#pragmatic-semiotic-information — Public Fediverse posts

Information = Comprehension × Extension • Comment 6

Returning to Peirce’s example of inductive inference in Comment 2, let’s try to get a clearer picture of why he connects it with disjunctive terms and indicial signs.

At this point in time I can’t say I’m entirely satisfied with my understanding of the relationship between disjunctive terms, indicial signs, and inductive inferences as presented by Peirce in his early accounts.  What follows is just one of the simplest and least question‑begging attempts at rational reconstruction I’ve been able to devise.

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

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

If there is any distinguishing feature shared by all the instances under the disjunctive description “neat, swine, sheep, deer” then sign users may take that feature as a predictor of being herbivorous, precisely because all the things under the disjunctive description are herbivorous.  But everything under the disjunctive description is cloven‑hoofed, so the cases under the disjunctive description serve to indicate, support, or witness the utility of the induction from cloven‑hoofed to herbivorous.

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Comment 5

Let’s stay with Peirce’s example of abductive inference a little longer and try to clear up the more troublesome confusions tending to arise.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

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

One thing needs to be stressed at this point.  It is important to recognize 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 with all six of the corresponding properties.

The symbol “spherical bright fragrant juicy tropical fruit” denotes objects which may be taken as icons of oranges by virtue of their bearing those six properties in common with oranges.  But there are no objects denoted by the symbol which 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 which have all the properties indicated.

The above analysis provides another way to understand the abductive inference from the Fact and the Rule to the Case   The lack of any cases which are and not is expressed by the implication   Taking that in conjunction with the Rule gives the logical equivalence   But that reduces the Case to the Fact 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 that form, or may there be weaker styles of abductive reasoning which enjoy their own levels of plausibility?  That must remain an open question at this point.

Remark

• Readers will notice I have slipped at this point from using symbol in the precise technical sense Peirce introduced at the beginning of this discussion to the more ordinary sense all of us, Peirce included, tend to use on other occasions.  Should it become a big problem we can always find a way to mark the distinction but so far it seems context has usually sufficed to resolve any likely confusion.

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Comment 4

Reflecting further on Comment 3, many things still puzzle me about Peirce’s account at this point.  The question marks I added to the Figures of that post indicate the node labels I have remaining doubts about.  For example, in Figure 3, is really an icon of object   Again, in Figure 4, is really an index of object   There is nothing for it but returning to Peirce’s text and trying once more to follow his 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.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

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

The relationship between conjunctive terms and iconic signs may be understood along the following lines.  If there is anything with 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 because it shares those properties with an orange.  But the only natural examples of things with all those properties are oranges themselves, so the only thing qualified to serve as a natural icon of an orange by virtue of those very properties is that orange itself or another orange.

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Comment 3

Peirce identifies inference with a process he describes as symbolization.  Let us consider what that might imply.

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.  (467).

Even if it were only a rough analogy between inference and symbolization, a principle of logical continuity, what is known in physics as 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 extending from exact to approximate (non‑demonstrative) inferences and from genuine to approximate (degenerate) symbols.

For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.

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.

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.

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.  (467–468).

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 leading up to genuine symbols, can all be recognized as pervasive Kantian themes.

In order to draw out those 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.

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

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

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Comment 2

Let’s examine Peirce’s second example of a disjunctive term — neat, swine, sheep, deer — within the style of lattice framework we used before.

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.  (468–469).

Accordingly, if we are engaged in symbolizing and we come to such a proposition as “Neat, swine, sheep, and deer are herbivorous”, we know firstly that the disjunctive term may be replaced by a true symbol.  But suppose we know of no symbol for neat, swine, sheep, and deer except cloven‑hoofed animals.  (469).

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

In view of the analogical symmetries the disjunctive term shares with the conjunctive case, we can run through this example in fairly short order.  We have the following four terms.

Suppose is the logical disjunction of the above four terms.

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

Here we have a situation which is dual to the structure of the conjunctive example.  There is a gap between the logical disjunction in lattice terminology, the least upper bound of the disjoined terms, and what we might regard as the natural disjunction or natural lub of those terms, namely, cloven‑hoofed.

Once again, the sheer implausibility of imagining the disjunctive term would ever be embedded exactly as such in a lattice of natural kinds leads to the evident naturalness of the induction to the implication namely, the rule that cloven‑hoofed animals are herbivorous.

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Comment 1

Selection 1 ends with Peirce drawing the following conclusion about the links between information, comprehension, inference, and symbolization.

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.

(Peirce 1866, p. 467)

At this point in his inventory of scientific reasoning, Peirce is relating the nature of inference, information, and inquiry to the character of the signs mediating the process in question, a process he describes as symbolization.

