Relation Theory
LoF Group,
Here's an introduction to Relation Theory geared to applications and taking a moderately general view at least as far as finite numbers of relational domains are concerned (kadic or kary relations). Relation Theory https://oeis.org/wiki/Relation_theory This article treats relations from the perspective of combinatorics, in other words, as a subject matter in discrete mathematics, with special attention to finite structures and concrete settheoretic constructions, many of which arise quite naturally in applications. This approach to relation theory, or the theory of relations, is distinguished from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other. Note. Relations include functions as a special case. Regards, Jon


James Bowery
Thanks for that very rigorous definition of "relation theory", Jon.
It's "trick" of including the name of the krelation in a (k+1)relation's tuples reminds me Etter's paper "Threeplace Identity" which was the result of some of our work at HP on dealing with identity (starting with the very practical need to identify individuals/corporations, etc. for the purpose of permitting metadata that attributed assertions of fact to certain identities aka "provenance" of data). The result of that effort threatens to upend set theory itself and was to be fully fleshed out in "Membership and Identity", the completion of which was stopped by Tom's tragic descent into dementia after his beloved wife died. We _were_ able to get a preliminary review of Threeplace Identity by a close associate of Ray Smullyan. It came back with a positive verdict. I believe I may still have that letter somewhere in my archives.


James Bowery
There is a typo in "Threeplace Identity" which is corrected below:
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(D3) x ∈ 0 y ⇐⇒ ¬((y ∈ x ∧ x ∈ x) ∨ (¬(y ∈ x) ∧ ¬(x ∈ x))) Also, a deficiency in this paper pointed out by a student of Tarski's (private communication) is as follows:


James Bowery
Unfortunately, I provided Tom's typo again rather than the correction of it. Here's the actual correction:
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(D3) y ∈' x ⇐⇒ ¬((y ∈ x ∧ x ∈ x) ∨ (¬(y ∈ x) ∧ ¬(x ∈ x))) In bold is the correct right hand side. This same correction should apply to every step of these subsequent steps of the proof: y ∈' x ⇔ ~( (y∈x & x∈x) OR (~ y∈x & ~ x∈x) ) y ∈' x ⇔ ~( (y∈x & F) OR (~ y∈x & T) ) y ∈' x ⇔ ~( (F OR (~ y∈x & T) ) y ∈' x ⇔ ~( ~ y∈x & T) y ∈' x ⇔ ~ ~ y∈x y ∈' x ⇔ y∈x Also, Tarski's student's critique needs to be fleshed out: If T(x,y,z) is the predicate (y ∈ x ∧ z ∈ x) ∨ (y ∉ x ∧ z ∉ x) then (with different definitions for x, y and z we can express the following formulas): T(x, y, y) T(x, y, z) ⇐⇒ T(x, z, y) T(x, w, y) ∧ T(x, y, z) =⇒ T(x, w, z) ¬(x ∈ x) =⇒ ¬T(x, x, y) ⇐⇒ x ∈ y This last formula proves that ¬(x ∈ x) can be expressed with T instead of ∈. The critique is that since no substitutivity properties are discussed for T(x,,) it is therefore:
"a bit premature to refer to T in any was as a kind of "identity" or "equality".


Dear James,
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Just about to kick back with a glass o' th' ol' fave chateau … or two … so I won't even try writing anything coherent for the rest of the day … and meetings all day tomorrow, so it will be a couple of days before I can make a better reply, but here's a few links that come to mind. Peirce has a fundamental concept he calls “teridentity” — a, b, c all identical, graphed as a node of degree three — and at first I thought you might be talking about that. But I see x(y = z) read as “x regards y as the same as z” — here I'd prefer writing [y = z]ₓ or [y]ₓ = [z]ₓ or y =ₓ z — is more like what I use to discuss “equivalence relations from a particular point of view”, following one of Peirce's more radical innovations from his 1870 “Logic of Relatives”. Cf: Peirce • On the Doctrine of Individuals https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_3#Selection_B I wrote this up in general somewhere or other I'll have to find later but here it is applied to the special case of “semiotic equivalence relations”. Semiotic Equivalence Relations (two links, no waiting) https://oeis.org/wiki/Sign_relation#Semiotic_equivalence_relations https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_1#Semiotic_Equivalence_Relations Regards, Jon
On 10/26/2021 1:09 PM, James Bowery wrote:
Thanks for that very rigorous definition of "relation theory", Jon.


