#mathematicallogic — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #mathematicallogic, aggregated by home.social.
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MVspa_ Stochastic processes
Стохастические процессы, Випадкові процеси
#Stochastic processes
#Стохастическиепроцессы, #Випадковіпроцеси
t.me/scilib_yura15cbx/230MVsa Statistics and applications
Статистика та додатки, Статистика и приложения
#Statistics
t.me/scilib_yura15cbx/229EM_ Mechanics of elastic materials
Механіка пружних матеріалів, Механика упругих материалов
t.me/scilib_yura15cbx/228MPd_ Dynamical systems
Динамические системы, Динамічні системи
#Dynamical systems
t.me/scilib_yura15cbx/227MNw_ Wavelets, signal processing
Вейвлети, обробка сигналів, Вейвлеты, обработка сигналов
#Wavelets, #signal processing
t.me/scilib_yura15cbx/226MNs_ Simulations of physics
Физические симуляции
t.me/scilib_yura15cbx/225MNl_ Numerical linear algebra
Числова лінійна алгебра, Численная линейная алгебра
t.me/scilib_yura15cbx/224MNf_ Finite elements
Конечные элементы, Скінченні елементи
t.me/scilib_yura15cbx/223MNd_ Numerical calculus
t.me/scilib_yura15cbx/222MDgt_ General topology
#topology
Общая топология, Загальна топологія
t.me/scilib_yura15cbx/221MDdg_ Differential geometry
Диференціальна геометрія, Дифференциальная геометрия
t.me/scilib_yura15cbx/220MDat_ Algebraic and differential topology
Алгебраическая и дифференциальная топология, Алгебраїчна та диференціальна топологія
t.me/scilib_yura15cbx/219MCv_ Variational calculus
Варіаційне числення, Вариационное исчисление
t.me/scilib_yura15cbx/218MCta_ Tensor calculus, forms
Тензорное исчисление, формы, Тензорне числення, форми
t.me/scilib_yura15cbx/217MCsf_ Special functions
Спеціальні функції, Специальные функции
t.me/scilib_yura15cbx/216MCf_ Functional analysis
Функциональный анализ, Функціональний аналіз
t.me/scilib_yura15cbx/215MCet_ Elementary calculus textbooks
Підручники з елементарного числення, Учебники по элементарному исчислению
t.me/scilib_yura15cbx/214MCde_ Differential equations
Дифференциальные уравнения, Диференціальні рівняння
#Differential equations
t.me/scilib_yura15cbx/213MCcf_ Continued fractions
Суцільні дроби, Непрерывные дроби
t.me/scilib_yura15cbx/212MCc_ Complex variable
Комплексная переменная, Комплексна змінна
t.me/scilib_yura15cbx/211MCat_ Advanced calculus
t.me/scilib_yura15cbx/210Asymptotics, perturbations
Асимптотика, збурення,
Асимптотика, возмущения
#Asymptotics, #perturbations
#Асимптотика, #збурення,
#Асимптотика, #возмущения
t.me/scilib_yura15cbx/209Group theory, Теория групп, Теорія груп
#Group theory, #Теориягрупп, #Теоріягруп
t.me/scilib_yura15cbx/208Algebra textbooks,
Підручники з алгебри,
Учебники по алгебре
t.me/scilib_yura15cbx/207Set theory,
Теория множеств, Теорія множин
#Set theory,
#Теория множеств,
t.me/scilib_yura15cbx/206Representation theory
Теорія репрезентації, Теория представлений
t.me/scilib_yura15cbx/205Quantum groups
#Quantum groups
Квантовые группы, Квантові групи
t.me/scilib_yura15cbx/204Mathematical logic
#Mathematical logic
Математична логіка, Математическая логика
t.me/scilib_yura15cbx/203Linear algebra
#Linear algebra
Линейная алгебра, Лінійна алгебра
t.me/scilib_yura15cbx/202Homology, Гомологія, Гомология
#Homology, #Гомологія, #Гомология
t.me/scilib_yura15cbx/201Algebraic geometry
Алгебраическая геометрия, Алгебраїчна геометрія
t.me/scilib_yura15cbx/200Algebraic geometry
#Algebraic geometry
Алгебраическая геометрия, Алгебраїчна геометрія
t.me/scilib_yura15cbx/200Finite groups
Скінченні групи, Конечные группы
t.me/scilib_yura15cbx/199Differential algebra
#Differential algebra
Дифференциальная алгебра, Диференціальна алгебра
t.me/scilib_yura15cbx/198Category theory
Теорія категорій, Теория категорий -
MVspa_ Stochastic processes
Стохастические процессы, Випадкові процеси
#Stochastic processes
#Стохастическиепроцессы, #Випадковіпроцеси
