C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • 1

Selection from C.S. Peirce, “On the Algebra of Logic : A Contribution to the Philosophy of Notation” (1885)

§1.  Three Kinds Of Signs

Any character or proposition either concerns one subject, two subjects, or a plurality of subjects.  For example, one particle has mass, two particles attract one another, a particle revolves about the line joining two others.  A fact concerning two subjects is a dual character or relation;  but a relation which is a mere combination of two independent facts concerning the two subjects may be called degenerate, just as two lines are called a degenerate conic.  In like manner a plural character or conjoint relation is to be called degenerate if it is a mere compound of dual characters.  (3.359).

A sign is in a conjoint relation to the thing denoted and to the mind.  If this triple relation is not of a degenerate species, the sign is related to its object only in consequence of a mental association, and depends upon a habit.  Such signs are always abstract and general, because habits are general rules to which the organism has become subjected.  They are, for the most part, conventional or arbitrary.  They include all general words, the main body of speech, and any mode of conveying a judgment.  For the sake of brevity I will call them tokens.  (3.360).

But if the triple relation between the sign, its object, and the mind, is degenerate, then of the three pairs

\begin{array}{lll}  \text{sign} & & \text{object}  \\  \text{sign} & & \text{mind}  \\  \text{object} & & \text{mind}  \end{array}

two at least are in dual relations which constitute the triple relation.  One of the connected pairs must consist of the sign and its object, for if the sign were not related to its object except by the mind thinking of them separately, it would not fulfill the function of a sign at all.  Supposing, then, the relation of the sign to its object does not lie in a mental association, there must be a direct dual relation of the sign to its object independent of the mind using the sign.  In the second of the three cases just spoken of, this dual relation is not degenerate, and the sign signifies its object solely by virtue of being really connected with it.  Of this nature are all natural signs and physical symptoms.  I call such a sign an index, a pointing finger being the type of this class.

The index asserts nothing;  it only says “There!”  It takes hold of our eyes, as it were, and forcibly directs them to a particular object, and there it stops.  Demonstrative and relative pronouns are nearly pure indices, because they denote things without describing them;  so are the letters on a geometrical diagram, and the subscript numbers which in algebra distinguish one value from another without saying what those values are.  (3.361).

The third case is where the dual relation between the sign and its object is degenerate and consists in a mere resemblance between them.  I call a sign which stands for something merely because it resembles it, an icon.  Icons are so completely substituted for their objects as hardly to be distinguished from them.  Such are the diagrams of geometry.  A diagram, indeed, so far as it has a general signification, is not a pure icon;  but in the middle part of our reasonings we forget that abstractness in great measure, and the diagram is for us the very thing.  So in contemplating a painting, there is a moment when we lose consciousness that it is not the thing, the distinction of the real and the copy disappears, and it is for the moment a pure dream — not any particular existence, and yet not general.  At that moment we are contemplating an icon.  (3.362).

References

  • Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
  • Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.  Volume 3 : Exact Logic (Published Papers), 1933.  CP 3.359–403.
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 5 (1884–1886), 1993.  Item 30, 162–190.

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Semiotics, Semiosis, Sign Relations • Discussion 19

Peirce Syllabus

Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.

❧ Charles Sanders Peirce • Collected Papers, CP 1.186 (1903)
Syllabus • Classification of Sciences (CP 1.180–202, G-1903-2b)

Re: Peirce ListJohn Sowa

JS:
Questions for everybody to consider:  In the 1903 classification of the sciences, Peirce did not mention semeiotic, the most important science that he introduced.  Why not?  Where does it belong in the classification?

The short schrift on this subject may be summed up in the following syllogism.

  • Corollary.  This leaves room for Descriptive Semiotic.

Additional Notes

  • Definition and Determination • (4)(5)

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Logical Graphs, Iconicity, Interpretation • 2

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.

C.S. Peirce (1866), Lowell Lecture 7, CE 1, 467–468

The Table in the previous post can now be sorted to bring out the “family resemblances”, likenesses, or symmetries among logical graphs and the boolean functions they denote, where the “orbits” or similarity classes are determined by the dual interpretation of logical graphs.  Performing the sort produces the following Table.  As we have seen in previous discussions, there are 10 orbits in all, 4 orbits of 1 point each and 6 orbits of 2 points each.

\text{Boolean Functions and Logical Graphs on Two Variables} \stackrel{_\bullet}{} \text{Orbit Order}

Boolean Functions and Logical Graphs on Two Variables

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Logical Graphs, Iconicity, Interpretation • 1

If exegesis raised a hermeneutic problem, that is, a problem of interpretation, it is because every reading of a text always takes place within a community, a tradition, or a living current of thought, all of which display presuppositions and exigencies — regardless of how closely a reading may be tied to the quid, to “that in view of which” the text was written.

