C.S. Peirce • Logic of Number (MS 229)
• https://inquiryintoinquiry.com/2012/09/01/c-s-peirce-logic-of-number-ms-229/
❝All other sciences without exception depend upon the principles of mathematics; and mathematics borrows nothing from them but hints.❞
— C.S. Peirce
#Peirce #Inquiry #Mathematics #Science
#FoundationsOfMathematics #Pragmatism
Higher Order Sign Relations • Discussion 1
• https://inquiryintoinquiry.com/2025/03/27/higher-order-sign-relations-discussion-1-a/
Re: FB | Charles S. Peirce Society • John Corcoran
• https://www.facebook.com/groups/peircesociety/posts/1768975423238442/
Questions about the proper treatment of use and mention from the standpoint of Peirce’s theory of signs came up recently in discussions on Facebook. In pragmatic semiotics the trade‑off between “signs-of-objects” and “signs-as-objects” opens up the wider space of Higher Order Sign Relations. In previous work on Inquiry Driven Systems I introduced the subject in the following way.
When interpreters reflect on their use of signs they require an appropriate technical language in which to pursue their reflections. They need signs referring to sign relations, signs referring to elements and components of sign relations, and signs referring to properties and classes of sign relations. The orders of signs developing as reflection evolves can be organized under the heading of “higher order signs” and the reflective sign relations involving them can be referred to as “higher order sign relations”.
References —
John Corcoran
• https://johncorcoran.academia.edu/
Schemata : The Concept of Schema in the History of Logic
• https://www.academia.edu/12691868/SCHEMATA_THE_CONCEPT_OF_SCHEMA_IN_THE_HISTORY_OF_LOGIC
Use And Mention, Use Without Mention, Mention Without Use
• https://www.academia.edu/s/ea64a3484e/schemata#comment_525151
Resources —
Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations
Survey of Inquiry Driven Systems
• https://inquiryintoinquiry.com/2024/02/28/survey-of-inquiry-driven-systems-6/
Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/
#Peirce #Inquiry #Logic #Mathematics #Reflection
#Semiotics #SignRelations #HigherOrderSignRelations
#InquiryDrivenSystems #ReflectiveInterpretiveFrameworks
#Arithmetization #GödelNumbers #Quotation #UseAndMention
Higher Order Sign Relations • 1
• https://inquiryintoinquiry.com/2025/03/22/higher-order-sign-relations-1-a/
Higher Order Sign Relations • Introduction —
When interpreters reflect on their use of signs they require an appropriate technical language in which to pursue their reflections. They need signs referring to sign relations, signs referring to elements and components of sign relations, and signs referring to properties and classes of sign relations. The orders of signs developing as reflection evolves can be organized under the heading of “higher order signs” and the reflective sign relations involving them can be referred to as “higher order sign relations”.
Some years ago I was formatting my old dissertation proposal on Inquiry Driven Systems for the web when the subject of “signs about signs” arose on the Peirce List. It called to mind the part of my document on Higher Order Sign Relations, on which basis Reflective Interpretive Frameworks are constructed, and the introduction to which begins as above.
Resources —
Inquiry Driven Systems
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview
Reflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#Reflective_Interpretive_Frameworks
Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations
Survey of Inquiry Driven Systems
• https://inquiryintoinquiry.com/2024/02/28/survey-of-inquiry-driven-systems-6/
Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/
#Peirce #Inquiry #Logic #Mathematics #Reflection
#Semiotics #SignRelations #HigherOrderSignRelations
#InquiryDrivenSystems #ReflectiveInterpretiveFrameworks
Signs Of Signs • 4
• https://inquiryintoinquiry.com/2025/03/20/signs-of-signs-4-a/
Re: Michael Harris • Language About Language
• https://mathematicswithoutapologies.wordpress.com/2015/05/23/language-about-language/comment-page-1/#comment-380
❝But then inevitably I find myself wondering whether a proof assistant, or even a formal system, can make the distinction between “technical” and “fundamental” questions. There seems to be no logical distinction. The formalist answer might involve algorithmic complexity, but I don't think that sheds any useful light on the question. The materialist answer (often? usually?) amounts to just‑so stories involving Darwin, and lions on the savannah, and maybe an elephant, or at least a mammoth. I don't find these very satisfying either and would prefer to find something in between, and I would feel vindicated if it could be proved (in I don't know what formal system) that the capacity to make such a distinction entails appreciation of music.❞
Peirce proposed a distinction between “corollarial” and “theorematic” reasoning in mathematics which strikes me as similar to the distinction Michael Harris seeks between “technical” and “fundamental” questions.
