Applications of Truth Tables

We have learned how to take sentences in English and translate them into logical statements using letters and the symbols for the logical connectives. And we have learned how to obtain a set of the truth values of a statement in every possible case by making a truth table. Now we can take these tools farther to do two important things:

1. We can prove that two different logical statements are equivalent or not equivalent to one another.
2. We can test the validity of the structure of logical arguments.

How to Tell if Two Different Logical Statements are Equivalent

To tell if two logical statements are equivalent or not, we need to create a truth table for each statement, and compare the truth values of the statements in each case. If both statements have truth tables with exactly the same truth values in the final column, then the two statements are logically equivalent, and one statement can be replaced with the other in a logical argument without changing the meaning.

Let's look at some examples:

Example 1

Is ~(p∧q) equivalent to ~p∧~q?

Because we know that in arithmetic 2(x+3)=2x+6, we may start to wonder if we can distribute the negation sign over a set of parentheses in logical statements. In other words, is ~(p∧q) equivalent to ~p∧~q?

The only way for us to find out is to make a truth table for ~(p∧q) and a truth table for ~p∧~q and then compare the truth values in each table:

In order for this comparison to work, we have to compare the same row in each truth table; for example, we must compare the row where p=T and q=T in the truth table for ~(p∧q) with the row where p=T and q=T in the truth table for ~p∧~q. We are interested in whether or not the two different statements have the same truth value in exactly the same conditions. So if we were to compare the row where p=T and q=T in the truth table for ~(p∧q) with the row where p=T and q=F in the truth table for ~p∧~q, we would be looking at two different sets of conditions, and so comparing the truth values for each statement in this case would lead to an error.

In the two truth tables I've created above, you can see that I've listed all the truth values of p and q in the same order. This is so that I can compare the values in the final column in the two truth tables without worrying about whether or not I am matching up the right rows - because the rows are already in the same order, I can just compare the final column of one table with the final column of the other like this:

In this case, the truth values for ~(p∧q) and ~p∧~q are not exactly the same, so we can conclude that the two statements are not equivalent!

So, this answers our question; if we have ~(p∧q), we cannot simply replace it with ~p∧~q. If we replace ~(p∧q) with ~p∧~q, we will actually be changing the meaning of our statement.

But is there some other way to get rid of the parentheses? In other words, is there some other statement which is equivalent to ~(p∧q) which does not contain any parentheses? Let's try another possibility:

Example 2:

Is ~(p∧q) equivalent to ~p∨~q?

To test this statement, we must make a truth table for ~(p∧q) and a truth table for ~p∨~q and then compare the truth values in each table:

In the two truth tables I've created above, you can see that I've listed all the truth values of p and q in the same order. This is so that I can compare the values in the final column in the two truth tables without worrying about whether or not I am matching up the right rows - because the rows are already in the same order, I can just compare the final column of one table with the final column of the other like this:

In this case, the truth values for ~(p∧q) and ~p∨~q are exactly the same, so we can conclude that the two statements are equivalent:

~(p∧q)≡~p∨~q

So, if we ever encounter ~(p∧q), we can replace it with ~p∨~q without changing the logical meaning of the statement!

Now let's try comparing two more complex statements to see if they are equivalent:

Example 3:

Is the statement (~r∧(p→~q))→p equivalent to r∨p?

To test this statement, we must make a truth table for (~r∧(p→~q))→p and a truth table for r∨p and then compare the truth values in each table.

Be careful - Since we want to compare (~r∧(p→~q))→p, which contains the letters p, q, and r, with r∨p, we must make sure that BOTH truth tables contain ALL THREE LETTERS p, q, and r (even though usually when we make a truth table of r∨p we would use only the two letters r and p). This is because, in order to COMPARE the two truth tables, they must have EXACTLY THE SAME ROWS.

In the two truth tables I've created above, you can see that I've listed all the truth values of p, q and r in the same order. This is so that I can compare the values in the final column in the two truth tables without worrying about whether or not I am matching up the right rows - because the rows are already in the same order, I can just compare the final column of one table with the final column of the other like this:

In this case, the truth values for (~r∧(p→~q))→p and r∨p are exactly the same, so we can conclude that the two statements are equivalent:

(~r∧(p→~q))→p≡r∨p

So, if we ever encounter(~r∧(p→~q))→p, we can replace it with r∨p without changing the logical meaning of the statement!

