Logical Musings #2

I promised that I would continue with my discoveries, but as I sketched out the post ahead of time, I realized that it would work better if I backtrack and do a little history. In 1847, the British mathematician George Boole published a pamphlet on the Mathematical Analysis of Logic, and in 1854, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. At a later point in my independent study, I went back and read them, and found, as with many pioneering tracts, hardly anybody reads them anymore, and for good reason. Boole had the bright and revolutionary idea of using algebraic variables for whole classes of objects instead of just numbers, using the number 0 to represent falsehood and 1 to represent truth, and the addition operation to mean “or” and multiplication to mean “and”. His original methods were quite cumbersome and have been vastly improved and simplified, but modern symbolic formal logic can be traced to his work. Symbolic, because it uses symbols instead of words, and formal, because it is more concerned with form than content.

Boole dealt with statements about classes of objects, such as “All men are mortal”, “Some roses are red”, and “No king is a fink”. The terms “all”, “some”, and “none” are nowadays referred to as quantifiers, and the rest of the sentence is the predicate, so this branch of the subject is now referred to as predicate logic. Mathematics employs this branch of the subject.

A simpler version, which deals with particular statements, such as “Socrates is mortal, “This rose is red”, or “The King of Id is a fink”, is called propositional logic. This was developed somewhat after Boole’s work.

It was discovered fairly early after Boole, that it vastly simplified the calculation if “or” were taken in the inclusive sense, so that she “She wore jeans or she wore a dress” would be true if she wore both (jeans under the dress, for instance), instead of the exclusive sense. (She wore either jeans or she wore a dress, but not both.) The inclusive sense has become conventional in logic, while English uses the exclusive one. We won’t talk about the logically literate having fun with using the inclusive sense when the exclusive is intended, by answering such questions as “Do you want cake or ice cream” with”Yes”.

So, in propositional logic, every statement is given a truth value, which is either true or false. Other possible values are simply not discussed, which is one of the limitations of logic.

The “arithmetic” of logic is quite simple. For “or”, you have “True or true equals true”. “True or false equals true”. “False or true equals true.” False or false equals false.”

For “and”, you have “true and true equals true”. “True and false equals false.” “False and true equals false.” false and false equals false.

There is also negation “Not true equals false”, and “not false equals true”.

These are often summarized in truth tables. There is nothing mysterious about truth table: they just like addition and multiplication tables, only much simpler.

On the other hand, the algebra of logic, Boolean algebra, it is called, after Boole, is a bit more complicated. There, as in ordinary algebra, you use letters to stand for statements, or more exactly, the truth values of statements, and there are rules for the combination and simplification of variables and expressions. The goal is not so much to “solve for the unknown as it to “make the expression as simple as possible.

Since this isn’t an algebra lesson, (aren’t you thankful)? that’s as far as I will go right now. Unless you want be a computer scientist, mathematician, or logician, you probably won’t need or use it. For those few who someday might, I will assign homework. No, I won’t collect or grade it. Like writing a textbook, you’d have to pay me for that. But mathematical logic is not a spectator sport and if you don’t flog your brain through the exercises, you won’t learn it.

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