Paul Ricoeur • The Conflict of Interpretations

If a picture is worth a thousand words, here’s my 48,000 words worth on the recurring question of logical graphs, their iconicity, and their interpretation, at least as concerns Peirce’s alpha graphs interpreted for propositional logic.  A few more actual words, literally speaking, may be called for.  I’ll return to that anon.

Referring to the Table —

  • Column 1 shows a conventional name f_{i} and a venn diagram for each of the sixteen boolean functions on two variables.
  • Column 2 shows the logical graph canonically representing the boolean function in Column 1 under the entitative interpretation.  This is the interpretation C.S. Peirce used in his earlier work on entitative graphs and the one Spencer Brown used in his book Laws of Form.
  • Column 3 shows the logical graph canonically representing the boolean function in Column 1 under the existential interpretation.  This is the interpretation C.S. Peirce used in his later work on existential graphs.

\text{Boolean Functions and Logical Graphs on Two Variables}

Boolean Functions and Logical Graphs on Two Variables

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Minimal Negation Operators • Discussion 2

Re: Minimal Negation Operators • (1)(2)(3)(4)
Re: Peirce List (1) (2) (3)Jerry Chandler

JC:
As a chemist, CSP often inscended hyle terminology into his logical corpse as he sought to extend the 15–17th century historical usages of the meaning of the concept of a “term”.

One particularity of chemical synthesis is the absence of the “negative” operators on the chemical elements.  Each element is a logical constant in the language of chemistry and hence can not be negated.  Yet, in the notation for chemistry it is necessary to assert and signify the absence of a chemical unit in a logical product.  This could be referred to as a minimal negation in a logically consistent semantics of a chemical syntax.

I have no information, either positive or negative, of the meaning Jon intends to infer logically with his usage of this non-standard semantics.  However, this semantics is obviously useful in attempting to give a logical semantics for the well‑established semiosis of hyle.

Dear Jerry,

I’ve been spending a lot of time lately thinking about how I first got into all the things I’ve gotten into over the years.  The thing that surprised me the most was how much of my life I’ve been immersed in raw data despite my best efforts to rise above it in flights of theory and just plain fancy.  The honors chemistry course I took my first year in college was pretty advanced — we “hit the ground running” as my Dad used to say from his paratroop days — moving from covalent bonding theory the first term to molecular orbital theory the second.

It was there I first encountered the triple interaction of theory, experiment, and electronic computation.  Aside from the routine programs we ran to analyze our data, drawing least squares lines through experimental scatterplots and all that, I began my first attempts to compute with symbolic forms, trying to get Fortran to place the electron dots around and between chemical symbols in various molecular combinations.  Mostly I learned to dislike Fortran — wrong tool for the job, I guess — and it would be years before I woke to Lisp.

At any rate, let me beg off on chemical logic or logical chemistry.  My experiences in that borderland are more a tale of fits and starts than anything conclusive and reconstructing the details would take a search through the darker corners of my basement archives.

The matter of “non-standard semantics”, however, is a timely and topical subject to address, one it would dispel a mass of obscurities about the link between logic and semiotics to clarify as much as we can.

To begin, we may pose the question as follows.

  • In what way does a propositional calculus based on minimal negation operators deviate from standard semantics?

I will take that up next time, perhaps under a different heading.

Regards,

Jon

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Minimal Negation Operators • Discussion 1

Re: Minimal Negation Operators • (1)(2)(3)(4)
Re: Peirce List (1) (2)Imran Makani

IM:
In his first post on this thread Jon clearly says that [minimal negation operators] were developed from Peirce’s alpha graphs for propositional calculus and that he has even outlined the history of this early development in a previous series of posts.

Dear Imran,

Welcome to the List and heartfelt thanks for your appreciation of my contribution to it.  I’m just a person who goes to sleep every night and wakes up every morning with sundry issues in Peirce’s work in the forefront of his mind.  It has been that way — no doubt with less persistence at first, there were other demands and diversions in the early days — since I chanced on Peirce’s work my first year in college and right up to the present time when my inquiries into the consequences of his work literally pervade my dreams and days.

If you’ll excuse my anecdotage, it took me nine years to complete my Bachelor of Arts — demands and diversions were abundant — matriculating first in Math and Physics, taking a break in Communication Arts where I tilted with Aristotle, at long last mustering out in a cross-cultural-cultivating radical-liberal arts college with a concentration I created myself in “Mathematical And Philosophical Method”.  The cornerstone of that first year and the capstone of my senior thesis, “Complications of the Simplest Mathematics”, compass the dark night and the dawn’s light of my Peirce Decade One.