t.me/scilib_yura15cbx/230MVsa Statistics and applications
Статистика та додатки, Статистика и приложения
#Statistics
t.me/scilib_yura15cbx/229EM_ Mechanics of elastic materials
Механіка пружних матеріалів, Механика упругих материалов
t.me/scilib_yura15cbx/228MPd_ Dynamical systems
Динамические системы, Динамічні системи
#Dynamical systems
t.me/scilib_yura15cbx/227MNw_ Wavelets, signal processing
Вейвлети, обробка сигналів, Вейвлеты, обработка сигналов
#Wavelets, #signal processing
t.me/scilib_yura15cbx/226MNs_ Simulations of physics
Физические симуляции
t.me/scilib_yura15cbx/225MNl_ Numerical linear algebra
Числова лінійна алгебра, Численная линейная алгебра
t.me/scilib_yura15cbx/224MNf_ Finite elements
Конечные элементы, Скінченні елементи
t.me/scilib_yura15cbx/223MNd_ Numerical calculus
t.me/scilib_yura15cbx/222MDgt_ General topology
#topology
Общая топология, Загальна топологія
t.me/scilib_yura15cbx/221MDdg_ Differential geometry
Диференціальна геометрія, Дифференциальная геометрия
t.me/scilib_yura15cbx/220MDat_ Algebraic and differential topology
Алгебраическая и дифференциальная топология, Алгебраїчна та диференціальна топологія
t.me/scilib_yura15cbx/219MCv_ Variational calculus
Варіаційне числення, Вариационное исчисление
t.me/scilib_yura15cbx/218MCta_ Tensor calculus, forms
Тензорное исчисление, формы, Тензорне числення, форми
t.me/scilib_yura15cbx/217MCsf_ Special functions
Спеціальні функції, Специальные функции
t.me/scilib_yura15cbx/216MCf_ Functional analysis
Функциональный анализ, Функціональний аналіз
t.me/scilib_yura15cbx/215MCet_ Elementary calculus textbooks
Підручники з елементарного числення, Учебники по элементарному исчислению
t.me/scilib_yura15cbx/214MCde_ Differential equations
Дифференциальные уравнения, Диференціальні рівняння
#Differential equations
t.me/scilib_yura15cbx/213MCcf_ Continued fractions
Суцільні дроби, Непрерывные дроби
t.me/scilib_yura15cbx/212MCc_ Complex variable
Комплексная переменная, Комплексна змінна
t.me/scilib_yura15cbx/211MCat_ Advanced calculus
t.me/scilib_yura15cbx/210Asymptotics, perturbations
Асимптотика, збурення,
Асимптотика, возмущения
#Asymptotics, #perturbations
#Асимптотика, #збурення,
#Асимптотика, #возмущения
t.me/scilib_yura15cbx/209Group theory, Теория групп, Теорія груп
#Group theory, #Теориягрупп, #Теоріягруп
t.me/scilib_yura15cbx/208Algebra textbooks,
Підручники з алгебри,
Учебники по алгебре
t.me/scilib_yura15cbx/207Set theory,
Теория множеств, Теорія множин
#Set theory,
#Теория множеств,
t.me/scilib_yura15cbx/206Representation theory
Теорія репрезентації, Теория представлений
t.me/scilib_yura15cbx/205Quantum groups
#Quantum groups
Квантовые группы, Квантові групи
t.me/scilib_yura15cbx/204Mathematical logic
#Mathematical logic
Математична логіка, Математическая логика
t.me/scilib_yura15cbx/203Linear algebra
#Linear algebra
Линейная алгебра, Лінійна алгебра
t.me/scilib_yura15cbx/202Homology, Гомологія, Гомология
#Homology, #Гомологія, #Гомология
t.me/scilib_yura15cbx/201Algebraic geometry
Алгебраическая геометрия, Алгебраїчна геометрія
t.me/scilib_yura15cbx/200Algebraic geometry
#Algebraic geometry
Алгебраическая геометрия, Алгебраїчна геометрія
t.me/scilib_yura15cbx/200Finite groups
Скінченні групи, Конечные группы
t.me/scilib_yura15cbx/199Differential algebra
#Differential algebra
Дифференциальная алгебра, Диференціальна алгебра
t.me/scilib_yura15cbx/198Category theory
Теорія категорій, Теория категорий -
In the mountains, I love exploring nature and climbing any little hidden corner.
My academic interests include mathematics (#mathematics) and mathematical logic ( #mathematicallogic #logic), particularly model theory ( #modeltheory), category theory ( #categorytheory), higher-order logic and metalogic ( #metalogic), as well as the philosophy of mathematics and logic ( #philosophyofmathematics #philosophyoflogic), epistemology and ontology of them.