In the interest of clarity let’s draw from Peirce’s account a couple of quick sketches, designed to show how the examples he gives of conjunctive terms and disjunctive terms might look if they were cast within a lattice‑theoretic framework.

Looking back on Selection 5, let’s first examine Peirce’s example of a conjunctive term — spherical, bright, fragrant, juicy, tropical fruit — within a lattice framework.  We have the following six terms.

Suppose is the logical conjunction of the above six terms.

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, consider the following statement.

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.  (Peirce 1866, p. 470).

In other words, if something is said to be then we may guess fairly surely is really an orange, in short, has all the additional features otherwise summed up quite succinctly in the much more constrained term where means an orange.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

What Peirce is saying about not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction in lattice terms, the greatest lower bound of the conjoined terms, and what we might regard as the natural conjunction or natural glb of those terms, namely, an orange.

In sum there is an extra measure of constraint which goes into forming the natural kinds lattice from the free lattice which logic and set theory would otherwise impose as a default background.  The local manifestations of that global information are meted out over the structure of the natural lattice by just such abductive gaps as the one we observe between and

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Selection 6

Selection 1 opens with Peirce proposing, “The information of a term is the measure of its superfluous comprehension”, and it closes with his offering the following promise.

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.

Summing up his account to this point, Peirce appears confident he’s kept his promise.  Promising on our own account to give it another pass, we’ll let him have the last word — for now.

We have now seen how the mind is forced by the very nature of inference itself to make use of induction and hypothesis.

But the question arises how these conclusions come to receive their justification by the event.  Why are most inductions and hypotheses true?  I reply that they are not true.  On the contrary, experience shows that of the most rigid and careful inductions and hypotheses only an infinitesimal proportion are never found to be in any respect false.

And yet it is a fact that all careful inductions are nearly true and all well‑grounded hypotheses resemble the truth;  why is that?  If we put our hand in a bag of beans the sample we take out has perhaps not quite but about the same proportion of the different colours as the whole bag.  Why is that?

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

(Peirce 1866, pp. 470–471)

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Selection 5

Peirce now turns to his example of a conjunctive term, which he uses to show the connection between iconic reference and abductive inference.

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:

spherical, bright, fragrant, juicy, tropical fruit.

Such a term, formed by the sum of the comprehensions of several terms, is called a conjunctive term.  A conjunctive term has no extension adequate to its comprehension.  Thus the only spherical bright fragrant juicy tropical fruit we know is the orange and that has many other characters besides these.  Hence, such a term is of no use whatever.

If it occurs in the predicate and something is said to be a spherical bright fragrant juicy tropical fruit, since there is nothing which is all this which is not an orange, we may say that this is an orange at once.

On the other hand, if the conjunctive term is subject and we know that every spherical bright fragrant juicy tropical fruit necessarily has certain properties, it must be that we know more than that and can simplify the subject.  Thus a conjunctive term may always be replaced by a simple one.

So if we find that light is capable of producing certain phenomena which could only be enumerated by a long conjunction of terms, we may be sure that this compound predicate may be replaced by a simple one.  And if only one simple one is known in which the conjunctive term is contained, this must be provisionally adopted.

(Peirce 1866, p. 470)

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Selection 4

Selection 3 showed how it was possible to combine symbols in such a way as to end up with species of representation outside the class of genuine symbols and introduced the concepts of conjunctive terms and disjunctive terms to describe two ways of doing that.  The essence of wit being quickly grasping the middle term, Peirce’s wit fastens on those terms to highlight the links between manners of representation and modes of inference.

Selection 4 finds Peirce in the middle of articulating the connection between indexical reference and inductive inference, using examples of disjunctive terms as pivotal cases.

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 altogether we must admit the validity of induction.  But even to doubt is to think.  So we cannot give up thinking and the validity of induction must be admitted.

(Peirce 1866, p. 469)

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Selection 3

Selection 3 opens with Peirce remarking a critical property of genuine symbols — the class of symbols is not closed under combinations.  In particular, there are logical conjunctions of symbols and logical disjunctions of symbols which are not themselves genuine symbols.

Applying this paradigm to terms, Peirce introduces two sets of examples under the headings of conjunctive terms and disjunctive terms designed to illustrate the correspondence between manners of representation and modes of inference.

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.

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.

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.

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, pp. 468–469)

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Preamble 1
• https://inquiryintoinquiry.com/2026/06/16/information-comprehension-x-extension-preamble-a/

Perhaps the best perspective from which to bring the connection between the theory of signs and the theory of inquiry into its proper focus is Peirce's own Theory of Information, which he began setting forth in lectures at Harvard and the Lowell Institute in 1865 and 1866. Peirce encapsulates the elements of his theory in the following formula.

• Information = Comprehension × Extension

In the Resources below I link to my study of Peirce's 1865–1866 Lectures on the Logic of Science, with selections from the lectures and my commentary on them.

