## The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations (1991)

Citations: | 40 - 2 self |

### BibTeX

@MISC{Maddux91theorigin,

author = {Roger D. Maddux},

title = {The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations},

year = {1991}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

211 |
On the caldus of relations
- Tarski
- 1941
(Show Context)
Citation Context ...t Peirce thought that "Professor Schroder attaches, it seems to me, too high a value on this alge94 [Lyndon1956], p. 307, footnote 13. 95 See [McKenzie1966], pp. 37--39, or [McKenzie1970], p. 286=-=. 96 [Tarski1941], p. -=-74. 25 bra." 97 Nevertheless, Schroder's work deserves much more attention than it has been given. (2) It was noted above that Lowenheim's Theorem, now conceived as the first theorem in model the... |

121 |
A Formalization of Set Theory Without Variables
- Tarski, Givant
- 1987
(Show Context)
Citation Context ...[DPP1911]. For another brief history of the calculus of relations, including remarks on the forty years following Peirce's summary, see the first four paragraphs of [Tarski1941], pp. 73--74. See also =-=[Tarski-Givant1987], p. xv, a-=-nd [Lewis1918]. 9 See [Peirce1933], Volume III, pp. 404--409, for Peirce's article on relatives. The historical sketch is paragraph 3.643. (References to [Peirce1933] of the form "n:m" refer... |

99 | Cylindric algebras - Henkin, Monk, et al. - 1971 |

98 |
Introduction to Mathematical Logic
- Church
- 1956
(Show Context)
Citation Context ...per, Peirce and Mitchell share credit with Frege for the introduction of quantifiers. 53 45 [Peirce1885]. 46 [Peirce1933], 3.366--370. The first explicit use of truth values appears [Peirce1885]; see =-=[Church1956]-=-, footnote 67, p. 25. 47 [Peirce1933] 3.376--384. 48 [Peirce1933], 3.387. 49 [Peirce1933], 3.396. 50 [Peirce1933], 3.398--400. 51 [Peirce1933], 3.398--400. 52 [Peirce1933], 3.393. 53 [Church1956], foo... |

82 |
Threshold Logic
- Lewis, Coates
- 1967
(Show Context)
Citation Context ...tory of the calculus of relations, including remarks on the forty years following Peirce's summary, see the first four paragraphs of [Tarski1941], pp. 73--74. See also [Tarski-Givant1987], p. xv, and =-=[Lewis1918]. 9 See [P-=-eirce1933], Volume III, pp. 404--409, for Peirce's article on relatives. The historical sketch is paragraph 3.643. (References to [Peirce1933] of the form "n:m" refer to paragraph number m i... |

65 |
The representation of relational algebras
- Lyndon
- 1950
(Show Context)
Citation Context ...y Peirce himself. 9 Literature: relatives have, since Aristotle, been a recognized topic of logic. The first germ of the modern doctrine appears in a somewhat trivial remark of Robert Leslie Ellis. 3 =-=[Lyndon1950]-=-, x8. 4 [Boole1847]. 5 [Lowenheim1915]. 6 See [Moore1987] for more details. 7 [Schroder1895]. 8 See [DPP1911]. For another brief history of the calculus of relations, including remarks on the forty ye... |

61 | The Origin of Relation Algebras - Maddux - 1991 |

56 |
Distributive and Modular Laws in the Arithmetic of Relation Algebras
- Chin, Tarski
- 1951
(Show Context)
Citation Context ...er two are different from relative addition. 30 The empty relation ; is a function since it contains no two pairs which begin with the same element and end with different elements. 31 It is stated in =-=[Chin-Tarski1951]-=- that the relative sum of two functions may not be a function. This mistake was noted in [Monk1961]. 32 [Peirce1983], p. 189, and [Peirce1933], 3.332. 33 [Peirce1983], p. 189, and [Peirce1933], 3.333.... |

49 |
Vorlesungen über die Algebra der Logik
- Schröder
- 1905
(Show Context)
Citation Context ...f logic. The first germ of the modern doctrine appears in a somewhat trivial remark of Robert Leslie Ellis. 3 [Lyndon1950], x8. 4 [Boole1847]. 5 [Lowenheim1915]. 6 See [Moore1987] for more details. 7 =-=[Schroder1895]-=-. 8 See [DPP1911]. For another brief history of the calculus of relations, including remarks on the forty years following Peirce's summary, see the first four paragraphs of [Tarski1941], pp. 73--74. S... |

