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  1. Inquiry Into Inquiry @[email protected] ·

    Sign Relations • Signs and Inquiry

    There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many congruences and parallels it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, “thinking” in its best sense, “is a term denoting the various ways in which things acquire significance” (Dewey, 38).

    Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between the converging modes of investigation.  Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject the theory of signs is specialized to treat from comparative and structural points of view.

    References

    • Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.
    • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  Archive.  Journal.  Online (doc) (pdf).

    Resources

    cc: Academia.edu • Laws of Form • Research Gate • Syscoi
    cc: Cybernetics • Structural Modeling • Systems Science

    #CSPeirce #Connotation #Denotation #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #SemioticEquivalenceRelations #Semiotics #SignRelations #TriadicRelations

    #cspeirce #connotation #denotation #inquiry #logic #logicofrelatives
  2. Inquiry Into Inquiry @[email protected] ·

    Sign Relations • Signs and Inquiry

    There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many congruences and parallels it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, “thinking” in its best sense, “is a term denoting the various ways in which things acquire significance” (Dewey, 38).

    Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between the converging modes of investigation.  Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject the theory of signs is specialized to treat from comparative and structural points of view.

    References

    • Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.
    • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  Archive.  Journal.  Online (doc) (pdf).

    Resources

    cc: Academia.edu • Laws of Form • Research Gate • Syscoi
    cc: Cybernetics • Structural Modeling • Systems Science

    #CSPeirce #Connotation #Denotation #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #SemioticEquivalenceRelations #Semiotics #SignRelations #TriadicRelations

    #cspeirce #connotation #denotation #inquiry #logic #logicofrelatives
  3. Inquiry Into Inquiry @[email protected] ·

    Sign Relations • Definition

    One of Peirce’s clearest and most complete definitions of a sign is one he gives in the context of providing a definition for logic, and so it is informative to view it in that setting.

    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.

    — C.S. Peirce, New Elements of Mathematics, vol. 4, 20–21

    In the general discussion of diverse theories of signs, the question arises whether signhood is an absolute, essential, indelible, or ontological property of a thing, or whether it is a relational, interpretive, and mutable role a thing may be said to have only within a particular context of relationships.

    Peirce’s definition of a sign defines it in relation to its objects and its interpretant signs, and thus defines signhood in relative terms, by means of a predicate with three places.  In that definition, signhood is a role in a triadic relation, a role a thing bears or plays in a determinate context of relationships — it is not an absolute or non‑relative property of a thing‑in‑itself, one it possesses independently of all relationships to other things.

    Some of the terms Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader.

    • Correspondence.  From the way Peirce uses the term throughout his work, it is clear he means what he elsewhere calls a “triple correspondence”, and thus this is just another way of referring to the whole triadic sign relation itself.  In particular, his use of the term should not be taken to imply a dyadic correspondence, like the kinds of “mirror image” correspondence between realities and representations bandied about in contemporary controversies about “correspondence theories of truth”.
    • Determination.  Peirce’s concept of determination is broader in several directions than the sense of the word referring 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 admitting a full spectrum of more and less determined relationships.
    • Non‑psychological.  Peirce’s “non‑psychological conception of logic” must be distinguished from any variety of anti‑psychologism.  He was quite interested in matters of psychology and had much of import to say about them.  But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a normative science where psychology is a descriptive science, and so they have very different aims, methods, and rationales.

    Reference

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

    Resources

    cc: Academia.edu • Laws of Form • Research Gate • Syscoi
    cc: Cybernetics • Structural Modeling • Systems Science

    #CSPeirce #Connotation #Denotation #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #SemioticEquivalenceRelations #Semiotics #SignRelations #TriadicRelations

    #cspeirce #connotation #denotation #inquiry #logic #logicofrelatives
  4. Inquiry Into Inquiry @[email protected] ·

    Sign Relations • Definition

    One of Peirce’s clearest and most complete definitions of a sign is one he gives in the context of providing a definition for logic, and so it is informative to view it in that setting.

    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.

    — C.S. Peirce, New Elements of Mathematics, vol. 4, 20–21

    In the general discussion of diverse theories of signs, the question arises whether signhood is an absolute, essential, indelible, or ontological property of a thing, or whether it is a relational, interpretive, and mutable role a thing may be said to have only within a particular context of relationships.

    Peirce’s definition of a sign defines it in relation to its objects and its interpretant signs, and thus defines signhood in relative terms, by means of a predicate with three places.  In that definition, signhood is a role in a triadic relation, a role a thing bears or plays in a determinate context of relationships — it is not an absolute or non‑relative property of a thing‑in‑itself, one it possesses independently of all relationships to other things.

    Some of the terms Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader.

    • Correspondence.  From the way Peirce uses the term throughout his work, it is clear he means what he elsewhere calls a “triple correspondence”, and thus this is just another way of referring to the whole triadic sign relation itself.  In particular, his use of the term should not be taken to imply a dyadic correspondence, like the kinds of “mirror image” correspondence between realities and representations bandied about in contemporary controversies about “correspondence theories of truth”.
    • Determination.  Peirce’s concept of determination is broader in several directions than the sense of the word referring 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 admitting a full spectrum of more and less determined relationships.
    • Non‑psychological.  Peirce’s “non‑psychological conception of logic” must be distinguished from any variety of anti‑psychologism.  He was quite interested in matters of psychology and had much of import to say about them.  But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a normative science where psychology is a descriptive science, and so they have very different aims, methods, and rationales.

    Reference

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

    Resources

    cc: Academia.edu • Laws of Form • Research Gate • Syscoi
    cc: Cybernetics • Structural Modeling • Systems Science

    #CSPeirce #Connotation #Denotation #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #SemioticEquivalenceRelations #Semiotics #SignRelations #TriadicRelations

    #cspeirce #connotation #denotation #inquiry #logic #logicofrelatives
  5. Inquiry Into Inquiry @[email protected] ·

    Survey of Semiotics, Semiosis, Sign Relations • 6

    C.S. Peirce defines logic as “formal semiotic”, using formal to highlight the place of logic as a normative science, over and above the descriptive study of signs and their role in wider fields of play.  Understanding logic as Peirce understands it thus requires a companion study of semiotics, semiosis, and sign relations.

    What follows is a Survey of blog and wiki resources on the theory of signs, variously known as semeiotic or semiotics, and the actions referred to as semiosis which transform signs among themselves in relation to their objects, all as based on C.S. Peirce’s concept of triadic sign relations.

    Elements

    Blog Series

    • Peircean Semiotics and Triadic Sign Relations • (1) • (2) • (3)

    Blog Dialogs

    Sources

    • C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • (1) • (2)
    • C.S. Peirce • Algebra of Logic 1885 • Selections • (1) • (2) • (3) • (4)

    Topics

    Excursions

    • Semiositis • (1)
    • Signspiel • (1)
    • Skiourosemiosis • (1)

    References

    • Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, 269–284.  Abstract.  Online.
    • Awbrey, S.M., and Awbrey, J.L. (September 1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal ‘Organization’, Re‑Organizing Knowledge, Trans‑Forming Institutions : Knowing, Knowledge, and the University in the 21st Century, University of Massachusetts, Amherst, MA.  Online.
    • 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).
    • Awbrey, J.L., and Awbrey, S.M. (1992), “Interpretation as Action : The Risk of Inquiry”, The Eleventh International Human Science Research Conference, Oakland University, Rochester, Michigan.

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

    #CSPeirce #IconIndexSymbol #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #Semiotics #SignRelations #TriadicRelations #Triadicity #Visualization

    #cspeirce #iconindexsymbol #inquiry #logic #logicofrelatives #mathematics
  6. Inquiry Into Inquiry @[email protected] ·

    Survey of Semiotics, Semiosis, Sign Relations • 6

    C.S. Peirce defines logic as “formal semiotic”, using formal to highlight the place of logic as a normative science, over and above the descriptive study of signs and their role in wider fields of play.  Understanding logic as Peirce understands it thus requires a companion study of semiotics, semiosis, and sign relations.

    What follows is a Survey of blog and wiki resources on the theory of signs, variously known as semeiotic or semiotics, and the actions referred to as semiosis which transform signs among themselves in relation to their objects, all as based on C.S. Peirce’s concept of triadic sign relations.

