Sinecure

Posted on January 22, 2012 by Jon Awbrey

Forgive me, Author, for I have signed
A sign that forges your original sign —
How I miss the indelible mark of thine!

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Forwarding Address for Knol Articles

Posted on December 25, 2011 by Jon Awbrey

It looks like the Annotum developers are following the fashion of the day in rolling out their platform first and testing it later. There doesn’t appear to be any way to edit the articles ported over from Knol to Annotum without reducing them to utter hash, so I copied my Knol articles to another blog, archiving them under the Annotum theme until such time as the Annotum developers get the bugs out of their system. In the meantime I will continue working on those and other articles here.

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I Am A Rock

Posted on December 21, 2011 by Jon Awbrey

I Am A Rock

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Peirce’s Law

Posted on October 6, 2008 by Jon Awbrey

A Curious Truth of Classical Logic

Peirce’s law is a propositional calculus formula which states a non‑obvious truth of classical logic and affords a novel way of defining classical propositional calculus.

Introduction

Peirce’s law is commonly expressed in the following form.

((p \Rightarrow q) \Rightarrow p) \Rightarrow p

Peirce’s law holds in classical propositional calculus, but not in intuitionistic propositional calculus.  The precise axiom system one chooses for classical propositional calculus determines whether Peirce’s law is taken as an axiom or proven as a theorem.

History

Here is Peirce’s own statement and proof of the law:

A fifth icon is required for the principle of excluded middle and other propositions connected with it.  One of the simplest formulae of this kind is:

\{ (x \,-\!\!\!< y) \,-\!\!\!< x \} \,-\!\!\!< x.

This is hardly axiomatical.  That it is true appears as follows.  It can only be false by the final consequent x being false while its antecedent (x \,-\!\!\!< y) \,-\!\!\!< x is true.  If this is true, either its consequent, x, is true, when the whole formula would be true, or its antecedent x \,-\!\!\!< y is false.  But in the last case the antecedent of x \,-\!\!\!< y, that is x, must be true.  (Peirce, CP 3.384).

Peirce goes on to point out an immediate application of the law:

From the formula just given, we at once get:

\{ (x \,-\!\!\!< y) \,-\!\!\!< \alpha \} \,-\!\!\!< x,

where the \alpha is used in such a sense that (x \,-\!\!\!< y) \,-\!\!\!< \alpha means that from (x \,-\!\!\!< y) every proposition follows.  With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of x follows the truth of x.  (Peirce, CP 3.384).

Note.  Peirce uses the sign of illation ``-\!\!\!<" for implication.  In one place he explains ``-\!\!\!<" as a variant of the sign ``\le" for less than or equal to;  in another place he suggests that A \,-\!\!\!< B is an iconic way of representing a state of affairs where A, in every way that it can be, is B.

Graphical Representation

Representing propositions in the language of logical graphs, and operating under the existential interpretation, Peirce’s law is expressed by means of the following formal equivalence or logical equation.

Peirce's Law

Graphical Proof

Using the axiom set given in the articles on logical graphs, Peirce’s law may be proved in the following manner.

Peirce's Law • Proof

The following animation replays the steps of the proof.

Peirce's Law • Proof Animation

Equational Form

A stronger form of Peirce’s law also holds, in which the final implication is observed to be reversible, resulting in the following equivalence.

((p \Rightarrow q) \Rightarrow p) \Leftrightarrow p

The converse implication p \Rightarrow ((p \Rightarrow q) \Rightarrow p) is clear enough on general principles, since p \Rightarrow (r \Rightarrow p) holds for any proposition r.

Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce’s law is expressed by the following equation.

Peirce's Law : Strong Form

Using the axioms and theorems listed in the entries on logical graphs, the equational form of Peirce’s law may be proved in the following manner.

Peirce's Law : Strong Form • Proof

The following animation replays the steps of the proof.

Peirce's Law : Strong Form • Proof Animation

References

  • Peirce, Charles Sanders (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7 (1885), 180–202.  Reprinted (CP 3.359–403), (CE 5, 162–190).
  • Peirce, Charles Sanders (1931–1935, 1958), 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.  Cited as (CP volume.paragraph).
  • Peirce, Charles Sanders (1981–), Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN.  Cited as (CE volume, page).

Resources

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

Posted in C.S. Peirce, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Peirce, Peirce's Law, Proof Theory, Propositional Calculus, Propositions As Types Analogy, Semiotics, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , | 15 Comments

Praeclarum Theorema

Posted on October 5, 2008 by Jon Awbrey

Introduction

The praeclarum theorema, or splendid theorem, is a theorem of propositional calculus noted and named by G.W. Leibniz, who stated and proved it in the following manner.