Mauro Bertani <Bertanimauro@...>
Hi James,
I'm not good in proof but I prove to follow your speech. I find some problem. It seems that my conclusions are different. I propose a little modification to step 4. Here my steps: (D3) y ∈' x ⇐⇒ ¬((y ∈ x ∧ x ∈ x) ∨ (¬(y ∈ x) ∧ ¬(x ∈ x))) y E' x <==> !((a&b)(!a&&!b)) y E' x <==> a XOR b y E' x <==> y XOR x y E' x <==> y 0+ x where 0+ symbol of albebric normal form If T(x,y,z) is the predicate (y ∈ x ∧ z ∈ x) ∨ (y ∉ x ∧ z ∉ x) then (with different definitions for x, y and z we can express the following formulas): 1) T(x, y, y) 2) T(x, y, z) ⇐⇒ T(x, z, y) 3) T(x, w, y) ∧ T(x, y, z) =⇒ T(x, w, z) 4) ¬(x ∈ x) =⇒ ¬T(x, x, y) ⇐⇒ x ∈ y This last formula proves that ¬(x ∈ x) can be expressed with T instead of ∈. 1) ((a&b)(!a&&!b)) 2) ((a&b)(!a&&!b)) <==> ((b&a)(!b&&!a)) 3) (((c&&a)(!c&&!a))&&((a&b)(!a&&!b)))=>((c&&b)(!c&&!b)) https://www.wolframalpha.com/input/?i=%28%28%28c%26%26a%29%7C%7C%28%21c%26%26%21a%29%29%26%26%28%28a%26b%29%7C%7C%28%21a%26%26%21b%29%29%29%3D%3E%28%28c%26%26b%29%7C%7C%28%21c%26%26%21b%29%29 4) !d=>!((d&&a)(!d&&!a)) non è una tautologia 5) !((d&&a)(!d&&!a))=>a non è una tautologia https://www.wolframalpha.com/input/?i=%21%28%28d%26%26a%29%7C%7C%28%21d%26%26%21a%29%29%3D%3Ea ???  Is the paradox of type?, x can't be member of himself?  ??? 1 e 2 . ((a&b)(!a&&!b)) && ((((a&b)(!a&&!b)) => ((b&a)(!b&&!a)))&&(((b&a)(!b&&!a))=>((a&b)(!a&&!b)))) (a&b)(!a&&!b) 1 e 2 e 3 . ((a&b)(!a&&!b) && (((c&&a)(!c&&!a))&&((a&b)(!a&&!b)))=>((c&&b)(!c&&!b))) (!a  !b c) 1e 2 e 3 e 4 . nel caso ¬(x ∈ x) =⇒( ¬T(x, x, y) ⇐⇒ x ∈ y) (((!a  !b c)) &&( !d=>!((d&&a)(!d&&!a)))&& ((!((d&&a)(!d&&!a))=>a)&&(a=>(!((d&&a)(!d&&!a))))) ) (a && !b && !d)(a && c &&!d) (yEx e z!Ex e x!Ex) o (yEx e wEx e x!Ex) 1 e 2 e 3 e 4 nel caso (¬(x ∈ x) =⇒ ¬T(x, x, y) )⇐⇒ x ∈ y (((!a  !b c))&&((!d=>!((d&&a)(!d&&!a)))=>a) && (a=>((!d=>!((d&&a)(!d&&!a)))))) (a && !b)  (a && c)  (!a && !d) (yEx e z!Ex) o (yEx e wEx) o (y!Ex e x!Ex) PROPOSTA: The 4 step can rewrite with: (a=>((b&&a)(!b&&!a)))=> (a=>b) ((a=>((b&&a)(!b&&!a)))=> (a=>b)) && ((b=>(((b&&a)(!b&&!a))))=>(b=>a)) y E x => (y E x => z E x) y E x <=> z E x So, x would be the equality that links the two differences y and z, similar to Etter's concept of "Membership and Identity". obviously y will not be a member of q (y!Eq) while z will be a member of q (zEq), thus forming the difference between y and z. it should be proved that: yEx <=> zEx y!Eq <=>zEq I found a very similar concept in "An Essay on Man" by Ernst Cassirer. Thanks in advance Mauro


