Truth Tables

The truth table test of validity

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:

1. The convict escaped either by crawling through the sewage pipes or by hiding out in the back of the delivery truck.
2. But the convict did not escape by crawling through the sewage pipes.
   
3. 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:

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:

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:

1. S v D
   
2. ~S
   
3. ∴ 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.

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:

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:

1. (A v B) v C
   
2. ~A
   
3. ∴ 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:

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:

So, our truth table for the above argument needs to have 8 rows. Here is how that truth table looks:

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).

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".

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.

Source: Matthew J. Van Cleave
This work is licensed under a Creative Commons Attribution 4.0 License.

Exercise

Use the truth table test of validity to determine whether or not the following arguments are valid or invalid.

1. 
   
   1. A v B
   2. B
   3. ∴ ~A
2. 
   
   1. A ⋅ B
   2. ∴ A v B
3. 
   
   1. ~C
   2. ∴ ~(C v A)
      
      
4. 
   
   1. (A v B) ⋅ (A v C)
   2. ~A
   3. ∴ B v C
5. 
   
   1. R ⋅ (T v S)
   2. T
   3. ∴ ~S
6. 
   
   1. A v B
   2. ∴ A ⋅ B
7. 
   
   1. ~(A ⋅ B)
   2. ∴ ~A v ~B
8. 
   
   1. ~(A v B)
   2. ∴ ~A v ~B
9. 
   
   1. (R v S) ⋅ ~D
   2. ~R
   3. ∴ S ⋅ ~D

Answers

1. Invalid
2. Valid
3. Invalid
4. Valid
   

   A	B	C	(A v B) · (A v C)	~A	B v C
   T	T	T	T     T     T	F	T
   T	T	F	T     T     T	F	T
   T	F	T	T     T     T	F	T
   T	F	F	T     T     T	F	F
   F	T	T	T     T     T	T	T
   F	T	F	T     F     F	T	T
   F	F	T	F     F     T	T	T
   F	F	F	F     F     F	T	F

   
   
   
   
5. Invalid
   

   R	T	S	R · (T v S)	T	~S
   T	T	T	T     T	T	F
   T	T	F	T     T	T	T
   T	F	T	T     T	F	F
   T	F	F	F     F	F	T
   F	T	T	F     T	T	F
   F	T	F	F     T	T	T
   F	F	T	F     T	F	F
   F	F	F	F     F	F	T

   
   
   
   
6. Invalid
7. Valid
8. Valid
   
   

   A	B	~(A v B)	~A v ~B
   T	T	F   T	F     F     F
   T	F	F   T	F     T    T
   F	T	F   T	T     T    F
   F	F	T   F	T    T    T

   
   
   
   
9. Valid
   

   D	R	S	(R v S)  · ~ D	~ R	S  · ~ D
   T	T	T	T     F     F	F	F
   T	T	F	T     F     F	F	F
   T	F	T	T     F     F	T	F
   T	F	F	F     F     F	T	F
   F	T	T	T     T     T	F	T
   F	T	F	T     T     T	F	F
   F	F	T	T     T     T	T	T
   F	F	F	F     F     T	F	F

   
   

Conditionals

So far, we have learned how to translate and construct truth tables for three truth functional connectives. However, there is one more truth functional connective that we have not yet learned: the conditional. The English phrase that is most often used to express conditional statements is "if...then". For example,

If it is raining then the ground it wet.

Like conjunctions and disjunctions, conditionals connect two atomic propositions. There are two atomic propositions in the above conditional:

The proposition that follows the "if" is called the antecedent of the conditional and the proposition that follows the "then" is call the consequent of the conditional. The conditional statement above is not asserting either of these atomic propositions. Rather, it is telling us about the relationship between them. Let's symbolize "it is raining" as "R" and "the ground is wet" as "G". Thus, our symbolization of the above conditional would be:

The "⊃" symbol is called the "horseshoe" and it represents what is called the "material conditional". A material conditional is defined as being true in every case except when the antecedent is true and the consequent is false. Below is the truth table for the material conditional. Notice that, as just stated, there is only one scenario in which we count the conditional false: when the antecedent is true and the consequent false.

Suppose that you pass all the exams and pass the class (first row). That would confirm my conditional statement E ⊃ C. Suppose, on the other hand, that although you passed all the exams, you did not pass the class (second row). This would should my statement is false (and you would have legitimate grounds for complaint!). How about if you don't pass all the exams and yet you do pass the course (third row)? My statement allows this to be true and it is important to see why. When I assert E ⊃ C I am not asserting anything about the situation in which E is false. I am simply saying that one way of passing the course is by passing all of the exams; but that doesn't mean there aren't other ways of passing the course. Finally, consider the case in which you do not pass all the exams and you also do not pass the course (fourth row). For the same reason, this scenario is compatible with my statement being true. Thus, again, we see that a material conditional is false in only one circumstance: when the antecedent is true and the consequent is false.

There are other English phrases that are commonly used to express conditional statements. Here are some equivalent ways of expressing the conditional, "if it is raining then the ground is wet":

It is raining only if the ground is wet

The ground is wet if is raining

Only if the ground is wet is it raining

That it is raining implies that the ground is wet

That it is raining entails that the ground is wet

As long as it is raining, the ground will be wet

So long as it is raining, the ground will be wet

The ground is wet, provided that it is raining

Whenever it is raining, the ground is wet

If it is raining, the ground is wet

All of these conditional statements are symbolized the same way, namely R ⊃ G. The antecedent of a conditional statement always lays down what logicians call a sufficient condition. A sufficient condition is a condition that suffices for some other condition to obtain. To say that x is a sufficient condition for y is to say that any time x is present, y will thereby be present. For example, a sufficient condition for dying is being decapitated; a sufficient condition for being a U.S. citizen is being born in the U.S. The consequent of a conditional statement always lays down a necessary condition. A necessary condition is a condition that must be in present in order for some other condition to obtain. To say that x is a necessary condition for y is to say that if x were not present, y would not be present either. For example, a necessary condition for being President of the U.S. is being a U.S. citizen; a necessary condition for having a brother is having a sibling. Notice, however, that being a U.S. citizen is not a sufficient condition for being President, and having a sibling is not a sufficient condition for having a brother. Likewise, being born in the U.S. is not a necessary condition for being a U.S. citizen (people can become "naturalized citizens"), and being decapitated is not a necessary condition for dying (one can die without being decapitated).