If a is b and d is c, then ad will be bc.
This is a fine theorem, which is proved in this way:
a is b, therefore ad is bd (by what precedes),
d is c, therefore bd is bc (again by what precedes),
ad is bd, and bd is bc, therefore ad is bc.  Q.E.D.

— Leibniz • Logical Papers, p. 41.

Expressed in contemporary logical notation, the theorem may be written as follows.

((a \Rightarrow b) \land (d \Rightarrow c)) \Rightarrow ((a \land d) \Rightarrow (b \land c))

Expressed as a logical graph under the existential interpretation, the theorem takes the shape of the following formal equivalence or propositional equation.

Praeclarum Theorema (Leibniz)

And here’s a neat proof of that nice theorem —

Praeclarum Theorema • Proof

The steps of the proof are replayed in the following animation.

Praeclarum Theorema • Proof Animation

Reference

  • Leibniz, Gottfried W. (1679–1686?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in G.H.R. Parkinson (ed., trans., 1966), Leibniz : Logical Papers, Oxford University Press, London, UK.

Readings

Resources

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

Posted in Abstraction, Animata, C.S. Peirce, Cactus Graphs, Deduction, Equational Inference, Form, Graph Theory, Laws of Form, Leibniz, Logic, Logical Graphs, Mathematics, Model Theory, Painted Cacti, Peirce, Praeclarum Theorema, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Semiotics, Spencer Brown | Tagged , , , , , , , , , , , , , , , , , , , , , | 17 Comments

Logical Graphs • Formal Development

Posted on September 19, 2008 by Jon Awbrey

Logical graphs are next presented as a formal system by going back to the initial elements and developing their consequences in a systematic manner.

Formal Development

Logical Graphs • First Impressions gives an informal introduction to the initial elements of logical graphs and hopefully supplies the reader with an intuitive sense of their motivation and rationale.

The next order of business is to give the precise axioms used to develop the formal system of logical graphs.  The axioms derive from C.S. Peirce’s various systems of graphical syntax via the calculus of indications described in Spencer Brown’s Laws of Form.  The formal proofs to follow will use a variation of Spencer Brown’s annotation scheme to mark each step of the proof according to which axiom is called to license the corresponding step of syntactic transformation, whether it applies to graphs or to strings.

Axioms

The formal system of logical graphs is defined by a foursome of formal equations, called initials when regarded purely formally, in abstraction from potential interpretations, and called axioms when interpreted as logical equivalences.  There are two arithmetic initials and two algebraic initials, as shown below.

Arithmetic Initials

Axiom I₁

Axiom I₂

Algebraic Initials

Axiom J₁

Axiom J₂

Logical Interpretation

One way of assigning logical meaning to the initial equations is known as the entitative interpretation (En).  Under En, the axioms read as follows.

\begin{array}{ccccc} \mathrm{I_1} & : & \text{true} ~ \text{or} ~ \text{true} & = & \text{true} \\[4pt] \mathrm{I_2} & : & \text{not} ~ \text{true} & = & \text{false} \\[4pt] \mathrm{J_1} & : & a ~ \text{or} ~ \text{not} ~ a & = & \text{true} \\[4pt] \mathrm{J_2} & : & (a ~ \text{or} ~ b) ~ \text{and} ~ (a ~ \text{or} ~ c) & = & a ~ \text{or} ~ (b ~ \text{and} ~ c) \end{array}

Another way of assigning logical meaning to the initial equations is known as the existential interpretation (Ex).  Under Ex, the axioms read as follows.

\begin{array}{ccccc} \mathrm{I_1} & : & \text{false} ~ \text{and} ~ \text{false} & = & \text{false} \\[4pt] \mathrm{I_2} & : & \text{not} ~ \text{false} & = & \text{true} \\[4pt] \mathrm{J_1} & : & a ~ \text{and} ~ \text{not} ~ a & = & \text{false} \\[4pt] \mathrm{J_2} & : & (a ~ \text{and} ~ b) ~ \text{or} ~ (a\ \text{and}\ c) & = & a ~ \text{and} ~ (b ~ \text{or} ~ c) \end{array}

Equational Inference

All the initials I_1, I_2, J_1, J_2 have the form of equations.  This means the inference steps they license are reversible.  The proof annotation scheme employed below makes use of double bars =\!=\!=\!=\!=\!= to mark this fact, though it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom.

