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Stream: deprecated: logic

Topic: sign relations

Jon Awbrey (Aug 09 2021 at 14:05):

Cf: Semiotics, Semiosis, Sign Relations • Comment 1

Opening a topic to work on Peirce's theory of triadic sign relations in a category-theoretic framework.

I had been reading Peirce for a decade or more before I found a math-strength definition of signs and sign relations.  A lot of the literature on semiotics takes almost any aperçu Peirce penned about signs as a “definition” but barely a handful of those descriptions are consequential enough to support significant theory.  For my part, the definition of a sign relation coming closest to the mark is one Peirce gave in the process of defining logic itself.   Two variants of that definition are linked and transcribed below.

Cf: C.S. Peirce • On the Definition of Logic

Selections from C.S. Peirce, “Carnegie Application” (1902)

No. 12.  On the Definition of Logic 

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, $A,$ which brings something, $B,$ its interpretant sign determined or created by it, into the same sort of correspondence with something, $C,$ its object, as that in which itself stands to $C.$  It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has virtually been quite generally held, though not generally recognized.  (NEM 4, 20–21).

No. 12.  On the Definition of Logic [Earlier Draft]

Logic is formal semiotic.  A sign is something, $A,$ which brings something, $B,$ its interpretant sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, $C,$ its object, as that in which itself stands to $C.$  This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time.  It is from this definition that I deduce the principles of logic by mathematical reasoning, and by mathematical reasoning that, I aver, will support criticism of Weierstrassian severity, and that is perfectly evident.  The word “formal” in the definition is also defined.  (NEM 4, 54).

Reference

• Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75), published in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.

Morgan Rogers (he/him) (Aug 09 2021 at 15:11):

Okay, this is hard to parse, but I've looked at it a few times now framed with discussion from a few different sources, and it seems that if we fix some sets $A$ of signs, $B$ of interpretants and $C$ of objects, and treating the sign relation as $R \subseteq A \times B \times C$, there are some reasonable restrictions/assumptions we could place on $R$. For example:
1a. $\forall a \in A, \, \forall b \in B, \, \exists c \in C, \, (a,b,c) \in R$, or "every sign means something to every interpretant",
1b. $\forall a \in A, \, \exists b \in B, \, \exists c \in C, \, (a,b,c) \in R$, a weaker alternative, "every sign means something to some interpretant",
2a. $\forall c \in C, \, \forall b \in B, \, \exists a \in A, \, (a,b,c) \in R$, "every interpretant has a name for every object",
2b. $\forall c \in C, \, \exists b \in B, \, \exists a \in A, \, (a,b,c) \in R$, a weaker alternative, "every object has at least one name assigned to it by each interpretant"

... and so on. However, none of these seem strictly necessary to me; there could be meaningless symbols or nameless objects. Does Peirce assume any of these things or similar? If not, I suspect the answer to my question regarding mathematical distinguishing features of sign relations is that there aren't any: that any ternary relation can be understood as a sign relation if one squints hard enough.

Jon Awbrey (Aug 09 2021 at 20:04):

Cf: Semiotics, Semiosis, Sign Relations • Comment 2

Definitions tend to call on other terms in need of their own definitions, and so on till the process terminates at the level of primitive terms.  The main two concepts requiring supplementation in Peirce’s definition of a sign relation are the ideas of correspondence and determination.  We can figure out fairly well what Peirce had in mind from things he wrote elsewhere, as I explained in the Sign Relation article I added to Wikipedia 15 years ago.  Not daring to look at what's left of that, here's the relevant section from the OEIS Wiki fork.

• Sign Relation • Definition

To be continued …

Morgan Rogers (he/him) (Aug 10 2021 at 08:29):

Okay, I may have mixed up the meanings of "object" and "interpretant" in my plain language translations above? Re determination, I read "B is determined by A" as meaning the conjunction of
$\forall a \in A, \, \exists b \in B, \, \exists c \in C, \, R(a,b,c)$ and
$\forall a \in A, \, \forall c \in C, \, R(a,b,c) \wedge R(a,b',c) \Rightarrow b = b'$?
Whether this is right depends on the answers to my previous questions.