Ten summers ago I hit on what struck me as a new insight into one of the most recalcitrant problems in Peirce's semiotics and logic of science, namely, the relation between “the manner in which different representations stand for their objects” and the way in which different inferences transform states of information. I roughed out a sketch of my epiphany in a series of blog posts then set it aside for the cool of later reflection. Now looks to be a choice moment for taking another look.

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Information = Comprehension × Extension • Selection 2

Over the course of Selection 1 Peirce introduces the ideas he needs to answer stubborn questions about the validity of scientific inference.  Briefly put, the validity of scientific inference depends on the ability of symbols to express superfluous comprehension, the measure of which Peirce calls information.

Selection 2 sharpens our picture of symbols as general representations, contrasting them with two species of representation whose characters fall short of genuine symbols.

For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.

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.

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.

The third and last kind of representations are symbols or general representations.  They connote attributes and so connote them as to determine what they denote.  To this class belong all words and all conceptions.  Most combinations of words are also symbols.  A proposition, an argument, even a whole book may be, and should be, a single symbol.

(Peirce 1866, pp. 467–468)

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Selection 1

I am going to begin with six selections from Peirce’s Lectures of 1866 where he addresses the roles of signs informing sign relations and their relation to the rules of inference governing inquiries.

⁂ ⁂ ⁂

Our first text comes from Peirce’s Lowell Lectures of 1866, titled “The Logic of Science, or, Induction and Hypothesis”.  I still remember the first time I read those words and the light that lit up the page and my mind.

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.

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.

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.

(Peirce 1866, p. 467)

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Preamble

Perhaps the best perspective from which to bring the connection between the theory of signs and the theory of inquiry into its proper focus is Peirce’s own Theory of Information, which he began setting forth in lectures at Harvard and the Lowell Institute in 1865 and 1866.  Peirce encapsulates the elements of his theory in the following formula.

Information = Comprehension × Extension

In the Resources below I link to my study of Peirce’s 1865–1866 Lectures on the Logic of Science, with selections from the lectures and my commentary on them.

Ten summers ago I hit on what struck me as a new insight into one of the most recalcitrant problems in Peirce’s semiotics and logic of science, namely, the relation between “the manner in which different representations stand for their objects” and the way in which different inferences transform states of information.  I roughed out a sketch of my epiphany in a series of blog posts then set it aside for the cool of later reflection.  Now looks to be a choice moment for taking another look.

A first pass through the variations of representation and reasoning detects the axes of iconic, indexical, and symbolic manners of representation on the one hand and the axes of abductive, inductive, and deductive modes of inference on the other.  Early and often Peirce suggests a natural correspondence between the main modes of inference and the main manners of representation but his early arguments differ from his later accounts in ways deserving close examination, partly for the extra points in his line of reasoning and partly for his explanation of indices as signs constituted by convening the variant conceptions of sundry interpreters.

References

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432.  Archive.  Online.

Resources

• Information = Comprehension × Extension
• Survey of Pragmatic Semiotic Information

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Information = Comprehension × Extension • Selection 2.3
• https://inquiryintoinquiry.com/2024/10/06/information-comprehension-x-extension-selection-2-a/

❝The third and last kind of representations are “symbols” or general representations. They connote attributes and so connote them as to determine what they denote. To this class belong all “words” and all “conceptions”. Most combinations of words are also symbols. A proposition, an argument, even a whole book may be, and should be, a single symbol.❞

(Peirce 1866, pp. 467–468)

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources —

Inquiry Blog • Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/

OEIS Wiki • Information = Comprehension × Extension
• https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension

C.S. Peirce • Upon Logical Comprehension and Extension
• https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Information = Comprehension × Extension • Selection 2.2
• https://inquiryintoinquiry.com/2024/10/06/information-comprehension-x-extension-selection-2-a/

❝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.

❝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.❞

(Peirce 1866, pp. 467–468)

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Information = Comprehension × Extension • Selection 2.1
• https://inquiryintoinquiry.com/2024/10/06/information-comprehension-x-extension-selection-2-a/

Re: Information = Comprehension × Extension • Selection 1
• https://inquiryintoinquiry.com/2024/10/05/information-comprehension-x-extension-selection-1-a/

Over the course of Selection 1 Peirce introduces the ideas he needs to answer stubborn questions about the validity of scientific inference. Briefly put, the validity of scientific inference depends on the ability of symbols to express “superfluous comprehension”, the measure of which Peirce calls “information”.

Selection 2 sharpens our picture of symbols as “general representations”, contrasting them with two species of representation whose characters fall short of genuine symbols.