    Elements

    Blog Series

    • Peircean Semiotics and Triadic Sign Relations • (1) • (2) • (3)

    Blog Dialogs

    Sources

    • C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • (1) • (2)
    • C.S. Peirce • Algebra of Logic 1885 • Selections • (1) • (2) • (3) • (4)

    Topics

    Excursions

    • Semiositis • (1)
    • Signspiel • (1)
    • Skiourosemiosis • (1)

    References

    • Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, 269–284.  Abstract.  Online.
    • Awbrey, S.M., and Awbrey, J.L. (September 1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal ‘Organization’, Re‑Organizing Knowledge, Trans‑Forming Institutions : Knowing, Knowledge, and the University in the 21st Century, University of Massachusetts, Amherst, MA.  Online.
    • 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).
    • Awbrey, J.L., and Awbrey, S.M. (1992), “Interpretation as Action : The Risk of Inquiry”, The Eleventh International Human Science Research Conference, Oakland University, Rochester, Michigan.

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

    #CSPeirce #IconIndexSymbol #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #Semiotics #SignRelations #TriadicRelations #Triadicity #Visualization

    #cspeirce #iconindexsymbol #inquiry #logic #logicofrelatives #mathematics
  7. Jon Awbrey @[email protected] ·

    Peircean Semiotics and Triadic Sign Relations • 3
    • inquiryintoinquiry.com/2024/08

    Having labored mightily to bring out a new edition of my primer on sign relations, including material on the pivotal concept of semiotic equivalence relations, I thought it worth the candle to post a notice of the new version here.

    Sign Relations
    • oeis.org/wiki/Sign_relation

    Semiotic Equivalence Relations
    • oeis.org/wiki/Sign_relation#Se

    #Peirce #Inquiry #Logic #LogicOfRelatives #RelationTheory
    #Semiotics #Semiosis #SignRelations #TriadicRelations

    #triadicrelations #signrelations #semiosis #semiotics #relationtheory #logicofrelatives
  8. Jon Awbrey @[email protected] ·

    Peircean Semiotics and Triadic Sign Relations • 3
    • inquiryintoinquiry.com/2024/08

    Having labored mightily to bring out a new edition of my primer on sign relations, including material on the pivotal concept of semiotic equivalence relations, I thought it worth the candle to post a notice of the new version here.

    Sign Relations
    • oeis.org/wiki/Sign_relation

    Semiotic Equivalence Relations
    • oeis.org/wiki/Sign_relation#Se

    #Peirce #Inquiry #Logic #LogicOfRelatives #RelationTheory
    #Semiotics #Semiosis #SignRelations #TriadicRelations

    #triadicrelations #signrelations #semiosis #semiotics #relationtheory #logicofrelatives
  9. Jon Awbrey @[email protected] ·

    Peircean Semiotics and Triadic Sign Relations • 2
    • inquiryintoinquiry.com/2024/08

    When I returned to graduate school for the third time around, this time in systems engineering, I had in mind integrating my long‑standing projects investigating the dynamics of information, inquiry, learning, and reasoning, viewing each as a process whose trajectory evolves over time through the medium which gives it concrete embodiment, namely, a triadic sign relation.

    Up until that time I don't believe I'd ever given much thought to sign relations that had anything smaller than infinite domains of objects, signs, and interpretant signs. Countably infinite domains are what come natural in logic, since that is the norm for the formal languages it uses. Continuous domains come first to mind when turning to physical systems, despite the fact that systems with a discrete or quantized character often enter the fray.

    So it came as a bit of a novelty to me when my advisor, following the motto of engineers the world over to “Keep It Simple, Stupid!” — affectionately known by the acronym KISS — asked me to construct the simplest non‑trivial finite example of a sign relation I could possibly come up with. The outcome of that exercise I wrote up in the following primer on sign relations.

    Inquiry Driven Systems • Sign Relations : A Primer
    • oeis.org/wiki/Inquiry_Driven_S

    Inquiry Driven Systems • Semiotic Equivalence Relations
    • oeis.org/wiki/Inquiry_Driven_S

    #Peirce #Inquiry #Logic #LogicOfRelatives #RelationTheory
    #Semiotics #Semiosis #SignRelations #TriadicRelations

    #triadicrelations #signrelations #semiosis #semiotics #relationtheory #logicofrelatives
  10. Jon Awbrey @[email protected] ·

    Peircean Semiotics and Triadic Sign Relations • 2
    • inquiryintoinquiry.com/2024/08

    When I returned to graduate school for the third time around, this time in systems engineering, I had in mind integrating my long‑standing projects investigating the dynamics of information, inquiry, learning, and reasoning, viewing each as a process whose trajectory evolves over time through the medium which gives it concrete embodiment, namely, a triadic sign relation.

    Up until that time I don't believe I'd ever given much thought to sign relations that had anything smaller than infinite domains of objects, signs, and interpretant signs. Countably infinite domains are what come natural in logic, since that is the norm for the formal languages it uses. Continuous domains come first to mind when turning to physical systems, despite the fact that systems with a discrete or quantized character often enter the fray.

    So it came as a bit of a novelty to me when my advisor, following the motto of engineers the world over to “Keep It Simple, Stupid!” — affectionately known by the acronym KISS — asked me to construct the simplest non‑trivial finite example of a sign relation I could possibly come up with. The outcome of that exercise I wrote up in the following primer on sign relations.

    Inquiry Driven Systems • Sign Relations : A Primer
    • oeis.org/wiki/Inquiry_Driven_S

    Inquiry Driven Systems • Semiotic Equivalence Relations
    • oeis.org/wiki/Inquiry_Driven_S

    #Peirce #Inquiry #Logic #LogicOfRelatives #RelationTheory
    #Semiotics #Semiosis #SignRelations #TriadicRelations

    #triadicrelations #signrelations #semiosis #semiotics #relationtheory #logicofrelatives
  11. Jon Awbrey @[email protected] ·

    Peircean Semiotics and Triadic Sign Relations • 1
    • inquiryintoinquiry.com/2024/08

    As a “guide for the perplexed”, at least when it comes to semiotics, I'll use this thread to collect a budget of resources I think have served to clarify the topic in the past.

    By way of a first offering, let me recommend the following most excellent paper, which I can say with all due modesty in light of the fact all its excellence is due to my most excellent co‑author.

    Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.
    • web.archive.org/web/2000121016
    • pdcnet.org/inquiryct/content/i
    • academia.edu/1266493/Interpret
    • academia.edu/57812482/Interpre

    #Peirce #Inquiry #Logic #LogicOfRelatives #RelationTheory
    #Semiotics #Semiosis #SignRelations #TriadicRelations

    #triadicrelations #signrelations #semiosis #semiotics #relationtheory #logicofrelatives
  12. Jon Awbrey @[email protected] ·

    Peircean Semiotics and Triadic Sign Relations • 1
    • inquiryintoinquiry.com/2024/08

    As a “guide for the perplexed”, at least when it comes to semiotics, I'll use this thread to collect a budget of resources I think have served to clarify the topic in the past.

    By way of a first offering, let me recommend the following most excellent paper, which I can say with all due modesty in light of the fact all its excellence is due to my most excellent co‑author.

    Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.
    • web.archive.org/web/2000121016
    • pdcnet.org/inquiryct/content/i
    • academia.edu/1266493/Interpret
    • academia.edu/57812482/Interpre

    #Peirce #Inquiry #Logic #LogicOfRelatives #RelationTheory
    #Semiotics #Semiosis #SignRelations #TriadicRelations

    #triadicrelations #signrelations #semiosis #semiotics #relationtheory #logicofrelatives
  13. Jon Awbrey @[email protected] ·

    Logic of Relatives
    • inquiryintoinquiry.com/2024/08

    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.
    • jstor.org/stable/25058006
    • archive.org/details/jstor-2505
    • books.google.com/books?id=fFnW

    Resources —

    Charles Sanders Peirce
    • mywikibiz.com/Charles_Sanders_

    Relation Theory
    • oeis.org/wiki/Relation_theory

    Survey of Relation Theory
    • inquiryintoinquiry.com/2024/03

    Peirce's 1870 Logic of Relatives
    • inquiryintoinquiry.com/2019/09
    • oeis.org/wiki/Peirce%27s_1870_

    #Peirce #Logic #LogicOfRelatives #MathematicalLogic
    #Mathematics #RelationTheory #Semiotics #SignRelations

    #signrelations #semiotics #relationtheory #mathematics #mathematicallogic #logicofrelatives
  14. Jon Awbrey @[email protected] ·

    Logic of Relatives
    • inquiryintoinquiry.com/2024/08

    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.
    • jstor.org/stable/25058006
    • archive.org/details/jstor-2505
    • books.google.com/books?id=fFnW

    Resources —

    Charles Sanders Peirce
    • mywikibiz.com/Charles_Sanders_

    Relation Theory
    • oeis.org/wiki/Relation_theory

    Survey of Relation Theory
    • inquiryintoinquiry.com/2024/03

    Peirce's 1870 Logic of Relatives
    • inquiryintoinquiry.com/2019/09
    • oeis.org/wiki/Peirce%27s_1870_

    #Peirce #Logic #LogicOfRelatives #MathematicalLogic
    #Mathematics #RelationTheory #Semiotics #SignRelations

    #signrelations #semiotics #relationtheory #mathematics #mathematicallogic #logicofrelatives
  15. Jon Awbrey @[email protected] ·

    @dearlove @JimPropp

    cf. Logical Involution

    Peirce’s 1870 “Logic of Relatives” • Selection 12
    • inquiryintoinquiry.com/2014/06

    • inquiryintoinquiry.com/2014/06

    #LogicOfRelatives #LogicalInvolution

    #logicalinvolution #logicofrelatives
  16. Jon Awbrey @[email protected] ·

    @dearlove @JimPropp

    cf. Logical Involution

    Peirce’s 1870 “Logic of Relatives” • Selection 12
    • inquiryintoinquiry.com/2014/06

    • inquiryintoinquiry.com/2014/06

    #LogicOfRelatives #LogicalInvolution

    #logicalinvolution #logicofrelatives
  17. Jon Awbrey @[email protected] ·

    Peirce's 1885 “Algebra of Logic” • Discussion 2
    • inquiryintoinquiry.com/2024/04

    Re: FB | Daniel Everett

    One thing I've been trying to understand for a very long time is the changes in Peirce's writing about math and logic from 1865 to 1885. If there's anything I've learned from reading Peirce in the often dim light of intellectual history it is to be wary of progressivist assumptions — but unlike many of his other fans I apply that caution also within the body of his own work. Long story short, from 1865 to 1885 I see progress on several fronts but also bits of backsliding from his more prescient early insights. So it's a puzzle … and it will take more study to ravel out the reasons why.

    Resources for reconciling Peirce's two accounts —
    1. The 1870 account of logical involution
    2. The 1885 account of universal quantification

    Peirce's 1870 “Logic of Relatives” • Selection 12 • The Sign of Involution
    • inquiryintoinquiry.com/2014/06
    Comments —
    (1) inquiryintoinquiry.com/2014/06
    (2) inquiryintoinquiry.com/2014/06
    (3) inquiryintoinquiry.com/2014/06
    (4) inquiryintoinquiry.com/2014/06
    (5) inquiryintoinquiry.com/2014/06

    Peirce's 1885 “Algebra of Logic” • Selections
    (1) inquiryintoinquiry.com/2024/03
    (2) inquiryintoinquiry.com/2024/03
    (3) inquiryintoinquiry.com/2024/03
    (4) inquiryintoinquiry.com/2024/04

    Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
    • jstor.org/stable/2369451

    #Peirce #Logic #AlgebraOfLogic #LogicOfRelatives #RelationTheory #CategoryTheory
    #Semiotics #PredicateCalculus #Quantification #LogicalInvolution #ComputerScience

    #computerscience #logicalinvolution #quantification #predicatecalculus #semiotics #categorytheory
  18. Jon Awbrey @[email protected] ·

    Peirce's 1885 “Algebra of Logic” • Discussion 2
    • inquiryintoinquiry.com/2024/04

    Re: FB | Daniel Everett

    One thing I've been trying to understand for a very long time is the changes in Peirce's writing about math and logic from 1865 to 1885. If there's anything I've learned from reading Peirce in the often dim light of intellectual history it is to be wary of progressivist assumptions — but unlike many of his other fans I apply that caution also within the body of his own work. Long story short, from 1865 to 1885 I see progress on several fronts but also bits of backsliding from his more prescient early insights. So it's a puzzle … and it will take more study to ravel out the reasons why.

    Resources for reconciling Peirce's two accounts —
    1. The 1870 account of logical involution
    2. The 1885 account of universal quantification

    Peirce's 1870 “Logic of Relatives” • Selection 12 • The Sign of Involution
    • inquiryintoinquiry.com/2014/06
    Comments —
    (1) inquiryintoinquiry.com/2014/06
    (2) inquiryintoinquiry.com/2014/06
    (3) inquiryintoinquiry.com/2014/06
    (4) inquiryintoinquiry.com/2014/06
    (5) inquiryintoinquiry.com/2014/06

    Peirce's 1885 “Algebra of Logic” • Selections
    (1) inquiryintoinquiry.com/2024/03
    (2) inquiryintoinquiry.com/2024/03
    (3) inquiryintoinquiry.com/2024/03
    (4) inquiryintoinquiry.com/2024/04

    Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
    • jstor.org/stable/2369451

    #Peirce #Logic #AlgebraOfLogic #LogicOfRelatives #RelationTheory #CategoryTheory
    #Semiotics #PredicateCalculus #Quantification #LogicalInvolution #ComputerScience

    #computerscience #logicalinvolution #quantification #predicatecalculus #semiotics #categorytheory
  19. Jon Awbrey @[email protected] ·

    Peirce's 1885 “Algebra of Logic” • Discussion 1
    • inquiryintoinquiry.com/2024/04

    Re: FB | Daniel Everett

    DE:
    ❝One of the most important papers in the history of logic. “On the Algebra of Logic” was the first to introduce the term “quantifier”.

    ❝Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
    • jstor.org/stable/2369451 ❞

    As far as quantification by any other word goes, Peirce had already introduced a more advanced and “functional” concept of quantification in his 1870 “Logic of Relatives”. The subsequent passage to Fregean styles of first order logic would turn out to be a retrograde movement toward syntacticism (a species of nominalism), as seen in the general run of what fol‑lowed in the fol‑lowing years.

    See ☞ Peirce's 1870 “Logic of Relatives”
    • inquiryintoinquiry.com/2019/09

    Especially ☞ “The Sign of Involution”
    • inquiryintoinquiry.com/2014/06

    The connection between logical involution and universal quantification which Peirce put to use in his 1870 Logic of Relatives will turn up again a century later with the application of category theory to computer science and both of those in turn to logic. Just one more time Peirce was that far ahead of it.

    See ☞ Lambek and Scott (1986), Introduction to Higher Order Categorical Logic, Cambridge University Press.
    • oeis.org/wiki/User:Jon_Awbrey/

    #Peirce #Logic #AlgebraOfLogic #LogicOfRelatives #RelationTheory #CategoryTheory
    #Semiotics #PredicateCalculus #Quantification #LogicalInvolution #ComputerScience

    #logicalinvolution #quantification #predicatecalculus #semiotics #relationtheory #logicofrelatives
  20. Jon Awbrey @[email protected] ·

    Peirce's 1885 “Algebra of Logic” • Discussion 1
    • inquiryintoinquiry.com/2024/04

    Re: FB | Daniel Everett

    DE:
    ❝One of the most important papers in the history of logic. “On the Algebra of Logic” was the first to introduce the term “quantifier”.

    ❝Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
    • jstor.org/stable/2369451 ❞

    As far as quantification by any other word goes, Peirce had already introduced a more advanced and “functional” concept of quantification in his 1870 “Logic of Relatives”. The subsequent passage to Fregean styles of first order logic would turn out to be a retrograde movement toward syntacticism (a species of nominalism), as seen in the general run of what fol‑lowed in the fol‑lowing years.

    See ☞ Peirce's 1870 “Logic of Relatives”
    • inquiryintoinquiry.com/2019/09

    Especially ☞ “The Sign of Involution”
    • inquiryintoinquiry.com/2014/06

    The connection between logical involution and universal quantification which Peirce put to use in his 1870 Logic of Relatives will turn up again a century later with the application of category theory to computer science and both of those in turn to logic. Just one more time Peirce was that far ahead of it.