I can't say I have a lot of insight into how the distinction might be drawn but I recall a number of traditions pointing to the etymology of “theorem” as having to do with the observation of objects and practices whose depth of detail always escapes full accounting by any number of partial views.
On the subject of music, all I have is the following incidental —
🙞 Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes
Perhaps it takes a number theorist to appreciate it …
Resource —
Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations
#Peirce #Inquiry #Logic #Mathematics #Reflection
#Semiotics #SignRelations #HigherOrderSignRelations
Signs Of Signs • 3
• https://inquiryintoinquiry.com/2025/03/19/signs-of-signs-3-a/
Re: Michael Harris • Language About Language
• https://mathematicswithoutapologies.wordpress.com/2015/05/23/language-about-language/comment-page-1/#comment-353
❝And if we don't [keep our stories straight], who puts us away?❞
One's answer, or at least one's initial response to that question will turn on how one feels about formal realities. As I understand it, reality is that which persists in thumping us on the head until we get what it's trying to tell us. Are there formal realities, forms which drive us in that way?
Discussions like those tend to begin by supposing we can form a distinction between external and internal. That is a formal hypothesis, not yet born out as a formal reality. Are there formal realities which drive us to recognize them, to pick them out of a crowd of formal possibilities?
Resources —
Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations
Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/
Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/
#Peirce #Inquiry #Logic #Mathematics #Reflection
#Semiotics #SignRelations #HigherOrderSignRelations
Signs Of Signs • 2
• https://inquiryintoinquiry.com/2025/03/18/signs-of-signs-2-a/
Re: Michael Harris • Language About Language
• https://mathematicswithoutapologies.wordpress.com/2015/05/23/language-about-language/comment-page-1/#comment-346
❝I compared mathematics to a “consensual hallucination”, like virtual reality, and I continue to believe that the aim is to get (consensually) to the point where that hallucination is a second nature.❞
I think that's called “coherentism”, normally contrasted with or complementary to “objectivism”. It's the philosophy of a gang of co‑conspirators who think, “We'll get off scot‑free so long as we all keep our stories straight.”
Resources —
Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations
Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/
Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/
#Peirce #Inquiry #Logic #Mathematics #Reflection
#Semiotics #SignRelations #HigherOrderSignRelations
Signs Of Signs • 1
• https://inquiryintoinquiry.com/2025/03/16/signs-of-signs-1-a/
Re: Michael Harris • Language About Language
• https://mathematicswithoutapologies.wordpress.com/2015/05/23/language-about-language/
There is a language and a corresponding literature treating logic and mathematics as related species of communication and information gathering, namely, the pragmatic-semiotic tradition transmitted through the lifelong efforts of C.S. Peirce. It is by no means a dead language but it continues to fly beneath the radar of many trackers in logic and math today. Nevertheless, the resource remains for those who wish to look into it.
Resources —
Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations
Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/
Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/
• https://www.academia.edu/community/LpWxoO
• https://www.researchgate.net/post/Signs_Of_Signs
#Peirce #Inquiry #Logic #Mathematics #Reflection
#Semiotics #SignRelations #HigherOrderSignRelations
The object of reasoning is to find out …
• https://inquiryintoinquiry.com/2024/01/23/the-object-of-reasoning-is-to-find-out-a/
❝No longer wondered what I would do in life but defined my object.❞
— C.S. Peirce (1861), “My Life, written for the Class-Book”, (CE 1, 3)
❝The object of reasoning is to find out, from the consideration of what we already know, something else which we do not know.❞
— C.S. Peirce (1877), “The Fixation of Belief”, (CP 5.365)
If the object of an investigation is to find out something we do not know then the clues we discover along the way are the signs which determine that object.
