Thursday, January 17, 2013

Boole and Uninterpretable Terms

George Boole's The Laws of Thought is one of the key early works in the development of mathematical logic. From the beginning, however, there have been a number of objections to Boole's own system. I think some of these objections are due to reading the book as a book on mathematical logic. Despite the book's importance to the early history of mathematical logic, however, The Laws of Thought is something else entirely. It is what Boole explicitly says it is: an investigation of the fundamental laws of thought as part of "the philosophy of intellectual powers", insofar as they can be expressed by algebraic operations.

One of the longstanding puzzles and criticisms of Boole is his permissive attitude toward uninterpretable symbols. In Boole's use of algebra, some expressions admit of interpretation. For instance, this is the Boolean form of the principle of noncontradiction:


In this expression we multiply x times (1-x), which is interpreted as saying "both x and not-x". This is equal to 0, which is interpreted as Nothing (the opposite of 1, which is Universe of Discourse, or, in other words, everything we are talking about at the moment). However, it quickly becomes clear when we solve any problems other than the most simple that there are many steps in our transformation that have no interpretation at all. For instance, both of these are uninterpretable in Boole's system:


Both of these could be elements in the analysis of a particular logical argument, but neither of them has any logical interpretation: 1+x is uninterpretable because 1 is everything in the Universe of Discourse, so there can't be something more than 1, while -1-x is uninterpretable because -1 has no logical meaning, violating as it does one of the cornerstones of Boole's entire approach, the index law, x2=x. (I think there's an argument that Boole could have relented on 1+x, interpreting it as 'irrelevant', but that would still leave -1-x, and there's nothing that strictly requires that he interpret it that way, anyway.)

Boole has no problem with these uninterpretable steps at all, citing the fact that mathematicians use imaginary numbers without requiring an interpretation (they are, it turns out, interpretable, as coordinates for rotations, for instance, and Boole in fact recognizes that, but he points out that the use of imaginary numbers does not require mathematicians to have any interpretation of what they stand for). The basic principles of the system don't depend on the uninterpretable steps; whether an operation is used depends only on whether it applies to interpretable expressions, and, if applied to interpretable expressions, yields correct interpretable expressions as a result.

And as Boole notes, even when the expressions are interpretable, we must keep in mind that we are doing "Algebra of Logic", not algebra simply speaking. Take the following expression:

x + y + z = 1

This is a perfectly intelligible expression in Boole's system. It tells us that the Universe of Discourse is divided into three parts, x, y, and z. But since Boole's algebra is an algebra of 0's and 1's, there is no numerical solution if we assume that we really are talking about three classes of things; none of the three can be 0 without being nothing, and none of the three can be 1 without being everything. The x, y, and z aren't numbers; they are signs of concepts that have certain algebraic features.

Thus Boole is not building a 'logical system' in our sense. He's investigating the laws of thought as a physicist investigates the laws of motion. Nothing about the way a physicist proceeds requires that every mathematical symbol he uses has a physical interpretation; he's just interested in kinds of calculation that yield the right conclusions. Boole himself insists that his investigation is a posteriori, based on observation of what thinking fundamentally requires, not a priori. He also occasionally uses the analogy to physics. Any mathematics used is tailored to fit the elements of logical reasoning. If algebraic operations have properties that don't fit these elements, they are ignored; if an assumption has to be made to get a logical interpretation of an equation, it is made; and if an expression has no logical interpretation, it is still used if it gives workable results. When Boole talks about "laws of thought", he means "laws" in pretty much exactly the same sense as a physicist does when he talks about "laws". In a sense, Boole's work is an attempt to build a mathematical theory of intellectual motion. It is therefore not surprising that he has no problem with uninterpretable terms; they are even, perhaps, to be expected.

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