Jon Awbrey (Aug 10 2021 at 21:45):

Cf: Semiotics, Semiosis, Sign Relations • Comment 3

It helps me to compare sign relations with my other favorite class of triadic relations, namely, groups.  Applications of mathematical groups came up just recently in the Laws of Form discussion group, so it will save a little formatting time to adapt the definition used there.

• Animated Logical Graphs • 60

Definition.  A group $(G, *)$ is a set $G$ together with a binary operation $* : G \times G \to G$ satisfying the following three conditions.

1. Associativity.  For any $x, y, z \in G,$
   we have $(x * y) * z = x * (y * z).$

2. Identity.  There is an identity element $1 \in G$ such that $\forall g \in G,$
   we have $1 * g = g * 1 = g.$

3. Inverses.  Each element has an inverse, that is, for each $g \in G,$
   there is some $h \in G$ such that $g * h = h * g = 1.$

Morgan Rogers (he/him) (Aug 11 2021 at 09:00):

You say it's your other favourite class of triadic relations, but you give the algebraic binary operation presentation of the theory! As a relational theory, it looks a bit different:

A group can be equivalently expressed as a set $G$ equipped with a ternary relation $R \subseteq G \times G \times G$ satisfying the properties required for this to be the graph of a binary function as described above, namely:
-1. Totality. For any $x,y \in G$ there exists $z \in G$ such that $R(x,y,z)$.

0. Well-defined. If $R(x,y,z)$ and $R(x,y,z')$, then $z=z'$.
1. Associativity. For any $u,v,w,x,y,z \in G$, with $R(u,v,w)$ and $R(v,x,y)$, we have $R(w,x,z)$ if and only if $R(u,y,z)$.
2. Identity. There exists $1 \in G$ such that for every $x \in G$, $R(1,x,x)$ and $R(x,1,x)$.
3. Inverses. For any $x \in G$ there exists $y \in G$ with $R(x,y,1)$ and $R(y,x,1)$.

The nice thing about this presentation is that you can get relational presentations of several related concepts, including groupoids, monoids and categories by dropping some of the axioms.

This brings me back to my question (again...) what are the axioms associated to sign operations?

Jon Awbrey (Aug 12 2021 at 18:06):

Cf: Semiotics, Semiosis, Sign Relations • Discussion 7

Morgan Rogers (he/him) said:

Okay, this is hard to parse, but I've looked at it a few times now framed with discussion from a few different sources, and it seems that if we fix some sets $A$ of signs, $B$ of interpretants and $C$ of objects, and treating the sign relation as $R \subseteq A \times B \times C$, there are some reasonable restrictions/assumptions we could place on $R$. For example:
1a. $\forall a \in A, \, \forall b \in B, \, \exists c \in C, \, (a,b,c) \in R$, or "every sign means something to every interpretant",
1b. $\forall a \in A, \, \exists b \in B, \, \exists c \in C, \, (a,b,c) \in R$, a weaker alternative, "every sign means something to some interpretant",
2a. $\forall c \in C, \, \forall b \in B, \, \exists a \in A, \, (a,b,c) \in R$, "every interpretant has a name for every object",
2b. $\forall c \in C, \, \exists b \in B, \, \exists a \in A, \, (a,b,c) \in R$, a weaker alternative, "every object has at least one name assigned to it by each interpretant"

... and so on. However, none of these seem strictly necessary to me; there could be meaningless symbols or nameless objects. Does Peirce assume any of these things or similar? If not, I suspect the answer to my question regarding mathematical distinguishing features of sign relations is that there aren't any: that any ternary relation can be understood as a sign relation if one squints hard enough.

As far as meaningless signs go, we do encounter them in theoretical analysis (“resolving conundra” and “steering around nonsense”) and empirical or computational applications (“missing data” and “error types”).  The defect of meaning can affect either denotative objects or connotative interpretants or both.  In those events we have to generalize sign relations to what are called sign relational complexes.

Signless objects are a different matter since cognitions and concepts count as signs in pragmatic semiotics and Peirce maintains we have no concept of inconceivable objects.

If you fancy indulging in a bit of cosmological speculation you could imagine the whole physical universe to be a sign of itself to itself, making $O = S = I,$ but this far downstream from the Big Bang we mortals usually have more pressing business to worry about.

