home.social
Sign in Create account

#algebraoflogic — Public Fediverse posts

Live and recent posts from across the Fediverse tagged #algebraoflogic, aggregated by home.social.

  1. 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
  2. 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
  3. 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

    #peirce #logic #algebraoflogic #logicofrelatives #relationtheory #categorytheory
  4. 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
  5. 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
  6. 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

    #categorytheory #computerscience #peirce #logic #algebraoflogic #logicofrelatives
  7. Jon Awbrey @[email protected] ·

    Peirce's 1885 “Algebra of Logic” • Selection 3
    • inquiryintoinquiry.com/2024/03

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

    ❝§1. Three Kinds Of Signs❞ (cont.)

    ❝For instance, take the syllogistic formula,
    \[\begin{array}{clcl}\text{All}&M&\text{is}&P\\&S&\text{is}&M\\\therefore&S&\text{is}&P.\end{array}\]

    ❝This is really a diagram of the relations of \(S,\) \(M,\) and \(P.\) The fact that the middle term occurs in the two premisses is actually exhibited, and this must be done or the notation will be of no value.

    ❝As for algebra, the very idea of the art is that it presents formulæ which can be manipulated, and that by observing the effects of such manipulation we find properties not to be otherwise discerned. In such manipulation, we are guided by previous discoveries which are embodied in general formulæ. These are patterns which we have the right to imitate in our procedure, and are the icons par excellence of algebra. The letters of applied algebra are usually tokens, but the \(x, y, z,\) etc. of a general formula, such as

    \[ (x + y)z = xz + yz, \]

    ❝are blanks to be filled up with tokens, they are indices of tokens. Such a formula might, it is true, be replaced by an abstractly stated rule (say that multiplication is distributive); but no application could be made of such an abstract statement without translating it into a sensible image.❞ (3.363).

    #Peirce #Logic #AlgebraOfLogic #RelationTheory
    #Semiotics #Icon #Index #Symbol #Quantification

    #quantification #symbol #index #icon #semiotics #relationtheory
  8. Jon Awbrey @[email protected] ·

    Peirce's 1885 “Algebra of Logic” • Selection 2.3
    • inquiryintoinquiry.com/2024/03

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

    ❝§1. Three Kinds Of Signs❞ (cont.)

    ❝With these two kinds of signs alone any proposition can be expressed; but it cannot be reasoned upon, for reasoning consists in the observation that where certain relations subsist certain others are found, and it accordingly requires the exhibition of the relations reasoned with in an icon.

    ❝It has long been a puzzle how it could be that, on the one hand, mathematics is purely deductive in its nature, and draws its conclusions apodictically, while on the other hand, it presents as rich and apparently unending a series of surprising discoveries as any observational science.

    ❝Various have been the attempts to solve the paradox by breaking down one or other of these assertions, but without success. The truth, however, appears to be that all deductive reasoning, even simple syllogism, involves an element of observation; namely, deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts.❞ (3.363).

    #Peirce #Logic #AlgebraOfLogic #RelationTheory
    #Semiotics #Icon #Index #Symbol #Quantification

    #quantification #symbol #index #icon #semiotics #relationtheory
  9. 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