The actual business of proof is a far more strategic affair than the routine cranking of inference rules might suggest.  Part of the reason for this lies in the circumstance that the customary types of inference rules combine the moving forward of a state of inquiry with the losing of information along the way that doesn’t appear immediately relevant, at least, not as viewed in the local focus and short run of the proof in question.  Over the long haul, this has the pernicious side‑effect that one is forever strategically required to reconstruct much of the information one had strategically thought to forget at earlier stages of proof, where “before the proof started” can be counted as an earlier stage of the proof in view.

This is just one of the reasons it can be very instructive to study equational inference rules of the sort our axioms have just provided.  Although equational forms of reasoning are paramount in mathematics, they are less familiar to the student of the usual logic textbooks, who may find a few surprises here.

Frequently Used Theorems

To familiarize ourselves with equational proofs in logical graphs let’s run though the proofs of a few basic theorems in the primary algebra.

C1.  Double Negation Theorem

The first theorem goes under the names of Consequence 1 (C1), the double negation theorem (DNT), or Reflection.

Double Negation Theorem

The proof that follows is adapted from the one George Spencer Brown gave in his book Laws of Form and credited to two of his students, John Dawes and D.A. Utting.

Double Negation Theorem • Proof

C2.  Generation Theorem

One theorem of frequent use goes under the nickname of the weed and seed theorem (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF it goes by the names of Consequence 2 (C2) or Generation.

Logical Graph Figure 27 (27)

Here is a proof of the Generation Theorem.

Logical Graph Figure 28 (28)

C3.  Dominant Form Theorem

The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as Consequence 3 (C3) or Integration. A better mnemonic might be dominance and recession theorem (DART), but perhaps the brevity of dominant form theorem (DFT) is sufficient reminder of its double-edged role in proofs.

Logical Graph Figure 29 (29)

Here is a proof of the Dominant Form Theorem.

Logical Graph Figure 30 (30)

Exemplary Proofs

Using no more than the axioms and theorems recorded so far, it is possible to prove a multitude of much more complex theorems. A couple of all-time favorites are listed below.

Posted in Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Deduction, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Propositional Calculus, Propositional Equation Reasoning Systems, Semiotics, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , , , , | 42 Comments

Hypostatic Abstraction

Posted on August 8, 2008 by Jon Awbrey

The Care and Breeding of Abstract Objects

Hypostatic Abstraction is a formal operation on a subject–predicate form which preserves its information while introducing a new subject and upping the “arity” of its predicate.  To cite a notorious example, hypostatic abstraction turns “Opium is drowsifying” into “Opium has dormitive virtue”.

Introduction

Hypostatic abstraction is a formal operation which takes an element of information, as expressed in a proposition X ~\text{is}~ Y, and conceives its information to consist in the relation between that subject and another subject, as expressed in the proposition X ~\text{has}~ Y\!\text{-ness}.  The existence of the abstract subject Y\!\text{-ness} consists solely in the truth of those propositions containing the concrete predicate Y.  Hypostatic abstraction is known under many names, for example, hypostasis, objectification, reification, and subjectal abstraction.  The object of discussion or thought thus introduced is termed a hypostatic object.

The above definition is adapted from the one given by Charles Sanders Peirce (CP 4.235, “The Simplest Mathematics” (1902), in Collected Papers, CP 4.227–323).

The way that Peirce describes it, the main thing about the formal operation of hypostatic abstraction, insofar as it can be observed to operate on formal linguistic expressions, is that it converts some part of a predicate into a number of additional subjects, at the same time creating a new predicate that tells how all of the subjects are related, at least, according to the information in the original proposition.

For example, a typical case of hypostatic abstraction occurs in the grammatical transformation which turns “honey is sweet” into “honey possesses sweetness”.  This transformation may be visualized in the following variety of ways.

Hypostatic Abstraction Figure 1

Hypostatic Abstraction Figure 2

Hypostatic Abstraction Figure 3

Hypostatic Abstraction Figure 4

The grammatical trace of the hypostatic transformation occurring in this case articulates a process that abstracts the adjective “sweet” from the main predicate “is sweet”, thus arriving at a new, increased-arity predicate “possesses”, and as a by-product of the reaction, as it were, precipitating out the substantive “sweetness” as a second subject of the new 2-place predicate, “possesses”.

References

  • 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.

Resources

Posted in Abstraction, Article, C.S. Peirce, Hypostatic Abstraction, Logic, Logic of Relatives, Logical Graphs, Mathematics, Molière, Peirce, Reification, Relation Theory | Tagged , , , , , , , , , , , | 5 Comments

Pragmatic Maxim

Posted on August 7, 2008 by Jon Awbrey

The pragmatic maxim is a guideline for the practice of inquiry formulated by Charles Sanders Peirce.  Serving as a practical recommendation or regulative principle in the normative science of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising the addressee on an optimal way of “attaining clearness of apprehension”.

