stoutfellow

Other than the calculus, Leibniz' most important work lay in the area of logic. I've neglected to talk about the history of logic, though, primarily because I know far less about it than about the history of mathematics. Still, I'd like to give at least a cursory treatment to the development of logic in Greece and in the Middle Ages. (There were interesting developments in India and China as well, but I know even less about them, and they don't appear to have had much influence on Western thought.)

Logic has two principal subfields, propositional logic and predicate logic. The former concerns itself with the ways in which statements can be combined or modified by logical operators, such as "and", "or", and "not". It does not deal with anything smaller than a full statement. Predicate logic, by contrast, is sometimes called the logic of noun expressions, and it does pay attention to units smaller than statements. It also, and crucially, concerns itself with quantifiers - expressions involving, e.g., "all", "no", or "some". (The actual range of conceivable quantifiers is much wider than this, but these three are the ones most often considered.)

Predicate logic is an extension of propositional logic; everything that can be done in the latter can also be done in the former, and more. Nonetheless, as a historical matter predicate logic was developed first, and has always drawn more attention. (Propositional logic did emerge soon afterward, but I won't be discussing that at this point.)

It was Aristotle who focused attention on what he called "categorical" statements, which were of four types, divided into affirmative and negative statements, and also into universal and particular statements:

Affirmative Universal: "All x is y"
Negative Universal: "No x is y"
Affirmative Particular: "Some x is y"
Negative Particular: "Some x is not y"

He then proceeded to describe various syllogisms, valid arguments constructed from categorical statements. A typical example:

All y is z
All x is y
Therefore, all x is z

The general structure of Aristotle's syllogisms is the same. There are three terms involved, major, minor, and middle. The major term (z) is the predicate of the conclusion; the minor (x) is the subject; the middle term (y) appears in both premises but not in the conclusion. The syllogism above is in the first figure, meaning that the major term is the predicate of the premise it appears in, and the minor is the subject of its premise. There were three other figures, depending on the location of the major and minor terms in their respective premises. (Not all of this can be traced to Aristotle; some of his proposals were rejected by his successors, and the fourth figure - major term in the subject position, minor in the predicate - was not fully accepted until later.)

In the Middle Ages, the four types of categorical statement were assigned letters - A, E, I, and O respectively - and the various syllogisms were given names. The syllogism above is the "syllogism in Barbara"; the three "a"'s indicate that all three statements are affirmative universal. The other first-figure syllogisms were called "Celarent", "Darii", and "Ferio" (and it shouldn't be too hard to work out their forms, from what I've said above). Syllogisms of the other figures had similar names, and elaborate rules were devised for identifying valid syllogisms and translating them into the first figure.

The study of the categorical syllogisms was a considerable achievement, but it was ultimately sterile. The symbolism did not open out towards further ideas (even though extended syllogisms with four or more statements had been studied), and the full richness of quantification was not recognized. (It is not hard to construct statements containing more than one quantifier - "You can fool all of the people some of the time" - but there are subtleties involved, and the medieval notation simply wasn't equipped to handle them. What subtleties? Let me point out that the sentence I just quoted has two quite distinct meanings; I'll leave it to you to figure them out.)

The first step towards a more adequate predicate logic came at Leibniz' hands, and that will be the subject of the next Ramble.

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