❝For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.❞

(Peirce 1866, pp. 467–468)

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Information = Comprehension × Extension • Selection 1.2
• https://inquiryintoinquiry.com/2024/10/05/information-comprehension-x-extension-selection-1-a/

❝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.

❝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”.❞

(Peirce 1866, p. 467)

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Information = Comprehension × Extension • Selection 1.1
• https://inquiryintoinquiry.com/2024/10/05/information-comprehension-x-extension-selection-1-a/

Our first text comes from Peirce's Lowell Lectures of 1866, titled “The Logic of Science, or, Induction and Hypothesis”. I still remember the first time I read these words and the light that lit up the page and my mind.

❝Let us now return to the information. The information of a term is the measure of its superfluous comprehension. That is to say that the proper office of the comprehension is to determine the extension of the term. For instance, you and I are men because we possess those attributes — having two legs, being rational, &c. — which make up the comprehension of “man”. Every addition to the comprehension of a term lessens its extension up to a certain point, after that further additions increase the information instead.❞

(Peirce 1866, p. 467)

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources —

Inquiry Blog • Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/

OEIS Wiki • Information = Comprehension × Extension
• https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension

C.S. Peirce • Upon Logical Comprehension and Extension
• https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Information = Comprehension × Extension • Preamble
• https://inquiryintoinquiry.com/2024/10/04/information-comprehension-x-extension-preamble/

Eight summers ago I hit on what struck me as a new insight into one of the most recalcitrant problems in Peirce’s semiotics and logic of science, namely, the relation between “the manner in which different representations stand for their objects” and the way in which different inferences transform states of information. I roughed out a sketch of my epiphany in a series of blog posts then set it aside for the cool of later reflection. Now looks to be a choice moment for taking another look.

A first pass through the variations of representation and reasoning detects the axes of iconic, indexical, and symbolic manners of representation on the one hand and the axes of abductive, inductive, and deductive modes of inference on the other. Early and often Peirce suggests a natural correspondence between the main modes of inference and the main manners of representation but his early arguments differ from his later accounts in ways deserving close examination, partly for the extra points in his line of reasoning and partly for his explanation of indices as signs constituted by convening the variant conceptions of sundry interpreters.

Resources —

Inquiry Blog • Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/

OEIS Wiki • Information = Comprehension × Extension
• https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension

C.S. Peirce • Upon Logical Comprehension and Extension
• https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Constraints and Indications • 1.2
• https://inquiryintoinquiry.com/2024/07/02/constraints-and-indications-1-a/

Re: Peirce List • Kaina Stoicheia and the Symbol Grounding Problem
• https://web.archive.org/web/20121231182129/http://comments.gmane.org/gmane.science.philosophy.peirce/9165

Reference —

• Ashby, W.R. (1956), Introduction to Cybernetics, Methuen, London, UK.

Resources —

Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/

Survey of Inquiry Driven Systems
• https://inquiryintoinquiry.com/2024/02/28/survey-of-inquiry-driven-systems-6/

Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/

#Peirce #Ashby #Cybernetics #InquiryDrivenSystems #PragmaticSemioticInformation
#Constraint #ControlSystems #Indication #Information #Inquiry #IntelligentSystems
#AdaptiveSystems #ErrorControlledRegulation #Feedback #SystemsTheory #Uncertainty
#Determination #Intentionality #Learning #Reasoning #SemioticInformation #Semiotics

Theme One Program • Motivation 1
• https://inquiryintoinquiry.com/2024/06/03/theme-one-program-motivation-1-b/

The main idea behind the Theme One program is the efficient use of graph‑theoretic data structures for the tasks of “learning” and “reasoning”.

I am thinking of “learning” in the sense of learning about an environment, in essence, gaining information about the nature of an environment and being able to apply the information acquired to a specific purpose.

Under the heading of “reasoning” I am simply lumping together all the ordinary sorts of practical activities which would probably occur to most people under that name.

There is a natural relation between the tasks. Learning the character of an environment leads to the recognition of laws which govern the environment and making full use of that recognition requires the ability to reason logically about those laws in abstract terms.

Resources —

Theme One Program • Overview
• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition
• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide
• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Pragmatic Semiotic Information • Comment 3
• https://inquiryintoinquiry.com/2024/03/22/pragmatic-semiotic-information-comment-3/

Memories are coming back to me more through the association of ideas than ordered by time or place. I can sense, almost touch a tangle of thoughts interlaced with each other — the “information first” approach to ontology, the “arrows only”, element‑free angle on category theory, Peirce's relativity of generals and individuals dispatching nominalism once and for all — but there is at core a hard knot of ideas so tightly wound it makes it difficult to articulate the links or see the untying if there is one to make.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Pragmatic Semiotic Information • Comment 2
• https://inquiryintoinquiry.com/2024/03/20/pragmatic-semiotic-information-comment-2/

I was at the time working as a “scanner” in the High Energy Physics Lab at Michigan State, sitting in a darkened room measuring tracks of particle interactions projected on a lighted scanning table from reels and reels of bubble chamber photographs gathered at CERN in a massive mad dash accelerator experiment some years before. For my part it was a menial job, 4pm to midnight every worklong day, but even a minion can imagine himself sharing in a hunt for the Ω⁻ particle, or whatever the Grail or Questying Beastie was at the time.