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Hi,
I'm an associate professor at Department of Engineering, University of Fukui. I'm interested in theoretical computer science, software engineering, mathematical logic, also related philosophical topics. If you want to study in Fukui, please let me know.
My recent papers:
Mathematics:
Beckmann, A., & Yamagata, Y. (2025). On proving consistency of equational theories in bounded arithmetic. The Journal of Symbolic LogicTheoretical Computer Science:
Ikeda, M., Yamagata, Y., & Kihara, T. (2024). On the Metric Temporal Logic for Continuous Stochastic Processes. Logical Methods in Computer Science,Software Engineering:
Yamagata, Y., Liu, S., Akazaki, T., Duan, Y., & Hao, J. (2020). Falsification of cyber-physical systems using deep reinforcement learning. IEEE Transactions on Software EngineeringPhilosophy:
Suzuki, U., & Yamagata, Y. (2023). Notion of validity for the bilateral classical logic. arXiv preprint arXiv:2310.13376.#Logic #MathematicalLogic #BoundedArithmetic #SoftwareEngineering
#Philosophy
#PhilosophicalLogic
#PhilosophyOfLanguage -
Mathematics is either inconsistent or incomplete.
What is your philosophical interpretation of Gödel’s incompleteness theorems?
#math #mathematics #maths #logic #MathematicalLogic #gödel #kurtgödel #godel #KurtGodel #Philosophy #PhilosophyOfMathematics #PhilosophicalLogic #philosophyoflogic #logic
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@LeoTsai14 While they do not actually call it so, set theoreticians do a lot of work in a category in which the objects are the models of set theory and the arrows are the elementary embeddings (https://en.wikipedia.org/wiki/Elementary_equivalence#Elementary_embeddings) between them.
Models of (ZFC-like) set theories have the interesting property that the maps between them are to some amount determined by the mappings between their classes of ordinals: If this map is an isomorphism, the whole map is one (https://en.wikipedia.org/wiki/Critical_point_(set_theory)).
You may also have a look at inner model theory (https://en.wikipedia.org/wiki/Inner_model_theory), I think. -
Do we have any Gödel experts in the house?
I'm trying to understand why Gödel used this encoding in the original Gödel numbering. To be clear, these are supposed to be the exponents in the unique factorisation 2ᵃ3ᵇ5ᶜ⋯, not the bases which are also coincidentally prime.
Why did Gödel pick prime exponents too?
And why did is the 0 assigned to the exponent 1 and not 2? Why is that first one not prime?
At first I thought this might be a typo in the inset figure in Wikipedia, but upon consulting the cited reference I saw that the same encoding is used in the original paper. https://en.wikipedia.org/wiki/G%C3%B6del_numbering#G%C3%B6del's_encoding
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Logic of Relatives
• https://inquiryintoinquiry.com/2024/08/05/logic-of-relatives-a/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.
• https://www.jstor.org/stable/25058006
• https://archive.org/details/jstor-25058006
• https://books.google.com/books?id=fFnWmf5oLaoCResources —
Charles Sanders Peirce
• https://mywikibiz.com/Charles_Sanders_PeirceRelation Theory
• https://oeis.org/wiki/Relation_theorySurvey of Relation Theory
• https://inquiryintoinquiry.com/2024/03/23/survey-of-relation-theory-8/Peirce's 1870 Logic of Relatives
• https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/
• https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview#Peirce #Logic #LogicOfRelatives #MathematicalLogic
#Mathematics #RelationTheory #Semiotics #SignRelations -
Peirce's 1870 “Logic of Relatives” • Selection 3.2
• https://inquiryintoinquiry.com/2014/01/30/peirces-1870-logic-of-relatives-selection-3/❝§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 -
Related question: Is Tarski's fixed point theorem constructive, predicative?
Since Tarski's fixed point theorem can be proved by a sort of transfinite induction, how the answer of this question telated to the previous question?
#mathematicallogic #settheory -
A question: is transfinite induction constructive?
Sure it would depend on the definition of ordinal, but how?
#mathematicallogic #settheory -
Peirce's 1870 “Logic of Relatives” • Selection 2.1
• https://inquiryintoinquiry.com/2014/01/29/peirces-1870-logic-of-relatives-selection-2/❝§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 -
Peirce's 1870 “Logic of Relatives” • Selection 1.2
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/❝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 -
Peirce's 1870 “Logic of Relatives” • Selection 1.1
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/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 -
Peirce's 1870 “Logic of Relatives” • Preliminaries 5
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/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 -
Hi,
I'm a researcher interested in theoretical computer science, software engineering, mathematical logic, also related philosophical topics.