    See ☞ Lambek and Scott (1986), Introduction to Higher Order Categorical Logic, Cambridge University Press.
    • oeis.org/wiki/User:Jon_Awbrey/

    #Peirce #Logic #AlgebraOfLogic #LogicOfRelatives #RelationTheory #CategoryTheory
    #Semiotics #PredicateCalculus #Quantification #LogicalInvolution #ComputerScience

    #logicalinvolution #quantification #predicatecalculus #semiotics #relationtheory #logicofrelatives
  21. Jon Awbrey @[email protected] ·

    Peirce's 1885 “Algebra of Logic” • Selection 1.1
    • inquiryintoinquiry.com/2024/03

    ❝On the Algebra of Logic❞
    ❝A Contribution to the Philosophy of Notation❞

    ❝§1. Three Kinds Of Signs❞

    ❝Any character or proposition either concerns one subject, two subjects, or a plurality of subjects. For example, one particle has mass, two particles attract one another, a particle revolves about the line joining two others. A fact concerning two subjects is a dual character or relation; but a relation which is a mere combination of two independent facts concerning the two subjects may be called “degenerate”, just as two lines are called a degenerate conic. In like manner a plural character or conjoint relation is to be called degenerate if it is a mere compound of dual characters.

    ❝A sign is in a conjoint relation to the thing denoted and to the mind. If this triple relation is not of a degenerate species, the sign is related to its object only in consequence of a mental association, and depends upon a habit. Such signs are always abstract and general, because habits are general rules to which the organism has become subjected. They are, for the most part, conventional or arbitrary. They include all general words, the main body of speech, and any mode of conveying a judgment. For the sake of brevity I will call them “tokens”.❞ [Note. Peirce more frequently calls these “symbols”.]

    #Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
    #MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
    #AlgebraOfLogic #PredicateCalculus #Quantification #Semiotics
    #RelationComposition #RelationConstruction #RelationReduction

    #relationreduction #relationconstruction #relationcomposition #semiotics #quantification #predicatecalculus
  22. Jon Awbrey @[email protected] ·

    Peirce's 1885 “Algebra of Logic” • Selection 1.1
    • inquiryintoinquiry.com/2024/03

    ❝On the Algebra of Logic❞
    ❝A Contribution to the Philosophy of Notation❞

    ❝§1. Three Kinds Of Signs❞

    ❝Any character or proposition either concerns one subject, two subjects, or a plurality of subjects. For example, one particle has mass, two particles attract one another, a particle revolves about the line joining two others. A fact concerning two subjects is a dual character or relation; but a relation which is a mere combination of two independent facts concerning the two subjects may be called “degenerate”, just as two lines are called a degenerate conic. In like manner a plural character or conjoint relation is to be called degenerate if it is a mere compound of dual characters.

    ❝A sign is in a conjoint relation to the thing denoted and to the mind. If this triple relation is not of a degenerate species, the sign is related to its object only in consequence of a mental association, and depends upon a habit. Such signs are always abstract and general, because habits are general rules to which the organism has become subjected. They are, for the most part, conventional or arbitrary. They include all general words, the main body of speech, and any mode of conveying a judgment. For the sake of brevity I will call them “tokens”.❞ [Note. Peirce more frequently calls these “symbols”.]

    #Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
    #MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
    #AlgebraOfLogic #PredicateCalculus #Quantification #Semiotics
    #RelationComposition #RelationConstruction #RelationReduction

    #relationreduction #relationconstruction #relationcomposition #semiotics #quantification #predicatecalculus
  23. Jon Awbrey @[email protected] ·

    Survey of Relation Theory
    • inquiryintoinquiry.com/2024/03

    In the present Survey of blog and wiki resources for Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set‑theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

    Please follow the above link for the full set of resources.
    A few basic articles are linked below.

    Relation Theory
    • oeis.org/wiki/Relation_theory

    Relation Composition
    • oeis.org/wiki/Relation_composi

    Relation Construction
    • oeis.org/wiki/Relation_constru

    Relation Reduction
    • oeis.org/wiki/Relation_reducti

    Relative Term
    • oeis.org/wiki/Relative_term

    Sign Relation
    • oeis.org/wiki/Sign_relation

    Triadic Relation
    • oeis.org/wiki/Triadic_relation

    Six Ways of Looking at a Triadic Relation ⌬ 1
    • inquiryintoinquiry.com/2015/02

    Mathematical Demonstration and the Doctrine of Individuals
    • inquiryintoinquiry.com/2023/05
    • inquiryintoinquiry.com/2023/05

    Peirce's 1870 “Logic of Relatives” —
    • inquiryintoinquiry.com/2019/09
    • inquiryintoinquiry.com/2014/01

    #Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
    #MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
    #PredicateCalculus #ContinuousPredicate #HypostaticAbstraction
    #RelationComposition #RelationConstruction #RelationReduction

    #relationreduction #relationconstruction #relationcomposition #hypostaticabstraction #continuouspredicate #predicatecalculus
  24. Jon Awbrey @[email protected] ·

    Survey of Relation Theory
    • inquiryintoinquiry.com/2024/03

    In the present Survey of blog and wiki resources for Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set‑theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

    Please follow the above link for the full set of resources.
    A few basic articles are linked below.

    Relation Theory
    • oeis.org/wiki/Relation_theory

    Relation Composition
    • oeis.org/wiki/Relation_composi

    Relation Construction
    • oeis.org/wiki/Relation_constru

    Relation Reduction
    • oeis.org/wiki/Relation_reducti

    Relative Term
    • oeis.org/wiki/Relative_term

    Sign Relation
    • oeis.org/wiki/Sign_relation

    Triadic Relation
    • oeis.org/wiki/Triadic_relation

    Six Ways of Looking at a Triadic Relation ⌬ 1
    • inquiryintoinquiry.com/2015/02

    Mathematical Demonstration and the Doctrine of Individuals
    • inquiryintoinquiry.com/2023/05
    • inquiryintoinquiry.com/2023/05

    Peirce's 1870 “Logic of Relatives” —
    • inquiryintoinquiry.com/2019/09
    • inquiryintoinquiry.com/2014/01

    #Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
    #MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
    #PredicateCalculus #ContinuousPredicate #HypostaticAbstraction
    #RelationComposition #RelationConstruction #RelationReduction

    #relationreduction #relationconstruction #relationcomposition #hypostaticabstraction #continuouspredicate #predicatecalculus
  25. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • Discussion 12
    • inquiryintoinquiry.com/2024/02

    Re: Sign Relations, Triadic Relations, Relation Theory • 1
    • inquiryintoinquiry.com/2024/02

    A note from a longtime correspondent points out a search of the available texts turns up no use of the plural form “semiotics” by Peirce and just one place where he uses the plural form “Semeiotics”. That prompts me to make the following excuse for my use or abuse of Peirce's terms, as the case may be.

    Peirce has always been one of my chief resources in the quest to understand how logic and math and science work. There is much to be gained by getting his distinctive ideas across to active practitioners in those fields. In doing that I find it better to tweak the words a bit, if that's what it takes to preserve the idea, than to hallow the words at the risk of losing the idea.

    As far as semiotics by any name goes, what seems to work best without too much clanging in modern ears is parsing “semiotics” in line with words like “mathematics” and “cybernetics”, plus we can now use the singular form as the adjective “semiotic”.

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #signrelations #triadicrelations #relationtheory #logicofrelatives #semiosis #semiotics
  26. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • Discussion 12
    • inquiryintoinquiry.com/2024/02

    Re: Sign Relations, Triadic Relations, Relation Theory • 1
    • inquiryintoinquiry.com/2024/02

    A note from a longtime correspondent points out a search of the available texts turns up no use of the plural form “semiotics” by Peirce and just one place where he uses the plural form “Semeiotics”. That prompts me to make the following excuse for my use or abuse of Peirce's terms, as the case may be.

    Peirce has always been one of my chief resources in the quest to understand how logic and math and science work. There is much to be gained by getting his distinctive ideas across to active practitioners in those fields. In doing that I find it better to tweak the words a bit, if that's what it takes to preserve the idea, than to hallow the words at the risk of losing the idea.

    As far as semiotics by any name goes, what seems to work best without too much clanging in modern ears is parsing “semiotics” in line with words like “mathematics” and “cybernetics”, plus we can now use the singular form as the adjective “semiotic”.