Cf: Relation Theory • 2
https://inquiryintoinquiry.com/2021/10/28/relationtheory2/ All, I am putting a new edition of my Relation Theory article on my blog and transcribing a plaintext copy for whatever discussion may arise, but please see the abovelinked blog post for betterformatted copy. Preliminaries ============= https://oeis.org/wiki/Relation_theory#Preliminaries Two definitions of the relation concept are common in the literature. Although it is usually clear in context which definition is being used at a given time, it tends to become less clear as contexts collide, or as discussion moves from one context to another. The same sort of ambiguity arose in the development of the function concept and it may save a measure of effort to follow the pattern of resolution that worked itself out there. When we speak of a function f : X → Y we are thinking of a mathematical object whose articulation requires three pieces of data, specifying the set X, the set Y, and a particular subset of their cartesian product X × Y. So far so good. Let us write f = (obj₁f, obj₂f, obj₁₂f) to express what has been said so far. When it comes to parsing the notation “f : X → Y”, everyone takes the part “X → Y” as indicating the type of the function, in effect defining type(f) as the pair (obj₁f, obj₂f), but “f” is used equivocally to denote both the triple (obj₁f, obj₂f, obj₁₂f) and the subset obj₁₂f forming one part of it. One way to resolve the ambiguity is to formalize a distinction between the function f = (obj₁f, obj₂f, obj₁₂f) and its “graph”, defining graph(f) = obj₁₂f. Another tactic treats the whole notation “f : X → Y” as a name for the triple, letting “f” denote graph(f). In categorical and computational contexts, at least initially, the type is regarded as an essential attribute or integral part of the function itself. In other contexts we may wish to use a more abstract concept of function, treating a function as a mathematical object capable of being viewed under many different types. Following the pattern of the functional case, let the notation “L ⊆ X × Y” bring to mind a mathematical object which is specified by three pieces of data, the set X, the set Y, and a particular subset of their cartesian product X × Y. As before we have two choices, either let L = (X, Y, graph(L)) or let “L” denote graph(L) and choose another name for the triple. Regards, Jon


Cf: Relation Theory • 3
http://inquiryintoinquiry.com/2021/10/29/relationtheory3/ All, It is convenient to begin with the definition of a kplace relation, where k is a positive integer. Definition. A kplace relation L ⊆ X₁ × … × Xₖ over the nonempty sets X₁, …, Xₖ is a (k+1)tuple (X₁, …, Xₖ, L) where L is a subset of the cartesian product X₁ × … × Xₖ. Several items of terminology are useful in discussing relations. • The sets X₁, …, Xₖ are called the “domains” of the relation L ⊆ X₁ × … × Xₖ, with Xₘ being the mth domain. • If all the Xₘ are the same set X then L ⊆ X₁ × … × Xₖ is more simply described as a kplace relation over X. • The set L is called the “graph” of the relation L ⊆ X₁ × … × Xₖ, on analogy with the graph of a function. • If the sequence of sets X₁, …, Xₖ is constant throughout a given discussion or is otherwise determinate in context then the relation L ⊆ X₁ × … × Xₖ is determined by its graph L, making it acceptable to denote the relation by referring to its graph. • Other synonyms for the adjective “kplace” are “kadic” and “kary”, all of which leads to the integer k being called the “dimension”, “adicity”, or “arity” of the relation L. Resources ========= • Survey of Relation Theory ( https://inquiryintoinquiry.com/2020/05/15/surveyofrelationtheory4/ ) Regards, Jon


James Bowery
On Tue, Oct 26, 2021 at 3:32 PM Jon Awbrey <jawbrey@...> wrote: ...But I see x(y = z) read as “x regards y as the same as z” — If we adopt the LoF notation convention of parentheses () as cleaving a space it reminds me of the notion that the unmarked "state", or the "void", is one's own identity relative to one's self  the primordial "I AM" prior to the first distinction. The way I interpret Tom's notation x(y = z), the x can then be taken as "a particular point of view" as it exists outside of that first distinction relative to itself. Of course, that implies that x has some sort of structure which renders it incapable of distinguishing y from z. Other points of view may have structure capable of distinguishing y from z.


