Talk:Abstract algebraic logic
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[edit] I question some of the history of ideas set out in this entry
Boole did not discover any isomorphism or model theoretic relation between the propositional connectives and his eponymous algebra, because those connectives were discovered only well after his lifetime. (Gergonne had something like the horseshoe for the conditional as early as 1815, but nobody noticed.) Peirce thought a lot about this topic, but almost no one read him. No one read Frege, period. Hugh McColl groped towards this connection, but he was poorly informed about Boolean algebra; he stumbled on sentential logic while investigating a problem in calculus! The connection may be in Schroeder, but even he did not have a clear idea of the connectives. Also, Schroeder's 3 volume treatise was published at his own expense, and intellectuals tend to discount strongly anything published by a vanity press.
I submit that Peano discovered the connectives, and Principia Mathematica made them well known. Hilbert's course on mathematical logic, 1917-22, which became the 1st ed. of Hilbert and Ackermann (1928), did much to make a variant of the PM notation known in central and eastern Europe. Hilbert and Ackermann was probably also the first undergraduate exposition of the connectives. When Quine began teaching intro undergrad logic in 1940, he could find no suitable text in English and so wrote the first edition of his Elementary Logic. The 1937 first edition of Suzanne Langer's logic text emphasized Boolean algebra over the notation of PM. Hilbert and Ackermann did not appear in English until 1950.
In any event, the masters of the connectives became the interwar Poles. A member of that schook, Tarski, was perhaps the first to be fully at ease with both the connectives and Boolean algebra. It was he and his student Adolf Lindenbaum that finally made the connection and the name Lindenbaum-Tarski algebra honors this fact.
It is insufficiently appreciated that Boole did not discover what we call Boolean algebra. The algebra of Boole is a muddle of only historical interest. For Boole, product and sum did not form a dual pair of operations, because he interpreted sum as exclusive OR. In the 1860s, Jevons and the young Charles Peirce both saw that interpreting sum as inclusive OR resulted in a much neater algebra, because in that case sum and product formed what we now call a dual pair of operations. Appreciation of the resulting Boolean algebra was slow in coming, with Schroeder probably being the first. Some years later, Peano, the British logician William Johnson (the author of 3 masterly articles in the 1892 Mind on algebraic logic), and A. N. Whitehead who wrote the 1898 treatise Universal Algebra all saw the light. In 1904, Huntington's postulates finally set this house in order.
The more I read about mathematical logic, the greater my admiration for Alfred Tarski.
Source: Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots. Princeton Univ. Press. Palnot (talk) 05:25, 21 January 2008 (UTC)
