Sunday, September 02, 2007

Logical Epochs

I find it interesting that philosophers often have a tendency to speak of logic as going through two epochs: the traditional, or Aristotelian, in which syllogistic reigns, and the modern, governed by the predicate calculus. Of course, there are many more epochs than this, but it's easy for people to do it. Thus when we get into other epochs in logic, like the splendid era started by Boole and De Morgan, we find that there's often an attempt to assimilate it to one or the other. Englebretsen, working in a modern extension of the traditional term logic, regards Carroll as the last great traditional logician; it's not uncommon for people used to the predicate calculus to treat Boole et al. as a sort of prolegomena to Frege.

Of course, the great algebraic logicians -- Boole, De Morgan, Jevons, John Neville Keynes, Lewis Carroll, John Venn -- are not so easily shunted to one side or another. They certainly saw themselves as making a sharp break with traditional logic -- which they would have understood in terms of the old staple of Aldrich's Artis logicae compendium, published in 1691 but still the standard reference for logic up until Whately published his Elements of Logic in 1826 (and in many places in England for yet longer). (The reader of George Eliot might recall poor Maggie trying to read Tom's school edition of Aldrich, and not being able to divine how the work related to the living world.) No one can read Venn's Symbolic Logic without recognizing that this is in his mind a radically different approach than anything traditional logic has to offer. And there is something to it. He regards the traditional emphasis on particular propositions (I and O) to be largely misguided -- one can put the traditional syllogistic, with I and O so prominent, in Boolean terms, but that is merely one permutation, and not an essential feature at all; logicians only work with universal propositions, and only deign to bother with particular propositions insofar as these can be considered incomplete universal propositions. One also finds that he reviewed Frege's work; it was a scathingly contemptuous review. He regarded Frege's notation as clumsy and generally not well-suited to application (which everyone agrees with) and his logical work to be far inferior to the algebraic logic being done in the wake of Boole (which is rather more controversial).

It's also true, I think, that there tends to be a complete difference in logical approach. It's dangerous to lump all the pre-Boolean work into the category 'traditional logic'; that ignores the fact that, for instance, the Ramists or the Cartesians tried to incite revolutions against what they perceived as the Aristotelian oppressor. And the sort of traditional logic the algebraic logicians would have known would have been a mix of simplified school manuals and Whately's adulterated revival version. But we can perhaps say to some extent that whereas logicians in the Aristotelian mode are intently focused on demonstration, the Booleans were not. They are very helpful in telling us what they are obsessed with, and it is not demonstration at all. Jevons puts it succinctly in his Philosophical Transactions: "Boole first put forth the problem of Logical Science in its complete generality: Given certain logical premisses or conditions, to determine the description of any class of objects under those conditions." As Keynes puts it in a passage in Studies and Exercises in Logic, where he quotes Jevons on this point:
Given any number of universal propositions involving any number of terms, to determine what is all the information they jointly afford with regard to any given term or combination of terms.

This is closely related to another obsession of algebraic logicians, the finding and eliminating of superfluous premises. It is not something we generally do much any more, at least as anything more than an occasional technical exercise; but for them the whole point of logical analysis is to determine precisely the most the premises give you and to determine precisely how you can get your conclusion without going over the same ground twice. But the predicate calculus is not used this way; I imagine that if there's an obsession it has brought about, it is identified by Quine in that passage in which he says that the whole point of logical grammar is to facilitate the tracing of truth conditions. One sees this difference in a number of ways they approach things. The algebraic logicians are generally dismissive of any attempt to reason about objects without specifying a domain of discourse; any symbolic system, for instance, that does not clearly specify a universe of discourse is regarded, by that very fact, as deficient. This does, in fact, tie in with their interest in the information afforded by terms in universal propositions. It is often difficult to get people trained in predicate calculus even to take universes of discourse seriously.

Logic, then, at least to a limited extent is not merely a set of formal considerations; it involves an approach, and what you are trying to do plays a key role in how the whole operation works. What counts as success, what counts as failure, what counts as even important, shifts depending on your view of the ends of logical thinking.

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