"If the world were a logical place, men would ride sidesaddle. " - Rita Mae Brown.
A simple explanation of logic -- suitable for the level we need here -- is given in Compton's Interactive Encyclopedia:
Because of the number of possible interpretations, the subject matter of logic has variously been defined as "the laws of thought," "the rules of right reasoning," "the principles of valid argumentation," and "the study of truths (true statements) based solely on the terms they contain."
The last definition, and probably the least known one, implies the curious notion that the "facts" of logic need not necessarily coincide with the everyday realities of life. If a statement, or proposition, is analyzed solely within itself, the statement may be logically true even when contradicted by sense, experience, or knowledge. Or it may be logically true, even if the facts it presents are doubtful or uncertain. This is true because logic depends, first of all, on the ability to move from a premise to a conclusion based on the premise.
. . . Such a use of so-called true statements, without regard for the everyday realities, may seem frivolous and insignificant. In fact, such logic, which moves from premise to premise in a strict and logical sequence, is quite useful. . .
Excerpted from Comptonís Interactive Encyclopedia
Copyright © 1994, 1995 Comptonís NewMedia, Inc.
Here is another definition of logic extracted from a text book (The Art and Science of Logic by Daniel Bonevac, Mayfield Publishing Co., 1990):
Logic is the study of correct reasoning. Aristotle (384-322 B.C.) founded the discipline of logic as a system of principles on which all other knowledge rests. Indeed, logic pertains to all subjects, since people can reason about anything they can think about. Politics, the arts, literature, business, the sciences, and everyday problems are all subjects open to reasoning. Sometimes the reasoning is good; sometimes not so good. People use logic to tell the difference.
. . .
The twentieth century has witnessed remarkable advances in science and technology that have improved the lives of vast numbers of people. These applications of scientific method required a great deal of good and highly sophisticated reasoning. But the twentieth century has also suffered the results of reasoning gone astray [political and social disasters].
. . .
Logic is not the study of how people do reason, but how they should reason. We might put this point differently by saying that logic does not describe the psychology of reasoning, with its flashes of insight and oversight; it prescribes methods of justifying reasoning, that is, for showing that a given bit of reasoning is proper. [emphasis added]
An example of such logical reasoning (deductive logic) is:
All cows are purple.
Wilma is a cow.
Therefore, Wilma is purple.
Note that the argument is true regardless of the truth of the two premises. The use of deductive logic is extremely powerful in science, mathematics and philosophy.
You do not have to be a mathematician to use logic. Mathematicians and scientists often use symbols to represent logic because it facilitates obtaining a conclusion and is not nearly as subject to error as ordinary languages are. It is important to realize, however, that whether an argument is stated in ordinary language or in mathematical symbols, has nothing to do with its truth or falsity. It is tragic that many intelligent people reject the rules of logic because they see it as simply a mathematical artifice.
Another example of "False Dilemma" that is having serious consequences for society is that "since science is not absolutely certain of many of its claims and does make mistakes, mysticism must be correct". No, the fact that science is sometimes in error does not in any way establish that mysticism is correct. Mysticism must be evaluated on its on claims. When this is done, we find that, relatively, science is far superior to mysticism in explaining the world around us.
Another aspect of this problem is the "easy out" we humans have taken in ignoring the consequences of taking no action. The consequences of taking no action are just as likely to do harm (or good, in some cases) as taking an action. For example, suppose you were driving on a dark night through a rough neighborhood. As you round a curve, you see a young child wandering in the roadway, possibly lost. At first you start to stop, but then decide to move on as it is a dangerous neighborhood and the whole thing could be a setup. You drive on. A few seconds later the child is hit by another car and killed.
If you had hit the child with your car, you could be tried for manslaughter and possibly go to prison. Yet, the law and possibly your conscience, generally will not hold you accountable for taking no action. Yet the consequences are the same. An interesting book that goes into some detail on this peculiarity of law in which we are usually not held accountable for consequences but only for certain actions is Bad Acts and Guilty Minds by Leon Katz (University of Chicago Press, 1987). An enlightening essay on this subject, temporarily accessible on the internet, is "Nonconsequentialist decisions" by Jonathon Baron.