49 | Some varieties containing relation algebras - Maddux - 1982 |

45 | On representable relation algebras - Monk - 1964 |

37 |
The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning
- Boole
(Show Context)
Citation Context ...Literature: relatives have, since Aristotle, been a recognized topic of logic. The first germ of the modern doctrine appears in a somewhat trivial remark of Robert Leslie Ellis. 3 [Lyndon1950], x8. 4 =-=[Boole1847]-=-. 5 [Lowenheim1915]. 6 See [Moore1987] for more details. 7 [Schroder1895]. 8 See [DPP1911]. For another brief history of the calculus of relations, including remarks on the forty years following Peirc... |

29 |
Über Möglichkeiten im Relativkalkül. Mathematische Annalen
- Löwenheim
- 1915
(Show Context)
Citation Context ...atives have, since Aristotle, been a recognized topic of logic. The first germ of the modern doctrine appears in a somewhat trivial remark of Robert Leslie Ellis. 3 [Lyndon1950], x8. 4 [Boole1847]. 5 =-=[Lowenheim1915]-=-. 6 See [Moore1987] for more details. 7 [Schroder1895]. 8 See [DPP1911]. For another brief history of the calculus of relations, including remarks on the forty years following Peirce's summary, see th... |

29 | On the algebra of logic - Peirce |

27 |
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 Sciences 9
- Peirce
- 1870
(Show Context)
Citation Context ...t, the logic of absolute terms, which, 10 [De Morgan1864b]. 11 Volume 10, pp. 331--358. 12 De Morgan's second and third papers are [De Morgan1856] and [De Morgan1864a]. 13 [De Morgan1966], p. 208. 14 =-=[Peirce1870]-=-. 15 See, for example, [Brink1978], [Brink1979], [Brunning1980], [Martin1976], [Martin1978], and [Merrill1984]. 16 See [Brink1981] and [Brink1988]. 3 when he wrote, was the only formal logic known. Th... |

26 | New sets of independent postulates for the algebra of logic - Huntington - 1933 |

25 |
Sets of independent postulates for the algebra of logic
- Huntington
- 1904
(Show Context)
Citation Context ...ns which is true in every domain of individuals is derivable from the axioms adopted under the second method? This problem presents some difficulties and remains open. I can only say that I am 65 See =-=[Huntington1904]-=- and [Tarski1941], p. 78, footnote 3. 17 practically sure that I can prove with the help of the second method all of the hundreds of theorems to be found in Schroder's Algebra und Logik der Relative. ... |

22 | Heijenoort, From Frege to Gödel. A Source Book - van - 1967 |

18 | Boolean modules - Brink - 1981 |

18 | Representations of integral relation algebras - McKenzie - 1970 |

18 | Principia Mathematica, volume I - Whitehead, Russell - 1910 |

14 | The representation of relation algebras - McKenzie - 1966 |

12 | On the algebra of logic: A contribution to the philosophy of notation - Peirce - 1885 |

10 | Algebras with Operators - Boolean - 1952 |

10 |
varieties containing relation algebras
- Some
- 1982
(Show Context)
Citation Context ... simplest conditions in this latter characterization which could possibly 90 This term is not used by Lyndon. It was borrowed from the theory of cylindric algebras and applied to relation algebras in =-=[Maddux1982]-=-. 91 See [J'onsson-Tarski1951], Definition 3.8. 92 [J'onsson-Tarski1951], Theorem 3.9. 93 See [Lyndon1950], p. 710. A slightly different first order characterization is given in [Maddux1978 ], Theorem... |

8 |
Studies in cylindric algebra. Doctoral Dissertation
- Monk
- 1961
(Show Context)
Citation Context ...o pairs which begin with the same element and end with different elements. 31 It is stated in [Chin-Tarski1951] that the relative sum of two functions may not be a function. This mistake was noted in =-=[Monk1961]-=-. 32 [Peirce1983], p. 189, and [Peirce1933], 3.332. 33 [Peirce1983], p. 189, and [Peirce1933], 3.333. 8 When the predicate is itself the subject of a relation, there may be a composition: thus if X::L... |

7 | Note B: the logic of relatives - PEIRCE |

6 |
Lattice theory, in: Colloquium Publications, vol
- Birkhoff
- 1967
(Show Context)
Citation Context ... result is reported in [Lowenheim1915], Theorem 1. See [vanHeijenoort], p. 233. 70 [Kwatinetz1981]. 71 Relation algebras have been defined in many different ways by various authors; see, for example, =-=[Birkhoff1948]-=-, Ch. XIII, x5, pp. 209--211. Such definitions are not pursued here, since the relation algebras of this paper are just the ones due to Tarski. 72 [J'onsson-Tarski1948], received October 21, 1947, pre... |