Meanwhile, in another part of the grove, I was spending my daylight hours checking off the final boxes for my Bachelor's degree, the main thing being to get my paper on Peirce, “Complications of the Simplest Mathematics”, approved as a substitute for a field study requirement. That had taken me two years' work in MSU's media library, poring through the microfilm reels of Peirce's Nachlass in search of enlightenment about a single puzzling paragraph I tripped over in his Collected Papers.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Pragmatic Semiotic Information • Comment 1
• https://inquiryintoinquiry.com/2024/03/18/pragmatic-semiotic-information-comment-1/

I remember it was back in ’76 when I began to notice a subtle shift of focus in the computer science journals I was reading, from discussing “X” to discussing “Information About X”, a transformation I noted mentally as \( X \to \mathrm{Info}(X) \) whenever I ran across it. I suppose that small arc of revolution had been building for years but it struck me as crossing a threshold to a more explicit, self‑conscious stage about that time.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Pragmatic Semiotic Information • 2.1
• https://inquiryintoinquiry.com/2024/03/04/pragmatic-semiotic-information-2/

What is information that a sign may bear it?

Three more questions arise at this juncture.

• How is a sign empowered to contain information?
• What is the practical context of communication?
• Why do we care about these bits of information?

A very rough answer to these questions might begin as follows.

Human beings are initially concerned solely with their own lives but then a world obtrudes on their subjective existence and so they find themselves forced to take an interest in the objective realities of its nature.

In pragmatic terms our initial aim, concern, interest, object, or “pragma” is expressed by the verbal infinitive “to live”, but the infinitive is soon reified into the derivative substantial forms of “nature”, “reality”, “the world”, and so on. Against that backdrop we find ourselves cast as the protagonists on a “scene of uncertainty”.

The situation may be pictured as a juncture from which a manifold of options fan out before us. It may be an issue of “truth”, “duty”, or “hope”, the last codifying a special type of uncertainty as to “what regulative principle has any chance of success”, but the chief uncertainty is that we are called on to make a choice and all too often we have very little clue which of the options is most fit to pick.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Pragmatic Semiotic Information • 1.2
• http://inquiryintoinquiry.com/2024/03/03/pragmatic-semiotic-information-1/

A question of what's true is a “descriptive question” and there exist what are called “descriptive sciences” devoted to answering descriptive questions about any domain of phenomena one might care to name.

A question of what's to do, in other words, what must be done by way of achieving a given aim, is a “normative question” and there exist what are called “normative sciences” devoted to answering normative questions about any domain of problems one might care to address.

Since information plays its role on a stage set by uncertainty, a big part of saying what information is will necessarily involve saying what uncertainty is. There is little chance the vagaries of a word like “uncertainty”, given the nuances of its ordinary, poetic, and technical uses, can be corralled by a single pen, but there do exist established models and formal theories which manage to address definable aspects of uncertainty and these do have enough uses to make them worth looking into.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Pragmatic Semiotic Information • 1.1
• http://inquiryintoinquiry.com/2024/03/03/pragmatic-semiotic-information-1/

Information • What's it good for?

The good of information is its use in reducing our uncertainty about an issue which comes before us. But uncertainty comes in many flavors and so the information which serves to reduce uncertainty can be applied in several ways. The situations of uncertainty human agents commonly find themselves facing have been investigated under many headings, literally for ages, and the categories subtle thinkers arrived at long before the dawn of modern information theory still have their uses in setting the stage of an introduction.

Picking an example of a subtle thinker almost at random, the philosopher‑scientist Immanuel Kant surveyed the questions of human existence within the span of the following three axes.

• What's true?
• What's to do?
• What's to hope?

The third question is a bit too subtle for the present frame of discussion but the first and second are easily recognizable as staking out the two main axes of information theory, namely, the dual dimensions of “information” and “control”. Roughly the same space of concerns is elsewhere spanned by the dual axes of competence and performance, specification and optimization, or just plain knowledge and skill.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Survey of Pragmatic Semiotic Information • 8
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/

This is a Survey of blog and wiki posts on a theory of information which grows out of pragmatic semiotic ideas. This line of inquiry is more open‑ended than most. The question is —

• What is information and how does it impact the spectrum of activities answering to the name of inquiry?

