Logic

Overview

Logic is a set of rules by which decisions and conclusions are either derived or inferred from a set of statements. Logic can be mathematical or predicate (dealing with statements and sentences). Logic is also a set of rules by which computers handle data, and circuit logic dictates how many devices operate.

However, a logical decision or a belief may or may not be correct. Logic is more of a set of rules to follow in reaching a decision. An example of a logical, but incorrect, bit of reasoning is the following: If I believe that sheep have a wool coat and that all sheep are mammals, then it could make logical sense for me to believe that all mammals have wool coat. The conclusion is incorrect, but it is logically drawn.

Fundamental Mathematical Concepts and Terms

Over twenty-four centuries ago, the idea of logic was explored and developed about the same time in China, India, and Greece. The Greek philosopher Aristotle (384 B.C.–322 B.C.) was important in the creation of logical systems.

REASONING

Logic does not necessarily lead to the truth. What logic does do is to allow us to look at an argument and to decide if the reasoning is valid or not valid. Logic also points out how we can come to believe something that is not true (even though that sounds illogical).

PROPOSITION AND CONCLUSION

The starting point of a logical line of thought is called the proposition (or the statement). A proposition is the real meaning of the sentence (or the equation, as it can be written in mathematical language also). The meaning can be expressed in different ways and still mean the same thing. For example "Today is Friday" and "Yesterday was Thursday" are the same proposition, while "My name is Brian" is a different proposition.

A proposition is always true or false, although it is sometimes unknown which proposition is true and which is false. "There is life on Mars" is an example of a proposition that may or may not be true; we have yet to find out.

Logic proceeds from the starting point of the proposition to the conclusion in a series of steps that are related

Proposition	Steps	Conclusion
True	Support the conclusion	Always true
True	Do not support the conclusion	Can be true or false
False	Support the conclusion	Can be true or false
False	Do not support the conclusion	Can be true or false

to each other. That is, one step is followed by a step that supports it.

Here is an example of a logical series of steps:

Today is Friday.
My library books were due Thursday.
My library books are overdue.

Here is an example of a series of steps that is not logical:

The moon is full.
There are clouds in the sky.
My cat has a hairball.

From the proposition, the steps that proceed to the conclusion can be set up so that the steps guarantee that the conclusion is true. This is a good style to use when debating. There is no middle ground with this type of approach. Either all the steps lead to a single conclusion or they do not.

A number of different outcomes can still result, depending on whether the steps from the proposition to the conclusion support this conclusion. Table 1 summarizes these various possibilities.

A less rigid style is when the steps from the proposition to the conclusion support the likelihood of the conclusion. In this style a conclusion does not have to be true, it is just likely to be true. Points can be presented that support the conclusion, but the conclusion could still be debatable. This style of logic is used in many courtrooms by lawyers trying to defend their clients from charges brought against them.

Real-life Applications

BOOLEAN LOGIC

Many persons do most of their banking while sitting at their desk. This is possible since they can hook up to the local bank's Web site, research bank accounts, and then use the computer directions built into the site to shift money from one account to another, pay bills, and look at the action in each account over whatever time period is desired.

These activities are pretty human-like. How can computers do them? The answer is something called Boolean logic.

Boolean logic is named after the Irish mathematician George Boole (1815–1864). From an early age, Boole showed a talent for languages and teaching. When he was 20, Boole began to teach himself mathematics. He proved to be talented at this as well, publishing papers in the leading math journals of the day. When he was 34 years old, he was appointed chair of mathematics at Queens College in Cork, Ireland. He taught there for the rest of his life.

In 1854, when he was only 39 years old, Boole published a paper called "An Investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilites." The ideas in this paper became the basis of Boolean logic.

One niche that Boolean logic has filled beautifully is the task of sifting through vast amounts of information to find those bits of information that are desired.

FUZZY LOGIC

Fuzzy logic is a way of making computers behave in a way that is similar to the way humans think. Often, we are able to use information that is not really clear or precise to make decisions that are definite. We can relate the imprecise (fuzzy) information with what we already know to make a decision.

Here is an example. You are driving your car on a crowded, four-lane freeway. The speed limit is 65 mph (105 km/h). As is usually the case, traffic is moving faster, at an average speed 70 mph (113 km/h). You know that it is safest for you and those around you to drive "with the traffic." But what exactly does driving "with the traffic" mean?

Watching other drivers, you realize that driving "with traffic" is done different ways. Some drivers will drive more slowly and stay in the right hand lane. Other drivers will speed and zig-zag their way between cars and lanes. Usually, the different styles mesh together to make a smooth flow of traffic. When they do not, there a traffic accident can occur.

Fuzzy logic was conceived by Lotfi Zadeh, a professor of electrical engineering at the University of California at Berkley, and was first proposed in a 1965 paper. From its humble beginnings, fuzzy logic has expanded to assume an important role in our daily lives. For example, because of fuzzy logic, the computer inside a video camera is able to keep focusing even when the camera is jostled. As another example, fuzzy logic makes it possible to program a microwave oven to cook differently sized and types of foods perfectly with the touch of one button.

The logic of fuzzy logic can be summed up as IF X AND Y THEN Z. It is the 'if' and 'and' that makes things less precise.

The following example may help to make this fuzziness clearer. A conventional oven operates on the basis of exact temperature. A thermometer in the oven can cut off the power to the oven's heater when the oven reaches whatever temperature has been selected, and will kick the heater back into action when the temperature falls below another set value. This occurs no matter what is in the oven.

A microwave with a fuzzy logic temperature control does not rely on exact temperatures. Instead, the process is like this: "IF (the process is too cool) AND (the process is getting colder) THEN (add more heat)", or "IF (the process is too hot) AND (the process is getting colder) THEN (heat it up now)."

Boolean Logic and Computer Searches

Boolean logic links the common parts of different pieces of information. This feature makes Boolean logic widely used in Internet search engines. For example, if there was no Boolean logic and information from the Internet on the trigonometry and homework problems was desired, the Internet search for every word would show all the documents that separately mention "trigonometry" or "homework." This would probably result in a huge number of sites to search, making the search nearly meaningless. Because of Boolean logic, however, a search can be done to look for those documents that contain "trigonometry" AND "homework." This number of sites will be much less, and the sites will be more likely to have something to do with homework related to trigonometry rather than homework related to all subjects.

Boolean logic even allows a search to focus on one word and not another. To use the above example, the following search could be done: "trigonometry" AND "homework" NOT "advanced." This would allow the search engine to zero in on those site that were about teaching methods of trigonometry at a basic level as opposed to sites that discussed advanced trigonometry.

Companies have leapt on fuzzy logic as a way of making products that will perform better for people. Self-focusing cameras and video recorders, washing machines that can adjust the strength of cleaning power to how much dirt is in the clothes being washed, the controls to car engines, anti-lock braking systems in vehicles, banking programs, programs that allow people to do stock market trades—all these would not exist if not for fuzzy logic.