | Line 1,074: | ||
<p>We may sum up then by the rule that the value of facts is in proportion to their number; and that from given facts the best inference when all possible retrenchment has been made, is the one which being inductive has the most comprehensive subject and which being hypothetic has the most extensive predicate.</p> | <p>We may sum up then by the rule that the value of facts is in proportion to their number; and that from given facts the best inference when all possible retrenchment has been made, is the one which being inductive has the most comprehensive subject and which being hypothetic has the most extensive predicate.</p> | ||
| − | <p>This seems to complete the logical theory of inference | + | <p>This seems to complete the logical theory of inference …</p> |
<p>(Peirce 1865, CE 1, 294).</p> | <p>(Peirce 1865, CE 1, 294).</p> | ||
| Line 1,121: | Line 1,089: | ||
<p>There are also principles of the Judgment corresponding to these conceptions of which we have instances in the laws that all things, forms, symbols are symbolizable.</p> | <p>There are also principles of the Judgment corresponding to these conceptions of which we have instances in the laws that all things, forms, symbols are symbolizable.</p> | ||
| − | <p>All the principles that can be so derived from the forms of logic must be valid for all experience. For experience has used logic. Everything else admits of speculative doubt. | + | <p>All the principles that can be so derived from the forms of logic must be valid for all experience. For experience has used logic. Everything else admits of speculative doubt.</p> |
| + | |||
| + | <p>(Peirce 1865, CE 1, 302).</p> | ||
|} | |} | ||
| − | == | + | ==Commentary Notes== |
| + | |||
| + | ===Commentary Note 1=== | ||
| − | + | 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: | |
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | + | <p>Let us now return to the information.</p> | |
| + | |||
| + | <p>The information of a term is the measure of its superfluous comprehension.</p> | ||
|} | |} | ||
| − | === | + | Today we would say that information has to do with constraint, law, redundancy. I think that Peirce is talking about more or less the same thing under the theme of ''superfluous comprehension'', where the comprehension of a term or expression is the collection of properties, also known as ''intensions'', that it implies about the things to which it applies. |
| + | |||
| + | ===Commentary Note 2=== | ||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | + | <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>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'' — ''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.</p> | ||
|} | |} | ||
| − | + | When we set about comprehending the comprehension of a sign, say, a term or expression, we run into a very troublesome issue as to how many intensions (predicates, properties, qualities) an object of that sign has. For how do we quantify the number of qualities a thing has? Without some more or less artificial strait imposed on the collection of qualities, the number appears without limit. | |
| − | + | Let's pass this by, as Peirce does, for now, and imagine that we have fixed on some way of speaking sensibly about ''the'' comprehension of a sign in a particular set of signs, the collection of which we may use as a language or a medium. | |
| − | |||
| − | |||
| − | |||
| − | + | Then we can begin to talk about the amount of redundancy, the superfluidity of comprehension, if you will, as Peirce does, that belongs to a given sign, and thus to its object. | |
| − | + | ===Commentary Note 3=== | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | + | 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''. | |
|} | |} | ||
| − | + | In a sense of primal innocence, logical laws bind the vacuum state of any medium that is capable of bearing, delivering, nurturing, and preserving signal meanings. In other words, when we use symbols, not simple signs, in a channel, language, or medium that is constrained by logical laws, these laws do more than strain, they also exact the generation of symbols upon symbols to fill the requisite logical forms, and so there will always be lots more ways than one to say any given thing you might choose to say. | |
| − | + | Alternate Version — | |
| − | |||
| − | |||
| − | + | In a sense, logical laws bind the initial state of any medium possessed of a capacity to bear, deliver, maintain, and nurture the meanings of signals. In other words, when we use symbols, not mere signs, in a channel, language, or medium constrained by logical laws, these laws do more than confine, they also beget the generation of symbols upon symbols to fill the requisite forms, guaranteeing there will always be ample ways to say any thing that can be said. | |
| − | + | ===Commentary Note 4=== | |
| − | |||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | <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> |
| + | |||
| + | <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> | + | <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.</p> | |
| − | |||
| − | |||
|} | |} | ||
| − | + | In order to speak of the meandering channel, the abdundancy of language, the superfluidity of media, the play in the wheel of symbolism, then, it is necessary to classify the different kinds of signs, the varied ways that signs up to and including symbols, namely, those that are interpretive by dint of their very essence, can be interpreted as being referential to their objects. | |
| − | + | On running through this familiar yet ever strange refrain for another time, I see that I have scarcely begun to trace the sinews of the linkages among the three types of signs, "the differences there are in the manner in which different representations stand for their objects", the matter of extension and comprehension, and the whole life-cycle of inquiry that engages me most. | |
| − | + | ===Commentary Note 5=== | |
| − | + | Signs, inquiry, and information. Let's focus on that for a while. | |
| − | + | To put Peirce's examples more in line with the order of his three categories, let us renumber them in the following way: | |
| − | + | {| cellpadding=4 | |
| + | |- | ||
| + | | || 1. || The conjunctive term "spherical bright fragrant juicy tropical fruit". | ||
| + | |- | ||
| + | | || 2.1. || The disjunctive term "man or horse or kangaroo or whale". | ||
| + | |- | ||
| + | | || 2.2. || The disjunctive term "neat or swine or sheep or deer". | ||
| + | |} | ||
| − | + | Peirce suggests an analogy or a parallelism between the corresponding elements of the following triples: | |
| − | + | {| cellpadding=4 | |
| − | + | |- | |
| − | + | | || 1. || Conjunctive Term || : || Iconical Sign || : || Abductive Case | |
| + | |- | ||
| + | | || 2. || Disjunctive Term || : || Indicial Sign || : || Inductive Rule | ||
| + | |} | ||
| − | + | Here is an overview of the two patterns of reasoning, along with the first steps of an analysis in sign-theoretic terms: | |
| − | + | 1. Conjunctive term "spherical bright fragrant juicy tropical fruit". | |
| − | |||
| − | |||
| + | {| align="center" cellspacing="10" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
| − | o-------------------------------------------------o | + | o-----------------------------o-----------------------------o |
| − | | | + | | Objective Framework | Interpretive Framework | |
| − | + | o-----------------------------o-----------------------------o | |
| − | + | | | | |
| − | | | + | | t_1 t_2 ... t_5 t_6 | |
| − | | | + | | o o o o | |
| − | | | + | | * * * * | |
| − | | | + | | * * * * | |
| − | | | + | | * * * * | |
| − | | | + | | ** ** | |
| − | | | + | | z o | |
| − | | | + | | |\ | |
| − | | | + | | | \ Rule | |
| − | | | + | | | \ y=>z | |
| − | | | + | | | \ | |
| − | | | + | | Fact | \ | |
| − | | | + | | x=>z | o y | |
| − | | | + | | | / | |
| − | | | + | | | / | |
| − | | | + | | | / Case | |
| − | | | + | | | / x=>y | |
| − | + | | |/ | | |
| − | | | + | | x o | |
| − | | | + | | | |
| − | | | + | o-----------------------------------------------------------o |
| − | | | + | | Conjunctive Predicate z, Abduction to the Case x => y | |
| − | | | + | o-----------------------------------------------------------o |
| − | | | + | | | |
| − | | | + | | !S! = !I! = {t_1, t_2, t_3, t_4, t_5, t_6, x, y, z} | |
| − | | | + | | | |
| − | | | + | | t_1 = "spherical" | |
| − | | | + | | t_2 = "bright" | |
| − | | | + | | t_3 = "fragrant" | |
| − | o-------------------------------------------------o | + | | t_4 = "juicy" | |
| − | + | | t_5 = "tropical" | | |
| + | | t_6 = "fruit" | | ||
| + | | | | ||
| + | | x = "subject" | | ||
| + | | y = "orange" | | ||
| + | | z = "spherical bright fragrant juicy tropical fruit" | | ||
| + | | | | ||
| + | o-----------------------------------------------------------o | ||
</pre></font> | </pre></font> | ||
| + | |} | ||
| − | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | |
| + | | | ||
| + | <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> | ||
| − | + | <center>"spherical, bright, fragrant, juicy, tropical fruit".</center> | |
| − | + | <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, CE 1, 470).</p> | |
| + | |} | ||
| − | + | 2. Disjunctive term "neat or swine or sheep or deer". | |
| − | |||
| − | |||
| + | {| align="center" cellspacing="10" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
| − | o-------------------------------------------------o | + | o-----------------------------o-----------------------------o |
| − | | | + | | Objective Framework | Interpretive Framework | |
| − | + | o-----------------------------o-----------------------------o | |
| − | | | + | | | |
| − | | | + | | w o | |
| − | | | + | | |\ | |
| − | | | + | | | \ Rule | |
| − | | | + | | | \ v=>w | |
| − | | | + | | | \ | |
| − | | | + | | Fact | \ | |
| − | | | + | | u=>w | o v | |
| − | | | + | | | / | |
| − | | | + | | | / | |
| − | | | + | | | / Case | |
| − | | | + | | | / u=>v | |
| − | | | + | | |/ | |
| − | | | + | | u o | |
| − | | | + | | ** ** | |
| − | | | + | | * * * * | |
| − | | | + | | * * * * | |
| − | | | + | | * * * * | |
| − | | | + | | o o o o | |
| − | | | + | | s_1 s_2 s_3 s_4 | |
| − | | | + | | | |
| − | | | + | o-----------------------------------------------------------o |
| − | | | | + | | Disjunctive Subject u, Induction to the Rule v => w | |
| − | | | + | o-----------------------------------------------------------o |
| − | | | + | | | |
| − | o-------------------------------------------------o | + | | !S! = !I! = {s_1, s_2, s_3, s_4, u, v, w} | |
| − | + | | | | |
| + | | s_1 = "neat" | | ||
| + | | s_2 = "swine" | | ||
| + | | s_3 = "sheep" | | ||
| + | | s_4 = "deer" | | ||
| + | | | | ||
| + | | u = "neat or swine or sheep or deer" | | ||
| + | | v = "cloven-hoofed" | | ||
| + | | w = "herbivorous" | | ||
| + | | | | ||
| + | o-----------------------------------------------------------o | ||
</pre></font> | </pre></font> | ||
| + | |} | ||
| + | |||
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| + | | | ||
| + | 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). | ||