In short, what we need sign relations for is not for settling big questions about cosmology or metaphysics but for organizing our thinking about object domains and constructing models of what goes on and what might go better in practical affairs like communication, inquiry, learning, and reasoning.

Jon Awbrey (Aug 13 2021 at 02:36):

Sign relations are a class of mathematical structures useful in modeling a wide variety of phenomena and practices involved in communication, inquiry, learning, and reasoning.  They are distinguished by the extra degree of freedom triadic or ternary relations provide over dyadic or binary relations traditionally used in model theory and semantics of codes, languages, and programs.  We can use this extra bit of “elbow room” to deal with the pragmatic aspect of symbol systems and their use.

Jon Awbrey (Aug 14 2021 at 08:30):

Cf: Semiotics, Semiosis, Sign Relations • Discussion 8
Re: Peirce List • Robert Marty

RM:  Thank you for reminding me of the definition of a group that I have taught for 45 years …

Auld acquaintance is not forgot 🍻 I will convey your thanks to one who reminded me.

My reason for encoring mathematical groups as a class of triadic relations and elsewhere casting divisibility in the role of a dyadic relation was not so much for their own sakes as for the critical exercise my English Lit teachers used to style as “Compare and Contrast”.  For the sake of our immediate engagement, then, we tackle that exercise all the better to highlight the distinctive qualities of triadic relations and sign relations.

A critical point of the comparison is to grasp sign relations as collections of ordered triples — collections endowed with collective properties extending well beyond the properties of individual triples and their components.

Jon Awbrey (Aug 14 2021 at 09:08):

Cf: Semiotics, Semiosis, Sign Relations • Discussion 9

Morgan Rogers (he/him) said:

Okay, I may have mixed up the meanings of "object" and "interpretant" in my plain language translations above? Re determination, I read "B is determined by A" as meaning the conjunction of
$\forall a \in A, \, \exists b \in B, \, \exists c \in C, \, R(a,b,c)$ and
$\forall a \in A, \, \forall c \in C, \, R(a,b,c) \wedge R(a,b',c) \Rightarrow b = b'$?
Whether this is right depends on the answers to my previous questions.

Let's look at the gloss I gave for Determination.

Cf: Sign Relation • Definition

• Determination.  Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes.  First, and especially in this context, he is invoking a more general concept of determination, what is called a formal or informational determination, as in saying “two points determine a line”, rather than the more special cases of causal and temporal determinisms.  Second, he characteristically allows for what is called determination in measure, that is, an order of determinism that admits a full spectrum of more and less determined relationships.

Other words for this general order of determination are structure, pattern, law, form, and one coming up especially in cybernetics and systems theory, constraint.  It's what happens when not everything that might happen actually does.  (The stochastic mechanic or the quantum technician will probably quip at this point, At least, not with equal probability.)

Morgan Rogers (he/him) (Aug 14 2021 at 12:05):

Jon Awbrey said:

For the sake of our immediate engagement, then, we tackle that exercise all the better to highlight the distinctive qualities of triadic relations and sign relations.

I beg of you, please actually clearly state at least one "distinctive quality of sign relations". This discussion has been going on for most of a week and you still haven't done that, despite my best efforts in suggesting properties that these relations might be assumed to have.

Jon Awbrey (Aug 14 2021 at 14:45):

Cf: Semiotics, Semiosis, Sign Relations • Discussion 10

Sign relations are triadic relations.

Can any triadic relation be a sign relation?

I don't know.  I have pursued the question myself whether any triadic relation could be used somehow or other in a context of communication, information, inquiry, learning, reasoning, and so on where concepts of signs and their meanings are commonly invoked — there's the rub — it's not about what a relation is, “in itself”, intrinsically or ontologically, but a question of “suitability for a particular purpose”, as they say in all the standard disclaimers.

What Peirce has done is to propose a definition intended to capture an intuitive, pre-theoretical, traditional concept of signs and their uses.  To put it on familiar ground, it's like Turing's proposal of his namesake machine to capture the intuitive concept of computation.  That is not a matter to be resolved by à priori dictates but by trying out candidate models in the intended applications.

I gave you what I consider Peirce's best definition of a sign in relational terms and I pointed out where it needs filling out to qualify as a proper mathematical definition, most pointedly in the further definitions of correspondence and determination.