My recent papers:
Mathematics:
"On proving consistency of equational theories in Bounded Arithmetic". Arnold Beckmann and Yoriyuki Yamagata, preprint: https://arxiv.org/abs/2203.04832"Consistency proof of a fragment of PV with substitution in bounded arithmetic." Yoriyuki Yamagata, The Journal of Symbolic Logic 2018: https://arxiv.org/abs/1411.7087
Software Engineering:
(2020). "Falsification of cyber-physical systems using deep reinforcement learning", Yamagata, Y., Liu, S., Akazaki, T., Duan, Y., & Hao, J, IEEE Transactions on Software Engineering, 47(12), 2823-2840 (2021), https://staff.aist.go.jp/yoriyuki.yamagata/paper/falsify.pdfPhilosophy:
, "On the notion of validity for the bilateral classical logic", Suzuki, Ukyo & Yamagata, Yoriyuki, preprint: https://philpapers.org/rec/SUZOTNExtra: COVID19
"Individual-based epidemiological model of COVID19 using location data". In 2022 IEEE International Conference on Big Data (Big Data) (pp. 4434-4442). IEEE., https://staff.aist.go.jp/yoriyuki.yamagata/paper/covid19.pdf#Logic #MathematicalLogic #BoundedArithmetic #SoftwareEngineering
#Philosophy
#PhilosophicalLogic
#PhilosophyOfLanguage
#COVID19 #Introduction -
Peirce's 1870 “Logic of Relatives” • Preliminaries 4
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/Conjugative Terms (Higher Adic Relatives)
• https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-conjugative-terms-higher-adic-relatives-2.0.pngThe 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 -
Peirce's 1870 “Logic of Relatives” • Preliminaries 3
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/Simple Relative Terms (Dyadic Relatives)
• https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-simple-relative-terms-dyadic-relatives-2.0.pngThe Table displays the single-letter abbreviations and their verbal equivalents for the “simple relative terms” (or “dyadic relative terms”) used in Peirce's examples of logical formulas. Peirce used a distinctive typeface for the abbreviations of dyadic relative terms, rendered here as LaTeX “mathit” or Italics.
#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
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Peirce's 1870 “Logic of Relatives” • Preliminaries 2
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/Absolute Terms (Monadic Relatives)
• https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-absolute-terms-monadic-relatives-2.0.pngThe Table displays the single-letter abbreviations and their verbal equivalents for the “absolute logical terms” (or “monadic relative terms”) used in Peirce's examples of logical formulas throughout the rest of the paper. Peirce used a distinctive typeface for the absolute term abbreviations, rendered here as LaTeX “mathrm” or Roman.
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#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory -
Peirce's 1870 “Logic of Relatives” • Preliminaries 1
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/In the beginning was the three-pointed star,
One smile of light across the empty face;
One bough of bone across the rooting air,
The substance forked that marrowed the first sun;
And, burning ciphers on the round of space,
Heaven and hell mixed as they spun.— #DylanThomas • #InTheBeginning
Peirce’s text uses lower case letters for logical terms of general reference and upper case letters for logical terms of individual reference. General terms fall into types, namely, absolute terms, dyadic relative terms, and higher adic relative terms, which Peirce distinguishes through the use of different typefaces. The following Tables show the typefaces used in the present transcript for Peirce's examples of general terms. (I'll post just the image links for now, then the full images and texts in the next three posts.)
Absolute Terms (Monadic Relatives)
• https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-absolute-terms-monadic-relatives-2.0.pngSimple Relative Terms (Dyadic Relatives)
• https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-simple-relative-terms-dyadic-relatives-2.0.pngConjugative Terms (Higher Adic Relatives)
• https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-conjugative-terms-higher-adic-relatives-2.0.png#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory -
Peirce’s 1870 “Logic of Relatives” • Overview
• https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/My long ago encounter with Peirce’s 1870 paper, “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, was one of the events precipitating my return from the hazier heights of philosophy to the solid plains of mathematics below. Over the years I copied out various drafts of my study notes to the web, consisting of selections from Peirce’s paper along with my running commentary. A few years back I serialized what progress I had made so far to this blog and this Overview consists of links to those installments.
#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory -
Peirce's 1870 “Logic of Relatives” • Selection 1.8
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/❝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.❞
(Peirce, CP 3.63)
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#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs -
#Peirce's 1870 “#LogicOfRelatives” • Selection 1.7
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/❝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.❞
#Logic #RelationTheory #LOR1870
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Peirce's 1870 “Logic of Relatives” • Selection 1.6
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/❝The conjugative term involves the conception of “third”, the relative that of second or “other”, the absolute term simply considers “an” object.❞
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#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs -
Peirce's 1870 “Logic of Relatives” • Selection 1.5
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/❝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.❞
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#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs -
Peirce's 1870 “Logic of Relatives” • Selection 1.4
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/❝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.❞
#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs