A. De Morgan. On the syllogism, no. IV, and on the logic of relations. Trans. Cambridge Phil. Soc., 10:331--358, 1864.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....R, together with a ternary relation (x; a; y) expressing that elements x and y of X are related by the binary relation a, an element of R. We require that if a 6= b then there exist x; y such that exactly one of (x; a; y) and (x; b; y) hold; this ensures that each element of R acts 1 De Morgan [DM64] writes ab, ab 0 , and a 0 b for a; b, a b, and a, b respectively, construing them as an a of a b of , an a of every b of , and an a of none but b s of, and asserting their sufficiency. Peirce [Pei33, 3.242,1880] observed their interdefinability, notating them ab, a b , a b and ....

.... We will find it convenient later on to break down (RKR) and (LKL) into four universal Horn formulas, a; b c b anc (KR) b anc a; b c (RK) a; b c a c=b (KL) a c=b a; b c: LK) The letter K in the names of these formulas connotes the theorem De Morgan refers to as Theorem K [DM64]. This theorem, which was brought to my attention by Roger Maddux, amounts to the assertion of KR and KL; RK and LK can be derived from these given a = a. The L and R refer to the left and right residuals respectively, and their placement relative to K indicates the direction of the implication. ....

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A. De Morgan. On the syllogism, no. IV, and on the logic of relations. Trans. Cambridge Phil. Soc., 10:331--358, 1864.

....LL, closely parallels that of relation algebras, RA. The latter amounts to two copies of the logical connectives or, false, and, true, not, and implies, distinguished as the logical and relative (relational) forms of those connectives, due to Peirce but anticipated to some extent by by De Morgan [DM60]. To these Schroder [Sch95] added reflexive transitive closure a 0 , nowadays a , and its De Morgan dual a 1 . Combining the separate involutory logical and relative duals, a Gamma and a, as a single involutory (a = a) dual a Gamma = a [Pra92a, p.252] weakens the Boolean ....

A. De Morgan. On the syllogism, no. IV, and on the logic of relations. Trans. Cambridge Phil. Soc., 10:331--358, 1860.

....calculus of binary relations. Chu spaces amount to K valued binary relations, which for K = 2 n we show generalize n ary relational structures. We also exhibit a four stage unique factorization system for Chu transforms that illuminates their operation. 1 Introduction In 1860 A. De Morgan [DM60] introduced a calculus of binary relations equivalent in expressive power to one whose formulas, written in today s notation, are inequalities a b between terms a; b; built up from variables with the operations of composition a; b, converse a, and complement a Gamma . In 1870 C.S. Peirce ....

A. De Morgan. On the syllogism, no. IV, and on the logic of relations. Trans. Cambridge Phil. Soc., 10:331--358, 1860.

....to the language. These are the Ajdukiewicz monoids [Ajd37] brought to greater prominence two decades later by Lambek [Lam58] The term residuation was coined by Ward and Dilworth [WD39] That the binary relations on a set formed a residuated ISR was first observed by De Morgan as his Theorem K [DM60]. The subclass BSR of Boolean semirings has the property that the semilattice structure forms a Boolean algebra; for this it suffices for an antimonotone operation of negation :a to exist and satisfy : a = a. This yields classical logic with a second conjunction : a :b) distinct (in general) ....

A. De Morgan. On the syllogism, no. IV, and on the logic of relations. Trans. Cambridge Phil. Soc., 10:331--358, 1860.

....the basic logical or implication order. Modal logic originated with Aristotle c.330 BC, making it the earliest 2D logic by a substantial margin. The next 2D logic to appear, and the first with two conjunctions, is the calculus of binary relations, introduced in primitive form by De Morgan in 1860 [DM60]. The two dimensions of this calculus are what Peirce [Pei33] subsequently called its logical and relative parts. Conjunction (and dually disjunction) took two forms, logical conjunction as the intersection of relations L and M and relative conjunction as their composition, notated L; M by Peirce ....

A. De Morgan. On the syllogism, no. IV, and on the logic of relations. Trans. Cambridge Phil. Soc., 10:331--358, 1860.

....and the virtually unknown principle of what we call pure induction used by Ng and Tarski to adjoin transitive closure to the language of relation algebras to obtain a finitely based variety RAT. 1 Residuation was noticed early on by De Morgan in his study of the calculus of binary relations [DM64]. He called the phenomenon Theorem K in remembrance of the office of that letter in [the syllogistic forms] Baroko and Bokardo. Ward and Dilworth christened and studied residuation [WD39] the concluding chapters of Birkhoff s Lattice Theory gave it considerable coverage [Bir67] and it has ....

....logics. In the realm of propositional logics, action logic is a close cousin to classical logic, intuitionistic logic, relevance logic, linear logic, relation algebras, regular expressions, and dynamic logic. Of these, perhaps its closest cousin is relation algebra, begun in 1860 by De Morgan [DM64] and further developed by Peirce [Pei33] Schroder [Sch95] Tarski [Tar41] and J onsson [JT51] and many others since. Classical and intuitionistic logics are pure static logics, while regular expressions constitute a pure action logic 6 . Relevance logic, linear logic, relation algebras, ....

A. De Morgan. On the syllogism, no. IV, and on the logic of relations. Trans. Cambridge Phil. Soc., 10:331--358, 1864.

....a century later Tarski, J onsson, Lyndon, and Monk further developed the calculus from the perspective of modern model theory. 1 The Calculus The origins of the calculus of binary relations go back to 1860 in a paper by Augustus De Morgan, On the Syllogism: IV; and on the Logic of Relations [DM60]. De Morgan begins his paper by categorizing Aristotle as rather too much the expositor of common language, too little the expositor of common thought. Aristotle had denied, in the 4th century BC, that every binary relation has a converse. His example was that the rudder of the ship lacked the ....

....(x; z) in b, and is the greatest c such that c; a b. Setting b to 0 leads to the definition of converse in terms of either residual, namely a = a Gamma n0 = 0 =a Gamma . 2 A Brief Chronology As noted above De Morgan was the founder of the calculus, writing a single paper on it in 1860 [DM60]. Though he did not develop an equational calculus he nevertheless completely identified the essence of residuation as we shall see later. Peirce turned to the subject in 1870, writing several papers and a good many unpublished manuscripts 1 The custom now is to take it to be the maximal binary ....

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A. De Morgan. On the syllogism, no. IV, and on the logic of relations. Trans. Cambridge Phil. Soc., 10:331--358, 1860.

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