5 | On the symbols of logic, the theory of the syllogism, and in particular of the copula, and the application of the theory of probabilities to some questions in the theory of evidence, (read February 25, 1850 - MORGAN |

5 | Representation problems for relation algebras, Abstract 89 - JONSSON, TARSKI - 1948 |

5 | Problems of Expressibility in Finite Lan guages - KWATINETZ - 1981 |

4 |
On Peirce's notation for the logic of relatives
- Brink
- 1978
(Show Context)
Citation Context ...ch, 10 [De Morgan1864b]. 11 Volume 10, pp. 331--358. 12 De Morgan's second and third papers are [De Morgan1856] and [De Morgan1864a]. 13 [De Morgan1966], p. 208. 14 [Peirce1870]. 15 See, for example, =-=[Brink1978]-=-, [Brink1979], [Brunning1980], [Martin1976], [Martin1978], and [Merrill1984]. 16 See [Brink1981] and [Brink1988]. 3 when he wrote, was the only formal logic known. The object of this paper is to show ... |

4 | Contributions to the theory of models. III, Koninklijke Nederlandse Akademie van Wetenschappen - Tarski |

3 |
The representation of relation algebras, Doctoral Dissertation
- McKenzie
(Show Context)
Citation Context ...mmary. Schroder developed Peirce's algebra from Note B, but Peirce thought that "Professor Schroder attaches, it seems to me, too high a value on this alge94 [Lyndon1956], p. 307, footnote 13. 95=-= See [McKenzie1966], pp.-=- 37--39, or [McKenzie1970], p. 286. 96 [Tarski1941], p. 74. 25 bra." 97 Nevertheless, Schroder's work deserves much more attention than it has been given. (2) It was noted above that Lowenheim's ... |

3 | A Chronological Edition, edited by Edward - Peirce - 1984 |

3 |
to the theory of models, III, Koninklijkle Nederlandse Akademie van Wetenschappen
- Contributions
- 1955
(Show Context)
Citation Context ...7 See [J'onsson-Tarski1951], p. 897, for further comments on this situation. 78 See [Chin-Tarski1951], p. 352, footnote 10, p. 354, footnote 12, and [J'onsson-Tarski1952], p. 128, footnote 15. 79 See =-=[Tarski1955]-=-, p. 60, and [Tarski1956], x3. 80 [Tarski-Givant1987], Definition 8.1. 81 [Huntington1933]. 20 However, Peirce made no mistake, for (B16)--(B19) do hold in ReU , which is the relation algebra of all b... |

3 | Equationally complete rings and relation algebras, Koninklijke Nederlandse Akademie van Wetenschappen - Tarski |

2 |
sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell's Principia
- New
- 1933
(Show Context)
Citation Context ...hin-Tarski1951], p. 352, footnote 10, p. 354, footnote 12, and [J'onsson-Tarski1952], p. 128, footnote 15. 79 See [Tarski1955], p. 60, and [Tarski1956], x3. 80 [Tarski-Givant1987], Definition 8.1. 81 =-=[Huntington1933]-=-. 20 However, Peirce made no mistake, for (B16)--(B19) do hold in ReU , which is the relation algebra of all binary relations on U . 82 The second distinguishing feature of the relation algebraic appr... |

2 | algebra. A correction - Boolean - 1933 |

2 |
A house divided against itself: The emergence of first-order logic as the basis for mathematics
- Moore
- 1987
(Show Context)
Citation Context ...totle, been a recognized topic of logic. The first germ of the modern doctrine appears in a somewhat trivial remark of Robert Leslie Ellis. 3 [Lyndon1950], x8. 4 [Boole1847]. 5 [Lowenheim1915]. 6 See =-=[Moore1987]-=- for more details. 7 [Schroder1895]. 8 See [DPP1911]. For another brief history of the calculus of relations, including remarks on the forty years following Peirce's summary, see the first four paragr... |

2 | Studies in Logic by Members of the Johns Hopkins University (1883), edited by - PEIRCE - 1983 |

1 |
Peirce's Development of the Algebra of Relations, Doctoral dissertation
- Brunning
- 1980
(Show Context)
Citation Context ...1 Volume 10, pp. 331--358. 12 De Morgan's second and third papers are [De Morgan1856] and [De Morgan1864a]. 13 [De Morgan1966], p. 208. 14 [Peirce1870]. 15 See, for example, [Brink1978], [Brink1979], =-=[Brunning1980]-=-, [Martin1976], [Martin1978], and [Merrill1984]. 16 See [Brink1981] and [Brink1988]. 3 when he wrote, was the only formal logic known. The object of this paper is to show that an affirmative answer ca... |

