Symbolic Logic

Symbolic logic is a body of rules that allows you to objectively determine the truth value of an arbitrarily complex series of declarative statements. For example, the statement “The world is flat and pigs fly” is false. The statement “Hydrogen atoms have 4 protons and Nietzsche is dead” is false despite the moribund state of Nietzsche. On the other hand, the statement “Hydrogen atoms have 4 protons or Nietzsche is dead” is true. Symbolic logic is a formalisation of these rules.

For the purposes of this class, you will be expected to be able to manipulate 4 very simple rules to determine the truth value of a variety of lengthy statements. Why? Because this type of formalized logical thinking lies at the heart of scientific thinking, even if it is rarely made explicit. Further, I believe the exercise of solving complicated looking problems by recognizing simple, soluble parts and proceeding slowly and consistently will help you order your thoughts and gain confidence in the method. And what I say goes for the rest of the millennium (let’s not quibble about the silly 2000 or 2001 thing. Those who cling to the ‘the new millennium begins in 2001’ thing are welcome to obey my every whim for the additional year).

Now for the brass tacks. There are only operators you will need to be familiar with: AND, OR, IF...THEN and NOT. Note that the use of and in that sentence means you need to know ALL of them, not just 1. See? You already know this stuff. What you may not know are the symbols that are sometimes used to replace them. Let’s look at each in turn:

NOT A (~A)
A
(~A)
True
False
False
True

Very simple: if A is true, “NOT A” is false. Henceforth, I’ll use T and F for the truth values of the variables.

 

A and B (A & B)
A
B
A & B
T
T
True
T
F
False
F
T
False
F
F
False

In other words, the statement “A and B”, which can be symbolized “A & B” is TRUE if A and B are both true, and FALSE if A is false, B is false, or A and B are both false.

A or B (A V B)
A
B
A V B
T
T
True
T
F
True
F
T
True
F
F
False

The statement “A or B” (A V B) is FALSE only if A and B are both false. By the way, symbolic logic,encompasses such statements as “if and only if”, “there exists an A such that”, “Every A...”. But you won’t need to deal with these.

If P then Q (P … Q)
P
Q
P … Q
T
T
True
T
F
False
F
T
True
F
F
True

This makes sense to you in plain English: if I say “If it is Friday, then there is no class today”, the remark is true if it is Friday and no class is had, but false if it is Friday AND class occurs. On the other hand, if it is not Friday, the statement is deemed true regardless of the occurrence of class. Note that this function can actually be re-stated as

~P V ~(P & ~ Q)

It may take you a couple moments, but look at that carefully until you see the sense of it. Here’s a verbalization: “the statement ’If P then Q’ is true IF P is false (i.e. ~P is true) OR if it is not the case that P is TRUE and Q is false”. That’s just a restatement of the table, if you look at it.

PUTTING IT ALL TOGETHER

By using these simple rules, it is trivial to assign a truth value to arbitrarily complex statements like:

((P & Q) V ((P … Q) & (P V Q))) & R

once you are told whether P, Q and R are true or false. You do need to know one more simple rule: solve things INSIDE the parens first. Thus you would first solve (P & Q) and (P … Q) and (P V Q). You would then use those solutions to solve ((P … Q) V (P … Q)). That, in turn, would allow you to solve ((P & Q) V ((P … Q) & (P V Q))), etc. Of course, you can also look to simplify matters. The outermost statement is (Blah blah blah) & R. You know that for an AND statement to be true, both components must be true, so if R is false, you’re done—the statement is false! Let’s solve the equation for some values of P, Q and R:

((P & Q) V ((P … Q) & (P V Q))) & R;
You are given that P is True, Q is False, R is True. So the statement becomes

((T & F) V ((T … F) & (T V F))) & T

Now, solving the innermost parentheses: (T & F) is F, (T … F) is F and (T V F) is T. Thus:

(F V (F & T)) & T

We again solve the innermost statement, (F & T) is F, thus

(F V F) & T

Solve in the parentheses (F V F) is F, yielding

F & T

Which everybody knows is false. So the big, intimidating looking statement

((T & F) V ((T … F) & (T V F))) & T is FALSE if P and R are True and Q is False.

The only other twist to all this is to use actual statements, which we will do. Thus instead of (P & Q) you will sometimes be confronted with “The moon is made of blue cheese and gobbledygook is gerflicktenschnaken”. This statement you can deduce to be false, regardless of whether I ever inform you of the truth value of gobbledygook. Ah, the power of careful scientific reasoning! Many examples will soon be appearing for you to try you hand at. Of course, now that you know the trick of it, you can make up your own examples and determine their truth values on your own and with your friends, who will find you to be an exceptionally engaging dinner companion.