stoutfellow

Gottfried Leibniz worked on a mathematized logic off and on for much of his career. His most successful work in that direction came in a series of essays which went unpublished in his lifetime; they did not see print, in fact, until late in the nineteenth century. They reflect some of the medieval work on logic, but they also foreshadow some of the ideas which were to fully emerge a century or two later. More under the cut.

The fundamental units that Leibniz worked with were what he called "terms", i.e., nouns and noun phrases - "triangle", "isosceles triangle", "God", "soul". These he represented by capital letters. He began by defining sameness and difference:

Terms which can be substituted for one another wherever we please without altering the truth of any statement are the same or coincident... Terms which are not the same, that is, terms which cannot always be substituted for one another, are different.

He indicated sameness with a conventional symbol for equality (not the familiar equals sign, but an alternative which has since largely disappeared). So far, there is nothing remarkable; nor is the next definition, of containment, anything new:

A is in L, or L contains A, is the same as to say that L can be made to coincide with a plurality of terms, taken together, of which A is one.

This begins to look like modern set theory, but it's not quite the same: square contains quadrilateral, not the other way around, by Leibniz' definition. It is not a question of sets containing one another, but of concepts - and the effect is to reverse the inclusions that we would expect.

Leibniz' notion of containment is essentially equivalent to the medieval notion of an affirmative universal; to say that square contains quadrilateral is to say that every square is a quadrilateral. But with his next step Leibniz does something new: he defines logical addition. If A and B are two terms, A+B denotes the combination of those terms: if A means "equilateral polygon" and B means "triangle", then A+B means "equilateral triangle". (Recognizing that this is not the familiar addition, Leibniz uses a different symbol to represent it, enclosing the usual plus sign in a circle. Unfortunately, that symbol doesn't seem available in HTML.) Leibniz proceeds to prove a number of interesting algebraic properties of this logical addition, including such familiar-looking facts as commutativity (A+B=B+A), but also odder properties such as absorption (A+A=A).

[It is worth mentioning that Leibniz admits the possibility of logically adding terms which seem unrelated - "God, soul, body, point, and heat" is his example - or contradictory - "quadrilateral" and "trilateral". He comes close to something like the later concept of empty set, thus, but seems to shy away at the last minute.]

Medieval logic dealt with relations between terms; Leibniz took the step of introducing operations on terms, opening the door to analogies between logic and algebra. His work, in some respects, prefigures the later set theory (which is the appropriate venue for predicate logic), but also the algebraic logic that George Boole and Augustus DeMorgan were to develop (and this was propositional in nature). His adherence to the idea of terms rendered some of his work a bit ambiguous (for example, he attempted to define a sort of "logical subtraction", under which, e.g., "equilateral triangle" - "triangle" = "equilateral polygon"), and ultimately a different approach proved more workable, but the key idea, of an algebra of logic, was his. This was to prove enormously fruitful.

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