Exercise

Translate the following English sentences into symbolic logic sentences using the constants indicated. Make sure you write out what the atomic propositions are. In some cases this will be straightforward, but not in every case. Remember: atomic propositions never contain any truth functional connectives - and that includes negation! Note: although many of these sentences can be translated using only the horseshoe, others require truth functional connectives other than the horseshoe.

1. The Tigers will win only if the Indians lose their star pitcher. (T, I)
   
2. Tom will pass the class provided that he does all the homework. (P, H)
   
3. The car will run only if it has gas. (R, G)
   
4. The fact that you are asking me about your grade implies that you care about your grade. (A, C)
   
5. Although Frog will swim without a bathing suit, Toad will swim only if he is wearing a bathing suit. (F, T, B)
   
6. If Obama isn't a U.S. citizen, then I'm a monkey's uncle. (O, M)
7. If Toad wears his bathing suit, he doesn't want Frog to see him in it. (T, F)
   
8. If Tom doesn't pass the exam, then he is either stupid or lazy. (P, S, L)
   
9. Bekele will win the race as long as he stays healthy. (W, H)
   
10. If Bekele is either sick or injured, he will not win the race. (S, I, W)
    
11. Bob will become president only if he runs a good campaign and doesn't say anything stupid. (P, C, S)
    
12. If that plant has three leaves then it is poisonous. (T, P)
    
13. The fact that the plant is poisonous implies that it has three leaves. (T, P)
    
14. The plant is poisonous only if it has three leaves. (T, P)
    
15. The plant has three leaves if it is poisonous. (T, P)
    
16. Olga will swim in the open water as long as there is a shark net present. (O, N)
    
17. Olga will swim in the open water only if there is shark net. (O, N)
    
18. The fact that Olga is swimming implies that she is wearing a bathing suit. (O, B)
    
19. If Olga is in Nice, she does not wear a bathing suit. (N, B)
    
20. If Terrence pulls Philip's finger, something bad will happen. (T, B)
    

Answers

1. T ⊃ I (T = The Tigers will win; I = The Indians will lose their star pitcher)
2. H ⊃ P (P = Tom will pass the class; H = Tom does all of his homework)
3. R ⊃ G (R = The car will run; G = The car has gas)
4. A ⊃ C (A = You are asking me about your grade; C = You care about your grade)
5. F ⋅ (T ⊃ B) (F = Frog will swim without his bathing suit; T = Toad will swim; B = Toad is wearing a bathing suit)
6. ~O ⊃ M (O = Obama is not a U.S. citizen; M = I am a monkey's uncle)
7. T ⊃ ~F (T = Toad wears his bathing suit; F = Toad wants Frog to see him in his bathing suit)
8. ~P ⊃ (S v L) (P = Tom passes his exam; S = Tom is stupid; L = Tom is lazy)
9. H ⊃ W (W = Bekele will win the race; H = Bekele stays healthy)
10. (S v I) ⊃ ~W (S = Bekele is sick; I = Bekele is injured; W = Bekele will win the race)
11. P ⊃ (C ⋅ ~S) (P = Bob will become president; C = Bob runs a good campaign; S = Bob says something stupid)
12. T ⊃ P (T = That plant has three leaves; P = That plant is poisonous)
13. P ⊃ T (T = That plant has three leaves; P = That plant is poisonous)
14. P ⊃ T (T = That plant has three leaves; P = That plant is poisonous)
15. P ⊃ T (T = That plant has three leaves; P = That plant is poisonous)
16. N ⊃ O (O = Olga will swim in the open water; N = There is a shark net present)
17. O ⊃ N (O = Olga will swim in the open water; N = There is a shark net present)
18. O ⊃ B (O = Olga is swimming; B = Olga is wearing a bathing suit)
19. N ⊃ ~B (N = Olga is in Nice; B = Olga wears a bathing suit)
20. T ⊃ B (T = Terrence pulls Philip's finger; B = Something bad will happen)

"Unless"

The English term "unless" can be tricky to translate. For example,

If we use the constant "R" to stand for the atomic proposition, "the Reds will win" and "S" to stand for the atomic proposition, "the Reds' starting pitcher is injured," how would we translate this sentence using truth functional connectives? Think about what the sentence is saying (think carefully). Is the sentence asserting that the Reds will win? No; it is only saying that

"As long as" denotes a conditional statement. In particular, what follows the "as long as" phrase is a sufficient condition, and as we have seen, a sufficient condition is always the antecedent of a conditional. But notice that the sufficient condition also contains a negation. Thus, the correct translation of this sentence is:

One simple trick you can use to translate sentences which use the term "unless" is just substitute the phrase "if it's not the case that" for the "unless". But another trick is just to substitute an "or" for the "unless". Although it may sound strange in English, a disjunction will always capture the truth functional meaning of "unless". Thus, we could also correctly translate the sentence like this:

In the next section we will show how we can prove that these two sentences are equivalent using a truth table.