This section is about logic- knowing how to construct logical arguments, and put them into proofs. Logic is the art of showing if something is true, or if it is false. The first part of this section is mainly text, which explains the basic concepts. After you have some idea of how these work, you can go on to the next part, which gives you an opportunity to test your skills.

Connectors

The first concepts are those of the four major connectors. These are the foundations of everything in logic.

And- in logic, and means both, or all, as it does in everyday language. For example, the statement I have a dog, and a cat, is only true if I own both a dog and a cat. If I owned just one of the animals, of neither, the statement would be false. This is more easily seen in a table (match T Or F in vertical column with T or F in horizontal column, box where they meet is what the truth value is):

 And True False True True False False False False

Or- has two different forms: the inclusive or, and the exclusive or. Either way, or means you have one thing or the other. For example, I will have the chicken, or the steak. The statement is true if I get either the chicken or if I get the steak. It is only false if I get neither, or sometimes it is false if I get both. This is where the inclusive/exclusive comes in. The inclusive includes both, so if I got the chicken and the steak the statement would be true. The exclusive says you can have one thing, or the other but not both. Unless otherwise stated, this page assumes that all ors are inclusive. Here are the tables:

 In Or True False True True True False True False

 Ex Or True False True False True False True False

Implies- is probably the most useful logical connector. Implies means that one thing leads to another. For example, I am going to high school implies that I am between the ages of 13 and 19. This statement is true if you are going to high school, and you are between the ages of 13 and 19, or you are not going to high school. If the first part of an implication is false, the statement is true no matter what. The only way for a implication to be false is if the first part is true, and the second part is false. So the statement would be false if I am going to high school, but I am 20 years old. Implications are also (and more commonly) written using the If, Then convention- If I am going to high school, then I am between the ages of 13 and 19. Here is the table for implies:

 Implies True False True True False False True True

If and only if- is sometimes called the biconditional, and is really just a composition of two other connectors. A if and only if B is the same as saying A implies B, and B implies A. Therefore the biconditional, is true only if it is true biconditional true, or false biconditional false. A statement is false when it is true biconditional false, or false biconditional true. Here is the table, which should clarify this a bit:

 Biconditional True False True True False False False True

Analyzing A Statement

Now that you know the four basic connectors, the next step is finding out what you can do with them. One thing is analyzing someone else's statement. But before you can find the truth value of a statement, you have to break it down into various parts. For example, if a politician said, "I'm going to lower taxes, and improve life for the average citizen, forsaking having a balanced national budget...and I'm going increase federal spending to preserve the environment, and help the planet we live on. But if I'm going to improve the environment, then I will have to raise taxes just a little This can be broken down into statements as follows:

P. Lower taxes

Q. Improve life for average citizen

R. Forsake balanced budget

S. Increase spending on planetary conditions

P. Raise taxes (Read Not P, means the negation, or opposite of P)

P And Q. Lower taxes makes better conditions

P And Q Implies R. Lower taxes and improved conditions means less balanced budget

S Implies P. Increase spending means increasing taxes

[(P and Q) implies R] and [S implies P]. Full statement

Now that a statement is broken down, the final step is assigning a truth value to the overall statement. The easiest way to do this is in a truth table. To do this, first assign a truth value to each statement. In this case assume all of the politicians statements are true. Now, start filling in the table. This is done by putting each individual statement in a column, and then putting its value below it. When you get to more complex statements, you have to assign them a value using what you know about the individuals, and the rules of connectors:

 P Q R S P P @ Q P @ Q Imp. R S Imp. P [(P @ Q) Imp R] @ [S Imp. P] T T T T T?!

STOP!!! By the time you get to here, if you haven't noticed all ready, you should see that P and P have the same value (raising and lowering taxes). This is a logical contradiction that is enough by itself to discredit all of this politician's claims. So this statement is proven false. But if we overlook this, and instead pretend the politician did not claim to both raise and lower taxes, let's see how the statement turns out.

 P Q R S P P @ Q P @ Q Imp. R S Imp. P [(P @ Q) Imp R] @ [S Imp. P] T T T T F T T T T

So, if our friend had not claimed to both raise and lower taxes, his argument would have been logically sound. There. Now you have seen you to break down and analyze a statement, using what you know about connectors. Go find a statement that seems like it doesn't make sense, and try your hand at logic. But first...

Constructing An Argument

The hardest part of logic is taking a few facts or ideas, and using what you know to construct an argument around them. But that is what logic is really about- being able to make your own argument. When you are constructing an argument, the most important thing to remember is that each idea has to follow logically from the previous one. You can't have A implies B, and them jump to E. Also, remember to avoid contradicting yourself at all costs. One contradiction will kill your entire argument. Here is one sample argument.

Let P, Q, R be statements. Say you wanted to prove statement R was true. You could do this if you knew that [(P and Q) Implies R], and that both P and Q were true. Because both P and Q are true, it follows that [P and Q] would be true. Since you know that [P and Q] is true, and that [(P and Q) Implies R], you can show that R is true. This is because what [(P and Q) Implies R] is really saying is that if [P and Q] is true, then R is also true.

Let's apply this argument to some real statements:

Let P mean I am responsible

Let Q mean I am a hard worker

Let R mean You should hire me

Suppose that you are an employer who is looking for an employee who is both hard working and responsible. If I proved that I work hard, and am responsible, it follows logical that I am both responsible and hard working (this little step may seem unnecessary, but you must make sure that everything follows directly. It is better to be redundant than to miss something, or make a jump that is hard to follow). Since you are looking for someone who is hard working and responsible, and I fit these characteristics, it follows logically that I should get the job.

There. Now that you have seen how to construct an argument, its time for you to try it. Go ask someone for a statement, and then break it down and see if everything is logical. Or, go find something to argue, and construct an argument around it. After you have done what you think is your best job, e-mail it to me here. I will look at all of them, and post some of the most complete and correct ones on future editions of this page. So, why are you still reading this? Get going!

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