Well, I’ve run out of time for now …
I’ll continue this memoir tomorrow …

Regards,

Jon

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Semiotics, Semiosis, Sign Relations • Discussion 18

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Edward Buckner raised a few questions about the sign relations implicit in Aristotle’s treatise “On Interpretation”, prompting the following thoughts on my part.

On Pragmata

The object of a sign is any object of discussion or thought.  It is relational not ontological.  This is the beginning of pragmatic semiotics.

On Homoiomata

The likeness theory of reference has the same problem as the correspondence theory of truth, namely, the concepts of likeness and correspondence used in those theories both refer to dyadic relations and dyadic relations are not adequate to the task of accounting for the complex of activities composing the intellect, for example, inquiry, learning, reasoning, speech, thought, in short, Information Development/Exchange Activities.

In actuality, Aristotle comes closer to recognizing the triadic relation of Objects, Signs, and Ideas than the majority of later writers before Peirce.  Here is the figure Susan Awbrey and I cut in our first hack at the matter.

The Sign Relation in Aristotle

Figure 1.  The Sign Relation in Aristotle

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Relations & Their Relatives • Discussion 23

Re: Ontolog ForumRoberto RovettoAlex Shkotin

Having lost my concentration to another round of home reconstruction disruption, let me loop back to the texts from Roberto and Alex which drew me into this discussion last week.

RR:

What’s your view on:

When to create a greater-than-binary relation rather than a binary relation?

Consider:  You want to represent some information, statement, or knowledge, without necessarily being forced to limit to binary relations.  A common example is when wanting to reference time.  And “between” is greater than binary.  What are other pieces of knowledge that you’d want assert a ternary, or greater than binary relation to capture it accurately?

Do you have any rules of thumb for knowing when to assert n-ary relations greater than binary?

AS:
Let me underline an important point:  first of all, we have found in nature and society one or another relation and ask how many members each example of this relation can have?  i.e. arity is a feature of relation itself.  So […] we come here to the logic of relations and its discovery.  For me, examples of relations of different arity from one or another domain would be great.

I will take up k-adic or k-ary relations from a mathematical perspective and I will treat them from the standpoint of one whose “customers” over his actually getting paid years were academic, education, health, and research science units or investigators engaged in gathering data by means of experiments, empirical studies, or survey instruments and analyzing those data according to the protocols of qualitative observation methods or quantitative statistical hypothesis testing, all toward the purpose of discovering reproducible facts about their research domains and subject populations.

A sidelong but critically necessary reflection on the research scene comes from the Peircean perspective on scientific inquiry, in which triadic relations and especially triadic sign relations are paramount.  I will develop Peirce’s pragmatic, semiotic, information-theoretic viewpoint in tandem with the treatment of relation theory.

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Susan Awbrey • I Looked Up

I Looked Up

I looked up and I was old
      The loose, wrinkled skin
      The lines that leave their traces
I looked up and I was old
      The scars of former illness written there
I looked up and I was old
      I ask my body why it has forsaken me
      It said it has been there through it all
I looked up and I was old
      But my spirit is the same
      Youth but a breath away
I looked up and I was one
      With ancient earth and new born robin
I looked up and my gaze rose
      Beyond my mollusk shell to the infinity that is now

❧ Susan Awbrey
     September 19, 2021

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Differential Logic • Discussion 15

Re: Differential Logic • Comment 7
Re: Laws of FormLyle Anderson

LA:
Differentials and partial differentials over the real numbers work because one can pick two real numbers that are arbitrarily close to one another.  The difference between any two real numbers can be made as small as desired.  If you have a real function of a real variable then you characterize the change in the function’s real value as the value of the argument changes.  This can be represented as the tangent to the curve representing the function at a given point.

In the Boolean domain there are only two values and they are always one step, unity, apart.  There is nothing to differentiate.  There is no variation in the spacing of the arguments or the function values.  There are no curves in the Boolean domain.  There is nothing to differentiate.

Dear Lyle,

My last post is really just a note-to-self reminding me to get back to work on differential logic, my memory being jogged by a number of posts on the Azimuth Blog.  But if I could nudge a few people to reflect on what the logical analogue of differential calculus ought to look like, that would be a plus.

The short answer to your objection is we don’t need limits in discrete spaces.  We follow the example of the finite difference calculus, using logical analogues of the enlargement operator \mathrm{E} and the difference operator \mathrm{D}.

A differential is a locally linear approximation to a function, that is, a linear function which approximates another function at a point.  In boolean spaces, we know what the functions are, we know what the linear functions are, and all we need is a notion of approximation to define differentials.  Yes, there are numerous tricky bits to work out in boolean spaces, but I worked those out in the array of expositions at many different levels of abstraction and detail to which I have linked before, as again below.

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