People will continue to be confused about determination so long as they can think of no other forms but analytic-behaviorist-causal-dyadic-temporal, object-as-stimulus, sign-as-response varieties. It’s true ordinary language biases us toward billiard‑ball styles of dyadic determination but there are triadic forms of constraint, determination, and interaction not captured by S‑R chains of that order.
Pragmatic objects of signs and concepts are anything we talk or think about and semiosis does not conduct its transactions within the bounds of object as cue, sign as cue ball, and interpretants as solids, stripes, and pockets.
References —
• Peirce, C.S. (1859–1861), “My Life, written for the Class-Book”, pp. 1–3 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1877), “The Fixation Of Belief”, Popular Science Monthly 12 (Nov 1877), pp. 1–15. Reprinted in Collected Papers, CP 5.358–387.
• https://www.cspeirce.com/menu/library/bycsp/fixation/fx-frame.htm
Cactus Language • Overview 3.2
• https://inquiryintoinquiry.com/2025/03/07/cactus-language-overview-3/
Given a body of conceivable propositions we need a way to follow the threads of their indications from their object domain to their values for the mind and a way to follow those same threads back again. Moreover, we need to implement both ways of proceeding in computational form. Thus we need programs for tracing the clues sentences provide from the universe of their objects to the signs of their values and, in turn, from signs to objects. Ultimately, we need to render propositions so functional as indicators of sets and so essential for examining the equality of sets as to give a rule for the practical conceivability of sets. Tackling that task requires us to introduce a number of new definitions and a collection of additional notational devices, to which we now turn.
Resources —
Cactus Language • Overview
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#AutomataTheory #FormalLanguages #FormalGrammars #GraphTheory
Cactus Language • Overview 3.1
• https://inquiryintoinquiry.com/2025/03/07/cactus-language-overview-3/
In the development of Cactus Language to date the following two species of graphs have been instrumental.
• Painted And Rooted Cacti (PARCAI).
• Painted And Rooted Conifers (PARCOI).
It suffices to begin with the first class of data structures, developing their properties and uses in full, leaving discussion of the latter class to a part of the project where their distinctive features are key to developments at that stage. Partly because the two species are so closely related and partly for the sake of brevity, we'll always use the genus name “PARC” to denote the corresponding cacti.
To provide a computational middle ground between sentences seen as syntactic strings and propositions seen as indicator functions the language designer must not only supply a medium for the expression of propositions but also link the assertion of sentences to a means for inverting the indicator functions, that is, for computing the “fibers” or “inverse images” of the propositions.
#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#AutomataTheory #FormalLanguages #FormalGrammars #GraphTheory
Cactus Language • Overview 2
• https://inquiryintoinquiry.com/2025/03/06/cactus-language-overview-2/
In order to facilitate the use of propositions as indicator functions it helps to acquire a flexible notation for referring to propositions in that light, for interpreting sentences in a corresponding role, and for negotiating the requirements of mutual sense between the two domains. If none of the formalisms readily available or in common use meet all the design requirements coming to mind then it is necessary to contemplate the design of a new language especially tailored to the purpose.
In the present application, there is a pressing need to devise a general calculus for composing propositions, computing their values on particular arguments, and inverting their indications to arrive at the sets of things in the universe which are indicated by them.
For computational purposes it is convenient to have a middle ground or an intermediate language for negotiating between the “koine” of sentences regarded as strings of literal characters and the realm of propositions regarded as objects of logical value, even if that makes it necessary to introduce an artificial medium of exchange between the two domains.