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #signrelations #triadicrelations #relationtheory #logicofrelatives #semiosis #semiotics
  27. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • 4.2
    • inquiryintoinquiry.com/2024/02

    Here's an earlier draft version of Peirce's 1902 definition of a sign, which he gives in the process of defining logic.

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

    ❝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 —

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

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #signrelations #triadicrelations #relationtheory #logicofrelatives #semiosis #semiotics
  28. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • 4.2
    • inquiryintoinquiry.com/2024/02

    Here's an earlier draft version of Peirce's 1902 definition of a sign, which he gives in the process of defining logic.

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

    ❝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 —

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

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #signrelations #triadicrelations #relationtheory #logicofrelatives #semiosis #semiotics
  29. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • 4.1
    • inquiryintoinquiry.com/2024/02

    For ease of reference, here are two variants of Peirce's 1902 definition of a sign, which he gives in the process of defining 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).

    Reference —

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

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #signrelations #triadicrelations #relationtheory #logicofrelatives #semiosis #semiotics
  30. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • 4.1
    • inquiryintoinquiry.com/2024/02

    For ease of reference, here are two variants of Peirce's 1902 definition of a sign, which he gives in the process of defining 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).

    Reference —

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

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #signrelations #triadicrelations #relationtheory #logicofrelatives #semiosis #semiotics
  31. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • 3
    • inquiryintoinquiry.com/2024/02

    The middle ground between relations in general and the sign relations informing logic, inquiry, and communication is occupied by triadic relations, also called ternary or 3‑place relations.

    Triadic relations are some of the most pervasive in mathematics, over and above the importance of sign relations for logic.

    For a primer on triadic relations, with examples from mathematics and semiotics, see the article linked below.

    Triadic Relations
    • OEIS Wiki ( oeis.org/wiki/Triadic_relation )
    • Wikiversity ( en.wikiversity.org/wiki/Triadi )

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #signrelations #triadicrelations #relationtheory #logicofrelatives #semiosis #semiotics
  32. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • 3
    • inquiryintoinquiry.com/2024/02

    The middle ground between relations in general and the sign relations informing logic, inquiry, and communication is occupied by triadic relations, also called ternary or 3‑place relations.

    Triadic relations are some of the most pervasive in mathematics, over and above the importance of sign relations for logic.

    For a primer on triadic relations, with examples from mathematics and semiotics, see the article linked below.

    Triadic Relations
    • OEIS Wiki ( oeis.org/wiki/Triadic_relation )
    • Wikiversity ( en.wikiversity.org/wiki/Triadi )

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #semiosis #semiotics #scientificmethod #mathematics #logic #peirce
  33. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • 2
    • inquiryintoinquiry.com/2024/02

    I always have trouble deciding whether to start with the genus and drive down to the species or begin with concrete examples and accompany Sisyphus up Mt. Abstraction.

    To start at the wide end of the funnel, the following article takes up relations in general, focusing on the discrete mathematical varieties we find most useful in applications, for example, as background for empirical data sets and relational data bases.

    Relation Theory
    • OEIS Wiki ( oeis.org/wiki/Relation_theory )
    • Wikiversity ( en.wikiversity.org/wiki/Relati )

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #signrelations #triadicrelations #relationtheory #logicofrelatives #semiosis #semiotics
  34. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • 2
    • inquiryintoinquiry.com/2024/02

    I always have trouble deciding whether to start with the genus and drive down to the species or begin with concrete examples and accompany Sisyphus up Mt. Abstraction.

    To start at the wide end of the funnel, the following article takes up relations in general, focusing on the discrete mathematical varieties we find most useful in applications, for example, as background for empirical data sets and relational data bases.

    Relation Theory
    • OEIS Wiki ( oeis.org/wiki/Relation_theory )
    • Wikiversity ( en.wikiversity.org/wiki/Relati )

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #signrelations #triadicrelations #relationtheory #logicofrelatives #semiosis #semiotics
  35. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • 1
    • inquiryintoinquiry.com/2024/02

    To understand how signs work in Peirce's theory of triadic sign relations, or “semiotics”, we have to understand, in order of increasing generality, sign relations, triadic relations, and relations in general, each as conceived in Peirce's logic of relative terms and the corresponding mathematics of relations.

    Toward that understanding, here are the current versions of articles I long ago contributed to Wikipedia and Wikiversity and continue to develop at a number of other places.

    Sign Relations
    • oeis.org/wiki/Sign_relation

    Triadic Relations
    • oeis.org/wiki/Triadic_relation

    Relation Theory
    • oeis.org/wiki/Relation_theory

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #signrelations #triadicrelations #relationtheory #logicofrelatives #semiosis #semiotics
  36. Jon Awbrey @[email protected] ·

    Sign Relations, Triadic Relations, Relation Theory • 1
    • inquiryintoinquiry.com/2024/02

    To understand how signs work in Peirce's theory of triadic sign relations, or “semiotics”, we have to understand, in order of increasing generality, sign relations, triadic relations, and relations in general, each as conceived in Peirce's logic of relative terms and the corresponding mathematics of relations.

    Toward that understanding, here are the current versions of articles I long ago contributed to Wikipedia and Wikiversity and continue to develop at a number of other places.

    Sign Relations
    • oeis.org/wiki/Sign_relation

    Triadic Relations
    • oeis.org/wiki/Triadic_relation

    Relation Theory
    • oeis.org/wiki/Relation_theory

    #Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
    #LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

    #signrelations #triadicrelations #relationtheory #logicofrelatives #semiosis #semiotics
  37. Jon Awbrey @[email protected] ·

    Survey of Relation Theory
    • inquiryintoinquiry.com/2023/07

    In this Survey of blog and wiki posts on Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set‑theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

    Please follow the above link for the full set of resources.
    A few basic articles are linked below.

    Elements —
    • Relation Theory ( oeis.org/wiki/Relation_theory )

    Relational Concepts —
    • Relation Composition ( oeis.org/wiki/Relation_composi )
    • Relation Construction ( oeis.org/wiki/Relation_constru )
    • Relation Reduction ( oeis.org/wiki/Relation_reducti )
    • Relative Term ( oeis.org/wiki/Relative_term )
    • Sign Relation ( oeis.org/wiki/Sign_relation )
    • Triadic Relation ( oeis.org/wiki/Triadic_relation )
    • Logic of Relatives ( oeis.org/wiki/Logic_of_relativ )
    • Hypostatic Abstraction ( oeis.org/wiki/Hypostatic_abstr )
    • Continuous Predicate ( oeis.org/wiki/Continuous_predi )

    Illustrations —

    Six Ways of Looking at a Triadic Relation ⌬ 1
    • inquiryintoinquiry.com/2015/02

    Information‑Theoretic Perspective (Escape from Nominalism)

    • Mathematical Demonstration and the Doctrine of Individuals
    • inquiryintoinquiry.com/2023/05
    • inquiryintoinquiry.com/2023/05

    Peirce's 1870 “Logic of Relatives” —

    Overview
    • inquiryintoinquiry.com/2019/09

    Preliminaries
    • inquiryintoinquiry.com/2014/01

    #Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
    #MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
    #RelationComposition #RelationConstruction #RelationReduction

    #relationreduction #relationconstruction #relationcomposition #signrelation #triadicrelation #dyadicrelation
  38. Jon Awbrey @[email protected] ·

    Survey of Relation Theory
    • inquiryintoinquiry.com/2023/07

    In this Survey of blog and wiki posts on Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set‑theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

    Please follow the above link for the full set of resources.
    A few basic articles are linked below.