Mauro Bertani <Bertanimauro@...>
Hi,
I have better structured the speech of my previous email, which was very confusing. The basic concept is this: This definition of difference is based on the simple principle that we can say that Islam is different from Christianity only and exclusively because both are religion, so two concepts must first be the same in one aspect to be perceived different on another. Thus the difference between two elements y and z, is between three sets X, Q, S. X is common to y and z, y is a member of Q but not of S, z is a member of S but not of Q. The Etter's symbol x(y!=z) in set theory. Here the link: https://docs.google.com/document/d/1gMXicxgMJHjNjS1qyXx5WAEr6QlzzJiFxZ3j4xkRQn8/edit?usp=sharing Thanks in advance Mauro


JerrySwatez
Hehheh. Good, James. Lots of deep philosophy, throughout the ages and across the planet, muses on such notions of the sort, “I am nothing, and, realizing that, something is created in contrast.” As the universal formal applicability of LoF to all the objects of human comprehension —the universal interpretability of LoF as any object of human apprehension— so the so pervasive “selfabnegation transforms into selfrealization” dictum may also interpret reentry. Image: falling backward into the abyss is the only . . .
On Oct 29, 2021, at 2:48 PM, James Bowery <jabowery@...> wrote:If we adopt the LoF notation convention of parentheses () as cleaving a space it reminds me of the notion that the unmarked "state", or the "void", is one's own identity relative to one's self  the primordial "I AM" prior to the first distinction.


James Bowery
"may also interpret reentry"  Yes. This is why I tend to think of time as prior to space. There is something of the inevitable about reentry in the first distinction. As I wrote to Bernie Lewin several years ago (borrowing from a sequence I found in Gorham's "The Pagan Bible"):  Forwarded message  From: James Bowery <jabowery@...> Date: Tue, Aug 22, 2017 at 11:20 AM Subject: Re: Wrapup of higher degree arithmetic To: Bernie Lewin <bernardjlewin@...> The thing that is changing is awareness which, in the primordial sense, is an oscillation between "I am." and "I am not." However, each stage of oscillation contains within it all prior stages as increasing order: 0 Joy "I am." (Before distinction.) 1 Despair "What am I? Nothing!" (First distinction _is_ reentry into the form.) 2 Joy "But I feel joy when I conceive being. If I feel, I am something. Even the pain I feel at the through that I am nothing is an awareness which convinces me that 'I am.'" (Second distinction/reentry into the form corresponds to 0 "I am." but at a higher order of awareness including _memory_ of Despair  Time has passed.) 4 Despair "All illusion. All this is sophistry without reality." (Third distinction, reentering into the higher order form  remembering both Joy and Despair of lower orders.) 5 Joy "But to create what was not, even though only an illusion, is something (Fourth distinction, reentering into an even higher order form containing within it all prior reentries/awareness.) ... etc. The contrasting _emotional_ states of ever higher order awareness/memory of prior _emotional_ states creates a selfsimilar perspective that, I suspect, brings us to a consideration of the relationship between GSB's 'i' and Pythagoras's 'phi'. Musical composition, then, corresponds to a resonant dynamical memory of emotional states. This is the Time.
On Sat, Oct 30, 2021 at 9:55 AM JerrySwatez <geraldswatez@...> wrote:


Cf: Relation Theory • 4
http://inquiryintoinquiry.com/2021/10/30/relationtheory4/ All, The next few definitions of “local incidence properties” of relations are given at a moderate level of generality in order to show how they apply to kplace relations. In the sequel we'll see what light they throw on a number of more familiar 2place relations and functions. A “local incidence property” of a relation L is a property which depends in turn on the properties of special subsets of L known as its “local flags”. The local flags of a relation are defined in the following way. Let L be a kplace relation L ⊆ X₁ × … × Xₖ. Pick a relational domain Xₘ and a point x in Xₘ. The “flag of L with x at m”, written Lₓ@ₘ and also known as the “x@mflag of L”, is a subset of L with the following definition. • Lₓ@ₘ = {(x₁, …, xₘ …, xₖ) ∈ L : xₘ = x}. Any property C of the local flag Lₓ@ₘ is said to be a “local incidence property of L with respect to the locus x @ m”. A kadic relation L ⊆ X₁ × … × Xₖ is said to be “Cregular at m” if and only if every flag of L with x at m has the property C, where x is taken to vary over the “theme” of the fixed domain Xₘ. Expressed in symbols, L is Cregular at m if and only if C(Lₓ@ₘ) is true for all x in Xₘ. Regards, Jon


James Bowery
It's unclear where you're going with "regularity"  which seems to motivate the entities "theme", "local incidence" and "flag". What does "regularity" buy us in terms of parsimony? How is the word "regular" used in other contexts that your definition clarifies?
On Sat, Oct 30, 2021 at 4:05 PM Jon Awbrey <jawbrey@...> wrote: Cf: Relation Theory • 4


Cf: Relation Theory • 5
https://inquiryintoinquiry.com/2021/10/31/relationtheory5/ All, Two further classes of incidence properties will prove to be of great utility. Regional Incidence Properties ============================= https://oeis.org/wiki/Relation_theory#Regional_incidence_properties The definition of a local flag can be broadened from a point to a subset of a relational domain, arriving at the definition of a “regional flag” in the following way. Let L be a kplace relation L ⊆ X₁ × … × Xₖ. Choose a relational domain Xₘ and one of its subsets S ⊆ Xₘ. Then L_{S @ m} is a subset of L called the flag of L with S at m, or the (S @ m)flag of L, a mathematical object with the following definition. • L_{S @ m} = {(x₁, …, xₘ …, xₖ) ∈ L : xₘ ∈ S}. Numerical Incidence Properties ============================== https://oeis.org/wiki/Relation_theory#Numerical_incidence_properties A “numerical incidence property” of a relation is a local incidence property predicated on the cardinalities of its local flags. For example, L is said to be “cregular at m” if and only if the cardinality of the local flag Lₓ@ₘ is c for all x in Xₘ — to write it in symbols, if and only if Lₓ@ₘ = c for all x ∈ Xₘ. In a similar fashion, one may define the numerical incidence properties, (<c)regular at m, (>c)regular at m, and so on. For ease of reference, a few definitions are recorded below. Numerical Incidence Properties https://inquiryintoinquiry.files.wordpress.com/2021/10/numericalincidencepropertiesalt.png Regards, Jon


Cf: Relation Theory • Discussion 2
https://inquiryintoinquiry.com/2021/11/01/relationtheorydiscussion2/ Re: Relation Theory • 4 https://inquiryintoinquiry.com/2021/10/30/relationtheory4/ Re: FB  Charles S. Peirce Society https://www.facebook.com/groups/peircesociety/posts/2457168684419109/ ::: Joseph Harry https://www.facebook.com/groups/peircesociety/posts/2457168684419109?comment_id=2457724677696843 <QUOTE JH:> These are iconic representations dealing with logical symbolic relations, and so of course are semiotic in Peirce's sense, since logic is semiotic. But couldn't a logician do all of this meticulous formalization and understand all of the discrete logical consequences of it without having any inkling of semiotics or of Peirce? Dear Joseph, As I noted at the top of the article and blog series — https://inquiryintoinquiry.com/2021/10/27/relationtheory1/ “This article treats relations from the perspective of combinatorics, in other words, as a subject matter in discrete mathematics, with special attention to finite structures and concrete settheoretic constructions, many of which arise quite naturally in applications. This approach to relation theory, or the theory of relations, is distinguished from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.” Of course one can always pull a logical formalism out of thin air, with no inkling of its historical sources, and proceed in a blithely syntactic and deductive fashion. But if we hew more closely to applications, original or potential, and even regard logic and math as springing from practice, we must take care for the semantic and pragmatic grounds of their use. From that perspective, models come first, well before the deductive theories whose consistency they establish. Regards, Jon


