So far, we have learned how to translate certain English sentences into our symbolic language, which consists of a set of constants (i.e., the capital letters that we use to represent different atomic propositions) and the truth-functional connectives. But what is the payoff of doing so? In this section we will learn what the payoff is. In short, the payoff will be that we will have a purely formal method of determining the validity of a certain class of arguments—namely, those arguments whose validity depends on the functioning of the truth-functional connectives. This is what logicians call “propositional logic” or “sentential logic.”
In the first chapter, we learned the informal test of validity, which required us to try to imagine a scenario in which the premises of the argument were true and yet the conclusion false. We saw that if we can imagine such a scenario, then the argument is invalid. On the other hand, if it is not possible to imagine a scenario in which the premises are true and yet the conclusion is false, then the argument is valid. Consider this argument:
- The convict escaped either by crawling through the sewage pipes or by hiding out in the back of the delivery truck.
- But the convict did not escape by crawling through the sewage pipes.
- Therefore, the convict escaped by hiding out in the back of the delivery truck.
Using the informal test of validity, we can see that if we imagine that the first premise and the second premise are true, then the conclusion must follow. However, we can also prove this argument is valid without having to imagine scenarios and ask whether the conclusion would be true in those scenarios. We can do this by a) translating this sentence into our symbolic language and then b) using a truth table to determine whether the argument is valid. Let’s start with the translation. The first premise contains two atomic propositions. Here are the propositions and the constants that I’ll use to stand for them:
S = The convict escaped through the sewage pipes
D = The convict escaped by hiding out in the back of the delivery van
As we can see, the first premise is a disjunction and so, using the constants indicated above, we can translate that first premise as follows:
S v D
The second premise is simply the negation of S:
Finally, the conclusion is simply the atomic sentence, D. Putting this all together in standard form, we have:
- S v D
- ∴ D
We will use the symbol “∴“ to denote a conclusion and will read it “therefore.”
The next thing we have to do is to construct a truth table. We have already seen some examples of truth tables when I defined the truth-functional connectives that I have introduced so far (conjunction, disjunction, and negation). A truth table (as we saw in section 2.2) is simply a device we use to represent how the truth value of a complex proposition depends on the truth of the propositions that compose it in every possible scenario. When constructing a truth table, the first thing to ask is how many atomic propositions need to be represented in the truth table. In this case, the answer is “two,” since there are only two atomic propositions contained in this argument (namely, S and D). Given that there are only two atomic propositions, our truth table will contain only four rows—one row for each possible scenario. There will be one row in which both S and D are true, one row in which both S and D are false, one row in which S is true and D is false, and one row in which S is false and D is true.
|D||S||S v D||~S||D|
The two furthest left columns are what we call the reference columns of the truth table. Reference columns assign every possible arrangement of truth values to the atomic propositions of the argument (in this case, just D and S). The reference columns capture every logically possible scenario. By doing so, we can replace having to use your imagination to imagine different scenarios (as in the informal test of validity) with a mechanical procedure that doesn’t require us to imagine or even think very much at all. Thus, you can think of each row of the truth table as specifying one of the possible scenarios. That is, each row is one of the possible assignments of truth values to the atomic propositions. For example, row 1 of the truth table (the first row after the header row) is a scenario in which it is true that the convict escaped by hiding out in the back of the delivery van, and is also true that the convict escaped by crawling through the sewage pipes. In contrast, row 4 is a scenario in which the convict did neither of these things.
The next thing we need to do is figure out what the truth values of the premises and conclusion are for each row of the truth table. We are able to determine what those truth values are because we understand how the truth value of the compound proposition depends on the truth value of the atomic propositions. Given the meanings of the truth functional connectives (discussed in previous sections), we can fill out our truth table like this:
|D||S||S v D||~S||D|
To determine the truth values for the first premise of the argument (“S v D”) we just have to know the truth values of S and D and the meaning of the truth functional connective, the disjunction. The truth table for the disjunction says that a disjunction is true as long as at least one of its disjuncts is true. Thus, every row under the “S v D” column should be true, except for the last row since on the last row both D and S are false (whereas in the first three rows at least one or the other is true). The truth values for the second premise (~S) are easy to determine: we simply look at what we have assigned to “S” in our reference column and then we negate those truth values—the Ts becomes Fs and the Fs becomes Ts. That is just what I’ve done in the fourth column of the truth table above. Finally, the conclusion in the last column of the truth table will simply repeat what we have assigned to “D” in our reference column, since the last conclusion simply repeats the atomic proposition “D.”
The above truth table is complete. Now the question is: How do we use this completed truth table to determine whether or not the argument is valid? In order to do so, we must apply what I’ll call the “truth table test of validity.” According to the truth table test of validity, an argument is valid if and only if for every assignment of truth values to the atomic propositions, if the premises are true then the conclusion is true. An argument is invalid if there exists an assignment of truth values to the atomic propositions on which the premises are true and yet the conclusion is false. It is imperative that you understand (and not simply memorize) what these definitions mean. You should see that these definitions of validity and invalidity have a similar structure to the informal definitions of validity and invalidity (discussed in chapter 1). The similarity is that we are looking for the possibility that the premises are true and yet the conclusion is false. If this is possible, then the argument is invalid; if it isn’t possible, then the argument is valid. The difference, as I’ve noted above, is that with the truth table test of validity, we replace having to use your imagination with a mechanical procedure of assigning truth values to atomic propositions and then determining the truth values of the premises and conclusion for each of those assignments.
