1_1 Logic: Resolving the recursive nature of truth.

"Every statement in this lesson is false."
A: The car is red.
C: Statement B is true.
B: Statement A is false.
D: Statement C is false.
...

Why do we need logic?

Life is full of choices. Bagel or cereal? 2% or skim? Shorts or pants? College or work force? From the moment you wake up you have decisions to make, from the trivial to the life-changing. There is no way to avoid having to make choices, since even the decision to stay in bed all day and avoid facing reality is a choice in itself. It is how one responds to these choices that shapes your life, day-in, day-out, and over a lifetime. How do we make decisions, though?

There are lots of ways to make a decision. You could flip a coin, for example. That is a good method when you really do not care about the outcome, since it gives you roughly a 50/50 shot at each option. It would not really be a smart way to make ALL your decisions, though. Imagine deciding whether or not to stop at a stop sign based on a coin flip? Asking other people's opinions is another way to make a decision, say about what outfit to wear for an evening. It would not be a very good way to decide on something really personal, though, or something that was really important.

Logic involves
  1. making statements about statements: Are they true, false, relevant, objective, subjective, ...,
  2. making new statements which are true, objective, ..., and
  3. making decision.

The way we make most of our decisions is by considering all the factors in a situation and deciding what seems best. Every day we examine statements we see and hear for truth. To judge means to examine and decide the trustworthiness of something. Hardly a day goes by without the need to examine facts, draw a conclusion, and make a decision based on those facts. We rely on our own evaluation and the evaluations of others to make choices, good or bad.

Has someone ever told you that you made “the logical choice″? Or have you heard that something was “the only logical conclusion″? Whenever you examine a situation and choose the decision that seems best, you are using logic in its most basic form. The process for thinking about all the facts and choices in a situation is called reasoning. It’s like a recipe for making a decision. Often the are several different methods for decision making. Logic is a collection of reasoning methods.

Have you ever followed a recipe to cook something? When you follow the recipe precisely the cake or roast you have at the end of the process is usually yummy, like you expected it to be. Occasionally, things will go wrong unexpectedly, but there is a better chance of things coming out right when you follow the process. Logic works the same way for making decisions: it is a process you may not follow precisely, but when you do things usually come out right.

Logic is the study of valid reasoning (we’ll get to what valid means in a few minutes). It is a valuable tool to have in your decision-making tool box because it so often leads to the right choices. A person who uses logic can analyze complex, confusing situations and find the best option. People who think logically make successful students, valued employees, and savvy leaders. Thinking logically is a life-long skill.

Usually, however, whenever scientists or mathematicians talk about logic they don’t mean this kind of everyday, rational decision making, they are talking about the science of logic, which you will learn about in this course. Learning about scientific logic will give you the tools to apply it in your everyday life.

Why Does it Have to Seem so Complicated?

When you read a recipe in a cook book you have to be able to understand some special terms in order to make sense of it. These special terms have consistent, standardized meanings to make the instructions clear. Imagine if every recipe called for “a bit″ of one thing and a “dash″ of another? It would be impossible to follow the recipe because those terms mean different things to different people. Instead, recipes use cups, teaspoons, ounces and other measures that everyone can agree on. In the same way, there are special terms in logic so that everyone can follow the process without getting confused.

All the special rules and symbols in logic may seem intimidating and even unnecessary at first but with a little practice it becomes easy. In order to make good decisions we need to have good ways to talk about the facts so that we can evaluate them. Everyday language works fine for most of us in our daily lives, but when things start getting complicated it is not ideal because it is vague and imprecise. This is because it uses both objective and subjective topics.

Objective and Subjective Reasoning

Objective statements are factual; they are not up for interpretation or debate. “There are 8 US fluid ounces in a US cup is an objective statement. Subjective statements are more complicated in that they are neither true nor false. The meaning of a subjective statement can change depending on who says it and who hears it (remember “a dash″ or “a bit″?).

Take, for instance, the following subjective statement:

″$100 is a lot of money.″

Do you think this is true? Some people would say it is true because to them $100 represents many hours of work. It takes someone on minimum wage about a day and a half to earn $100. Others would say that $100 is not very much money and that they spend much more than that every day. This is the challenge with everyday language; everyday language includes many statements that mean different things to different people. When the meaning of something is open for interpretation that is called ambiguity, meaning that it is unclear. Ambiguity can cause a lot of problems when trying to evaluate facts and make decisions. How can you decide what it true when different people have different opinions? There are some statements that are not either true or false, which brings up another question − what is true? We’ll get to that in a few minutes.

Logic is an old science. Wait, what is old? That statement sounds very subjective. Let’s look at the facts. The first person to study logic systematically was the Greek philosopher Aristotle (no last name, apparently), who lived from 384 − 322 B.C. The revival of science and technology during the Industrial Revolution connected the classical logic developed by Aristotle with more recent developments in modern mathematics. From this union came the field of mathematical logic, which has experienced rapid progress ever since.i

If we evaluate the fact that logic got its start around 300 BC we could reasonably arrive at the conclusion that logic is indeed a very old science. Still, it would be very hard to prove because the word old means different things to different people. To most of us 300 BC (about 2,300 years ago) seems very old, but when you consider that dinosaurs went extinct roughly 65 million years ago, it seems very recent. As we explored before, subjective statements like this can be complicated and do not always lead to clear answers or definitive conclusions. Wouldn’t it be nice to have certainty − a clear yes or no?