| + | |} | ||
| − | + | ===Commentary Note 6=== | |
| − | + | Before we return to Peirce's description of a near duality between icons and indices, involving a reciprocal symmetry between intensions and extensions of concepts that becomes perturbed to the breaking and yet the growing point by the receipt of a fresh bit of information, I think that it may help to recall a few pieces of technical terminology that Peirce introduced into this discussion. | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | '''Passage 1''' | |
| − | + | {| align="center" cellspacing="6" 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. …</p> | |
| − | + | <p>Now the sphere considered as a quantity is called the Extension; and the content considered as quantity is called the Comprehension. Extension and Comprehension are also termed Breadth and Depth. So that a wider term is one which has a greater extension; a narrower one is one which has a less extension. A higher term is one which has a less Comprehension and a lower one has more.</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> | |
| − | |||
| − | |||
| + | {| align="center" cellspacing="6" style="text-align:center; width:70%" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| − | | | + | | | | |
| + | | What is 'denoted' | What is 'connoted' | | ||
| + | | | | | ||
| + | | Sphere | Content | | ||
| + | | | | | ||
| + | | Extension | Comprehension | | ||
| + | | | | | ||
| + | | ( wider | ( lower | | ||
| + | | Breadth < | Depth < | | ||
| + | | ( narrower | ( higher | | ||
| + | | | | | ||
| + | | What is contained 'under' | What is contained 'in' | | ||
| + | | | | | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
</pre></font> | </pre></font> | ||
| + | |} | ||
| − | + | <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> | ||
| − | <p> | + | <p>or:</p> |
| − | <p> | + | <p>Whatever is contained ''in'' a word belongs to whatever is contained under it.</p> |
| − | <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> | + | <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> |
| − | |||
| − | |||
| − | |||
| − | |||
|} | |} | ||
| − | + | '''Passage 2''' | |
| − | |||
| − | |||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | <p>The | + | <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> |
| − | + | {| align="center" cellspacing="6" style="text-align:center; width:60%" | |
| + | | | ||
| + | <font face="courier new"><pre> | ||
| + | o------------------------o------------------------o | ||
| + | | | | | ||
| + | | Being | Nothing | | ||
| + | | | | | ||
| + | | All breadth | All depth | | ||
| + | | | | | ||
| + | | No depth | No breadth | | ||
| + | | | | | ||
| + | o------------------------o------------------------o | ||
| + | </pre></font> | ||
|} | |} | ||
| − | We | + | <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> |
| + | {| align="center" cellspacing="6" style="text-align:center; width:60%" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
| − | o | + | o------------------------o------------------------o |
| − | | | + | | | | |
| − | + | | Simple terms | Individual terms | | |
| − | + | | | | | |
| − | + | o------------------------o------------------------o | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | | | ||
| − | | | ||
| − | |||
| − | |||
| − | o------------------------ | ||
</pre></font> | </pre></font> | ||
| + | |} | ||
| − | + | <p>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.</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> | ||
| + | |||
| + | <p><center>Some ''A'' is ''B''</center> | ||
| + | and | ||
| + | <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, 460–461).</p> | |
| + | |} | ||
| − | + | '''Passage 3''' | |
| − | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | |
| − | + | | | |
| − | + | <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> | |
| + | |} | ||
| − | + | ===Commentary Note 7=== | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | I find one more patch of material from Peirce's early lectures that we need to cover the subject of indices. I include a piece of the context, even if it overlaps a bit with fragments that still live in recent memory. | |
| − | + | '''Passage 4''' | |
| − | |||
| − | |||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | <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> |
| + | |||
| + | <center>man and horse and kangaroo and whale,</center> | ||
| − | <p> | + | <p>and secondly, cases like:</p> |
| − | < | + | <center>spherical bright fragrant juicy tropical fruit.</center> |
| − | <p>In this | + | <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> | + | <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> | ||
|} | |} | ||
| − | + | ===Commentary Note 8=== | |
| + | |||
| + | I'm going to make yet another try at following the links that Peirce makes among conventions, disjunctive terms, indexical signs, and inductive rules. For this purpose, I'll break the text up into smaller pieces, and pick out just those parts of it that have to do with the indexical aspect of things. | ||
| + | |||
| + | Before I can get to this, though, I will need to deal with the uncertainty that I am experiencing over the question as to whether a ''connotation'' is just another ''notation'', and thus belongs to the interpretive framework, that is, the ''SI''-plane, or whether it is an objective property, a quality of objects of terms. I have decided to finesse the issue by forcing my own brand of interpretation on the next text, where the trouble starts: | ||
| + | |||
| + | '''Passage 1''' | ||
| + | |||
| + | {| align="center" cellspacing="6" 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. …</p> | ||
| + | |||
| + | <p>Now the sphere considered as a quantity is called the Extension; and the content considered as quantity is called the Comprehension. Extension and Comprehension are also termed Breadth and Depth. So that a wider term is one which has a greater extension; a narrower one is one which has a less extension. A higher term is one which has a less Comprehension and a lower one has more.</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> | |
| + | {| align="center" cellspacing="6" style="text-align:center; width:70%" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| − | | | + | | | | |
| + | | What is 'denoted' | What is 'connoted' | | ||
| + | | | | | ||
| + | | Sphere | Content | | ||
| + | | | | | ||
| + | | Extension | Comprehension | | ||
| + | | | | | ||
| + | | ( wider | ( lower | | ||
| + | | Breadth < | Depth < | | ||
| + | | ( narrower | ( higher | | ||
| + | | | | | ||
| + | | What is contained 'under' | What is contained 'in' | | ||
| + | | | | | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
</pre></font> | </pre></font> | ||
| + | |} | ||
| + | |||
| + | <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> | |
| − | + | |} | |
| − | + | ||
| − | + | 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''. | |
| − | + | ||
| − | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | |
| − | + | | | |
| − | + | <p>Now the sphere considered as a quantity is called the Extension;</br> | |
| − | | | + | and the content considered as quantity is called the Comprehension.</p> |
| − | + | |} | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | | | ||
| − | |||
| − | |||
| − | </ | ||
| − | I | + | Taking this point of view, then, I will consider the Extensions of terms and the Comprehensions of terms, to be ''quantities'', in effect, objective formal elements that are subject to being compared with one another within their respective domains. In particular, I will view them as elements of partially ordered sets. On my reading of Peirce's text, the word ''content'' is still ambiguous from context of use to context of use, but I will simply let that be as it may, hoping that it will suffice to fix the meaning of the more technical term ''comprehension''. |
| − | + | This is still experimental — I'll just have to see how it works out over time. | |
| − | + | ===Commentary Note 9=== | |
| − | + | 2. Conventions, Disjunctive Terms, Indexical Signs, Inductive Rules | |
| − | + | ||
| − | + | 2.1. "man and horse and kangaroo and whale" (intensional conjunction). | |
| − | + | ||
| − | + | '''Nota Bene.''' In this particular choice of phrasing, Peirce is using the intensional "and", meaning that the compound term has the intensions that are shared by all of the component terms, in this way producing a term that bears the ''greatest common intension'' of the terms that are connected in it. This is formalized as the ''greatest lower bound'' in a lattice of intensions, dual to the union of sets or ''least upper bound'' in a lattice of extensions. | |
| − | + | ||
| − | + | It is perhaps more common today to use the extensional "or" in order to express the roughly equivalent compound concept: | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | 2.1. "men or horses or kangaroos or whales" (extensional disjunction). | |
| − | === | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| + | | | ||
| + | <p>Yet there are combinations of words and combinations of conceptions which are not strictly speaking symbols.</p> | ||
| − | + | <p>These are of two kinds of which I will give you instances.</p> | |
| − | + | <p>We have first cases like: "man and horse and kangaroo and whale" ...</p> | |
| − | < | + | <p>[This term] has no comprehension which is adequate to the limitation of the extension.</p> |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | <p>In fact, men, horses, kangaroos, and whales have no attributes in common which are not possessed by the entire class of mammals.</p> | |
| − | + | <p>For this reason, this disjunctive term, "man and horse and kangaroo and whale", is of no use whatever.</p> | |
| − | |||
| − | <p> | ||
| − | <p> | + | <p>For suppose it is the subject of a sentence; suppose we know that men and horses and kangaroos and whales have some common character.</p> |
| − | <p> | + | <p>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.</p> |
| − | <p> | + | <p>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;</p> |
| − | <p> | + | <p>then, the person who has found out this, knows more about this thing than that it is a mammal;</p> |
| − | <p> | + | <p>he therefore knows which of these four it is for these four have nothing in common except what belongs to all other mammals.</p> |
| − | <p> | + | <p>Hence in this case the particular one may be substituted for the disjunctive term.</p> |
| − | <p> | + | <p>A disjunctive term, then, — one which aggregates the extension of several symbols, — may always be replaced by a simple term.</p> |
| − | <p> | + | <p>C.S. Peirce, 'Chronological Edition', CE 1, 468.</p> |
| + | |} | ||
| − | + | Let us first assemble a minimal syntactic domain ''S'' that is sufficient to begin discussing this example: | |
| − | + | : ''S'' = {"m", "h", "k", "w", "S", "M", "P"} | |
| − | + | Here, I have introduced the abbreviations: | |
| − | + | : "m" = "man" | |
| + | : "h" = "horse" | ||
| + | : "k" = "kangaroo" | ||
| + | : "w" = "whale" | ||
| − | + | : "S" = "man or horse or kangaroo or whale" | |