Applying these definitions to the above truth table, we can see that the argument is valid because there is no assignment of truth values to the atomic propositions (i.e., no row of our truth table) on which all the premises are true and yet the conclusion is false. Look at the first row. Is that a row in which all the premises are true and yet the conclusion false? No, it isn’t, because not all the premises are true in that row. In particular, “~S” is false in that row. Look at the second row. Is that a row in which all the premises are true and yet the conclusion false? No, it isn’t; although both premises are true in that row, the conclusion is also true in that row. Now consider the third row. Is that a row in which all the premises are true and yet the conclusion false? No, because it isn’t a row in which both the premises are true. Finally, consider the last row. Is that a row in which all the premises are true and yet the conclusion false? Again, the answer is “no” because the premises aren’t both true in that row. Thus, we can see that there is no row of the truth table in which the premises are all true and yet the conclusion is false. And that means the argument is valid.
Since the truth table test of validity is a formal method of evaluating an argument’s validity, we can determine whether an argument is valid just in virtue of its form, without even knowing what the argument is about! Here is an example:
- (A v B) v C
- ∴ C
Here is an argument written in our symbolic language. I don’t know what A, B, and C mean (i.e., what atomic propositions they stand for), but it doesn’t matter because we can determine whether the argument is valid without having to know what A, B, and C mean. A, B, and C could be any atomic propositions whatsoever. If this argument form is invalid then whatever meaning we give to A, B, and C, the argument will always be invalid. On the other hand, if this argument form is valid, then whatever meaning we give to A, B, and C, the argument will always be valid.
The first thing to recognize about this argument is that there are three atomic propositions, A, B, and C. And that means our truth table will have 8 rows instead of only 4 rows like our last truth table. The reason we need 8 rows is that it takes twice as many rows to represent every logically possible scenario when we are working with three different propositions. Here is a simple formula that you can use to determine how many rows your truth table needs:
2n (where n is the number of atomic propositions)
You read this formula “two to the n-th power.” So if you have one atomic proposition (as in the truth table for negation), your truth table will have only two rows. If you have two atomic propositions, it will have four rows. If you have three atomic propositions, it will have 8 rows. The number of rows needed grows exponentially as the number of atomic propositions grows linearly. The table below represents the same relationship that the above formula does:
|Number of atomic propositions||Number of rows in the truth table|
So, our truth table for the above argument needs to have 8 rows. Here is how that truth table looks:
|A||B||C||(A v B) v C||~A||C|
Here is an important point to note about setting up a truth table. You need to make sure that your reference columns capture each distinct possible assignment of truth values. One way to make sure you do this is by following the same pattern each time you construct a truth table. There is no one right way of doing this, but here is how I do it (and recommend that you do it too). Construct the reference columns so that the atomic propositions are arranged alphabetically, from left to right. Then on the right-most reference column (the C column above), alternate true and false each row, all the way to the bottom. On the reference column to the left of that (the B column above), alternate two rows true, two rows false, all the way to the bottom. On the next column to the left (the A column above), alternate 4 true, 4 false, all the way to the bottom.
The next step is to determine the truth values of the premises and conclusion. Note that our first premise is a more complex sentence that consists of two disjunctions. The main operator is the second disjunction since the two main grouping, denoted by the parentheses, are “A v B” and “C”. Notice, however, that we cannot figure out the truth values of the main operator of the sentence until we figure out the truth values of the left disjunct, “A v B.” So that is where we need to start. Thus, in the truth table below, I have filled out the truth values directly underneath the “A v B” part of the sentence by using the truth values I have assigned to A and B in the reference columns. As you can see in the truth table below, each line is true except for the last two lines, which are false, since a disjunction is only false when both of the disjuncts are false. (If you need to review the truth table for disjunction, please see section 2.3.)
|A||B||C||(A v B) v C||~A||C|
Now, since we have figured out the truth values of the left disjunct, we can figure out the truth values under the main operator (which I have emphasized in bold in the truth table below). The two columns you are looking at to determine the truth values of the main operator are the “A v B” column that we have just figured out above and the “C” reference column to the left. It is imperative to understand that the truth values under the “A v B” are irrelevant once we have figured out the truth values under the main operator of the sentence. That column was only a means to an end (the end of determining the main operator) and so I have grayed those out to emphasize that we are no longer paying any attention to them. (When you are constructing your own truth tables, you may even want to erase these subsidiary columns once you’ve determined the truth values of the main operator of the sentence. Or you may simply want to circle the truth values under the main operator to distinguish them from the rest.)
|A||B||C||(A v B) v C||~A||C|
Finally, we will fill out the remaining two columns, which is very straightforward. All we have to do for the “~A” is negate the truth values that we have assigned to our “A” reference column. And all we have to do for the final column “C” is simply repeat verbatim the truth values that we have assigned to our reference column “C.”
|A||B||C||(A v B) v C||~A||C|
The above truth table is now complete. The next step is to apply the truth table test of validity in order to determine whether the argument is valid or invalid. Remember that what we’re looking for is a row in which the premises are true and the conclusion is false. If we find such a row, the argument is invalid. If we do not find such a row, then the argument is valid. Applying this definition to the above truth table, we can see that the argument is invalid because of the 6th row of the table (which I have highlighted). Thus, the explanation of why this argument is invalid is that the sixth row of the table shows a scenario in which the premises are both true and yet the conclusion is false.
Use the truth table test of validity to determine whether or not the following arguments are valid or invalid.
1. A v B
3. ∴ ~A
1. A ⋅ B
2. ∴ A v B
2. ∴ ~(C v A)
1. (A v B) ⋅ (A v C)
3. ∴ B v C
1. R ⋅ (T v S)
3. ∴ ~S
1. A v B
2. ∴ A ⋅ B
1. ~(A ⋅ B)
2. ∴ ~A v ~B
1. ~(A v B)
2. ∴ ~A v ~B
1. (R v S) ⋅ ~D
3. ∴ S ⋅ ~D