Boolean Logic

In an effort to achieve certainty whenever possible, mathematicians and scientists turn to logic to reach definitive conclusions. Logic is a tool for being careful, precise, and exact. The most common logic used by mathematicians and scientists is Boolean logic. This is the kind of logic that this course focuses on so we’ll be talking about Boolean logic from here on. Boolean logic is a type of reasoning in which statements are determined to always be either true or false. Boolean logic considers only statements that are either true or false.

Because of the ambiguity in everyday language, Boolean logic does not deal with all the statements that can be formulated in ordinary language. That is, not everything which is said can be analyzed with logic. Logic only looks at objective statements: the ones that can be judged true or false. With logic, scientists make one simple assumption: that a statement is either true or false, for sure. To make an assumption means to believe that something is true, whether it is or not. Logic is built on assumptions, you’ll learn more about them later, but the first and most basic one is this. If a statement it’s not clearly true or false, logic ignores it.

Which of the following statements could be used in Boolean logic?

  1. Chocolate tastes good. (no)
  2. A water molecule consists of two atoms of hydrogen and one of oxygen. (yes)
  3. The Hunger Games is a great movie. (no)
  4. J. K. Rowling is listed as the author of the Harry Potter book series. (yes)
  5. This book has 128 pages. (yes)
  6. The area of a rectangle is its width times its height. (yes)
  7. It snows during the winter time. (no)

When dealing with logic, we sometimes have to ignore a lot of knowledge, uncertain points, and our gut feeling. It may feel wrong at first, but because of this we can also be sure that logic leads to reliable results, something we can trust. Using Boolean logic allows you to organize information in sets, thus allowing you to analyze and interpret the information.i Related is Boolean Algebra, the symbolic system of mathematical logic that represents relationships between entities − either ideas or objects.ii Boolean logic has proven to be a very effective method of modeling the world around us and, in this way, of predicting the future.

Imagine again that someone tells you ″$100 is a lot of money.″

Remember the discussion above. Is this true? Is this false? Will this be true or false tomorrow? The value of money is subjective and relative to whom you ask but there is no ambiguity, no “sometimes ″, no “it depends on whom you ask″; in Boolean logic. There is a clear cut, true or false answer each time.

Which of the following statements is always true?

  1. Whenever it rains, I forget an umbrella.
  2. Whatever number you write down, you can always write down a bigger one.

Statements

Statements have a name, a content and a truth value. The content specifies the property of something, such as "The car is red." or "The value of the variable x is 10.". The truth value of the statement is "true" or "false". In this example:

Name of the statementContent of the statementWhat is known about this statement?
A The car is red. It is true.

"A" is the name of the statement. "The car is red.". is the content of the statement. "true" is the truth value o the statement. Similarly one could write:

1: x is 5. (true)
2: y is 7. (true)
3: x plus y is 10. (false)
Here names of the statements are the numbers 1,2,3. The content of statement 1 is "x is 5." and its truth value is true. The content of statement 2is "y is 7." and its truth value is true. The content of statement 3 is "x plus y is 10." and its truth value is false.

If a statement is false, the opposite statement is true. The opposite of "true" is "false". The opposite of "is" is "is not".

Examples:

(1) Given are the following statement

The car is red. (false)
is equivalent to the statement
The car is not red. (true)
(2) Given are the following statement
The car is not red. (false)
is equivalent to the statement
The car is red. (true)

Inconsistent Statements

A statement can be true or false but not both. A set of statements is in consistent if there is statement with opposite truth values. If we assume that statement B and C are true, the following set of statement is inconsistant, because statement A cannot be both true and false.
A: The car is red. (true)
B: A is false. (true)
This set of statement is inconsistent, because statement A cannot be both true and false.

Reasoning: Finding the truth value of a statement

The correct truth value of a statement makes a set of statements consistent. For example for the following set of statements:
A: The car is red.
B: A is false. (true)
The correct truth value for statement A is "false". The solution to the task "Find the truth value." is
A: The car is red.(false)
B: A is false. (true)
A systematic method for finding solutions is:
  1. Convert all false statements into true statements.
  2. Assign a truth value to all unknown truth values.
  3. Check all statements for inconsistences.
  4. If there are ...
Here is an example. Find the solution of:
A: The car is red.(true)
B: A is false.
The solution is:
A: The car is red.(true)
B: A is false. (false)
The following example has two solution:
A: B is true.
B: A is true.
The solutions are:
A: B is true. (true)
B: A is true. (true)
and
A: B is true. (false)
B: A is true. (false)
The statements A and B are both false or both true. Examples
Exercises

Index