C.S. Peirce • Syllabus • Selection 2

But round about the castle there began to grow a hedge of thorns, which every year became higher, and at last grew close up round the castle and all over it, so that there was nothing of it to be seen, not even the flag upon the roof.

Grimm’s Fairy Tales • Little Briar-Rose

Selection from C.S. Peirce, “A Syllabus of Certain Topics of Logic” (1903)

Section “Sundry Logical Conceptions”  •  Subsection “Speculative Grammar”

The second trichotomy of representamens is [divided] into:

  • first, simple signs, substitutive signs, or Sumisigns;
  • second, double signs, informational signs, quasi-propositions, or Dicisigns;
  • third, triple signs, rationally persuasive signs, arguments, or Suadisigns.

Of these three classes, the one whose nature is, by all odds, the easiest to comprehend, is the second, that of quasi-propositions, despite the fact that the question of the essential nature of the “judgment” is today quite the most vexed of all questions of logic.

The truth is that all these classes are of very intricate natures;  but the problem of the day is needlessly complicated by the attention of most logicians, instead of extending to propositions in general, being confined to “judgments”, or acts of mental acceptance of propositions, which not only involve characters additional to those of propositions in general, — characters required to differentiate them as propositions of a particular kind, — but which further involve, beside the mental proposition itself, the peculiar act of assent.

The problem is difficult enough, when we merely seek to analyze the essential nature of the Dicisign, in general, that is, the kind of sign that conveys information, in contradistinction to a sign from which information may be derived.

The readiest characteristic test showing whether a sign is a Dicisign or not, is that a Dicisign is either true or false, but does not directly furnish reasons for its being so.

This shows that a Dicisign must profess to refer or relate to something as having a real being independently of the representation of it as such, and further that this reference or relation must not be shown as rational, but must appear as a blind Secondness.  But the only kind of sign whose Object is necessarily existent is the genuine Index.  This Index might, indeed, be part of a Symbol;  but in that case the relation would appear as rational.  Consequently a Dicisign necessarily represents itself to be a genuine Index, and to be nothing more.

At this point let us discard all other considerations, and see what sort of a sign a sign must be that in any way represents itself to be a genuine Index of its Object, and nothing more.

Substituting for “represents_____to be” a clearer interpretation, the statement is that the Dicisign’s Interpretant represents an identity of the Dicisign with a genuine Index of the Dicisign’s real Object.  That is, the Interpretant represents a real existential relation, or genuine Secondness, as subsisting between the Dicisign and its real Object.

But the Interpretant of a Sign can represent no other Object than that of the Sign itself.

Hence, this same existential relation must be an Object of the Dicisign, if the latter have any real Object.

This represented existential relation, in being an Object of the Dicisign, makes that real Object which is the correlate of this relation also an Object of the Dicisign.  This latter Object may be distinguished as the Primary Object, the other being termed the Secondary Object.

The Dicisign, in so far as it is the relate of the existential relation which is the Secondary Object of the Dicisign, can evidently not be the entire Dicisign.  It is at once a part of the Object and a part of the Interpretant of the Dicisign.

Since the Dicisign is represented in its Interpretant to be an Index of a complexus as such, it must be represented in that same Interpretant to be composed of two parts, corresponding respectively to its Object and to itself.

That is to say, in order to understand the Dicisign, it must be regarded as composed of two such parts whether it be in itself so composed or not.  It is difficult to see how this can be, unless it really have two such parts;  but perhaps this may be possible.

Let us consider these two represented parts separately.

The part which is represented to represent the Primary Object, since the Dicisign is represented to be an Index of its Object, must be represented as an Index, or some Representamen of an Index, of the Primary Object.

The part which is represented to represent a part of the Dicisign, is represented as at once part of the Interpretant and part of the Object.  It must, therefore, be represented as such a sort of Representamen (or to represent such a sort) as can have its Object and its Interpretant the same.

Now, a Symbol cannot even have itself as its Object;  for it is a law governing its Object.

For example, if I say “This proposition conveys information about itself”, or “Let the term ‘sphinx’ be a general term to denote any thing of the nature of a symbol that is applicable to every ‘sphinx’ and to nothing else”, I shall talk unadulterated nonsense.

But a Representamen mediates between its Interpretant and its Object, and that which cannot be the Object of the Representamen cannot be the Object of the Interpretant.

Hence, a fortiori, it is impossible that a Symbol should have its Object as its Interpretant.

An Index can very well represent itself.

Thus, every number has a double;  and thus the entire collection of even numbers is an Index of the entire collection of numbers, and so this collection of even numbers contains an Index of itself.

But it is impossible for an Index to be its own Interpretant, since an Index is nothing but an individual existence in a Secondness with something;  and it only becomes an Index by being capable of being represented by some Representamen as being in that relation.  Could this Interpretant be itself, there would be no difference between an Index and a Second.

An Icon, however, is strictly a possibility, involving a possibility, and thus the possibility of its being represented as a possibility is the possibility of the involved possibility.  In this kind of Representamen alone, then, the Interpretant may be the Object.  Consequently, that constituent of the Dicisign which is represented in the Interpretant as being a part of the Object, must be represented by an Icon or by a Representamen of an Icon.

The Dicisign, as it must be understood in order to be understood at all, must contain those two parts.  But the Dicisign is represented to be an Index of the Object, in that the latter involves something corresponding to these parts;  and it is this Secondness that the Dicisign is represented to be the Index of.

Hence the Dicisign must exhibit a connection between these parts of itself, and must represent this connection to correspond to a connection in the Object between the Secundal Primary Object and Firstness indicated by the part corresponding to the Dicisign.

We conclude, then, that, if we have succeeded in threading our way through the maze of these abstractions, a Dicisign, defined as a Representamen whose Interpretant represents it as an Index of its Object, must have the following characters.

First, it must, in order to be understood, be considered as containing two parts.  Of these, the one, which may be called the Subject, is or represents an Index of a Second existing independently of its being represented, while the other, which may be called the Predicate, is or represents an Icon of a Firstness.

Second, these two parts must be represented as connected;  and that in such a way that if the Dicisign has any Object, it must be an Index of a Secondness subsisting between the real Object represented in one represented part of the Dicisign to be indicated, and a Firstness represented in the other represented part of the Dicisign to be iconized.

Let us now examine whether these conclusions, together with the assumption from which they proceed, hold good of all signs which profess to convey information without furnishing any rational persuasion of it;  and whether they fail alike for all signs which do not convey information as well as for all those which furnish evidence of the truth of their information, or reasons for believing it.  If our analysis sustains these tests, we may infer that the definition of the Dicisign on which they are founded, holding, at least, within the sphere of signs, is presumably sound beyond that sphere.

(Peirce, EP 2.275–277, CP 2.309–313)

Notes

Collected Papers 1

  • A Syllabus of Certain Topics of Logic, 1903, Alfred Mudge & Son, Boston, bearing the following preface:  “This syllabus has for its object to supplement a course of eight lectures to be delivered at the Lowell Institute, by some statements for which there will not be time in the lectures, and by some others not easily carried away from one hearing.  It is to be a help to those who wish seriously to study the subject, and to show others what the style of thought is that is required in such study.  Like the lectures themselves, this syllabus is intended chiefly to convey results that have never appeared in print;  and much is omitted because it can be found elsewhere.”

Essential Peirce 2(a)(b)(c)

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.  Volume 2 : Elements of Logic, 1932.
  • Peirce Edition Project (eds., 1998), The Essential Peirce, Selected Philosophical Writings, Volume 2 (1893–1913), Indiana University Press, Bloomington and Indianapolis, IN.
Posted in Assertion, Foundations of Mathematics, Icon Index Symbol, Logic, Mathematics, Metaphysics, Normative Science, Peirce, Phenomenology, Philosophy, Pragmatism, Propositions, References, Relation Theory, Semiosis, Semiotics, Sign Relations, Sources, Triadic Relations, Triadicity, Truth | Tagged , , , , , , , , , , , , , , , , , , , , | 1 Comment

What’s The Use?

What’s the use of getting up in the morning?

Never mind that now, I’m already up.

Be constructive.  Try to focus on something positive.

Okay, then, what’s the use of logic?

You call that focused?  Be more specific!

So what’s the use of a logical system, if you think of logic as embodied in systems?

Or what’s the use of a logical organ, if you think of logic as embodied in bodies?

To be continued …

Posted in Anthem, Inquiry, Inquiry Into Inquiry, Knowledge, Logic, Meditation, Science, Truth, Use, Utility | Tagged , , , , , , , , , | Leave a comment

Frankl, My Dear : 4

Re: Dick Lipton & Ken Regan(1)(2)

Let’s go back to the “key lemma” from (2) and try it out on a simple example, just to get a sense of what the terms mean.

Lemma.  Let {f(x_{1}, \dots, x_{n})} be a Boolean function and let {x_{i}} be fixed.  Then the Boolean inputs can be partitioned into six sets:

\displaystyle A_{0}, A_{1}, B_{0}, B_{1}, C_{0}, C_{1}.

These sets have the following properties:

  1. The variable {x_{i}} is equal to {0} on {A_{0} \cup B_{0} \cup C_{0}}.
  2. The variable {x_{i}} is equal to {1} on {A_{1} \cup B_{1} \cup C_{1}}.
  3. The union {A_{0} \cup A_{1}} is equal to {J_{i}}.
  4. The function is always {0} on {B_{0} \cup B_{1}}.
  5. The function is always {1} on {C_{0} \cup C_{1}}.
  6. Finally {|A_{0}| = |A_{1}|}  and  {|B_{0}| = |B_{1}|}  and  {|C_{0}| = |C_{1}|}.

Example 1


Venn Diagram PQR
(1)

Consider the boolean function {f(x_1, x_2, x_3) = f(p, q, r) = pqr} pictured in Figure 1 and fix on the variable {x_1 = p}.