Introduction

The “pragmatic maxim”, also known as the “maxim of pragmatism” or the “maxim of pragmaticism”, is a maxim of logic formulated by Charles Sanders Peirce.  Serving as a practical recommendation or regulative principle in the normative science of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising the addressee on an optimal way of “attaining clearness of apprehension”.

Seven Ways of Looking at a Pragmatic Maxim

Peirce stated the pragmatic maxim in many different ways over the years, each of which adds its own bit of clarity or correction to their collective corpus.

  • The first excerpt appears in the form of a dictionary entry, intended as a definition of pragmatism.

    Pragmatism.  The opinion that metaphysics is to be largely cleared up by the application of the following maxim for attaining clearness of apprehension:

    “Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have.  Then, our conception of these effects is the whole of our conception of the object.”  (Peirce, CP 5.2, 1878/1902).

  • The second excerpt gives another version of the pragmatic maxim, a recommendation about a way of clarifying meaning that can be taken to stake out the general philosophy of pragmatism.

    Pragmaticism was originally enounced in the form of a maxim, as follows:  Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have.  Then, your conception of those effects is the whole of your conception of the object.

    I will restate this in other words, since ofttimes one can thus eliminate some unsuspected source of perplexity to the reader. This time it shall be in the indicative mood, as follows: The entire intellectual purport of any symbol consists in the total of all general modes of rational conduct which, conditionally upon all the possible different circumstances and desires, would ensue upon the acceptance of the symbol.  (Peirce, CP 5.438, 1878/1905).

  • The third excerpt puts a gloss on the meaning of a practical bearing and provides an alternative statement of the maxim.

    Such reasonings and all reasonings turn upon the idea that if one exerts certain kinds of volition, one will undergo in return certain compulsory perceptions.  Now this sort of consideration, namely, that certain lines of conduct will entail certain kinds of inevitable experiences is what is called a “practical consideration”.  Hence is justified the maxim, belief in which constitutes pragmatism;  namely:

    In order to ascertain the meaning of an intellectual conception one should consider what practical consequences might conceivably result by necessity from the truth of that conception;  and the sum of these consequences will constitute the entire meaning of the conception.  (Peirce, CP 5.9, 1905).

  • The fourth excerpt illustrates one of Peirce’s many attempts to get the sense of the pragmatic philosophy across by rephrasing the pragmatic maxim in an alternative way.  In introducing this version, he addresses an order of prospective critics who do not deem a simple heuristic maxim, much less one that concerns itself with a routine matter of logical procedure, as forming a sufficient basis for a full-grown philosophy.

    On their side, one of the faults that I think they might find with me is that I make pragmatism to be a mere maxim of logic instead of a sublime principle of speculative philosophy.  In order to be admitted to better philosophical standing I have endeavored to put pragmatism as I understand it into the same form of a philosophical theorem.  I have not succeeded any better than this:

    Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood.  (Peirce, CP 5.18, 1903).

  • The fifth excerpt is useful by way of additional clarification, and was aimed to correct a variety of historical misunderstandings that arose over time with regard to the intended meaning of the pragmatic maxim.

    The doctrine appears to assume that the end of man is action — a stoical axiom which, to the present writer at the age of sixty, does not recommend itself so forcibly as it did at thirty.  If it be admitted, on the contrary, that action wants an end, and that that end must be something of a general description, then the spirit of the maxim itself, which is that we must look to the upshot of our concepts in order rightly to apprehend them, would direct us towards something different from practical facts, namely, to general ideas, as the true interpreters of our thought.  (Peirce, CP 5.3, 1902).

  • A sixth excerpt is useful in stating the bearing of the pragmatic maxim on the topic of reflection, namely, that it makes all of pragmatism boil down to nothing more or less than a method of reflection.

    The study of philosophy consists, therefore, in reflexion, and pragmatism is that method of reflexion which is guided by constantly holding in view its purpose and the purpose of the ideas it analyzes, whether these ends be of the nature and uses of action or of thought. … It will be seen that pragmatism is not a Weltanschauung but is a method of reflexion having for its purpose to render ideas clear.  (Peirce, CP 5.13 note 1, 1902).