How to Tell if the Structure of a Logical Argument is Valid

We can use truth tables to determine if the structure of a logical argument is valid.To tell if the structure of a logical argument is valid, we first need to translate our argument into a series of logical statements written using letters and logical connectives. Once we have done this, then we can create truth tables for each statement in the argument.

Then we need to create a truth table for each premise statement and a truth table for the conclusion statement, and compare the truth values of the premise statements versus the conclusion statement in each case. If the truth value of the conclusion is true in EVERY case in which ALL of the premises are true, then the argument has a valid structure. If the there is even one case in which all the premises are true but the conclusion is false, then it is clear that the truth of the conclusion does not follow directly from the premises, and therefore the argument is invalid.

Note: If there are NO cases in which all the premises are true, then the argument is valid by default. Because if it is never possible for all of the premises to be true at the same time, then it is not possible to find a circumstance in which all the premises are true and the conclusion is false, so it is impossible to prove that the argument is invalid; therefore, the argument becomes valid by default.

Let's look at some examples of arguments and let's determine if they have a valid structure or not:

Example Argument 1:

If is it October, then classes are in session.

It is October.

Therefore classes are in session.

If we let p="It is October" and q="Classes are in session," then we can rewrite this argument using letters and logical connectives like this:

p → q

p

Therefore, q.

This argument has two premises:

1. p → q
2. p

And the conclusion is: q.

We then create truth tables for both premises and for the conclusion. Again, since our argument contains two letters: p and q, all of our truth tables should contain both p and q and should have all the rows in the same order:

In the three truth tables I've created above, you can see that I've listed all the truth values of p and q in the same order. This is so that I can compare the values in the final column in each of the truth tables without worrying about whether or not I am matching up the right rows - because the rows are already in the same order, I can just compare the final column of each table with the final column of the other tables like this:

Here we have separated the premises from the first letter columns and from the conclusion column by dark black lines. Now all we need to do is to look at any cases in which ALL of the premises are TRUE. We highlight those instances in yellow on the truth table:

So we have only one row in which all of the premises are true. We recall that in order for the argument to be valid, whenever all the premises are true, the conclusion must also be true. Is the conclusion also true in this row? Yes!

So we can conclude that this argument is valid!

Example Argument 2:

If is it October, then classes are in session.

Classes are in session.

Therefore it is October.

If we let p="It is October" and q="Classes are in session," then we can rewrite this argument using letters and logical connectives like this:

p → q

q

Therefore, p.

This argument has two premises:

1. p → q
2. q

And the conclusion is: p.

We then create truth tables for both premises and for the conclusion. Again, since our argument contains two letters: p and q, all of our truth tables should contain both p and q and should have all the rows in the same order:

In the three truth tables I've created above, you can see that I've listed all the truth values of p and q in the same order. This is so that I can compare the values in the final column in each of the truth tables without worrying about whether or not I am matching up the right rows - because the rows are already in the same order, I can just compare the final column of each table with the final column of the other tables like this:

Here we have separated the premises from the first letter columns and from the conclusion column by dark black lines. Now all we need to do is to look at any cases in which ALL of the premises are TRUE. We highlight those instances in yellow on the truth table:

So we have two rows in which all of the premises are true. We recall that in order for the argument to be valid, whenever all the premises are true, the conclusion must also be true. Is the conclusion also true in the first yellow row? Yes! Is the conclusion also true in the second yellow row? No! This means that a case exists in which all the premises are true, but the conclusion is false, so this argument must be invalid!

So we can conclude that this argument is invalid!

Example Argument 3:

I have a passing grade in this class.

I did not turn in any homework late and I passed all the tests.

I am failing my chemistry class or this class.

Therefore, it is false that I am failing chemistry, only if I turned in some of my homework late.

If we let p="I am passing this class", q="I turn some homework in late," r="I pass all the tests," and s="I am passing my chemistry class," then we can rewrite this argument using letters and logical connectives like this:

p

~q∧r

~s∨~p

Therefore, ~~s→q.

Note: This argument is tricky, because we have to rewrite so many of the sentences before we can put them into symbols as we have done above. Here is how we would rewrite this argument so that it is easier to recognize the correct logical connectives:

I am passing this class.

It is not true that I turned some homework in late and I passed all the tests.

I am not passing my chemistry class or I am not passing this class.

Therefore, If it is false that I am not passing chemistry, then I turned some homework in late.

Look back at the original argument and see if you can understand why we can rewrite each of the sentences in the argument this way without changing their meaning. (If you are having trouble with this, you should take another look at logic lecture 1).

This argument has three premises:

1. p
2. ~q∧r
3. ~s∨~p

And the conclusion is:~~s→q.

We then create truth tables for both premises and for the conclusion. Again, since our argument contains four letters: p, q, r, and s, all of our truth tables should contain both all four of these letters and should have all the rows in the same order:

In the FOUR truth tables I've created above, you can see that I've listed all the truth values of p, q, r, and s in the same order. This is so that I can compare the values in the final column in each of the truth tables without worrying about whether or not I am matching up the right rows - because the rows are already in the same order, I can just compare the final column of each table with the final column of the other tables like this:

Here we have separated the premises from the first letter columns and from the conclusion column by dark black lines. Now all we need to do is to look at any cases in which ALL of the premises are TRUE. We highlight those instances in yellow on the truth table:

So we have only one row in which all of the premises are true. We recall that in order for the argument to be valid, whenever all the premises are true, the conclusion must also be true. Is the conclusion also true in the yellow row? Yes! This means that NO case exists in which all the premises are true but the conclusion is false, so this argument must be valid!

So we can conclude that this argument is valid!

Example Argument 4:

x≠y.

x=10 if and only if x=y.

If y is not an even number, then x=10.

Therefore, y is an odd number.

If we let p be "x=y", q be "x =10," and r be "y is an even number," then we can rewrite this argument using letters and logical connectives like this:

~p

q↔p

~r→q

Therefore, ~r.

Note that the statements "y is not an even number" and "y is an odd number" mean the same thing, and are therefore equivalent.

This argument has three premises:

1. ~p
2. q↔p
3. ~r→q

And the conclusion is:~r.

We then create truth tables for both premises and for the conclusion. Again, since our argument contains three letters: p, q, and r, all of our truth tables should contain both all three of these letters and should have all the rows in the same order:

In the three truth tables I've created above, you can see that I've listed all the truth values of p, q, and r, in the same order. This is so that I can compare the values in the final column in each of the truth tables without worrying about whether or not I am matching up the right rows - because the rows are already in the same order, I can just compare the final column of each table with the final column of the other tables like this:

Here we have separated the premises from the first letter columns and from the conclusion column by dark black lines. Now all we need to do is to look at any cases in which ALL of the premises are TRUE. We highlight those instances in yellow on the truth table:

So we have only one row in which all of the premises are true. We recall that in order for the argument to be valid, whenever all the premises are true, the conclusion must also be true. Is the conclusion also true in the yellow row? No! This means that a case exists in which all the premises are true but the conclusion is false, so this argument must be invalid!

So we can conclude that this argument is invalid!

Example Argument 5:

I got the highest grade in the class on the last test, and I have perfect attendance.

If I get a cold, then I always miss at least one class.

I came down with a cold last week.

Therefore, If I missed at least one class, then I did not get the highest grade in the class on the last test.

If we let p="I got the highest grade in the class on the last test", q="I have perfect attendance," and r be "I get a cold," then we can rewrite this argument using letters and logical connectives like this:

p∧q

r→~q

r

Therefore, ~r→~p.

Note that the statements "I do not have perfect attendance" and "I miss at least one class" mean the same thing, and are therefore equivalent.

This argument has three premises:

1. p∧q
2. r→~q
3. r

And the conclusion is: ~r→~p.

We then create truth tables for both premises and for the conclusion. Again, since our argument contains three letters: p, q, and r, all of our truth tables should contain both all three of these letters and should have all the rows in the same order:

In the three truth tables I've created above, you can see that I've listed all the truth values of p, q, and r, in the same order. This is so that I can compare the values in the final column in each of the truth tables without worrying about whether or not I am matching up the right rows - because the rows are already in the same order, I can just compare the final column of each table with the final column of the other tables like this:

Here we have separated the premises from the first letter columns and from the conclusion column by dark black lines. Now all we need to do is to look at any cases in which ALL of the premises are TRUE. We highlight those instances in yellow on the truth table:

But there are NO instances in which all of the premises are true! We recall that in order for the argument to be valid, whenever all the premises are true, the conclusion must also be true. Since we never have any instance in which all of the premises are true, this means that NO case can exist in which all the premises are true but the conclusion is false, so this argument must be valid by default.

So we can conclude that this argument is valid!

Be careful! If there are NO instances in which ALL the premises are true, then we can never have an instance when all the premises are true but the conclusion is false, so we cannot say that the argument is invalid. Therefore, in this instance, we always say that the argument is valid (by default).

Now return to Brightspace to answer the Logic Lecture questions 3: Evaluating Arguments and Equivalent Statements!