Cf: Relation Theory • Discussion 3
https://inquiryintoinquiry.com/2021/11/03/relationtheorydiscussion3/ Re: Laws of Form https://groups.io/g/lawsofform/topic/relation_theory/86544796 ::: James Bowery ( https://groups.io/g/lawsofform/message/1116 <QUOTE JB:> Thanks for that very rigorous definition of “relation theory”. Its “trick” of including the name of the krelation in a (k+1)relation’s tuples reminds me Etter’s paper “ThreePlace Identity” which was the result of some of our work at HP on dealing with identity (starting with the very practical need to identify individuals/corporations, etc. for the purpose of permitting metadata that attributed assertions of fact to certain identities aka “provenance” of data). The result of that effort threatens to upend set theory itself and was to be fully fleshed out in “Membership and Identity” […] We were able to get a preliminary review of ThreePlace Identity by a close associate of Ray Smullyan. It came back with a positive verdict. I believe I may still have that letter somewhere in my archives. </QUOTE> Dear James, The article on Relation Theory ( https://oeis.org/wiki/Relation_theory ) represents my attempt to bridge the two cultures of weak typing and strong typing approaches to functions and relations. Weak typing was taught in those halcyon Halmos days when functions and relations were nothing but subsets of cartesian products. Strong typing came to the fore with category theory, its arrows from source to target domains, and the need for closely watched domains in computer science. Peirce recognizes a fundamental triadic relation he calls “teridentity” where three variables a, b, c denote the same object, represented in his logical graphs as a node of degree three, and at first I thought you might be talking about that. But I see x(y = z) read as “x regards y as the same as z” is more like the expressions I use to discuss “equivalence relations from a particular point of view”, following one of Peirce’s more radical innovations from his 1870 “Logic of Relatives”. • C.S. Peirce • On the Doctrine of Individuals https://inquiryintoinquiry.com/2015/02/23/mathematicaldemonstrationthedoctrineofindividuals2/ Using square brackets in the form [a]_e for the equivalence class of an element a in an equivalence relation e we can express the above idea in one of the following forms. • [y = z]_x • [y]_x = [z]_x • y =_x z I wrote this up in general somewhere but there’s a fair enough illustration of the main idea in the following application to “semiotic equivalence relations”. • Semiotic Equivalence Relations https://oeis.org/wiki/Sign_relation#Semiotic_equivalence_relations The rest of your remarks bring up a wealth of associations for me, as seeing the triadic unity in the multiplicity of dyadic appearances is a lot of what the Peircean perspective is all about. I’ll have to dig up a few old links to fill that out … Regards, Jon


Cf: Relation Theory • 6
https://inquiryintoinquiry.com/2021/11/04/relationtheory6/ All, Let's take a look at a few old friends from the class of 2place relations as they appear against the backdrop of our current view on relation theory. Species of Dyadic Relations =========================== https://oeis.org/wiki/Relation_theory#Species_of_dyadic_relations Returning to 2adic relations, it is useful to describe several familiar classes of objects in terms of their local and numerical incidence properties. Let L ⊆ S × T be an arbitrary 2adic relation. The following properties of L can be defined. Display 1. Dyadic Relations • Total • Tubular https://inquiryintoinquiry.files.wordpress.com/2021/11/dyadicrelationse280a2totale280a2tubular.png If L ⊆ S × T is tubular at S then L is called a “partial function” or a “prefunction” from S to T. This is sometimes indicated by giving L an alternate name, for example, “p”, and writing L = p : S ⇀ T. Just by way of formalizing the definition: • L = p : S ⇀ T if and only if L is tubular at S. If L is a prefunction p : S ⇀ T which happens to be total at S, then L is called a “function” from S to T, indicated by writing L = f : S → T. To say a relation L ⊆ S × T is totally tubular at S is to say it is 1regular at S. Thus, we may formalize the following definition. • L = f : S → T if and only if L is 1regular a S. In the case of a function f : S → T, we have the following additional definitions. Display 2. Dyadic Relations • Surjective, Injective, Bijective https://inquiryintoinquiry.files.wordpress.com/2021/11/dyadicrelationse280a2surjectiveinjectivebijective.png Resources ========= https://inquiryintoinquiry.com/2020/05/15/surveyofrelationtheory4/ Regards, Jon