If the necessary computations are to be carried out in an organized fashion, and ultimately or partially by familiar classes of machines, then the strings expressing logical propositions are likely to find themselves parsed into tree‑like data structures at some stage of the game. As far as their abstract structures as graphs are concerned, there are several species of graph‑theoretic data structures fitting the task in a reasonably effective and efficient way.
#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#FormalLanguages
Cactus Language • Overview 1.2
• https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/
Resource —
For readers interested and intrepid enough to read ahead, here’s an outline of my work in progress on the OEIS Wiki, which I’ll be revising and serializing to my Inquiry blog.
Part 1
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Syntax
Part 2
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2
Generalities About Formal Grammars
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2#Generalities
Part 3
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3
Cactus Language • Stylistics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stylistics
Cactus Language • Mechanics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Mechanics
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Semantics
Stretching Exercises
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stretching_Exercises
References
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_References
Document History
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Document_History
#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#Automata #FormalLanguages #FormalGrammars #GraphTheory
Cactus Language • Overview 1.1
• https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/
❝Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas. If our intellectual gaze could shift slightly, it would alter each quill’s direction, and suddenly our entire reality would change.❞
— Herbert J. Bernstein • “Idols of Modern Science”
The following report describes a calculus for representing propositions as sentences, that is, as syntactically defined sequences of signs, and for working with those sentences in light of their semantically defined contents as logical propositions. In their computational representation the expressions of the calculus parse into a class of graph‑theoretic data structures whose underlying graphs are called “painted cacti”.
Painted cacti are a specialization of what graph‑theorists refer to as “cacti”, which are in turn a generalization of what they call “trees”. The data structures corresponding to painted cacti have especially nice properties, not only useful in computational terms but interesting from a theoretical standpoint. The remainder of the present Overview is devoted to motivating the development of the indicated family of formal languages, going under the generic name of Cactus Language.
#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#Automata #FormalLanguages #FormalGrammars #GraphTheory
Ouch❢
• https://inquiryintoinquiry.com/2025/02/27/ouch%e2%9d%a2-a/
❝A child hears it said that the stove is hot. But it is not, he says; and, indeed, that central body is not touching it, and only what that touches is hot or cold. But he touches it, and finds the testimony confirmed in a striking way. Thus, he becomes aware of ignorance, and it is necessary to suppose a self in which this ignorance can inhere. …
❝In short, error appears, and it can be explained only by supposing a self which is fallible.
❝Ignorance and error are all that distinguish our private selves from the absolute ego of pure apperception.❞
🙞 C.S. Peirce • “Questions Concerning Certain Faculties Claimed For Man”
• https://www.cspeirce.com/menu/library/bycsp/question/qu-frame.htm
Resource —
Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/
Theory and Therapy of Representations • 5
• https://inquiryintoinquiry.com/2025/02/25/theory-and-therapy-of-representations-5-b/
Re: R.J. Lipton and K.W. Regan • Legal Complexity
• https://rjlipton.com/2022/09/04/legal-complexity/
❝I do not pretend to understand the moral universe;
the arc is a long one, my eye reaches but little ways;
I cannot calculate the curve and complete the figure by
the experience of sight; I can divine it by conscience.
And from what I see I am sure it bends towards justice.❞
🙞 Theodore Parker
• https://web.archive.org/web/20200302045624/https://books.google.com/books?id=eHgYAAAAYAAJ&pg=PA48#v=onepage&q&f=false
The arc of the moral universe may bend toward justice — there's hope it will.
For the logic of laws to converge on justice may take some doing on our part.
Resources —
Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/
Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7/
Survey of Inquiry Driven Systems
• https://inquiryintoinquiry.com/2024/02/28/survey-of-inquiry-driven-systems-6/
#AdaptiveSystems #Cybernetics #SystemsTheory #Governance #Democracy
#Plato #Peirce #MaxWeber #Accountability #Representation #Statistics
#Inquiry #InquiryDrivenSystems #Observation #Expectation #Intention
Theory and Therapy of Representations • 4
• https://inquiryintoinquiry.com/2025/02/24/theory-and-therapy-of-representations-4-b/
Re: Ontolog Forum • Paola Di Maio
• https://groups.google.com/g/ontolog-forum/c/Ek_7cCCyFkQ/m/qI0kQv4UAgAJ
JA: What are the forces distorting our representations of what's observed, what's expected, and what's intended?
