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© Copyright 2001, Jim Loy
In Symbolic Logic, a variable (A or some other letter) can have one of two values, true or false. And we have three operators, and (usually shown as an upside-down "v"), or (shown as a "v"), and not (A' is the negation of A, in other words it is true when A is false, and false when A is true). I will spell out the bold words, as I do not have all of the symbols. The following laws (theorems) hold:
Unlike the same words in English, these operators always produce a value, true or false. As you can see from the above laws, or is the "inclusive or." In other words, A or B is true if either A or B is true, or if both are true.
There are a couple other operators that amount to abbreviations of slightly more complicated expressions. A implies B is the equivalent of (not A) or B. This usage is similar to "if...then" in English, but not exactly. When A is false, A implies B is always true. The other operator is A if and only if B (or A is equivalent to B), which is true when A and B are both true or both false. These operators have symbols, which I cannot type here.
As in regular algebra, operators have precedence:
This means that we evaluate what is in parentheses (inner pair) first. Then we do the nots: (not A) and B can be written as not A and B, and so on. This helps get rid of parentheses. Of course (((A)))=A. Symbolic logic allows us to simplify very complicated statements: [A implies (not B if and only if C)] implies (A and C implies not B).
The variables (A or B) that I have been using stand for logical statements. B could be the statement "Jim is a man." Such statements can be true or false; in this case it happens to be true. A typical logic problem involves many such statements, strung together (sometimes they must all be true). So we can use the above laws to simplify. We can also use truth tables to do that. I will finish this article later.
Symbolic Logic is a form of Boolean Algebra. I took three courses in Symbolic Logic in college, from the Mathematics Department, from the Philosophy Department, and from the Computer Science Department, as I wanted to compare the various approaches. All were very similar. The mathematics version involved many more proofs. Also, Set Theory covered some of the same territory.
In Set Theory, and becomes the intersection of two sets (drawn as a large upside-down U); or becomes the union of two sets (drawn as a U); and not becomes the complement of a set (set A' is the complement of set A). A variable is not limited to two values, but the same laws hold.
In switching circuits (in Computer Science), true becomes 1, false becomes 0, the operators are x (two switches in series), + (two switches in parallel), and ' (A' is the complement of A, in other words it is 1 when A is 0, and 0 when A is 1). In this Computer Science class, we delved into the alternate operators NAND and NOR, which allowed the use of only two operators, instead of three.