That is the current state of the inquiry as it stands on this site …

Jon Awbrey (Aug 22 2021 at 18:30):

Cf: Semiotics, Semiosis, Sign Relations • Discussion 11

Re: Peirce List • Robert Marty

Robert Marty writes:

You evoke many concepts with their relations, the explanation of which would take a considerable amount of time, to the point that you are reduced to answering yourself.  I want to question you on the point that interests me particularly, which concerns your entry into Peirce’s semiotics.  I found it among all your links here:

• Sign Relation

You will tell me if this is the right reference.  If it is so, then I think you have made a bad choice, and of course, I explain myself.  To be clear and precise, I must reproduce the entirety of your “Definition” paragraph:

• Definition

I’m just beginning to get out from under the deluge of tasks put off by the pandemic … I think I can finally return to your remarks of August 12 on my sketch of Peirce’s theory of signs for the general reader interested in semiotics.

Your message to the List had many detailed quotations, so I’m in the process of drafting an easier-on-the-eyes blog version.  When I get done with that — it may be a day — I’ll post my reply on the thread dealing with Semiotics, Semiosis, Sign Relations, so as to keep focused on signs.

Jon Awbrey (Aug 25 2021 at 20:44):

The following exchange serves to set up a number of issues in the theory of signs.

Cf: Semiotics, Semiosis, Sign Relations • Discussion 12

Robert Marty writes:

I persist in the idea that in your six combinations [O, S, I] only one is relevant for semiotics, the others being out of the field …

I take it you are referring to the section of the Sign Relation article titled “Six Ways of Looking at a Sign Relation” which begins as follows.

In the context of 3-adic relations in general, Peirce provides the following illustration of the six converses of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation:

So in a triadic fact, say, for example

• $A ~\text{gives}~ B ~\text{to}~ C$

we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object.  We say that the proposition has three logical subjects.  We regard it as a mere affair of English grammar that there are six ways of expressing this:

$\begin{array}{lll} A ~\text{gives}~ B ~\text{to}~ C & \quad & A ~\text{benefits}~ C ~\text{with}~ B \\[4pt] B ~\text{enriches}~ C ~\text{at expense of}~ A & \quad & C ~\text{receives}~ B ~\text{from}~ A \\[4pt] C ~\text{thanks}~ A ~\text{for}~ B & \quad & B ~\text{leaves}~ A ~\text{for}~ C \end{array}$

These six sentences express one and the same indivisible phenomenon.
(C.S. Peirce, “The Categories Defended”, MS 308 (1903), EP 2, 170–171).

“These six sentences express one and the same indivisible phenomenon.”

It’s a statement telling of the difference between affairs of grammar and affairs of logic, mathematics, and phenomena.

To be continued …

Peiyuan Zhu (Aug 29 2021 at 07:23):

Anyone seen this paper? https://arxiv.org/pdf/1605.03099.pdf

Henry Story (Aug 29 2021 at 08:06):

@Peiyuan Zhu you mean: "How Logic Interacts with Geometry: Infinitesimal Curvature of Categorical Spaces" by Michael Heller · Jerzy Kr ́ol. The introduction is very readable. It is about how conceptions of space and physics affect logic.

Classical logic is a formalization of our everyday patterns of reasoning. In doing science and trying to understand the universe, we extend this way of reasoning to the entire realm of reality. The first serious warning that this huge extrapolation can be misleading came from quantum mechanics in which some time honoured principles of classical logic turned out to be invalid. An attempt to cope with this “deviation” led to what one calls quantum logic...

Jon Awbrey (Aug 29 2021 at 12:04):

Cf: Semiotics, Semiosis, Sign Relations • Discussion 13

Way back during my first foundations + identity crisis I explored every alternative, deviant, non-standard version of logic and set theory I could scrape up — I remember saying to one of my professors, “How come we're still talking about logical atoms in the quantum era?” — and he sent me off to read about quantum logics, which had apparently already fallen out of fashion at the time.  Remarkably enough, I did find one Peircean scholar who had done a lot of work on them, but they didn't seem to be what I needed right then.