Setting out on what would become his lifelong quest to explore and explain the “Logic of Science”, C.S. Peirce pierced the veil of historical confusions obscuring the issue and fixed on what he called the “laws of information” as the key to solving the puzzle.

The first hints of the Information Revolution in our understanding of scientific inquiry may be traced to Peirce’s lectures of 1865–1866 at Harvard University and the Lowell Institute. There Peirce took up “the puzzle of the validity of scientific inference” and claimed it was “entirely removed by a consideration of the laws of information”.

Please follow the above link for the full set of resources.
Articles and blog posts on the core ideas are linked below.

Information = Comprehension × Extension
• https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension

{ Information = Comprehension × Extension }
• https://inquiryintoinquiry.com/2016/05/18/information-comprehension-x-extension/

{ Information = Comprehension × Extension } • Revisited
• https://inquiryintoinquiry.com/2019/01/23/information-comprehension-x-extension-revisited/

Pragmatic Semiotic Information • Ψ
• https://inquiryintoinquiry.com/2023/07/22/pragmatic-semiotic-information-%cf%88-a/

Pragmatic Semiotic Information
• https://oeis.org/wiki/Pragmatic_Semiotic_Information

Peirce's Logic Of Information
• https://oeis.org/wiki/User:Jon_Awbrey/Peirce%27s_Logic_Of_Information

Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”
• https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm

#Peirce #Logic #LogicOfScience #ScientificMethod #InformationTheory
#Pragmatism #Semiotics #SignRelations #PragmaticSemioticInformation

Survey of Pragmatic Semiotic Information • 7
• http://inquiryintoinquiry.com/2023/07/23/survey-of-pragmatic-semiotic-information-7/

This is a Survey of blog and wiki posts on a theory of information which grows out of pragmatic semiotic ideas. All my projects are exploratory in character but this line of inquiry is more open‑ended than most. The question is —

• What is information and how does it impact the spectrum of activities answering to the name of inquiry?

Setting out on what would become his lifelong quest to explore and explain the “Logic of Science”, C.S. Peirce pierced the veil of historical confusions obscuring the issue and fixed on what he called the “laws of information” as the key to solving the puzzle.

The first hints of the Information Revolution in our understanding of scientific inquiry may be traced to Peirce's lectures of 1865–1866 at Harvard University and the Lowell Institute. There Peirce took up “the puzzle of the validity of scientific inference” and claimed it was “entirely removed by a consideration of the laws of information”.

Fast forward to the present and I see the Big Question as follows. Having gone through the exercise of comparing and contrasting Peirce's theory of information, however much it yet remains in a rough‑hewn state, with Shannon's paradigm so pervasively informing the ongoing revolution in our understanding and use of information, I have reason to believe Peirce's idea is root and branch more general and has the potential, with due development, to resolve many mysteries still bedeviling our grasp of inference, information, and inquiry.

#Peirce #Logic #LogicOfScience #ScientificMethod #InformationTheory
#Pragmatism #Semiotics #SignRelations #PragmaticSemioticInformation

Pragmatic Semiotic Information • Ψ
• http://inquiryintoinquiry.com/2023/07/22/pragmatic-semiotic-information-%cf%88-2/

I remember it was back in ’76 when I began to notice a subtle shift of focus in the computer science journals I was reading, from discussing X to discussing “Information About X”, or \(X \to \mathrm{Info}(X)\) as I came to notate the transformation. I suppose that small arc of revolution had been building for years but it struck me as crossing a threshold to a more explicit, self‑conscious stage about that time.

#Peirce #Logic #LogicOfScience #ScientificMethod #InformationTheory
#Pragmatism #Semiotics #SignRelations #PragmaticSemioticInformation

Functional Logic • Inquiry and Analogy • 5

Inquiry and Analogy • Aristotle’s “Paradigm” • Reasoning by Analogy

Aristotle examines the subject of analogical inference or “reasoning by example” under the heading of the Greek word παραδειγμα, from which comes the English word paradigm.  In its original sense the word suggests a kind of “side‑show”, or a parallel comparison of cases.

We have an Example (παραδειγμα, or analogy) when the major extreme is shown to be applicable to the middle term by means of a term similar to the third.  It must be known both that the middle applies to the third term and that the first applies to the term similar to the third.

E.g., let A be “bad”, B “to make war on neighbors”, C “Athens against Thebes”, and D “Thebes against Phocis”.  Then if we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad.  Evidence of this can be drawn from similar examples, e.g., that war by Thebes against Phocis is bad.  Then since war against neighbors is bad, and war against Thebes is against neighbors, it is evident that war against Thebes is bad.

Aristotle, “Prior Analytics” 2.24, Hugh Tredennick (trans.)