1 |
Dictionary of Philosophy and Psychology, second edition, edited by
- Martin
- 1911
(Show Context)
Citation Context ...m of the modern doctrine appears in a somewhat trivial remark of Robert Leslie Ellis. 3 [Lyndon1950], x8. 4 [Boole1847]. 5 [Lowenheim1915]. 6 See [Moore1987] for more details. 7 [Schroder1895]. 8 See =-=[DPP1911]-=-. For another brief history of the calculus of relations, including remarks on the forty years following Peirce's summary, see the first four paragraphs of [Tarski1941], pp. 73--74. See also [Tarski-G... |

1 |
Problems of expressibility in finite languages, Doctoral dissertation
- Kwatinetz
- 1981
(Show Context)
Citation Context ...bols for the distinguished 66 [Tarski1941], pp. 87--88. 67 [Lyndon1950], x8. 68 See [Tarski1941], p. 88. 69 Korselt's result is reported in [Lowenheim1915], Theorem 1. See [vanHeijenoort], p. 233. 70 =-=[Kwatinetz1981]-=-. 71 Relation algebras have been defined in many different ways by various authors; see, for example, [Birkhoff1948], Ch. XIII, x5, pp. 209--211. Such definitions are not pursued here, since the relat... |

1 |
Some comments on De Morgan, Peirce, and the logic of relations
- MARTIN
- 1976
(Show Context)
Citation Context ... 331--358. 12 De Morgan's second and third papers are [De Morgan1856] and [De Morgan1864a]. 13 [De Morgan1966], p. 208. 14 [Peirce1870]. 15 See, for example, [Brink1978], [Brink1979], [Brunning1980], =-=[Martin1976]-=-, [Martin1978], and [Merrill1984]. 16 See [Brink1981] and [Brink1988]. 3 when he wrote, was the only formal logic known. The object of this paper is to show that an affirmative answer can be given to ... |

1 |
servants, lovers, and benefactors: Peirce's algebra of relatives of 1870
- Of
- 1978
(Show Context)
Citation Context ...De Morgan's second and third papers are [De Morgan1856] and [De Morgan1864a]. 13 [De Morgan1966], p. 208. 14 [Peirce1870]. 15 See, for example, [Brink1978], [Brink1979], [Brunning1980], [Martin1976], =-=[Martin1978]-=-, and [Merrill1984]. 16 See [Brink1981] and [Brink1988]. 3 when he wrote, was the only formal logic known. The object of this paper is to show that an affirmative answer can be given to this question.... |

1 |
The 1870 logic of relatives memoir
- MERRILL
(Show Context)
Citation Context ... and third papers are [De Morgan1856] and [De Morgan1864a]. 13 [De Morgan1966], p. 208. 14 [Peirce1870]. 15 See, for example, [Brink1978], [Brink1979], [Brunning1980], [Martin1976], [Martin1978], and =-=[Merrill1984]-=-. 16 See [Brink1981] and [Brink1988]. 3 when he wrote, was the only formal logic known. The object of this paper is to show that an affirmative answer can be given to this question. I think there can ... |

1 |
description of the algebra of relatives, privately printed
- Brief
(Show Context)
Citation Context ... gleaned from Peirce's Note B. In this section, any reference of the form "(Bn)", with 1sns50, refers to that list. 17 [Peirce1933], 3.45. 18 The most important papers are [Peirce1870], [Pei=-=rce1880], [Peirce1882], [Pe-=-irce1883b], [Peirce1885], [Peirce1892] and [Peirce1897]. 19 This is the second footnote in [Peirce1903]; see [Peirce1933], 3.574. 4 Let U be an arbitrary nonempty set, called the "universe of dis... |

1 | in Logic by Members of the Johns Hopkins University, edited by - Studies |

1 |
B: the logic of relatives
- Note
(Show Context)
Citation Context ...08A02, 08A40. Key words and phrases. Calculus of relations, relation algebras, algebraic logic, first order logic, history of logic, De Morgan, Peirce, Schroder, Tarski, Lyndon. 1 [De Morgan1864b]. 2 =-=[Peirce1883b]-=-. Typeset by A M S-T E X 1 to the question of whether there are relation algebras which are not representable. Both questions were answered by Roger Lyndon 3 , as we shall see in x6. The theory of rel... |