| + | : "M" = "Mammal" | ||
| + | : "P" = "Predicate shared by man, horse, kangaroo, whale" | ||
| − | + | Let's attempt to keep tabs on things by using angle brackets for the comprehension of a term, and square brackets for the extension of a term. | |
| − | + | For brevity, let x = ["x"], in general. | |
| − | |||
| − | |||
| − | + | Here is an initial picture of the situation, so far as I can see it: | |
| − | |||
| − | |||
| + | {| align="center" cellspacing="10" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| − | | | + | | Objective Framework | Interpretive Framework | |
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| | | | | | ||
| − | | | + | | P <------------o------------ "P" | |
| − | | | + | | = \ |\ | |
| − | | | + | | = \ | \ | |
| − | | | + | | = \ | \ | |
| − | | | + | | = \ | \ | |
| − | | | + | | = \ | \ | |
| − | | | + | | P M <------o--------------|--- "M" | |
| − | | | + | | \ = | / | |
| − | | | + | | \ = | / | |
| − | | | + | | \ = | / | |
| − | | | + | | \ = | / | |
| + | | \ = |/ | | ||
| + | | S <------------o------------ "S" | | ||
| + | | ** ** ** ** | | ||
| + | | * * * * * * * * | | ||
| + | | * * * * * * * * | | ||
| + | | * * * * * * * * | | ||
| + | | o o o o o o o o | | ||
| + | | m h k w "m" "h" "k" "w" | | ||
| + | | | | ||
| + | o-----------------------------------------------------------o | ||
| + | | Disjunctive Subject "S" and Inductive Rule "M => P" | | ||
| + | o-----------------------------------------------------------o | ||
| + | | | | ||
| + | | !S! = !I! = {"m", "h", "k", "w", "S", "M", "P"} | | ||
| + | | | | ||
| + | | "m" = "man" | | ||
| + | | "h" = "horse" | | ||
| + | | "k" = "kangaroo" | | ||
| + | | "w" = "whale" | | ||
| | | | | | ||
| + | | "S" = "man or horse or kangaroo or whale" | | ||
| + | | "M" = "Mammal" | | ||
| + | | "P" = "Predicate shared by man, horse, kangaroo, whale" | | ||
| | | | | | ||
| − | |||
o-----------------------------------------------------------o | o-----------------------------------------------------------o | ||
</pre></font> | </pre></font> | ||
| + | |} | ||
| + | |||
| + | In effect, relative to the lattice of natural (non-phony) kinds, any property ''P'', predicated of ''S'', can be "lifted" to a mark of ''M''. | ||
| + | |||
| + | ===Commentary Note 10=== | ||
| − | + | 2. Conventions, Disjunctive Terms, Indexical Signs, Inductive Rules (cont.) | |
| − | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | |
| − | + | | | |
| − | + | <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>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>The reason why this takes place is worthy of notice. Every addition to the information which is incased in a term, results in making some term equivalent to that term. …</p> | |
| − | + | <p>Thus, every addition to our information about a term is an addition to the number of equivalents which that term has. Now, in whatever way a term gets to have a new equivalent, whether by an increase in our knowledge, or by a change in the things it denotes, this always results in an increase either of extension or comprehension without a corresponding decrease in the other quantity.</p> | |
| − | < | + | <p>(Peirce 1866, Lowell Lecture 7, CE 1, 462–464).</p> |
| − | + | |} | |
| − | |||
| − | |||
| − | | | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | 2.1. "man and horse and kangaroo and whale" (aggregarious animals). | |
| − | + | It seems to me now that my previous explanation of the use of "and" in this example was far too complicated and contrived. So let's just say that the conjunction "and" is being used in its ''aggregational'' sense. | |
| − | + | I will also try an alternate style of picture for the ''lifting property'', by means of which, relative to the lattice of natural (non-ad-hoc) kinds, a property ''P'', naturally predicated of ''S'', can be ''elevated'' to apply to ''M''. | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| + | {| align="center" cellspacing="6" style="text-align:center; width:70%" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| − | | | + | | Objective Framework | Interpretive Framework | |
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| | | | | | ||
| − | | | + | | P <------------o------------ "P" | |
| − | | | + | | |\ |\ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | M <------o--------------|--- "M" | |
| + | | | = | / | | ||
| + | | | = | / | | ||
| + | | | = | / | | ||
| + | | | = | / | | ||
| + | | |= |/ | | ||
| + | | S <------------o------------ "S" | | ||
| + | | ** ** ** ** | | ||
| + | | * * * * * * * * | | ||
| + | | * * * * * * * * | | ||
| + | | * * * * * * * * | | ||
| + | | o o o o o o o o | | ||
| + | | m h k w "m" "h" "k" "w" | | ||
| | | | | | ||
| + | o-----------------------------------------------------------o | ||
| + | | Disjunctive Subject "S" and Inductive Rule "M => P" | | ||
| + | o-----------------------------------------------------------o | ||
| | | | | | ||
| − | | | + | | !S! = !I! = {"m", "h", "k", "w", "S", "M", "P"} | |
| − | | | + | | | |
| − | | | + | | "m" = "man" | |
| − | | | + | | "h" = "horse" | |
| − | | | + | | "k" = "kangaroo" | |
| − | | | + | | "w" = "whale" | |
| − | | | + | | | |
| + | | "S" = "man or horse or kangaroo or whale" | | ||
| + | | "M" = "Mammal" | | ||
| + | | "P" = "Predicate shared by man, horse, kangaroo, whale" | | ||
| | | | | | ||
| − | |||
| − | |||
o-----------------------------------------------------------o | o-----------------------------------------------------------o | ||
</pre></font> | </pre></font> | ||
| + | |} | ||
| − | + | I believe that we can now begin to see the linkage to inductive rules. When a sample ''S'' is ''fairly'' or ''randomly'' drawn from the membership ''M'' of some population and when every member of ''S'' is observed to have the property ''P'', then it is naturally rational to expect that every member of ''M'' will also have the property ''P''. This is the principle behind all of our more usual statistical generalizations, giving us the leverage that it takes to lift predicates from samples to a membership sampled. | |
| + | |||
| + | Now, the aggregate that is designated by "man, horse, kangaroo, whale", even if it's not exactly a random sample from the class of mammals, is drawn by design from sufficiently many and sufficiently diverse strata within the class of mammals to be regarded as a quasi-random selection. Thus, it affords us with a sufficient basis for likely generalizations. | ||
| + | |||
| + | ===Commentary Note 11=== | ||
| + | |||
| + | 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). | ||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | + | 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. | |
| − | The | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| + | | | ||
| + | <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> | ||
| + | |||
| + | : [Abduction of a Case] | ||
| − | + | : ''M'' is, for instance, ''P''<sub>1</sub>, ''P''<sub>2</sub>, ''P''<sub>3</sub>, and ''P''<sub>4</sub>; | |
| − | |||
| − | + | : ''S'' is ''P''<sub>1</sub>, ''P''<sub>2</sub>, ''P''<sub>3</sub>, and ''P''<sub>4</sub>: | |
| − | + | : Therefore, ''S'' is ''M''. | |
| + | |||
| + | <p>Here the first premiss amounts to this, that "''P''<sub>1</sub>, ''P''<sub>2</sub>, ''P''<sub>3</sub>, and ''P''<sub>4</sub>" is a likeness of ''M'', and thus the premisses are or represent a likeness of the conclusion. That it is different with induction another example will show:</p> | ||
| + | |||
| + | : [Induction of a Rule] | ||
| − | + | : ''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 ''P'': |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | < | ||
| − | + | : Therefore, All ''M'' is ''P''. | |
| − | < | + | <p>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.</p> |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | </ | ||
| − | + | <p>(Peirce 1867, CP 1.559).</p> | |
| + | |} | ||
| − | + | 1. Abductive Inference and Iconic Signs | |
| − | : | + | Peirce's analysis of the patterns of abductive argument can be understood according to the following paraphrase: |
| − | |||
| − | |||
| − | : | + | * Abduction of a Case: |
| − | |||
| − | |||
| − | : | + | : Fact: ''S'' ⇒ ''P''<sub>1</sub>, ''S'' ⇒ ''P''<sub>2</sub>, ''S'' ⇒ ''P''<sub>3</sub>, ''S'' ⇒ ''P''<sub>4</sub> |
| − | : | + | : Rule: ''M'' ⇒ ''P''<sub>1</sub>, ''M'' ⇒ ''P''<sub>2</sub>, ''M'' ⇒ ''P''<sub>3</sub>, ''M'' ⇒ ''P''<sub>4</sub> |
| − | : | + | : ------------------------------------------------- |
| + | : Case: ''S'' ⇒ ''M'' | ||
| + | |||
| + | : If ''X'' ⇒ each of ''A'', ''B'', ''C'', ''D'', …, | ||
| − | + | : then we have the following equivalents: | |
| − | + | : 1. ''X'' ⇒ the ''greatest lower bound'' (''glb'') of ''A'', ''B'', ''C'', ''D'', … | |
| − | + | : 2. ''X'' ⇒ the logical conjunction ''A'' ∧ ''B'' ∧ ''C'' ∧ ''D'' ∧ … | |
| − | + | : 3. ''X'' ⇒ ''Q'' = ''A'' ∧ ''B'' ∧ ''C'' ∧ ''D'' ∧ … | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | More succinctly, letting ''Q'' = ''P''<sub>1</sub> ∧ ''P''<sub>2</sub> ∧ ''P''<sub>3</sub> ∧ ''P''<sub>4</sub>, the argument is summarized by the following scheme: | |
| − | + | * Abduction of a Case: | |
| − | : | + | : Fact: ''S'' ⇒ ''Q'' |
| − | : | + | : Rule: ''M'' ⇒ ''Q'' |
| − | : | + | : -------------- |
| + | : Case: ''S'' ⇒ ''M'' | ||
| − | + | In this piece of Abduction, it is the ''glb'' or the conjunction of the ostensible predicates that is the operative predicate of the argument, that is, it is the predicate that is common to both the Fact and the Rule of the inference. | |
| − | + | Finally, the reason why one can say that ''Q'' is an iconic sign of the object ''M'' is that ''Q'' can be taken to denote ''M'' by virtue of the qualities that they share, namely, ''P''<sub>1</sub>, ''P''<sub>2</sub>, ''P''<sub>3</sub>, ''P''<sub>4</sub>. | |
| − | + | Notice that the iconic denotation is symmetric, at least in principle, that is, icons are icons of each other as objects, at least potentially, whether or not a particular interpretive agent is making use of their full iconicity during a particular phase of semeiosis. | |
| − | + | The abductive situation is diagrammed in Figure 11.1. | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | {| align="center" cellspacing="6" style="text-align:center; width:60%" | |
| − | |||
| − | |||
| − | |||
| − | {| align="center" cellspacing="6" | ||
| | | | ||
| − | + | <font face="courier new"><pre> | |
| − | | | + | o-------------------------------------------------o |
| − | + | | | | |
| − | + | | P_1 P_2 P_3 P_4 | | |
| − | + | | o o o o | | |
| − | + | | * * * * | | |
| − | + | | * * * * | | |
| − | + | | * * * * | | |
| − | + | | * * * * | | |
| − | + | | ** ** | | |
| + | | Q o | | ||
| + | | |\ | | ||
| + | | | \ | | ||
| + | | | \ | | ||
| + | | | \ | | ||
| + | | | \ | | ||
| + | | | o M | | ||
| + | | | / | | ||
| + | | | | | ||
| + | | | / | | ||
| + | | | | | ||
| + | | |/ | | ||
| + | | S o | | ||
| + | | | | ||
| + | o-------------------------------------------------o | ||
| + | | Figure 1. Abduction of the Case S => M | | ||
| + | o-------------------------------------------------o | ||
| + | </pre></font> | ||
| + | |} | ||
| − | + | In a diagram like this, even if one does not bother to show all of the implicational or the subject-predicate relationships by means of explicit lines, then one may still assume the ''[[transitive closure]]'' of the relations that are actually shown, along with any that are noted in the text that accompanies it. | |
| + | |||
| + | 2. Inductive Inference and Indexic Signs | ||
| − | Peirce's | + | Peirce's analysis of the patterns of inductive argument can be understood according to the following paraphrase: |
| − | + | * Induction of a Rule: | |
| − | |||
| − | |||
| − | < | + | : Case: ''S''<sub>1</sub> ⇒ ''M'', ''S''<sub>2</sub> ⇒ ''M'', ''S''<sub>3</sub> ⇒ ''M'', ''S''<sub>4</sub> ⇒ ''M'' |
| − | + | : Fact: ''S''<sub>1</sub> ⇒ ''P'', ''S''<sub>2</sub> ⇒ ''P'', ''S''<sub>3</sub> ⇒ ''P'', ''S''<sub>4</sub> ⇒ ''P'' | |
| + | : ------------------------------------------------- | ||
| + | : Rule: ''M'' ⇒ ''P'' | ||
| − | + | : If ''X'' <= each of ''A'', ''B'', ''C'', ''D'', …, | |
| − | + | : then we have the following equivalents: | |
| − | + | : 1. ''X'' <= the ''least upper bound'' (''lub'') of ''A'', ''B'', ''C'', ''D'', … | |
| − | + | ||
| − | + | : 2. ''X'' <= the logical disjunction ''A'' ∨ ''B'' ∨ ''C'' ∨ ''D'' ∨ … | |
| − | + | : 3. ''X'' <= ''L'' = ''A'' ∨ ''B'' ∨ ''C'' ∨ ''D'' ∨ … | |
| − | |||
| − | + | More succinctly, letting ''L'' = ''P''<sub>1</sub> ∨ ''P''<sub>2</sub> ∨ ''P''<sub>3</sub> ∨ ''P''<sub>4</sub>, the argument is summarized by the following scheme: | |
| − | + | * Induction of a Rule: | |
| − | + | : Case: ''L'' ⇒ ''M'' | |
| + | : Fact: ''L'' ⇒ ''P'' | ||
| + | : -------------- | ||
| + | : Rule: ''M'' ⇒ ''P'' | ||
| − | + | In this bit of Induction, it is the ''lub'' or the disjunction of the ostensible subjects that is the operative subject of the argument, to wit, the subject that is common to both the Case and the Fact of the inference. | |
| − | + | Finally, the reason why one can say that ''L'' is an indexical sign of the object ''M'' is that ''L'' can be taken to denote ''M'' by virtue of the instances that they share, namely, ''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, ''S''<sub>4</sub>. | |
| − | |||
| − | |||
| − | |||
| − | + | Notice that the indexical denotation is symmetric, at least in principle, that is, indices are indices of each other as objects, at least potentially, whether or not a particular interpretive agent is making use of their full indiciality during a particular phase of semeiosis. | |
| − | + | The inductive situation is diagrammed in Figure 11.2. | |
| − | {| align="center" cellspacing="6" | + | {| align="center" cellspacing="6" style="text-align:center; width:60%" |
| | | | ||
| − | < | + | <font face="courier new"><pre> |
| − | + | o-------------------------------------------------o | |
| − | + | | | | |
| − | + | | P o | | |
| − | + | | |\ | | |
| − | + | | | | | |
| − | < | + | | | \ | |
| + | | | | | ||
| + | | | \ | | ||
| + | | | o M | | ||
| + | | | / | | ||
| + | | | / | | ||
| + | | | / | | ||
| + | | | / | | ||
| + | | |/ | | ||
| + | | L o | | ||
| + | | ** ** | | ||
| + | | * * * * | | ||
| + | | * * * * | | ||
| + | | * * * * | | ||
| + | | * * * * | | ||
| + | | o o o o | | ||
| + | | S_1 S_2 S_3 S_4 | | ||
| + | | | | ||
| + | o-------------------------------------------------o | ||
| + | | Figure 2. Induction of the Rule M => P | | ||
| + | o-------------------------------------------------o | ||
| + | </pre></font> | ||
|} | |} | ||
| − | + | ===Commentary Note 12=== | |
| − | + | Let's redraw the ''New List'' pictures of Abduction and Induction in a way that is a little less cluttered, availing ourselves of the fact that logical implications or lattice subsumptions obey a transitive law to leave unmarked what is thereby understood. | |
| − | === | + | {| align="center" cellspacing="6" style="text-align:center; width:60%" |
| − | + | | | |
| − | + | <font face="courier new"><pre> | |
| − | + | o-------------------------------------------------o | |
| − | + | | | | |
| − | + | | P_1 ... P_k | | |
| − | + | | o o o | | |
| − | | | + | | \ | / | |
| − | | | + | | \ | / | |
| − | | | + | | \ | / | |
| − | | | + | | \ | / | |
| − | | | + | | \|/ | |
| − | | | + | | Q o | |
| + | | |\ | | ||
| + | | | \ | | ||
| + | | | \ | | ||
| + | | | \ | | ||
| + | | | \ | | ||
| + | | | o M | | ||
| + | | | ^ | | ||
| + | | | / | | ||
| + | | | / | | ||
| + | | | / | | ||
| + | | |/ | | ||
| + | | S o | | ||
| + | | | | ||
| + | o-------------------------------------------------o | ||
| + | | Icon Q of Object M, Abduction of Case "S is M" | | ||
| + | o-------------------------------------------------o | ||
| + | </pre></font> | ||
|} | |} | ||
| − | + | {| align="center" cellspacing="6" style="text-align:center; width:60%" | |
| − | + | | | |
| − | + | <font face="courier new"><pre> | |
| − | | | + | o-------------------------------------------------o |
| − | | | + | | | |
| − | | | + | | P o | |
| − | | | + | | |^ | |
| + | | | \ | | ||
| + | | | \ | | ||
| + | | | \ | | ||
| + | | | \ | | ||
| + | | | o M | | ||
| + | | | / | | ||
| + | | | / | | ||
| + | | | / | | ||
| + | | | / | | ||
| + | | |/ | | ||
| + | | L o | | ||
| + | | /|\ | | ||
| + | | / | \ | | ||
| + | | / | \ | | ||
| + | | / | \ | | ||
| + | | / | \ | | ||
| + | | o o o | | ||
| + | | S_1 ... S_k | | ||
| + | | | | ||
| + | o-------------------------------------------------o | ||
| + | | Index L of Object M, Induction of Rule "M is P" | | ||
| + | o-------------------------------------------------o | ||
| + | </pre></font> | ||
|} | |} | ||
| − | + | The main problem that I have with these pictures in their present form is that they do not sufficiently underscore the distinction in roles between signs and objects, and thus we may find it a bit jarring that the middle term of a syllogistic figure is described as an ''object'' of iconic and indexic signs. | |
| + | |||
| + | I will try to address that issue when I return to Peirce's earlier lectures. | ||
| + | |||
| + | ===Commentary Note 13=== | ||
| − | + | In the process of rationalizing Peirce's account of induction to myself I find that I have now lost sight of the indexical sign relationships, so let me go back to the drawing board one more time to see if I can get the indexical and the inductive aspects of the situation back into the very same picture. Here is how we left off last time: | |
| + | {| align="center" cellspacing="6" style="text-align:center; width:70%" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| Line 2,006: | Line 1,936: | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| | | | | | ||
| − | | | + | | P <------------------------- "P" | |
| − | | | + | | |\ |\ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | M <---------------------|--- "M" | |
| − | | | + | | | = | / | |
| − | | | + | | | = | / | |
| − | | | + | | | = | / | |
| − | | | + | | | = | / | |
| − | | | + | | |= |/ | |
| − | | | + | | S <------------------------- "S" | |
| − | | | + | | ** ** ** ** | |
| − | | | + | | * * * * * * * * | |
| − | | | + | | * * * * * * * * | |
| − | | | + | | * * * * * * * * | |
| − | | | + | | o o o o o o o o | |
| − | | | + | | m h k w "m" "h" "k" "w" | |
| + | | S_1 S_2 S_3 S_4 "S_1" "S_2" "S_3" "S_4" | | ||
| | | | | | ||
o-----------------------------------------------------------o | o-----------------------------------------------------------o | ||
| − | | | + | | Disjunctive Subject "S" and Inductive Rule "M => P" | |
o-----------------------------------------------------------o | o-----------------------------------------------------------o | ||
| | | | | | ||
| − | | !S! = !I! = { | + | | !S! = !I! = {"m", "h", "k", "w", "S", "M", "P"} | |
| | | | | | ||
| − | | | + | | "m" = "man" | |
| − | + | | "h" = "horse" | | |
| − | | | + | | "k" = "kangaroo" | |
| − | + | | "w" = "whale" | | |
| − | | | ||
| − | | | ||
| | | | | | ||
| − | | | + | | "S" = "man or horse or kangaroo or whale" | |
| − | | | + | | "M" = "Mammal" | |
| − | | | + | | "P" = "Predicate shared by man, horse, kangaroo, whale" | |
| | | | | | ||
o-----------------------------------------------------------o | o-----------------------------------------------------------o | ||
</pre></font> | </pre></font> | ||
| + | |} | ||
| + | |||
| + | In this double-entry account we are more careful to distinguish the signs that belong to the ''interpretive framework'' (IF) from the objects that belong to the ''objective framework'' (OF). One benefit of this scheme is that it immediately resolves many of the conceptual puzzles that arise from confusing the roles of objects and the roles of signs in the relevant sign relation. | ||
| + | |||
| + | For example, we observe the distinction between the objects ''S'', ''M'', ''P'' and the signs "''S''", "''M''", "''P''". The objects may be regarded as extensive classes or as intensive properties, as the context demands. The signs may be regarded as sentences or as terms, in accord with the application and the ends in view. | ||
| + | |||
| + | It is as if we collected a stratified sample ''S'' of the disjoint type "man, horse, kangaroo, whale" from the class ''M'' of mammals, and observed the property ''P'' to hold true of each of them. Now we know that this could be a statistical fluke, in other words, that ''S'' is just an arbitrary subset of the relevant universe of discourse, and that the very next ''M'' you pick from outside of ''S'' might not have the property ''P''. But that is not very likely if the sample was ''fairly'' or ''randomly'' drawn. So the objective domain is not a lattice like the power set of the universe but something more constrained, of a kind that makes induction and learning possible, a lattice of ''natural kinds'', you might say. In the natural kinds lattice, then, the ''lub'' of ''S'' is close to ''M''. | ||
| + | |||
| + | Now that I have this much of the picture assembled in one frame, it occurs to me that I might be confusing myself about what are the sign relations of actual interest in this situation. After all, samples and signs are closely related, as evidenced by the etymological connection between them that goes back at least as far as Hippocrates. | ||
| + | |||
| + | I need not mention any further the more obvious sign relations that we use just to talk about the objects in the example, for the signs and the objects in these relations of denotation are organized according to their roles in the diptych of objective and interpretive frames. But there are, outside the expressly designated designations, the ways that samples of species tend to be taken as signs of their genera, and these sign relations are discovered internal to the previously marked object domain. | ||
| + | |||
| + | Let us look to Peirce's ''New List'' of the next year for guidance: | ||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | <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> |
| + | |||
| + | : [Induction of a Rule, where the premisses are an index of the conclusion.] | ||
| + | |||
| + | : ''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 ''P'': | ||