Jon Awbrey (Aug 29 2021 at 12:25):

My present, still pressing applications require me to start from much more elementary grounds, stuff I can build up from boolean sources and targets, universes with coordinate spaces of type $(\mathbb{B}^k, \mathbb{B}^k \to \mathbb{B}).$

Jon Awbrey (Aug 31 2021 at 20:00):

Cf: Semiotics, Semiosis, Sign Relations • Discussion 14

Topics arising in various circles I traverse on the web are flashing me back to my earliest influences in the ways of inquiry driven systems.  Dick Lipton and Ken Regan brought to mind the generative power of negative operations and the specific limits of perceptrons.  @Peiyuan Zhu and @Henry Story discussed a paper by Michael Heller and Jerzy Król titled “How Logic Interacts with Geometry : Infinitesimal Curvature of Categorical Spaces”.  It was over my head, just a bit, but it reminded me of early questions about logical atoms, individuals, nominalism vs. realism, and quantum logics, not to mention current pursuits in differential logic, all of which feedback into the ouroborian ampheckbaena of NAND and NNOR among negative ops.

It will be interesting to see what evolves …

Resources

• Survey of Animated Logical Graphs
• Survey of Differential Logic
• Survey of Inquiry Driven Systems
• Survey of Semiotics, Semiosis, Sign Relations

Jon Awbrey (Sep 01 2021 at 20:40):

Cf: Semiotics, Semiosis, Sign Relations • Discussion 15

Re: Peirce List • Robert Marty

Returning to our discussion of $3$-place relations and the $6$ conversions they enjoy under the action of the symmetric group $\mathrm{S}_3$ permuting their places, it’s been a while so I’ll extract the substance of my last reply and continue from there.

We had been contemplating Peirce’s variations on a theme of giving as presented in the section of the Sign Relation article titled “Six Ways of Looking at a Sign Relation”.  That section begins as follows.

In the context of 3-adic relations in general, Peirce provides the following illustration of the six converses of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation:

So in a triadic fact, say, for example

• $A ~\text{gives}~ B ~\text{to}~ C$

we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object.  We say that the proposition has three logical subjects.  We regard it as a mere affair of English grammar that there are six ways of expressing this:

$\begin{array}{lll} A ~\text{gives}~ B ~\text{to}~ C & \quad & A ~\text{benefits}~ C ~\text{with}~ B \\[4pt] B ~\text{enriches}~ C ~\text{at expense of}~ A & \quad & C ~\text{receives}~ B ~\text{from}~ A \\[4pt] C ~\text{thanks}~ A ~\text{for}~ B & \quad & B ~\text{leaves}~ A ~\text{for}~ C \end{array}$

These six sentences express one and the same indivisible phenomenon.
(C.S. Peirce, “The Categories Defended”, MS 308 (1903), EP 2, 170–171).

I called attention to the moral Peirce draws.

• “These six sentences express one and the same indivisible phenomenon.”

With that one statement Peirce draws the clearest possible line of demarcation between affairs of grammar and affairs of logic, mathematics, and phenomena.

The same lesson applies to any relation whose places are not in general reserved for fixed types of entities, in particular, it applies to triadic sign relations.  As we say, “objects, signs, and interpretants are roles not essences”.

Jon Awbrey (Sep 03 2021 at 11:42):

Cf: Semiotics, Semiosis, Sign Relations • Discussion 16

Marius Constantin asked a series of questions which allow me to clear up a number of points.

Re: FB | Semeiotics • Marius V. Constantin

• MVC:

Have you taken into consideration the difference between weak negation and strong negation?

I always begin classically where logic is concerned — I guess that means “strong” negation — we make a stronger start and get better mileage on that basis before we run into the specialized circumstances, mainly in computational and generalized semiotic settings, which force us to weaken our logic.

• MVC:

It is so-called semiotic negation, which, by the way, was an aspect, for me, in so-called resolution logic (Ch. Sanders Peirce is mentioned on that one).

I took a computer science course on resolution-unification theorem provers at U. Illinois in the mid 1980s.  If that’s the same sort of resolution, it generalizes the modus ponens inference rule, all of which exemplify implicational inference.  Peirce’s logical graphs allow a degree of equational or information-preserving inference, a fact which Spencer Brown drew out and made more clear.