    Elements —
    • Relation Theory ( oeis.org/wiki/Relation_theory )

    Relational Concepts —
    • Relation Composition ( oeis.org/wiki/Relation_composi )
    • Relation Construction ( oeis.org/wiki/Relation_constru )
    • Relation Reduction ( oeis.org/wiki/Relation_reducti )
    • Relative Term ( oeis.org/wiki/Relative_term )
    • Sign Relation ( oeis.org/wiki/Sign_relation )
    • Triadic Relation ( oeis.org/wiki/Triadic_relation )
    • Logic of Relatives ( oeis.org/wiki/Logic_of_relativ )
    • Hypostatic Abstraction ( oeis.org/wiki/Hypostatic_abstr )
    • Continuous Predicate ( oeis.org/wiki/Continuous_predi )

    Illustrations —

    Six Ways of Looking at a Triadic Relation ⌬ 1
    • inquiryintoinquiry.com/2015/02

    Information‑Theoretic Perspective (Escape from Nominalism)

    • Mathematical Demonstration and the Doctrine of Individuals
    • inquiryintoinquiry.com/2023/05
    • inquiryintoinquiry.com/2023/05

    Peirce's 1870 “Logic of Relatives” —

    Overview
    • inquiryintoinquiry.com/2019/09

    Preliminaries
    • inquiryintoinquiry.com/2014/01

    #Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
    #MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
    #RelationComposition #RelationConstruction #RelationReduction

    #relationreduction #relationconstruction #relationcomposition #signrelation #triadicrelation #dyadicrelation
  39. Jon Awbrey @[email protected] ·

    Logic Syllabus • 4
    • inquiryintoinquiry.com/logic-s

    Relational Concepts
    • oeis.org/wiki/Logic_Syllabus#R

    Continuous Predicate • oeis.org/wiki/Continuous_predi
    Hypostatic Abstraction • oeis.org/wiki/Hypostatic_abstr
    Logic of Relatives • oeis.org/wiki/Logic_of_relativ
    Logical Matrix • oeis.org/wiki/Logical_matrix
    Relation • oeis.org/wiki/Relation
    Relation Composition • oeis.org/wiki/Relation_composi
    Relation Construction • oeis.org/wiki/Relation_constru
    Relation Reduction • oeis.org/wiki/Relation_reducti
    Relation Theory • oeis.org/wiki/Relation_theory
    Relative Term • oeis.org/wiki/Relative_term
    Sign Relation • oeis.org/wiki/Sign_relation
    Triadic Relation • oeis.org/wiki/Triadic_relation

    #Logic #LogicSyllabus #ContinuousPredicate #HypostaticAbstraction #LogicOfRelatives
    #LogicalMatrix #Relation #RelationComposition #RelationConstruction #RelationReduction
    #RelationTheory #RelativeTerm #SignRelation #TriadicRelation

    #triadicrelation #signrelation #relativeterm #relationtheory #relationreduction #relationconstruction
  40. Jon Awbrey @[email protected] ·

    Logic Syllabus • 4
    • inquiryintoinquiry.com/logic-s

    Relational Concepts
    • oeis.org/wiki/Logic_Syllabus#R

    Continuous Predicate • oeis.org/wiki/Continuous_predi
    Hypostatic Abstraction • oeis.org/wiki/Hypostatic_abstr
    Logic of Relatives • oeis.org/wiki/Logic_of_relativ
    Logical Matrix • oeis.org/wiki/Logical_matrix
    Relation • oeis.org/wiki/Relation
    Relation Composition • oeis.org/wiki/Relation_composi
    Relation Construction • oeis.org/wiki/Relation_constru
    Relation Reduction • oeis.org/wiki/Relation_reducti
    Relation Theory • oeis.org/wiki/Relation_theory
    Relative Term • oeis.org/wiki/Relative_term
    Sign Relation • oeis.org/wiki/Sign_relation
    Triadic Relation • oeis.org/wiki/Triadic_relation

    #Logic #LogicSyllabus #ContinuousPredicate #HypostaticAbstraction #LogicOfRelatives
    #LogicalMatrix #Relation #RelationComposition #RelationConstruction #RelationReduction
    #RelationTheory #RelativeTerm #SignRelation #TriadicRelation

    #triadicrelation #signrelation #relativeterm #relationtheory #relationreduction #relationconstruction
  41. Jon Awbrey @[email protected] ·

    Survey of Relation Theory
    • inquiryintoinquiry.com/2023/04

    In this Survey of blog and wiki posts on Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

    Please follow the above link for the full set of resources.
    A few basic articles are linked below.

    Elements —
    • Relation Theory ( oeis.org/wiki/Relation_theory )

    Relational Concepts —
    • Relation Construction ( oeis.org/wiki/Relation_constru )
    • Relation Composition ( oeis.org/wiki/Relation_composi )
    • Relation Reduction ( oeis.org/wiki/Relation_reducti )
    • Relative Term ( oeis.org/wiki/Relative_term )
    • Sign Relation ( oeis.org/wiki/Sign_relation )
    • Triadic Relation ( oeis.org/wiki/Triadic_relation )
    • Logic of Relatives ( oeis.org/wiki/Logic_of_relativ )
    • Hypostatic Abstraction ( oeis.org/wiki/Hypostatic_abstr )
    • Continuous Predicate ( oeis.org/wiki/Continuous_predi )

    Illustrations —

    Six Ways of Looking at a Triadic Relation ⌬ 1
    • inquiryintoinquiry.com/2015/02

    Peirce's 1870 “Logic of Relatives” —

    Overview
    • inquiryintoinquiry.com/2019/09

    Preliminaries
    • inquiryintoinquiry.com/2014/01

    #Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
    #MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
    #PredicateCalculus #ContinuousPredicate #HypostaticAbstraction
    #RelationComposition #RelationConstruction #RelationReduction

    #hypostaticabstraction #continuouspredicate #predicatecalculus #signrelation #triadicrelation #dyadicrelation
  42. Jon Awbrey @[email protected] ·

    Survey of Relation Theory
    • inquiryintoinquiry.com/2023/04

    In this Survey of blog and wiki posts on Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

    Please follow the above link for the full set of resources.
    A few basic articles are linked below.

    Elements —
    • Relation Theory ( oeis.org/wiki/Relation_theory )

    Relational Concepts —
    • Relation Construction ( oeis.org/wiki/Relation_constru )
    • Relation Composition ( oeis.org/wiki/Relation_composi )
    • Relation Reduction ( oeis.org/wiki/Relation_reducti )
    • Relative Term ( oeis.org/wiki/Relative_term )
    • Sign Relation ( oeis.org/wiki/Sign_relation )
    • Triadic Relation ( oeis.org/wiki/Triadic_relation )
    • Logic of Relatives ( oeis.org/wiki/Logic_of_relativ )
    • Hypostatic Abstraction ( oeis.org/wiki/Hypostatic_abstr )
    • Continuous Predicate ( oeis.org/wiki/Continuous_predi )

    Illustrations —

    Six Ways of Looking at a Triadic Relation ⌬ 1
    • inquiryintoinquiry.com/2015/02

    Peirce's 1870 “Logic of Relatives” —

    Overview
    • inquiryintoinquiry.com/2019/09

    Preliminaries
    • inquiryintoinquiry.com/2014/01

    #Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
    #MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
    #PredicateCalculus #ContinuousPredicate #HypostaticAbstraction
    #RelationComposition #RelationConstruction #RelationReduction

    #hypostaticabstraction #continuouspredicate #predicatecalculus #signrelation #triadicrelation #dyadicrelation
  43. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 3.3
    • inquiryintoinquiry.com/2014/01

    Comment on ❝The Signs of Inclusion, Equality, Etc.❞

    Peirce's use of a square bracket \([t]\) to indicate a mapping from logical terms to numbers provides a basis for the computation of frequencies, probabilities, and other statistical measures constructed from them, thus affording a “principle of correspondence” between probability theory and its limiting case in the forms of logic.

    This brings us once again to the relativity of contingency and necessity, as one way of approaching necessity is through the avenue of probability, describing necessity as a probability of \(1,\) but the whole apparatus of probability theory only figures in if it is cast against the backdrop of probability space axioms, the reference class of distributions, and the sample space we cannot help but abduce on the scene of observations.

    Aye, there's the snake eyes. And with them we can see there is always an irreducible quantum of facticity to all our necessities. More plainly spoken, it takes a fairly complex conceptual infrastructure just to begin speaking of probabilities, and that setting can only be set up by means of abductive, fallible, hypothetical, and inherently risky mental acts.

    Pragmatic thinking is the logic of abduction, which is another way of saying it addresses the question: What may be hoped? We have to face the possibility it may be just as impossible to speak of absolute identity with any hope of making practical philosophical sense as it is to speak of absolute simultaneity with any hope of making operational physical sense.

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870

    #lor1870 #relationtheory #logicofrelatives #logic #peirce
  44. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 3.3
    • inquiryintoinquiry.com/2014/01

    Comment on ❝The Signs of Inclusion, Equality, Etc.❞

    Peirce's use of a square bracket \([t]\) to indicate a mapping from logical terms to numbers provides a basis for the computation of frequencies, probabilities, and other statistical measures constructed from them, thus affording a “principle of correspondence” between probability theory and its limiting case in the forms of logic.