  • The seventh excerpt is a late reflection on the reception of pragmatism.  With a sense of exasperation that is almost palpable, Peirce tries to justify the maxim of pragmatism and to correct its misreadings by pinpointing a number of false impressions that the intervening years have piled on it, and he attempts once more to prescribe against the deleterious effects of these mistakes.  Recalling the very conception and birth of pragmatism, he reviews its initial promise and its intended lot in the light of its subsequent vicissitudes and its apparent fate.  Adopting the style of a post mortem analysis, he presents a veritable autopsy of the ways that the main idea of pragmatism, for all its practicality, can be murdered by a host of misdissecting disciplinarians, by what are ostensibly its most devoted followers.

    This employment five times over of derivates of concipere must then have had a purpose.  In point of fact it had two.  One was to show that I was speaking of meaning in no other sense than that of intellectual purport.  The other was to avoid all danger of being understood as attempting to explain a concept by percepts, images, schemata, or by anything but concepts.  I did not, therefore, mean to say that acts, which are more strictly singular than anything, could constitute the purport, or adequate proper interpretation, of any symbol.  I compared action to the finale of the symphony of thought, belief being a demicadence.  Nobody conceives that the few bars at the end of a musical movement are the purpose of the movement.  They may be called its upshot.  (Peirce, CP 5.402 note 3, 1906).

References

  • 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.  Cited as CP n.m for volume n, paragraph m.

Readings

  • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.  Archive.  Journal.  Online (doc) (pdf).

Resources

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

Posted in C.S. Peirce, Logic, Method, Peirce, Philosophy, Pragmatic Maxim, Pragmatism, References, Sources | Tagged , , , , , , , , | 29 Comments

Logic of Relatives

Posted on July 31, 2008 by Jon Awbrey

Relations Via Relative Terms

The logic of relatives is the study of relations as represented in symbolic forms known as rhemes, rhemata, or relative terms.

Introduction

The logic of relatives, more precisely, the logic of relative terms, is the study of relations as represented in symbolic forms called rhemes, rhemata, or relative terms.  The treatment of relations by way of their corresponding relative terms affords a distinctive perspective on the subject, even though all angles of approach must ultimately converge on the same formal subject matter.

The consideration of relative terms has its roots in antiquity but it entered a radically new phase of development with the work of Charles Sanders Peirce, beginning with his paper “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic” (1870).

References

  • Peirce, C.S., “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 1870. Reprinted, Collected Papers CP 3.45–149. Reprinted, Chronological Edition CE 2, 359–429.
    Online (1) (2) (3).

Readings

  • Aristotle, “The Categories”, Harold P. Cooke  (trans.), pp. 1–109 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
  • Aristotle, “On Interpretation”, Harold P. Cooke (trans.), pp. 111–179 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
  • Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
  • Boole, George, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Macmillan, 1854. Reprinted with corrections, Dover Publications, New York, NY, 1958.
  • 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. Cited as CP volume.paragraph.
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Volume 2, 1867–1871, Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1984. Cited as CE 2.

Resources

Posted in C.S. Peirce, Logic, Logic of Relatives, Mathematical Logic, Mathematics, Relation Theory, Semiotics | Tagged , , , , , , | 5 Comments

Semeiotic

Posted on July 30, 2008 by Jon Awbrey

Theory of Signs

Semeiotic is one of the terms C.S. Peirce used for his theory of triadic sign relations and it serves to distinguish his theory of signs from other approaches to the same subject matter, more generally referred to as semiotics.

Types of Signs

There are three principal ways a sign may denote its objects.  The modes of representation are often referred to as kinds, species, or types of signs, but it is important to recognize they are not ontological species, that is, they are not mutually exclusive features of description, since the same thing can be a sign in several different ways.

Beginning very roughly, the three main ways of being a sign can be described as follows.

  • An icon denotes its objects by virtue of a quality it shares with its objects.
  • An index denotes its objects by virtue of an existential connection it has to its objects.
  • A symbol denotes its objects solely by virtue of being interpreted to do so.

One of Peirce’s early delineations of the three types of signs is still quite useful as a first approach to understanding their differences and their relationships to each other.

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, Lowell Lecture № 7, Writings 1, 467–468.

References

  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1982.
  • Peirce, C.S. (1865), “On the Logic of Science”, Harvard University Lectures, Writings 1, 161–302.
  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Institute Lectures, Writings 1, 357–504.

Readings

  • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52. Archive. Journal.
    Online (doc) (pdf).

Resources

Posted in C.S. Peirce, Icon Index Symbol, Logic, Logic of Relatives, Logic of Science, Mathematics, Peirce, Pragmatics, Relation Theory, Semantics, Semeiosis, Semeiotic, Semiosis, Semiotics, Sign Relations, Syntax, Triadic Relations | Tagged , , , , , , , , , , , , , , , , | 9 Comments