Truth Tables

What Are Truth Tables?

Truth Tables enable philosophers to tell if an argument is valid or invalid. They make it easy to identify what the truth values are and how well-formed an argument is.

Truth tables are mechanical ways to work through an argument's statements. They provide an easy visualization that is easy to analyze and understand statements. They work by assigning values to statements and arguments, and then comparing the values in the argument with the values in the conclusion.

The Method

    Truth tables are used for something called Statement Logic. Statement logic is a type of deductive reasoning where an argument is broken down into its base units (called statements) and then represented by something called a statement letter. (A statement letter is just a fancy way of saying a capital letter that represents a statement.)

    Let's look at this example:

1. If you are a politician, then you are lying
2. You are not lying
3. Therefore you are not a politician. 

    This is the valid form modus tollens. If you remember, that is 

1. If A, then B.
2. Not B
3. Therefore not A.

    Now, let's practice the method. First, we have to break our argument about politicians down into statements. In logic, this is often (but not always) done by identifying the phrases and sentences. The argument above is very easy. It only contains two statements; 'You are a politician", and 'You are lying."

    Most arguments are not so easy, but I thought we'd start small. So, now we need to assign the two statements their statement letters. Traditionally the letters have something to do with the statements themselves. So we'll assign 'You are a politician' the letter P and 'You are lying' L. 

    From here, what we need to do is plug the statement letters into the argument. That would look like 

1. If P, then L
2. Not L
3. Therefore not P

    So we've figured out what the statements are and how to to assign them. What we need to do now is symbolize the argument. I'll these all in a page for reference, but in short there 6 symbols that all represent different logical operators. (A logic operator is philosophy's version of math symbol.) 

    The symbols are;

    I'll put more information on what exactly all of those mean in that page on symbols. Now that we have that information we fully symbolize our argument. A modus tollens argument is a simple if/then argument with negations, so let's use that symbol. It would look like

1. P→L
2. ~L
3. :. ~P

    After all of that we are now ready to create a truth table. How this works is we list the statement letters followed by the argument. This will look like the following.

    After that we simply assign the statement letters on the left all possible truth values. We are making all possible combinations of truth values. When there are just 2 statements, there are a total of 4 possible combinations to make. 

    Now we just shift the the truth values on the left to the statement letters on the right.

    Allow me to point out that we have not yet filled in the logical operator in the first statement. That is because we haven't discussed how to do that yet. Allow me to also point out that the ~ negates the original truth value by flipping it. So, if L is true, then ~L is false. 

    This is easier to understand in English terms. Remember, L represents 'you are lying.' So, if L represents 'you are lying,' then ~L represents 'you are not lying.' Another way of saying this would be, ~L represents 'the statement "you are lying" is false.' In other words, ~L means 'not L.'

    So we can fill out the columns under ~L and ~P simply because we know that a ~ flips the truth value.

    The next step is to figure out the truth value of →. To do this we need to familiarize ourselves with the definition of an if/then statement. 

    An if/then statement means that the first statement brings about the second statement. That means that the second statement is necessary for the existence of the first statement. I'm not going to go into it too much here, but what this means is that an if/then statement is false in only 1 way. Basically, if the first statement is true, the second has to be true, but the value of the second statement only affects the first if it is false. 

    What this means is that an if/then statement is true unless the truth value of the second statement is false and the truth value of the first statement is true. We can now fill out the final column of the table.

    Now we can figure out if the argument has a valid form. The way we do this is by comparing the truth values of the premises with the truth values of the conclusion. If the truth value of the conclusion is false in a row where the truth values of the premises are all true, then it is an invalid argument. We'll go through the table one row at a time.

    In the first row we can see that ~P is false, ~L is false, and the if/then is true. So, because not all the premises are true while the conclusion is false this row is valid.

    In this row we see the opposite. ~P is false, ~L is true, and the if/then is false. So this row is also valid.

    Now here's a secret, because a valid argument only cares about a false conclusion, we can assume that if all the iterations of an argument where the conclusion is false are valid, then the argument is valid. Which we would expect by using a modus tollens argument. Modus tollens cannot be invalid so it was a safe example to use when demonstrating a simple truth table. 

Conclusion

    Hopefully this has made the method of using a truth table very clear and simple. I'll post a page of a more complicated truth table later. As always, please ask questions:)