PDM: The short answer is — the force behind all distortions is our own unenlightened mind, and all the shortfalls this comes with.
I think that's true, we have to keep reflecting on the state of our personal enlightenments. If we can do that without losing our heads and our systems thinking caps, there will be much we can do to promote the general Enlightenment of the State.
On both personal and general grounds we have a stake in the projects of self‑governing systems — whether it is possible for them to exist and what it takes for them to thrive in given environments. Systems on that order have of course been studied from many points of view and at many levels of organization. Whether we address them under the names of adaptive, cybernetic, error-correcting, intelligent, or optimal control systems they all must be capable to some degree of learning, reasoning, and self‑guidance.
Resources —
Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/
Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7/
Survey of Inquiry Driven Systems
• https://inquiryintoinquiry.com/2024/02/28/survey-of-inquiry-driven-systems-6/
#AdaptiveSystems #Cybernetics #SystemsTheory #Governance #Democracy
#Plato #Peirce #MaxWeber #Accountability #Representation #Statistics
#Inquiry #InquiryDrivenSystems #Observation #Expectation #Intention
Theory and Therapy of Representations • 3.2
• https://inquiryintoinquiry.com/2025/02/23/theory-and-therapy-of-representations-3-b/
Scene 2. Theory and Therapy of Representations • 1
• https://inquiryintoinquiry.com/2025/02/21/theory-and-therapy-of-representations-1-b/
Statistics were originally the data a ship of state needed for stationkeeping and staying on course. The Founders of the United States, like the Cybernauts of the Enlightenment they were, engineered a ship of state with checks and balances and error-controlled feedbacks for the sake of representing both reality and the will of the people. In that connection Max Weber saw how a state's accounting systems are intended as representations of realities its crew and passengers must observe or perish.
That brings us to Question 2 —
• What are the forces distorting our representations of what's observed, what's expected, and what's intended?
Resources ─
Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/
Pragmatic Theory Of Truth
• https://oeis.org/wiki/Pragmatic_Theory_Of_Truth
#AdaptiveSystems #Cybernetics #SystemsTheory #Governance #Democracy
#Plato #Peirce #MaxWeber #Accountability #Representation #Statistics
#Inquiry #InquiryDrivenSystems #Observation #Expectation #Intention
Theory and Therapy of Representations • 3.1
• https://inquiryintoinquiry.com/2025/02/23/theory-and-therapy-of-representations-3-b/
Representation is a concept we find at the intersection of cybernetics, epistemology, logic, mathematics, psychology, and sociology. In my studies it led me from math to psych and back again, with sidelong glances at the history of democratic governance. Its time come round again, I find myself returning to the scenes of two recurring questions.
Scene 1. Pragmatic Truth • Discussion 18
• https://inquiryintoinquiry.com/2019/11/14/pragmatic-truth-discussion-18/
We do not live in axiom systems. We do not live encased in languages, formal or natural. There is no reason to think we will ever have exact and exhaustive theories of what's out there, and the truth, as we know, is “out there”. Peirce understood there are more truths in mathematics than are dreamt of in logic — and Gödel’s realism should have put the last nail in the coffin of logicism — but some ways of thinking just never get a clue.
That brings us to Question 1 —
• What are formalisms and all their embodiments in brains and computers good for?
Resources ─
Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/
Pragmatic Theory Of Truth
• https://oeis.org/wiki/Pragmatic_Theory_Of_Truth
#AdaptiveSystems #Cybernetics #SystemsTheory #Governance #Democracy
#Plato #Peirce #MaxWeber #Accountability #Representation #Statistics
#Inquiry #InquiryDrivenSystems #Observation #Expectation #Intention