The lemma says that the boolean inputs can be partitioned into six sets:

\displaystyle A_{0}, A_{1}, B_{0}, B_{1}, C_{0}, C_{1}.

Let’s identify those six sets in the present example.

Back in a flash … in the meantime, exercise for the reader …

Later that day …

I had trouble with the term “Boolean input” in (2).  Sometimes people use it to mean one of the input wires to a logic gate, that is, one of the variables x_i.  Other times people use it to mean one of the coordinate elements x \in \mathbb{B}^n.  It’s always possible that I’m reading things wrong but it looks like the first sense is used to define the “influence” I_{i}(f) and the related set J_{i}(f) while the second sense is used to define the six sets of the Lemma.  At any rate, I will go with those two senses for now.

On that reading, the six sets named in the Lemma are shown in Figure 2.


Venn Diagram Frankl Figure 2
(2)

In other words:

\begin{matrix}  A_0 & = & \{ 011 \} & = & \tilde{p} q r  \\  A_1 & = & \{ 111 \} & = & p q r  \\  B_0 & = & \{ 000, 001, 010 \}  & = & \tilde{p} \tilde{q} \tilde{r} \lor \tilde{p} \tilde{q} r \lor \tilde{p} q \tilde{r}   \\  B_1 & = & \{ 100, 101, 110 \}  & = & p \tilde{q} \tilde{r} \lor p \tilde{q} r \lor p q \tilde{r}   \\  C_0 & = & \varnothing & = & 0  \\  C_1 & = & \varnothing & = & 0  \end{matrix}

Resources.  A few pages on differential logic, which may or may not be useful here.

Posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl | Tagged , , , , , , , , | Leave a comment

Character, Action, Discretion

 Character is revealed by action.          ~~ Aristotle
 Action is discrete.                       ~~ Planck
----------------------------------------------------------
 The better part of valour is discretion.  ~~ Shakespeare
Posted in Action, Anthem, Arete, Aristotle, Character, Discretion, Planck, Shakespeare, Syllogism, Valor, Valour, Virtue | Tagged , , , , , , , , , , , | Leave a comment

Frankl, My Dear : 3

Re: Dick Lipton & Ken Regan(1)(2)

Here’s a few pages on differential logic, whose ideas I’ll be trying out in the present setting:

I next need to look at the following “key lemma” from (2) and see if I can wrap my head, or at least my own formalism, around what it says.

Lemma.  Let {f(x_{1}, \dots, x_{n})} be a Boolean function and let {x_{i}} be fixed.  Then the Boolean inputs can be partitioned into six sets:

\displaystyle A_{0}, A_{1}, B_{0}, B_{1}, C_{0}, C_{1}.

These sets have the following properties:

  1. The variable {x_{i}} is equal to {0} on {A_{0} \cup B_{0} \cup C_{0}}.
  2. The variable {x_{i}} is equal to {1} on {A_{1} \cup B_{1} \cup C_{1}}.
  3. The union {A_{0} \cup A_{1}} is equal to {J_{i}}.
  4. The function is always {0} on {B_{0} \cup B_{1}}.
  5. The function is always {1} on {C_{0} \cup C_{1}}.
  6. Finally {|A_{0}| = |A_{1}|}  and  {|B_{0}| = |B_{1}|}  and  {|C_{0}| = |C_{1}|}.

This may take a while …

Posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl | Tagged , , , , , , , , | Leave a comment

Consequences of Triadic Relation Irreducibility : 2

From time to time I come to the realization that there are ways of reading Peirce that make no sense to me.  When I stop to think about the potential sources of that evident divergence from common sense, the first thing that comes to mind is the fact that people come to reading Peirce with so many different aims, backgrounds, and collateral experiences with the objects that he wraps his signs and ideas around.

Thinking about that leads to all sorts of questions that I see no way of beginning to address with any sense of coherence.  All I can do is try to give a good account of what makes sense to me and why.  My experience with failures to communicate over many trials tells me that the biggest and most numerous rifts in our several understandings of Peirce all point back to the visions of relations, triadic relations, and triadic sign relations that dance in our various and sundry heads.

So that is what I’ll take up first …

Posted in Logic, Logic of Relatives, Mathematics, Peirce, Peirce List, Philosophy, Pragmatism, Relation Theory, Semiotics, Sign Relations, Thirdness, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , | Leave a comment

Consequences of Triadic Relation Irreducibility : 1

2014 Sep 10

I will have to be out of the loop for some days, but this post will give me a peg on which I can hang a few thoughts via mobile device that have been tugging at the edge of my mind for a while.

2014 Sep 13

I am still a bit loopy from jumping through a triple of orthogonal loops and it’s taking me longer to get back in the saddle than I thought it might, so let me just paste in a passel of links to a collection of background materials I’ve referenced before, a lot of it work-in-hopeful-progress, the rest of it more polished, published, and with luck less perishable.

Basic Concepts

Related Readings

Posted in Logic, Logic of Relatives, Mathematics, Peirce, Peirce List, Philosophy, Pragmatism, Relation Theory, Semiotics, Sign Relations, Thirdness, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , | Leave a comment