Figure 6 shows the logical relationships involved in Aristotle’s example of analogy.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Theme One Program • Exposition 1.2
• https://inquiryintoinquiry.com/2022/06/15/theme-one-program-exposition-1-2/

The Idea↑Form Flag

The graph-theoretic data structures used by the program are built up from a basic data structure called an “idea-form flag”. That structure is defined as a pair of Pascal data types by means of the following specifications.

Figure 1. Type Idea = ^Form
• https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-type-idea-5eform.png

Figure 2. Code Box
• type idea = ^form;
• form = record
• sign: char;
• as, up, on, by: idea;
• code: numb
• end;

An “idea” is a pointer to a “form”.
• A “form” is a record consisting of:
• A “sign” of type “char”;
• Four pointers, “as”, “up”, “on”, “by”, of type “idea”;
• A “code” of type “numb”, that is, an integer in [0, max integer].

Represented in terms of “digraphs”, or directed graphs, the combination of an idea pointer and a form record is most easily pictured as an “arc”, or directed edge, leading to a node labeled with the other data, in this case, a letter and a number.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Exposition 1.1
• https://inquiryintoinquiry.com/2024/06/09/theme-one-program-exposition-1-b/

Theme One is a program for constructing and transforming a particular species of graph‑theoretic data structures, forms designed to support a variety of fundamental learning and reasoning tasks.

The program evolved over the course of an exploration into the integration of contrasting types of activities involved in learning and reasoning, especially the types of algorithms and data structures capable of supporting all sorts of inquiry processes, from everyday problem solving to scientific investigation. In its current state, Theme One integrates over a common data structure fundamental algorithms for one type of inductive learning and one type of deductive reasoning.

We begin by describing the class of graph-theoretic data structures used by the program, as determined by their local and global features. It will be the usual practice to shift around and view these graphs at many different levels of detail, from their abstract definition to their concrete implementation, and many points in between.

The main work of the Theme One program is achieved by building and transforming a single species of graph-theoretic data structures. In their abstract form these structures are closely related to the graphs called cacti and conifers in graph theory, so we’ll generally refer to them under those names.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 6
• https://inquiryintoinquiry.com/2022/08/20/theme-one-program-motivation-6-2/

Comments I made in reply to a correspondent’s questions about delimiters and tokenizing in the Learner module may be worth sharing here.

In one of the projects I submitted toward a Master’s in psychology I used the Theme One program to analyze samples of data from my advisor’s funded research study on family dynamics. In one phase of the study observers viewed video-taped sessions of family members (parent and child) interacting in various modes (“play” or “work”) and coded qualitative features of each moment’s activity over a period of time.

The following page describes the application in more detail and reflects on its implications for the conduct of scientific inquiry in general.

Exploratory Qualitative Analysis of Sequential Observation Data
• https://oeis.org/wiki/User:Jon_Awbrey/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data

In this application a “phrase” or “string” is a fixed-length sequence of qualitative features and a “clause” or “strand” is a sequence of such phrases delimited by what the observer judges to be a significant pause in the action.

In the qualitative research phases of the study one is simply attempting to discern any significant or recurring patterns in the data one possibly can.

In this case the observers are tokenizing the observations according to a codebook that has passed enough intercoder reliability studies to afford them all a measure of confidence it captures meaningful aspects of whatever reality is passing before their eyes and ears.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 5
• https://inquiryintoinquiry.com/2022/08/20/theme-one-program-motivation-5-2/

Since I’m working from decades-old memories of first inklings I thought I might peruse the web for current information about Zipf’s Law. I see there is now something called the Zipf–Mandelbrot (and sometimes –Pareto) Law and that was interesting because my wife Susan Awbrey made use of Mandelbrot’s ideas about self-similarity in her dissertation and communicated with him about it. So there’s more to read up on.

Just off-hand, though, I think my Learner is dealing with a different problem. It has more to do with the savings in effort a learner gets by anticipating future experiences based on its record of past experiences than the savings it gets by minimizing bits of storage as far as mechanically possible. There is still a type of compression involved but it’s more like Korzybski’s “time-binding” than space-savings proper. Speaking of old memories …

The other difference I see is that Zipf’s Law applies to an established and preferably large corpus of linguistic material, while my Learner has to start from scratch, accumulating experience over time, making the best of whatever data it has at the outset and every moment thereafter.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 4
• https://inquiryintoinquiry.com/2022/08/19/theme-one-program-motivation-4-2/

From Zipf’s Law and the category of “things that vary inversely with frequency” I got my first brush with the idea that keeping track of usage frequencies is part and parcel of building efficient codes.

In its first application the environment the Learner has to learn is the usage behavior of its user, as given by finite sequences of characters from a finite alphabet, which sequences of characters might as well be called “words”, together with finite sequences of those words which might as well be called “phrases” or “sentences”. In other words, Job One for the Learner is the job of constructing a “user model”.