    This brings us once again to the relativity of contingency and necessity, as one way of approaching necessity is through the avenue of probability, describing necessity as a probability of \(1,\) but the whole apparatus of probability theory only figures in if it is cast against the backdrop of probability space axioms, the reference class of distributions, and the sample space we cannot help but abduce on the scene of observations.

    Aye, there's the snake eyes. And with them we can see there is always an irreducible quantum of facticity to all our necessities. More plainly spoken, it takes a fairly complex conceptual infrastructure just to begin speaking of probabilities, and that setting can only be set up by means of abductive, fallible, hypothetical, and inherently risky mental acts.

    Pragmatic thinking is the logic of abduction, which is another way of saying it addresses the question: What may be hoped? We have to face the possibility it may be just as impossible to speak of absolute identity with any hope of making practical philosophical sense as it is to speak of absolute simultaneity with any hope of making operational physical sense.

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870

    #lor1870 #relationtheory #logicofrelatives #logic #peirce
  45. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 3.2
    • inquiryintoinquiry.com/2014/01

    ❝§3. Application of the Algebraic Signs to Logic❞

    ❝The Signs of Inclusion, Equality, Etc.❞

    ❝But not only do the significations of \(=\) and \(<\) here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes.

    ❝So, to write \(5 < 7\) is to say that \(5\) is part of \(7,\) just as to write \(\mathrm{f} < \mathrm{m}\) is to say that Frenchmen are part of men. Indeed, if \(\mathrm{f} < \mathrm{m},\) then the number of Frenchmen is less than the number of men, and if \(\mathrm{v} = \mathrm{p},\) then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.❞

    (Peirce, CP 3.66)

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus
  46. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 3.2
    • inquiryintoinquiry.com/2014/01

    ❝§3. Application of the Algebraic Signs to Logic❞

    ❝The Signs of Inclusion, Equality, Etc.❞

    ❝But not only do the significations of \(=\) and \(<\) here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes.

    ❝So, to write \(5 < 7\) is to say that \(5\) is part of \(7,\) just as to write \(\mathrm{f} < \mathrm{m}\) is to say that Frenchmen are part of men. Indeed, if \(\mathrm{f} < \mathrm{m},\) then the number of Frenchmen is less than the number of men, and if \(\mathrm{v} = \mathrm{p},\) then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.❞

    (Peirce, CP 3.66)

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus
  47. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 3.1
    • inquiryintoinquiry.com/2014/01

    ❝§3. Application of the Algebraic Signs to Logic❞

    ❝The Signs of Inclusion, Equality, Etc.❞

    ❝I shall follow Boole in taking the sign of equality to signify identity. Thus, if \(\mathrm{v}\) denotes the Vice-President of the United States, and \(\mathrm{p}\) the President of the Senate of the United States,

    \[\mathrm{v}=\mathrm{p}\]

    ❝means that every Vice-President of the United States is President of the Senate, and every President of the United States Senate is Vice-President.

    ❝The sign “less than” is to be so taken that

    \[\mathrm{f}<\mathrm{m}\]

    ❝means that every Frenchman is a man, but there are men besides Frenchmen. Drobisch has used this sign in the same sense. It will follow from these significations of \(=\) and \(<\) that the sign \(-\!\!\!<\) (or \(\leqq,\) “as small as”) will mean “is”. Thus,

    \[\mathrm{f}-\!\!\!<\mathrm{m}\]

    ❝means “every Frenchman is a man”, without saying whether there are any other men or not. So,

    \[\mathit{m}-\!\!\!<\mathit{l}\]

    ❝will mean that every mother of anything is a lover of the same thing; although this interpretation in some degree anticipates a convention to be made further on. These significations of \(=\) and \(<\) plainly conform to the indispensable conditions.

    ❝Upon the transitive character of these relations the syllogism depends, for by virtue of it, from \(\mathrm{f}-\!\!\!<\mathrm{m}\) and \(\mathrm{m}-\!\!\!<\mathrm{a}\) we can infer that \(\mathrm{f}-\!\!\!<\mathrm{a},\) that is, from every Frenchman being a man and every man being an animal, that every Frenchman is an animal.❞

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870

    #lor1870 #relationtheory #logicofrelatives #logic #peirce
  48. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 3.1
    • inquiryintoinquiry.com/2014/01

    ❝§3. Application of the Algebraic Signs to Logic❞

    ❝The Signs of Inclusion, Equality, Etc.❞

    ❝I shall follow Boole in taking the sign of equality to signify identity. Thus, if \(\mathrm{v}\) denotes the Vice-President of the United States, and \(\mathrm{p}\) the President of the Senate of the United States,

    \[\mathrm{v}=\mathrm{p}\]

    ❝means that every Vice-President of the United States is President of the Senate, and every President of the United States Senate is Vice-President.

    ❝The sign “less than” is to be so taken that

    \[\mathrm{f}<\mathrm{m}\]

    ❝means that every Frenchman is a man, but there are men besides Frenchmen. Drobisch has used this sign in the same sense. It will follow from these significations of \(=\) and \(<\) that the sign \(-\!\!\!<\) (or \(\leqq,\) “as small as”) will mean “is”. Thus,

    \[\mathrm{f}-\!\!\!<\mathrm{m}\]

    ❝means “every Frenchman is a man”, without saying whether there are any other men or not. So,

    \[\mathit{m}-\!\!\!<\mathit{l}\]

    ❝will mean that every mother of anything is a lover of the same thing; although this interpretation in some degree anticipates a convention to be made further on. These significations of \(=\) and \(<\) plainly conform to the indispensable conditions.

    ❝Upon the transitive character of these relations the syllogism depends, for by virtue of it, from \(\mathrm{f}-\!\!\!<\mathrm{m}\) and \(\mathrm{m}-\!\!\!<\mathrm{a}\) we can infer that \(\mathrm{f}-\!\!\!<\mathrm{a},\) that is, from every Frenchman being a man and every man being an animal, that every Frenchman is an animal.❞

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870

    #lor1870 #relationtheory #logicofrelatives #logic #peirce
  49. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 2.2
    • inquiryintoinquiry.com/2014/01

    Peirce's remarks at CP 3.65 are so replete with remarkable ideas, some of them so taken for granted in mathematical discourse as usually to escape explicit mention, others so suggestive of things to come in a future remote from his time of writing, and yet so smoothly slipped into the stream of thought it's all too easy to overlook their significance — that all I can do to highlight their impact is to dress them up in different words, whose main advantage is being more jarring to the mind's sensibilities.

    • Peirce's mapping of letters to numbers, or logical terms to mathematical quantities, is the very core of what quantification theory is all about, definitely more to the point than the mere “innovation” of using distinctive symbols for the so-called quantifiers.

    • The mapping of logical terms to numerical measures, to express it in current language, would probably be recognizable as a “morphism” or “functor” from a logical domain to a quantitative co-domain.

    • Peirce follows the mathematician's usual practice of making the status of being an individual or being a universal relative to a discourse in progress.

    • Peirce takes the plural denotation of terms for granted. Indeed. what's the number of a term for, if it could not vary apart from being one or nil?

    • Peirce takes the individual objects of a particular universe of discourse in a generative way, as opposed to a totalizing way, and thus its contingent individuals afford us with a basis for talking freely about collections, constructions, properties, qualities, subsets, and higher types built on them.

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870

    #lor1870 #relationtheory #logicofrelatives #logic #peirce
  50. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 2.2
    • inquiryintoinquiry.com/2014/01

    Peirce's remarks at CP 3.65 are so replete with remarkable ideas, some of them so taken for granted in mathematical discourse as usually to escape explicit mention, others so suggestive of things to come in a future remote from his time of writing, and yet so smoothly slipped into the stream of thought it's all too easy to overlook their significance — that all I can do to highlight their impact is to dress them up in different words, whose main advantage is being more jarring to the mind's sensibilities.

    • Peirce's mapping of letters to numbers, or logical terms to mathematical quantities, is the very core of what quantification theory is all about, definitely more to the point than the mere “innovation” of using distinctive symbols for the so-called quantifiers.