| − | + | : Therefore, All ''M'' is ''P''. | |
| − | <p> | + | <p>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. (Peirce 1867, CP 1.559).</p> |
|} | |} | ||
| − | 2. | + | There we see an abstract example with the same logical structure and almost precisely the same labeling. It is a premiss of this argument that "''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, ''S''<sub>4</sub>" is an index of ''M''. But we are left wondering if he means the objective class ''M'' or the sign "''M''". If we take the quotation marks of "''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, ''S''<sub>4</sub>" as giving the disjunctive term equal to "''S''", then we have the next picture: |
| + | {| align="center" cellspacing="6" style="text-align:center; width:70%" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| Line 2,062: | Line 2,012: | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| | | | | | ||
| − | | | + | | P <------------------------- "P" | |
| − | | | + | | |\ |\ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | M <---------------------|--- "M" | |
| − | | | + | | | = $ | / % | |
| − | | | + | | | = | / | |
| − | | | + | | | = | / | |
| − | | | + | | | = $ | / % | |
| − | | | + | | |= |/ | |
| − | | | + | | S <------------------------- "S" | |
| − | | | + | | ** ** $ $ ** *% % | |
| − | | | + | | * * * * $ * * * % | |
| − | | | + | | * * * * $ * * * % | |
| − | | | + | | * * * * $ $ * * * % % | |
| − | | | + | | o o o o o o o o | |
| − | | | + | | m h k w "m" "h" "k" "w" | |
| + | | S_1 S_2 S_3 S_4 "S_1" "S_2" "S_3" "S_4" | | ||
| | | | | | ||
o-----------------------------------------------------------o | o-----------------------------------------------------------o | ||
| − | | Disjunctive Subject | + | | Disjunctive Subject "S" and Inductive Rule "M => P" | |
o-----------------------------------------------------------o | o-----------------------------------------------------------o | ||
| − | | | + | </pre></font> |
| − | + | |} | |
| − | + | ||
| − | + | So we have two readings of what Peirce is saying: | |
| − | + | ||
| − | + | # The interpretation where "''S''" is an index of ''M'' by virtue of "''S''" being a property of each ''S''<sub>''j''</sub>, literally a generic sign of each of them, and by virtue of each ''S''<sub>''j''</sub> being an instance of ''M''. The "''S''" to ''S''<sub>4</sub> to ''M'' linkage is painted $ $ $. | |
| − | + | # The interpretation where "''S''" is an index of "''M''" by virtue of "''S''" being a property of each "''S''<sub>''j''</sub>", literally an implicit sign of each of them, and by dint of each "''S''<sub>''j''</sub>" being an instance of "''M''". The "''S''" to "''S''<sub>4</sub>" to "''M''" link is drawn as % % %. | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | < | ||
| − | + | On third thought, there is still the possibility of a sense in which ''S'' is literally an index of ''M'', that is, we might regard a fair sample from ''S'' as nothing less than a representative sample from ''M''. | |
| − | |||
| − | |||
| − | |||
| − | ===Commentary Note | + | ===Commentary Note 14=== |
| − | + | With the clarity afforded by a reflective interval, my third thought, the relatively ultimate, more perfect interpretant of the intervening struggle toward that final, hopefully neither dying nor raging light, begins to look ever more like the fitting icon of my first impression. | |
| − | + | 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: | |
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | <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> |
| − | + | : [Induction of a Rule, where the premisses are an index of the conclusion.] | |
| − | < | + | : ''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 ''P'': |
| − | + | : Therefore, All ''M'' is ''P''. | |
| − | < | + | <p>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. (Peirce 1867, CP 1.559).</p> |
| − | + | |} | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | </ | ||
| − | + | I've gotten as far as sketching this picture of the possible readings: | |
| − | + | {| align="center" cellspacing="6" style="text-align:center; width:70%" | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | {| align="center" cellspacing="6" | ||
| | | | ||
| − | |||
| − | |||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
| − | o------------------------o------------------------o | + | o-----------------------------o-----------------------------o |
| − | | | + | | Objective Framework | Interpretive Framework | |
| − | | | + | o-----------------------------o-----------------------------o |
| − | | | | + | | | |
| − | | | + | | P <------------------------- "P" | |
| − | | | + | | |\ |\ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | o------------------------o------------------------o | + | | | \ | \ | |
| + | | | \ | \ | | ||
| + | | | M <---------------------|--- "M" | | ||
| + | | | = * | / % | | ||
| + | | | = | / | | ||
| + | | | = | / | | ||
| + | | | = * | / % | | ||
| + | | |= |/ | | ||
| + | | S <------------------------- "S" | | ||
| + | | ** ** * $ ** *% % | | ||
| + | | * * * * $ * * * % | | ||
| + | | * * * * $ * * * % | | ||
| + | | * * * * * $ * * * % % | | ||
| + | | o o o o o o o o | | ||
| + | | m h k w "m" "h" "k" "w" | | ||
| + | | S_1 S_2 S_3 S_4 "S_1" "S_2" "S_3" "S_4" | | ||
| + | | | | ||
| + | o-----------------------------------------------------------o | ||
| + | | Disjunctive Subject "S" and Inductive Rule "M => P" | | ||
| + | o-----------------------------------------------------------o | ||
</pre></font> | </pre></font> | ||
| + | |} | ||
| + | |||
| + | In order of increasing ''objectivity'', here are three alternatives: | ||
| − | < | + | # The interpretation where "''S''" is an index of "''M''" by virtue of "''S''" being a property of each "''S''<sub>''j''</sub>", literally an implicit sign of each of them, and by dint of each "''S''<sub>''j''</sub>" being an instance of "''M''". The "''S''" to "''S''<sub>4</sub>" to "''M''" link is drawn [% % % %]. |
| + | # The interpretation where "''S''" is an index of ''M'' by virtue of "''S''" being a property of each ''S''<sub>''j''</sub>, literally a generic sign of each of them, and by virtue of each ''S''<sub>''j''</sub> being an instance of ''M''. The "''S''" to ''S''<sub>4</sub> to ''M'' link is a 2-tone [$ $ * *]. | ||
| + | # The interpretation where ''S'' is an index of ''M'' by virtue of ''S'' being a property of each ''S''<sub>''j''</sub>, literally a supersample of each of them, and by virtue of each ''S''<sub>''j''</sub> being an instance of ''M''. The ''S'' to ''S''<sub>4</sub> to ''M'' link is shown as [* * * *]. | ||
| − | + | Perhaps it is the nature of the sign situation that all three interpretations will persevere and keep some measure of merit. At the moment I am leaning toward the third interpretation as it manifests the possibility of a higher grade of objectivity. | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | ===Commentary Note 15=== | |
| − | + | I am going to stick with the Index-Induction side of the problem until I feel like I understand what's going on with this linkage between the faces of the sign relation and the phases of inquiry. | |
| − | + | The ''New List'' (1867) account of the relationship between the kinds of signs and the kinds of arguments says this: | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | + | 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 general, if one takes the components of an Argument to be its Conclusion, its Premisses taken collectively, and its Interpretant, then they can be seen to take up the following sign relational duties: | |
| − | + | : <Conclusion, Premisses, Interpretant> = <Object, Sign, Interpretant> | |
| − | + | This generality may be broken down according to the role of the premisses: | |
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | + | The premisses may afford a likeness, index, or symbol of the conclusion. | |
| + | |} | ||
| − | + | In the case of the inductive argument, we have the following role assigments: | |
| − | < | + | : <Conclusion, Premisses, Interpretant> = <Object, Index, Interpretant> |
| − | + | Marked out in greater detail, we have the following role assignments: | |
| − | + | Premisses (Index): | |
| − | <p> | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | <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> | ||
| + | |||
| + | <p>''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub> are ''P''.</p> | ||
|} | |} | ||
| − | + | Conclusion (Object): | |
| − | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | |
| + | | | ||
| + | All ''M'' is ''P''. | ||
| + | |} | ||
| − | + | Remark: | |
| − | |||
| − | |||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | < | + | 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. |
| + | |} | ||
| − | + | One of the questions that I have at this point is whether Peirce is speaking loosely or strictly when he refers to the conclusion and the premisses of the argument in question. Strictly speaking, the conclusion has the form ''M'' ⇒ ''P'' and the premisses have the forms ''S''<sub>''j''</sub> ⇒ ''M'' and ''S''<sub>''j''</sub> ⇒ ''P''. But taken more loosely, as often happens in contexts where the antecedent of a conditional statement is already assumed to hold true, people will sometimes refer to the consequent of a conditional conclusion as the conclusion and the consequents of conditional premisses as the premisses. In the present case, such a practice would lead to speaking of the predicate ''M'' as one of the premisses and the predicate ''P'' as the conclusion. So let us keep that interpretive option in mind as we go. | |
| − | + | ==Commentary Work Notes== | |
| − | + | ===Commentary Work Note 1=== | |
| − | + | Here is my current picture of the situation, so far as it goes: | |
| + | {| align="center" cellspacing="6" style="text-align:center; width:70%" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| − | | | + | | Objective Framework | Interpretive Framework | |
| − | | | + | o-----------------------------o-----------------------------o |
| − | | | + | | | |
| − | | | + | | P <------------@------------ "P" | |
| − | | | + | | |\ |\ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | \ | \ | |
| − | | | + | | | M <------@--------------|--- "M" | |
| − | | | + | | | = . | / # | |
| − | | | + | | | = | / | |
| − | | | + | | | = | / | |
| − | o----------------------------- | + | | | = . | / # | |
| + | | |= |/ | | ||
| + | | S <------------@------------ "S" | | ||
| + | | .. .. . * .. .# # | | ||
| + | | . . . . * . . . # | | ||
| + | | . . . . * . . . # | | ||
| + | | . . . . . * . . . # # | | ||
| + | | o o o o o o o o | | ||
| + | | m h k w "m" "h" "k" "w" | | ||
| + | | S_1 S_2 S_3 S_4 "S_1" "S_2" "S_3" "S_4" | | ||
| + | | | | ||
| + | o-----------------------------------------------------------o | ||
| + | | Disjunctive Subject "S" and Inductive Rule "M => P" | | ||
| + | o-----------------------------------------------------------o | ||
</pre></font> | </pre></font> | ||
| + | |} | ||
| − | + | I got as far as sketching a few readings of the penultimate sentence: | |
| − | < | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| + | | | ||
| + | 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. | |
| − | < | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | + | | | |
| − | + | Hence the premisses are an index of the conclusion. | |
| − | |||
| − | |||
|} | |} | ||
| − | + | The first premiss is this: | |
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | < | + | ''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''. |
| − | and | ||
|} | |} | ||
| − | + | We gather that it says that "''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, ''S''<sub>4</sub>" is an index of ''M''. | |
| − | + | Taking this very literally, I would guess that it holds by way of the paths from "''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, ''S''<sub>4</sub>" to <font size="+3">∪</font> ''S''<sub>''j''</sub> to ''M''. | |
| − | + | The second premiss is this: | |
| − | 2 | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| + | | | ||
| + | ''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: | |
| − | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | |
| + | | | ||
| + | All ''M'' is ''P''. | ||
| + | |} | ||
| − | + | And all of this is said to be so because: | |
| − | |||
| − | |||
{| align="center" cellspacing="6" width="90%" <!--QUOTE--> | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> | ||
| | | | ||
| − | + | 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. | |
| + | |} | ||
| − | + | And that is a bit that I will need to try to think about a bit before I even try to draw a picture of it. | |
| − | + | Let me advance a few words in prospect of how I plan to address the problem of objectivity. Although I've been using the bipartite scheme of objective and interpretive frameworks, that is only a matter of convenient organization, and embodies nothing like a claim to the invariant status of either objects or signs, as we have seen plenty of examples already of just how shifty those roles can be. | |
| − | + | As I suggested in my last note, one sort of evidence, the amassing of which tends to make me assign a matter to the objective side of my experience, is the possibility of viewing it from many diverse angles, of being able to describe it from manifold points of view, and being able to relate those angles and views in a sensible way. | |
| − | + | In the mathematical perspective known as "category theory", the question of objectivity is handled by way of what are derivatively enough nomenclated as "universal properties". | |
| − | + | Working within any given category, the things that are potentially worth caring about, called "objects" or "spaces", along with the corresponding metamorphoses among them, called "arrows" or "morphisms", can be treated as having an objective status to the extent that there are many distinct "views" of them, called "functors", that relate to each other in natural and especially nice ways called "natural transformations". That's it in a nutshell, very roughly, but I have forced a few details in prospect of the ways that I will have to change the setting a little for the sake of better accommodating semiotics in a suitably re-modelled category theory. | |
| − | + | ===Commentary Work Note 2=== | |
| − | + | 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: | |
| − | <p> | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| + | | | ||
| + | <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> | + | <p>[Deduction of a Fact]</p> |
| − | <p> | + | <p>In deductive argument, the conclusion is represented by the premisses as by a general sign under which it is contained.</p> |
| − | <p> | + | <p>[Abduction of a Case]</p> |
| − | <p> | + | <p>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> | + | <p>''M'' is, for instance, ''P''<sub>1</sub>, ''P''<sub>2</sub>, ''P''<sub>3</sub>, and ''P''<sub>4</sub>;</p> |
| − | |||
| − | + | <p>''S'' is ''P''<sub>1</sub>, ''P''<sub>2</sub>, ''P''<sub>3</sub>, and ''P''<sub>4</sub>:</p> | |
| − | + | <p>∴ ''S'' is ''M''.</p> | |
| − | Here, | + | <p>Here the first premiss amounts to this, that "''P''<sub>1</sub>, ''P''<sub>2</sub>, ''P''<sub>3</sub>, and ''P''<sub>4</sub>" is a likeness of ''M'', and thus the premisses are or represent a likeness of the conclusion.</p> |
| − | + | <p>[Induction of a Rule]</p> | |
| − | |||
| − | |||
| − | |||
| − | + | <p>That it is different with induction another example will show.</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> | |
| − | + | <p>''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, and ''S''<sub>4</sub> are ''P'':</p> | |
| − | + | <p>∴ All ''M'' is ''P''.</p> | |
| + | |||
| + | <p>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.</p> | ||
| + | |||
| + | <p>C.S. Peirce, "New List" CP 1.559 and CE 2, p. 58.</p> | ||
| + | |||
| + | <p>Charles Sanders Peirce, "On a New List of Categories" (1867), ''Collected Papers'' CP 1.545-567. ''Chronological Edition'' CE 2, 49-59.</p> | ||
| + | |||
| + | <p>http://www.peirce.org/writings/p32.html</p> | ||
| + | <p>http://members.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm</p> | ||
| + | |} | ||
| + | {| align="center" cellspacing="6" style="text-align:center; width:70%" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| Line 2,358: | Line 2,311: | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| | | | | | ||
| − | | | + | | P_1 P_2 P_3 P_4 "P_1" "P_2" "P_3" "P_4" | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| o o o o o o o o | | | o o o o o o o o | | ||
| − | | | + | | .. . . ... * .. . . #.# | |
| − | | | + | | . . . . . . * . . . . # . | |
| − | + | | . . . . * . . . # | | |
| − | | | + | | . .. .. . . . * .. .# . # | |
| − | + | | P <------------@------------ "P" | | |
| − | + | | . ^= . . ^^ . | | |
| − | | | + | | | = . | \ # | |
| − | | | + | | . | = . . | \ . | |
| − | | | + | | | = | \ | |
| − | | | + | | . | .= . . | .\ # | |
| − | | | + | | | M <------@--------------|--- "M" | |
| − | | | + | | . | . ^ . | . ^ | |
| − | | | + | | | / | / | |
| − | | | + | | . | ./ . | ./ | |
| − | | | + | | | / | / | |
| − | | " | + | | .|/ .|/ | |
| + | | S <------------@------------ "S" | | ||
| | | | | | ||
| + | o-----------------------------------------------------------o | ||
| + | | Conjunctive Predicate "P" and Abductive Case "S => M" | | ||
o-----------------------------------------------------------o | o-----------------------------------------------------------o | ||
</pre></font> | </pre></font> | ||
| + | |} | ||
| − | + | {| align="center" cellspacing="6" style="text-align:center; width:70%" | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | {| align="center" cellspacing="6" | ||
| | | | ||
| − | < | + | <font face="courier new"><pre> |
| − | + | o-----------------------------o-----------------------------o | |
| − | < | + | | Objective Framework | Interpretive Framework | |
| − | + | o-----------------------------o-----------------------------o | |
| − | + | | | | |
| − | + | | P <------------@------------ "P" | | |
| − | + | | ^^^ ^^^ | | |
| − | + | | | \ | \ | | |
| − | < | + | | . | .\ . | .\ | |
| + | | | \ | \ | | ||
| + | | . | . \ . | . \ | | ||
| + | | | M <------@--------------|--- "M" | | ||
| + | | . | .= . . | .^ # | | ||
| + | | | = | / | | ||
| + | | . | = . . | / . | | ||
| + | | | = . | / # | | ||
| + | | . |= . . |/ . | | ||
| + | | S <------------@------------ "S" | | ||
| + | | . .. .. . . . * .. .# . # | | ||
| + | | . . . . * . . . # | | ||
| + | | . . . . . . * . . . . # . | | ||
| + | | .. . . ... * .. . . #.# | | ||
| + | | o o o o o o o o | | ||
| + | | S_1 S_2 S_3 S_4 "S_1" "S_2" "S_3" "S_4" | | ||
| + | | | | ||
| + | o-----------------------------------------------------------o | ||
| + | | Disjunctive Subject "S" and Inductive Rule "M => P" | | ||
| + | o-----------------------------------------------------------o | ||
| + | </pre></font> | ||
|} | |} | ||
| − | + | ===Commentary Work Note 3=== | |
| − | + | Let me pause just a moment to knock out a couple of quick sketches of how I see the scene that Peirce is depicting here. | |
| − | + | Here is the rougher draft of the two, a diptych impaneled of an object fold and a sign fold, an interpretant being, after all, just another passing moment in the life cycle of a sign, and so there is an object ''o'', with its intensions ''p ''through ''q'', collectively constellating the comprehension of any sign ''s''<sub>''k''</sub>, above the instances, instantiations, or instants ''i'' through ''j'' of the object ''o'', aggregatively constituting the extension of any sign ''s''<sub>''k''</sub> that is said to denote ''o'' or its plurality below. | |
| + | {| align="center" cellspacing="6" style="text-align:center; width:70%" | ||
| + | | | ||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| Line 2,426: | Line 2,383: | ||
o-----------------------------o-----------------------------o | o-----------------------------o-----------------------------o | ||
| | | | | | ||
| − | | | + | | p ... q · s_1 | |
| − | + | | \ / · · | | |
| − | + | | ^ ^ · · · | | |
| − | + | | \ / · · · · | | |
| − | | | + | | o< · · · · · · · · · · · · s_k | |
| − | + | | / \ · · · · | | |
| − | + | | ^ ^ · · · | | |
| − | + | | / \ · · | | |
| − | | | + | | i ... j · s_n | |
| − | | | ||
| − | | | ||
| − | | | ||
| − | |||
| − | | | ||
| − | |||
| − | | | ||
| − | | | ||
| − | |||
| − | |||
| | | | | | ||
o-----------------------------------------------------------o | o-----------------------------------------------------------o | ||
| − | | | + | </pre></font> |
| + | |} | ||
| + | |||
| + | I should mention, in no uncertain terms, that Peirce's present account does not yet count this ''abstract hypostasis'' or ''hypostatic object'' ''o'' in any explicit way, though I think it is inherent in the very form of his thinking. So let us keep an eye out, as we proceed with the story, for when, if ever, this particular character first treads on the scene. | ||
| + | |||
| + | For future reference, I will go ahead and post here this advance notice of what may well be the next stage in the developmental differentiation of this, my embyronic autopoetics, but let's just see what we shall see. | ||
| + | |||
| + | {| align="center" cellspacing="6" style="text-align:center; width:70%" | ||
| + | | | ||
| + | <font face="courier new"><pre> | ||
o-----------------------------------------------------------o | o-----------------------------------------------------------o | ||
| + | | Higher Order Framework (HOF) | | ||
| + | o-------------------o-------------------o-------------------o | ||
| + | | Objective Operands, Operators Organon | | ||
| + | o-------------------o-------------------o-------------------o | ||
| | | | | | ||
| − | | | + | | implications incitations | |
| − | | | + | | higher intensions | |
| − | | | + | | (·········) · s_1 | |
| − | | | + | | \ / · · | |
| − | | | + | | ^ ^ · · · | |
| − | | | + | | \ / · · · · | |
| − | | | + | | o <·············· pomps < · · · · · · · · s_k | |
| − | | | + | | / \ · · · · | |
| − | | | + | | ^ ^ · · · | |
| − | | | + | | / \ · · | |
| + | | (·········) · s_n | | ||
| + | | implementations | | ||