In that frame of mind we are not seeking anything so grand as a Universal Induction Algorithm but simply looking for any approach to give us a leg up, complexity wise, in Interactive Real Time.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 3
• https://inquiryintoinquiry.com/2022/08/18/theme-one-program-motivation-3-2/

Sometime around 1970 John B. Eulenberg came from Stanford to direct Michigan State’s Artificial Language Lab, where I would come to spend many interesting hours hanging out all through the 70s and 80s. Along with its research program the lab did a lot of work on augmentative communication technology for limited mobility users and the observations I made there prompted the first inklings of my Learner program.

Early in that period I visited John’s course in mathematical linguistics, which featured Laws of Form among its readings, along with the more standard fare of Wall, Chomsky, Jackendoff, and the Unified Science volume by Charles Morris which credited Peirce with pioneering the pragmatic theory of signs. I learned about Zipf’s Law relating the lengths of codes to their usage frequencies and I named the earliest avatar of my Learner program XyPh, partly after Zipf and playing on the xylem and phloem of its tree data structures.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 2.2
• https://inquiryintoinquiry.com/2022/08/17/theme-one-program-motivation-2-2/

As I mentioned, work on those two projects proceeded in a parallel series of fits and starts through interwoven summers for a number of years, until one day it dawned on me how the Learner, one of whose aliases was Index, could be put to work helping with sundry substitution tasks the Modeler needed to carry out.

So I began integrating the functions of the Learner and the Modeler, at first still working on the two component modules in an alternating manner, but devoting a portion of effort to amalgamating their principal data structures, bringing them into convergence with each other, and unifying them over a common basis.

Another round of seasons and many changes of mind and programming style, I arrived at a unified graph-theoretic data structure, strung like a wire through the far‑flung pearls of my programmed wit. But the pearls I polished in alternate years maintained their shine along axes of polarization whose grains remained skew in regard to each other. To put it more plainly, the strategies I imagined were the smartest tricks to pull from the standpoint of optimizing the program’s performance on the Learning task I found the next year were the dumbest moves to pull from the standpoint of its performance on the Reasoning task. I gradually came to appreciate that trade-off as a discovery.

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#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 2.1
• https://inquiryintoinquiry.com/2022/08/17/theme-one-program-motivation-2-2/

A side-effect of working on the Theme One program over the course of a decade was the measure of insight it gave me into the reasons why empiricists and rationalists have so much trouble understanding each other, even when those two styles of thinking inhabit the very same soul.

The way it came about was this. The code from which the program is currently assembled initially came from two distinct programs, ones I developed in alternate years, at first only during the summers.

In the Learner program I sought to implement a Humean empiricist style of learning algorithm for the adaptive uptake of coded sequences of occurrences in the environment, say, as codified in a formal language. I knew all the theorems from formal language theory telling how limited any such strategy must ultimately be in terms of its generative capacity, but I wanted to explore the boundaries of that capacity in concrete computational terms.

In the Modeler program I aimed to implement a variant of Peirce’s graphical syntax for propositional logic, making use of graph-theoretic extensions I had developed over the previous decade.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 1
• https://inquiryintoinquiry.com/2022/08/16/theme-one-program-motivation-1-2/

The main idea behind the Theme One program is the efficient use of graph-theoretic data structures for the tasks of “learning” and “reasoning”.

I am thinking of learning in the sense of learning about an environment, in essence, gaining information about the nature of an environment and being able to apply the information acquired to a specific purpose.

Under the heading of reasoning I am simply lumping together all the ordinary sorts of practical activities which would probably occur to most people under that name.

There is a natural relation between the tasks. Learning the character of an environment leads to the recognition of laws which govern the environment and making full use of that recognition requires the ability to reason logically about those laws in abstract terms.

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#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

More Thoughts —
#PragmaticSemioticInformation
• https://oeis.org/wiki/Pragmatic_Semiotic_Information

Test ...

#DifferentialLogicAndDynamicSystems

#PragmaticSemioticInformation

#ThemeOneProgram • #SurveyPage
• https://inquiryintoinquiry.com/2022/06/12/survey-of-theme-one-program-4/

A program I worked on all through the 80s aimed to develop #Algorithms and #DataStructures for integrating #EmpiricalLearning and #LogicalReasoning. I had earlier developed programs for #Learning #TwoLevelFormalLanguages and for #PropositionalLogic #ConstraintSatisfaction based on C.S. #Peirce's #LogicalGraphs. Getting those 2 faculties to work together meant integrating the ways of #Empiricism and #Rationalism.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism
#ModelTheory #ProofTheory #Semiotics
#ParallelDistributedProcessing #PDP
#Grossberg #McClelland #Rumelhart