    • The mapping of logical terms to numerical measures, to express it in current language, would probably be recognizable as a “morphism” or “functor” from a logical domain to a quantitative co-domain.

    • Peirce follows the mathematician's usual practice of making the status of being an individual or being a universal relative to a discourse in progress.

    • Peirce takes the plural denotation of terms for granted. Indeed. what's the number of a term for, if it could not vary apart from being one or nil?

    • Peirce takes the individual objects of a particular universe of discourse in a generative way, as opposed to a totalizing way, and thus its contingent individuals afford us with a basis for talking freely about collections, constructions, properties, qualities, subsets, and higher types built on them.

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870

    #lor1870 #relationtheory #logicofrelatives #logic #peirce
  51. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 2.1
    • inquiryintoinquiry.com/2014/01

    ❝§3. Application of the Algebraic Signs to Logic❞

    ❝Numbers Corresponding to Letters❞

    ❝I propose to use the term “universe” to denote that class of individuals about which alone the whole discourse is understood to run. The universe, therefore, in this sense, as in Mr. De Morgan's, is different on different occasions. In this sense, moreover, discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains.

    ❝I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men \((\mathrm{men}),\) the number of “tooth of” would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus, \([t].\)❞

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus
  52. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 2.1
    • inquiryintoinquiry.com/2014/01

    ❝§3. Application of the Algebraic Signs to Logic❞

    ❝Numbers Corresponding to Letters❞

    ❝I propose to use the term “universe” to denote that class of individuals about which alone the whole discourse is understood to run. The universe, therefore, in this sense, as in Mr. De Morgan's, is different on different occasions. In this sense, moreover, discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains.

    ❝I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men \((\mathrm{men}),\) the number of “tooth of” would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus, \([t].\)❞

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus
  53. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 1.2
    • inquiryintoinquiry.com/2014/01

    ❝The conjugative term involves the conception of third, the relative that of second or other, the absolute term simply considers an object. No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship. Whether this reason for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory of not, the fact itself is made perfectly evident by the study of the logic of relatives.❞

    One thing that strikes me about the above passage is a pattern of argument I can recognize as invoking a closure principle. This is a figure of reasoning Peirce uses in three other places: his discussion of continuous predicates, his definition of a sign relation, and his formulation of the pragmatic maxim itself.

    One might also call attention to the following two statements:

    ❝Now logical terms are of three grand classes.❞

    ❝No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.❞

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus
  54. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 1.2
    • inquiryintoinquiry.com/2014/01

    ❝The conjugative term involves the conception of third, the relative that of second or other, the absolute term simply considers an object. No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship. Whether this reason for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory of not, the fact itself is made perfectly evident by the study of the logic of relatives.❞

    One thing that strikes me about the above passage is a pattern of argument I can recognize as invoking a closure principle. This is a figure of reasoning Peirce uses in three other places: his discussion of continuous predicates, his definition of a sign relation, and his formulation of the pragmatic maxim itself.

    One might also call attention to the following two statements:

    ❝Now logical terms are of three grand classes.❞

    ❝No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.❞

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus
  55. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 1.1
    • inquiryintoinquiry.com/2014/01

    We pick up Peirce's text at the following point.

    ❝§3. Application of the Algebraic Signs to Logic❞

    ❝Use of the Letters❞

    ❝The letters of the alphabet will denote logical signs.

    ❝Now logical terms are of three grand classes.

    ❝The first embraces those whose logical form involves only the conception of quality, and which therefore represent a thing simply as “a ──”. These discriminate objects in the most rudimentary way, which does not involve any consciousness of discrimination. They regard an object as it is in itself as such (quale); for example, as horse, tree, or man. These are absolute terms.

    ❝The second class embraces terms whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation. These discriminate objects with a distinct consciousness of discrimination. They regard an object as over against another, that is as relative; as father of, lover of, or servant of. These are simple relative terms.

    ❝The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness of its origin. They regard an object as medium or third between two others, that is as conjugative; as giver of ── to ──, or buyer of ── for ── from ──. These may be termed conjugative terms.❞

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus
  56. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Selection 1.1
    • inquiryintoinquiry.com/2014/01

    We pick up Peirce's text at the following point.

    ❝§3. Application of the Algebraic Signs to Logic❞

    ❝Use of the Letters❞

    ❝The letters of the alphabet will denote logical signs.

    ❝Now logical terms are of three grand classes.

    ❝The first embraces those whose logical form involves only the conception of quality, and which therefore represent a thing simply as “a ──”. These discriminate objects in the most rudimentary way, which does not involve any consciousness of discrimination. They regard an object as it is in itself as such (quale); for example, as horse, tree, or man. These are absolute terms.

    ❝The second class embraces terms whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation. These discriminate objects with a distinct consciousness of discrimination. They regard an object as over against another, that is as relative; as father of, lover of, or servant of. These are simple relative terms.

    ❝The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness of its origin. They regard an object as medium or third between two others, that is as conjugative; as giver of ── to ──, or buyer of ── for ── from ──. These may be termed conjugative terms.❞

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus
  57. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Preliminaries 5
    • inquiryintoinquiry.com/2014/01

    Individual terms are taken to denote individual entities falling under a general term. Peirce uses upper case Roman letters for individual terms, for example, the individual horses \(\mathrm{H}, \mathrm{H}^{\prime}, \mathrm{H}^{\prime\prime}\) falling under the general term \(\mathrm{h}\) for horse.

    The path to understanding Peirce's system and its wider implications for logic can be smoothed by paraphrasing his notations in a variety of contemporary mathematical formalisms, while preserving the semantics as much as possible. Remaining faithful to Peirce's orthography while adding parallel sets of stylistic conventions will, however, demand close attention to typography-in-context.

    Current style sheets for mathematical texts specify italics for mathematical variables, with upper case letters for sets and lower case letters for individuals. So we need to keep an eye out for the difference between the individual \(\mathrm{X}\) of the genus \(\mathrm{x}\) and the element \(x\) of the set \(X\) as we pass between the two styles of text.

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus
  58. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Preliminaries 5
    • inquiryintoinquiry.com/2014/01

    Individual terms are taken to denote individual entities falling under a general term. Peirce uses upper case Roman letters for individual terms, for example, the individual horses \(\mathrm{H}, \mathrm{H}^{\prime}, \mathrm{H}^{\prime\prime}\) falling under the general term \(\mathrm{h}\) for horse.

    The path to understanding Peirce's system and its wider implications for logic can be smoothed by paraphrasing his notations in a variety of contemporary mathematical formalisms, while preserving the semantics as much as possible. Remaining faithful to Peirce's orthography while adding parallel sets of stylistic conventions will, however, demand close attention to typography-in-context.

    Current style sheets for mathematical texts specify italics for mathematical variables, with upper case letters for sets and lower case letters for individuals. So we need to keep an eye out for the difference between the individual \(\mathrm{X}\) of the genus \(\mathrm{x}\) and the element \(x\) of the set \(X\) as we pass between the two styles of text.

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus
  59. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Preliminaries 4
    • inquiryintoinquiry.com/2014/01

    Conjugative Terms (Higher Adic Relatives)
    • inquiryintoinquiry.files.wordp

    The Table displays the single-letter abbreviations and their verbal equivalents for the “conjugative terms” (or “higher adic relative terms”) used in Peirce's examples of logical formulas. Peirce used a distinctive typeface for the abbreviations of higher adic relative terms, rendered here as LaTeX “mathfrak”, Fraktur, or Gothic.

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus
  60. Jon Awbrey @[email protected] ·

    Peirce's 1870 “Logic of Relatives” • Preliminaries 4
    • inquiryintoinquiry.com/2014/01

    Conjugative Terms (Higher Adic Relatives)
    • inquiryintoinquiry.files.wordp

    The Table displays the single-letter abbreviations and their verbal equivalents for the “conjugative terms” (or “higher adic relative terms”) used in Peirce's examples of logical formulas. Peirce used a distinctive typeface for the abbreviations of higher adic relative terms, rendered here as LaTeX “mathfrak”, Fraktur, or Gothic.

    #Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
    #Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
    #PropositionalCalculus #PredicateCalculus #CategoryTheory

    #categorytheory #predicatecalculus #propositionalcalculus #logicalgraphs #mathematicallogic #logicalcalculus