| + | | institutions inditations | | ||
| | | | | | ||
o-----------------------------------------------------------o | o-----------------------------------------------------------o | ||
</pre></font> | </pre></font> | ||
| + | |} | ||
| − | + | ===Commentary Work Note 4=== | |
| − | + | <font face="courier new"><pre> | |
| + | Lest we get totally lost in the devilish details, | ||
| + | I think that it might be a good idea to remember | ||
| + | what I observed once before about the main theme | ||
| + | of this whole part of Peirce's discussion: | ||
| − | + | At this point in his discussion, Peirce is relating the nature of | |
| + | inference, inquiry, and information to the character of the signs | ||
| + | that are invoked in support of the overall process in question, | ||
| + | a process that he is presently describing as "symbolization". | ||
| − | + | The links among the types of signs -- icons, indices, symbols -- | |
| + | the aspects of semiosis -- connotation, denotation, information -- | ||
| + | and the types of inference -- abduction, induction, deduction -- | ||
| + | are parts of a bigger picture that we should try to keep in view. | ||
| − | + | Outline of Peirce's Examples: | |
| − | |||
| − | |||
| − | |||
| − | + | 1. Conjunctive term "spherical bright fragrant juicy tropical fruit". | |
| + | 2.1. Disjunctive term "man or horse or kangaroo or whale". | ||
| + | 2.2. Disjunctive term "neat or swine or sheep or deer". | ||
| − | + | | Yet there are combinations of words and combinations of conceptions | |
| + | | which are not strictly speaking symbols. These are of two kinds | ||
| + | | of which I will give you instances. We have first cases like: | ||
| + | | | ||
| + | | 'man and horse and kangaroo and whale', | ||
| + | | | ||
| + | | and secondly, cases like: | ||
| + | | | ||
| + | | 'spherical bright fragrant juicy tropical fruit'. | ||
| | | | ||
| − | + | | (CSP, CE 1, 468-469). | |
| − | |||
| − | |||
| − | + | Returning to our present examples, the unmentioned elements of the | |
| + | universe of discourse X are the potential objects of !O!, and the | ||
| + | sign domain !S! will contain all of the terms that we want to put | ||
| + | on the sign relational Tables. What are the interpretant signs? | ||
| + | Well, in public discursions we cannot literally cut and paste our | ||
| + | mental signs into texts, so we are stuck with using the same sorts | ||
| + | of signs that we ordinarily assign to the sign domain !S! as proxies | ||
| + | for these concepts, or "mental symbols", and so this surrogation will | ||
| + | constitute the customary practice in formal, public discussions. But | ||
| + | the nice thing about the "formal" point of view, in other words, the | ||
| + | outlook that looks out for the forms of things above all, is that we | ||
| + | can be relatively sure that there is an approximate isomorph of the | ||
| + | one-foot-in-the-psyche sort of sign relation that can found among | ||
| + | the wholly unsubsidiary public domains. So nothing much is lost, | ||
| + | formally speaking, if we forget about the possible distincture | ||
| + | in essence between a sign in the mind and a sign on the wall. | ||
| − | + | Given that we are now working with sign relations of the form | |
| + | L c OxSxI = OxSxS, that is to say, where S = I is the common | ||
| + | syntactic domain, we can picture the situation as follows: | ||
| − | + | o-----------------------------o-----------------------------o | |
| + | | Objective Framework | Interpretive Framework | | ||
| + | o-----------------------------o-----------------------------o | ||
| + | | | | | ||
| + | | | "man" | | ||
| + | | | "horse" | | ||
| + | | | "kangaroo" | | ||
| + | | | "whale" | | ||
| + | | | "mammal" | | ||
| + | | | | | ||
| + | | | "spherical" | | ||
| + | | | "bright" | | ||
| + | | | "fragrant" | | ||
| + | | | "juicy" | | ||
| + | | | "tropical" | | ||
| + | | | "fruit" | | ||
| + | | | "orange" | | ||
| + | | | | | ||
| + | o-----------------------------o-----------------------------o | ||
| − | + | That is to say, in the beginning there is nothing but words, | |
| + | and the objective side of the universe is all form and void. | ||
| − | + | Example 2.1. "man or horse or kangaroo or whale" | |
| − | + | | 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. (CSP, CE 1, 468-469). | ||
| − | + | Suppose that one "attribute in common" or "common character" | |
| − | + | that comes to mind for men, horses, kangaroos, and whales | |
| + | is the descriptor "sucks when young" (SWY). Then we have | ||
| + | the following beginnings of a sign relational set-up: | ||
| − | + | o-----------------------------o-----------------------------o | |
| − | + | | Objective Framework | Interpretive Framework | | |
| − | + | o-----------------------------o-----------------------------o | |
| − | + | | | | |
| − | * | + | | w | |
| − | + | | o | | |
| − | + | | * * Rule | | |
| − | + | | * * v=>w | | |
| − | + | | * * | | |
| − | + | | * * | | |
| + | | Fact * * | | ||
| + | | u=>w * o v | | ||
| + | | * * | | ||
| + | | * * | | ||
| + | | * * Case | | ||
| + | | * * u=>v | | ||
| + | | * * | | ||
| + | | o u | | ||
| + | | .. .. | | ||
| + | | . . . . | | ||
| + | | . . . . | | ||
| + | | . . . . | | ||
| + | | o o o o | | ||
| + | | s_1 s_2 s_3 s_4 | | ||
| + | | | | ||
| + | | | | ||
| + | o-----------------------------------------------------------o | ||
| + | | Disjunctive Subject u, Induction to the Rule v => w | | ||
| + | o-----------------------------------------------------------o | ||
| − | + | S = I = {s_1, s_2, s_3, s_4, u, v, w} | |
| − | + | s_1 = "man" | |
| + | s_2 = "horse" | ||
| + | s_3 = "kangaroo" | ||
| + | s_4 = "whale" | ||
| − | + | u = "man or horse or kangaroo or whale" | |
| + | v = "sucks when young" | ||
| + | w = "mammal" | ||
| − | + | Still nothing happening on the objective side of the world. | |
| + | Our first inkling of a connection to the "world of matter" | ||
| + | comes with the notions of denotation and Peirce's "sphere". | ||
| − | + | | It is important to distinguish between the two functions of a word: | |
| − | + | | 1st to denote something -- to stand for something, and 2nd to mean | |
| − | + | | something -- or as Mr. Mill phrases it -- to 'connote' something. | |
| − | + | | | |
| − | + | | What it denotes is called its 'Sphere'. | |
| − | + | | What it connotes is called its 'Content'. | |
| − | + | | Thus the 'sphere' of the word 'man' is for | |
| − | + | | me every man I know; and for each of you it | |
| − | + | | is every man you know. The 'content' of 'man' | |
| − | + | | is all that we know of all men, as being two- | |
| − | + | | legged, having souls, having language, &c., &c. | |
| − | + | | It is plain that both the 'sphere' and the | |
| − | + | | 'content' admit of more and less. ... | |
| − | + | | | |
| − | + | | Now the sphere considered as a quantity is called the Extension; | |
| − | + | | and the content considered as quantity is called the Comprehension. | |
| − | + | | Extension and Comprehension are also termed Breadth and Depth. So that | |
| − | The | + | | a wider term is one which has a greater extension; a narrower one is |
| − | + | | one which has a less extension. A higher term is one which has a | |
| − | + | | less Comprehension and a lower one has more. | |
| − | o-------------------------------------------------o | + | | |
| − | | | + | | The narrower term is said to be contained under the wider one; |
| − | | | + | | and the higher term to be contained in the lower one. |
| − | | | + | | |
| − | | | + | | We have then: |
| − | | | + | | |
| − | | | + | | o-----------------------------o-----------------------------o |
| − | | | + | | | | | |
| − | | | + | | | What is 'denoted' | What is 'connoted' | |
| − | | | + | | | | | |
| − | | | + | | | Sphere | Content | |
| − | | | + | | | | | |
| − | | | + | | | Extension | Comprehension | |
| − | | | + | | | | | |
| − | + | | | ( wider | ( lower | | |
| − | | | + | | | Breadth < | Depth < | |
| − | | | + | | | ( narrower | ( higher | |
| − | + | | | | | | |
| − | | | + | | | What is contained 'under' | What is contained 'in' | |
| − | + | | | | | | |
| − | | | + | | o-----------------------------o-----------------------------o |
| − | | | + | | |
| − | | | + | | The principle of explicatory or deductive reasoning then is that: |
| − | o | + | | |
| − | + | | Every part of a word's Content belongs to | |
| − | o------------------------------ | + | | every part of its Sphere, |
| + | | | ||
| + | | or: | ||
| + | | | ||
| + | | Whatever is contained 'in' a word belongs to | ||
| + | | whatever is contained under it. | ||
| + | | | ||
| + | | CSP, CE 1, pages 459-460. | ||
</pre></font> | </pre></font> | ||
| − | + | ===Commentary Work Note 5=== | |
| − | + | <font face="courier new"><pre> | |
| + | Obeying some hobgoblin of consistency, I present Peirce's second example | ||
| + | of a {disjunctive term, indexical sign, inductive rule} configuration on | ||
| + | the same pattern as the first, and then I want to look more closely at a | ||
| + | tricky feature of his account of comprehension, connotation, and content. | ||
| − | + | Example 2.2. "neat or swine or sheep or deer" | |
| − | + | | 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. (CSP, CE 1, 468-469). | ||
| − | + | o-----------------------------o-----------------------------o | |
| − | + | | Objective Framework | Interpretive Framework | | |
| − | + | o-----------------------------o-----------------------------o | |
| − | + | | | | |
| − | + | | w | | |
| − | + | | o | | |
| + | | * * Rule | | ||
| + | | * * v=>w | | ||
| + | | * * | | ||
| + | | * * | | ||
| + | | Fact * * | | ||
| + | | u=>w * o v | | ||
| + | | * * | | ||
| + | | * * | | ||
| + | | * * Case | | ||
| + | | * * u=>v | | ||
| + | | * * | | ||
| + | | o u | | ||
| + | | .. .. | | ||
| + | | . . . . | | ||
| + | | . . . . | | ||
| + | | . . . . | | ||
| + | | o o o o | | ||
| + | | s_1 s_2 s_3 s_4 | | ||
| + | | | | ||
| + | | | | ||
| + | o-----------------------------------------------------------o | ||
| + | | Disjunctive Subject u, Induction to the Rule v => w | | ||
| + | o-----------------------------------------------------------o | ||
| − | + | Syntactic domain, !S! = {s_1, s_2, s_3, s_4, u, v, w}. | |
| − | + | s_1 = "neat" | |
| + | s_2 = "swine" | ||
| + | s_3 = "sheep" | ||
| + | s_4 = "deer" | ||
| − | + | u = "neat or swine or sheep or deer" | |
| + | v = "cloven-hoofed" | ||
| + | w = "herbivorous" | ||
| − | + | In this situation there is a gap between the logical disjunction u, | |
| + | or, expressed in lattice terminology, the "least upper bound" (lub) | ||
| + | of the disjoined terms, u = lub(s_j), and what we might well call | ||
| + | their "natural disjunction" v = "cloven-hoofed". | ||
| − | + | The sheer implausibility of imagining that the disjunctive term u | |
| + | would ever be embedded exactly per se in a lattice of natural kinds, | ||
| + | leads to the evident "naturalness" of the induction to v => w, namely, | ||
| + | the rule that cloven-hoofed animals are herbivorous. | ||
