Truth Table Basics

Our logical theory so far consists of a vocabulary of basic symbols, rules defining how to combine symbols into wffs, and rules defining how to construct proofs from wffs. All of this only concerns manipulating symbols. We now need to give these symbols some meanings.

We are going to give them just a little meaning. For the sentence letters, all that we are actually going to notice is that each of them must be either true or false. That's as far as we will go.

For the connectives, we will develop more of a theory. Each of them has a meaning that is defined in terms of how it affects the meanings of sentences that contain it.

Since a wff represents a sentence, it must be either true or false. We will call this its truth value: the truth value of a wff is "true" if the wff is true and "false" if the wff is false.

The truth values of atomic sentences are determined by whatever those sentences mean and what the world is like. For example, the truth value of "It is raining" is determined by what it means and whether or not it is raining. Likewise, the truth value of "Austin is the largest city in Texas" is determined by what it means and what the facts are about cities in Texas. These two sentences are about the weather and geography, respectively. Since this is not a course in meteorology or geography, we won't have anything else to say about the truth values of atomic sentences except that they have them.

For compound sentences, however, we do have a theory. Some compound sentences are truth functions of their constituents. Take the simple sentence "It's cold and it's snowing." Is it true or false? We can't tell without knowing something about the weather, but we can say how its truth value depends on the truth values of the two atomic sentences in it:

All that you need to know to determine whether or not "It's cold and it's snowing" is true or false is whether each of its constitutents is true or false. We describe this by saying that "It's cold and it's snowing" is a truth function of its constituents.

Notice that this sentence works like it does because of the meaning of the word "and". This word combines two sentences into a new sentence that has a truth value determined in a certain way as a function of the truth values of those two sentences. So, and is a truth functional connective.

We define each of the four connections using a table like the one above that shows, schematically, how the truth value of a wff made with that connective depends on the truth values of its constituents.

For the ampersand:

For the wedge:

For the arrow:

For the double arrow:

And the simplest of all, for the tilde:

These rules also define the meanings of more complex sentences. Consider this sentence:

~P → Q

This is a conditional (main connective →), but the antecedent of the conditional is a negation. To construct its truth table, we might do this:

However, ~P is also a truth function of P. So, to get a more complete truth table, we should consider the truth values of the atomic constituents.

A still more complicated example is the truth table for (P→Q)&(Q→P).

How is this table constructed? It will help to go through it step by step.

1. Determine the Columns for the Truth Table

The first step is to determine the columns of our truth table. To do that, we take the wff apart into its constituents until we reach sentence letters. As we do that, we add a column for each constituent.

This is a step-by-step process as well. The steps are these:

1. Find the main connective of the wff we are working on.
2. Determine the main constituents that go with this connective.
3. Add new columns to the left for each constituent.
4. Repeat for each new constituent.

To continue with the example(P→Q)&(Q→P), the first step is to set up a truth table with this statement as its its only column:

Next, we identify the main connective of this wff:

Now we identify the main constituents that go with this connective. To make it clear that these are part of a single step, they are identified with a "1" to indicate that this is the first step:

Next, we add columns under the constituents and the main connective:

We now repeat the process with the constituents we have just found, working down below each constituent. We start with P→Q:

We then proceed to the constituents of P→Q:

Next, Q→P

We've now reached sentence letters under each of the constituents. So, the next step is to add columns to the left for each sentence letter:

2. Determine the Rows of the Truth Table

What we are trying to construct is a table that shows what the truth value of the main wff is for any combination of truth values of its constituents. We will do this by constructing one row for each possible combination of truth values.

All that we have to consider is the combinations of truth values of the sentence letters, since everything else is determined by these. So, we want to include one row in our truth table for each combination of truth values of the sentence letters. In this case, there are two sentence letters, P and Q. What are the possible combinations of truth values for P and Q? Think about it this way:

• There are two possibilities, T and F, for P.
• For each of these cases, there are two possibilities: Q = T and Q = F.
• Therefore, there are 2 × 2 = 4 possibilities altogether.

An easy way to write these down is to begin by adding four rows to our truth table, since we know that there are four combinations:

Half of these will have P = T and half will have P = F:

For each of these halves, one will have Q = T and one will have Q = F:

3. Calculate the Truth Values for Each Row

The last step is to work across each row from left to right, calculating the truth value for each column based on the truth values of wffs to the left and the connective used in that column. So, we start with the first row and work across.

For each column in that row, we need to ask:

1. What is the main connective of the wff at the top of the column?
2. What previous column(s) are the main constituents in?

For the first column, the main connective is → and the previous columns are the first two columns:

Next, look at the truth value combination we find in those previous columns:

Now, substitute that combination of truth values for the constituents in the column we're working on and look up the value they produce using the truth table for the main connective. In this case, we want to use the combination P = T, Q = T in the wff (P→Q).

Now we need to look up the appropriate combination in the truth table for the arrow:

And we substitute this into the cell we are working on in our truth table:

That's one! We go on to the next column, headed by (Q→P). This depends on the same two columns as the previous column did, but not in the same order: here, Q is the antecedent and P is the consequent.

We can then substitute the value from the table for →:

Going on to the last column, we have a wff that is a conjunction (main connective &), with constituents (P → Q) and (Q → P):

We need to evaluate this combination:

That corresponds to this row of the truth table for the ampersand:

So, we complete the first row as follows:

Here's the next row. Notice that the values under (P → Q) and (Q → P) are not the same. Why?

Finally, here is the full truth table. Notice that what this shows, overall, is what the truth value of (P → Q) & (Q → P) is for each combination of truth values of its atomic constituents (sentence letters).