| − | + | On to the tricky feature of which I spoke. | |
| + | Let us focus the explanation of "content" | ||
| + | that was given in the following statement: | ||
| − | + | | It is important to distinguish between the two | |
| − | + | | functions of a word: 1st to denote something -- | |
| − | + | | to stand for something, and 2nd to mean something -- | |
| − | + | | or as Mr. Mill phrases it -- to 'connote' something. | |
| + | | | ||
| + | | What it denotes is called its 'Sphere'. | ||
| + | | What it connotes is called its 'Content'. | ||
| + | | | ||
| + | | Thus the 'sphere' of the word 'man' is for me every man | ||
| + | | I know; and for each of you it is every man you know. | ||
| + | | | ||
| + | | The 'content' of 'man' is all that we know of all men, as | ||
| + | | being two-legged, having souls, having language, &c., &c. | ||
| − | + | The question is: What sort of thing is a connotation? | |
| + | Is it a sign? That is to say, is it yet another term? | ||
| + | Or is it something like an abstract attribute, namely, | ||
| + | a character, an intension, a property, or a quality? | ||
| + | And while we're asking, does it really even matter? | ||
| + | </pre></font> | ||
| − | + | ===Commentary Work Note 6=== | |
| − | |||
| − | |||
| − | |||
| − | |||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
| − | + | | It is important to distinguish between the two | |
| − | + | | functions of a word: 1st to denote something -- | |
| − | | | + | | to stand for something, and 2nd to mean something -- |
| − | + | | or as Mr. Mill phrases it -- to 'connote' something. | |
| − | + | | | |
| − | + | | What it denotes is called its 'Sphere'. | |
| − | | | + | | What it connotes is called its 'Content'. |
| − | | | + | | |
| − | + | | Thus the 'sphere' of the word 'man' is for me every man | |
| − | | | + | | I know; and for each of you it is every man you know. |
| − | | | + | | |
| − | + | | The 'content' of 'man' is all that we know of all men, as | |
| − | | | + | | being two-legged, having souls, having language, &c., &c. |
| − | + | | | |
| − | + | | CSP, CE 1, page 459. | |
| − | |||
| − | |||
| − | | | ||
| − | |||
| − | | | ||
| − | | | ||
| − | |||
| − | | | ||
| − | |||
| − | | | ||
| − | |||
| − | |||
| − | + | The question is: What sort of thing is a connotation? | |
| + | Is it a sign? That is to say, is it yet another term? | ||
| + | Or is it something like an abstract attribute, namely, | ||
| + | a character, an intension, a property, or a quality? | ||
| + | And while we're asking, does it really even matter? | ||
| + | |||
| + | "No" is one answer worth considering. | ||
| + | But then: Why does it not matter? | ||
| + | What reason might be given that | ||
| + | would excuse the indifference? | ||
| + | |||
| + | This is a question that has exercised me since | ||
| + | my earliest studies of Peirce. I can remember | ||
| + | discussing it with my philosophy mentor at the | ||
| + | time and I distinctly recall having arrived at | ||
| + | some conclusion or other, but, alas, I haven't | ||
| + | the foggiest notion what exactly my revelation | ||
| + | amounted to. Perhaps that is all for the best, | ||
| + | as the vagrancy of memory is frequently better | ||
| + | than the vapidity of one's banalytic anamnesia. | ||
| − | + | These days, I usually try to finesse the trick | |
| + | under the trumped up rubric of a factorization. | ||
| + | So let me excavate my last attempts to explain | ||
| + | this business and see if I can improve on them. | ||
| − | + | I am gathering some wooly old links here just for my future reference. | |
| − | + | I would not chase these if I were you. Since I am not you, I have to, | |
| − | + | and when I have carded them all out I will get back to you, but later. | |
| − | + | ||
| − | + | Factorization Issues | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | 01. http://suo.ieee.org/email/msg02332.html | |
| − | + | 02. http://suo.ieee.org/email/msg02334.html | |
| − | + | 03. http://suo.ieee.org/email/msg02338.html | |
| − | + | 04. http://suo.ieee.org/email/msg02340.html | |
| − | + | 05. http://suo.ieee.org/email/msg02345.html | |
| − | + | 06. http://suo.ieee.org/email/msg02349.html | |
| − | + | 07. http://suo.ieee.org/email/msg02355.html | |
| − | + | 08. http://suo.ieee.org/email/msg02396.html | |
| − | + | 09. http://suo.ieee.org/email/msg02400.html | |
| − | + | 10. http://suo.ieee.org/email/msg02430.html | |
| − | + | 11. http://suo.ieee.org/email/msg02448.html | |
| − | + | 12. http://suo.ieee.org/email/msg04334.html | |
| − | + | 13. http://suo.ieee.org/email/msg04416.html | |
| − | + | 14. http://suo.ieee.org/email/msg07143.html | |
| − | + | 15. http://suo.ieee.org/email/msg07166.html | |
| − | + | 16. http://suo.ieee.org/email/msg07182.html | |
| − | + | 17. http://suo.ieee.org/email/msg07185.html | |
| − | + | 18. http://suo.ieee.org/email/msg07186.html | |
| − | + | ||
| − | + | http://suo.ieee.org/ontology/msg00007.html | |
| − | + | http://suo.ieee.org/ontology/msg00025.html | |
| − | + | http://suo.ieee.org/ontology/msg00032.html | |
| − | + | http://suo.ieee.org/ontology/msg01926.html | |
| − | + | http://suo.ieee.org/ontology/msg02008.html | |
| − | + | http://suo.ieee.org/ontology/msg03285.html | |
| − | |||
</pre></font> | </pre></font> | ||
| − | + | ===Commentary Work Note 7=== | |
| − | |||
| − | |||
| − | |||
| − | ===Commentary Note | ||
| − | |||
| − | |||
<font face="courier new"><pre> | <font face="courier new"><pre> | ||
| − | + | My head is spinning, and I need to try and get a grip on the matter | |
| − | + | of how we are supposed to understand words like attribute, character, | |
| − | + | intension, feature, mark, predicate, property, quality, and so on, not | |
| − | + | to say that all of them even belong to the same series of near synonyms, | |
| − | + | and when we have achieved a measure of clarity about these general issues | |
| − | + | of not quite common enough usage, then maybe we can begin to address the | |
| − | + | question of what Peirce intends for us to understand by way of the more | |
| − | + | specialized terms of art, to wit, comprehension, connotation, content. | |
| − | + | ||
| − | | | + | | It is important to distinguish between the two |
| − | | | + | | functions of a word: 1st to denote something -- |
| − | | | + | | to stand for something, and 2nd to mean something -- |
| − | + | | or as Mr. Mill phrases it -- to 'connote' something. | |
| − | + | | | |
| − | + | | What it denotes is called its 'Sphere'. | |
| − | + | | What it connotes is called its 'Content'. | |
| − | + | | | |
| − | + | | Thus the 'sphere' of the word 'man' is for me every man | |
| − | | | + | | I know; and for each of you it is every man you know. |
| − | | | + | | |
| − | + | | The 'content' of 'man' is all that we know of all men, as | |
| − | | | + | | being two-legged, having souls, having language, &c., &c. |
| − | | | + | | |
| − | + | | CSP, CE 1, page 459. | |
| − | | | + | |
| − | + | The question is: What sort of thing is a connotation? | |
| − | + | Is it a sign? That is to say, is it yet another term? | |
| − | + | Or is it something like an abstract attribute, namely, | |
| − | + | a character, an intension, a property, or a quality? | |
| − | + | And while we're asking, does it really even matter? | |
| − | |||
| − | | | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | In this | + | In this context, is the word's function to mean something, |
| + | to connote something, and all that we know by means of it, | ||
| + | is it meant to mean that it mean a thing quiet apart from | ||
| + | signs, or is it meant to mean just more of the same order | ||
| + | of signs, ever and again yet more signs, if yet new kinds? | ||
| − | + | Rather than just make up whatever comes into my head, | |
| + | let me go back to some passages that I cited earlier | ||
| + | and consider them more carefully in the light of all | ||
| + | that has transpired in the mean time. Also, for the | ||
| + | sake of equipping this discussion with a comparative | ||
| + | anchor in contemporary usage, let me first cite this: | ||
| − | + | | Predicate. | |
| − | + | | | |
| − | + | | The four traditional kinds of categorical propositions are: | |
| − | + | | All S is P, No S is P, Some S is P, Some S is not P. In each | |
| − | + | | of these the concept denoted by S is the 'subject' and that | |
| − | + | | denoted by P is the 'predicate'. | |
| − | + | | | |
| − | + | | Hilbert and Ackermann use the word 'predicate' for | |
| − | + | | a propositional function of one or more variables; | |
| + | | Carnap uses it for the corresponding syntactical entity, | ||
| + | | the name or designation of such a propositional function | ||
| + | | (i.e., of a property or relation). | ||
| + | | | ||
| + | | Alonzo Church, in Runes, page 248. | ||
| | | | ||
| − | + | | Dagobert Runes (ed.), 'Dictionary of Philosophy', | |
| + | | Littlefield, Adams, & Company, Totowa, NJ, 1972. | ||
| − | + | Well, that was a helpful verdict. Evidently a predicate | |
| + | is proclaimed to be anything from a syntactic signifier | ||
| + | to a propositional function (property or relation) to | ||
| + | a concept denoted by a term in a categorical premiss. | ||
| + | The Latin "predicamentum" is supposed to translate | ||
| + | the Greek "kategoria", a manner of speaking about | ||
| + | a subject, initially as a "put-down in public". | ||
| − | + | I tend to think of a predicate as a sign or utterance, | |
| + | but maybe it is better to avoid the word in favor of | ||
| + | object-sign pairs like "feature" and "feature name". | ||
| + | </pre></font> | ||
| − | + | ===Commentary Work Note 8=== | |
| − | + | <font face="courier new"><pre> | |
| + | Passage 1 | ||
| − | + | | 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? | ||
| + | | | ||
| + | | CSP, CE 1, pages 462-463. | ||
| − | + | Passage 2 | |
| − | + | | To explain this, we must remember that the process of induction is a | |
| − | + | | process of adding to our knowledge; it differs therein from deduction -- | |
| − | | | + | | which merely explicates what we know -- and is on this very account called |
| − | + | | scientific inference. Now deduction rests as we have seen upon the inverse | |
| − | | | + | | proportionality of the extension and comprehension of every term; and this |
| − | + | | principle makes it impossible apparently to proceed in the direction of | |
| − | | | + | | ascent to universals. But a little reflection will show that when our |
| − | + | | knowledge receives an addition this principle does not hold. | |
| − | + | | | |
| − | | | + | | Thus suppose a blind man to be told that no red things are |
| − | + | | blue. He has previously known only that red is a color; | |
| − | + | | and that certain things 'A', 'B', and 'C' are red. | |
| − | + | | | |
| − | + | | The comprehension of red then has been for him 'color'. | |
| − | + | | Its extension has been 'A', 'B', 'C'. | |
| − | | | ||
| − | | | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | | | ||
| − | |||
| − | | | ||
| − | | | ||
| − | | | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||