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.2 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:
It is raining.
The ground it wet.
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:
R ⊃ G
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.
|p||q||p ⊃ q|
Let’s see how this applies to the above conditional, “if it is raining, then the ground is wet.” As before, we can think about the meaning of the truth functional connectives by asking whether the sentences containing those connectives would be true or false in the four possible scenarios. The first two are pretty easy. If I assert the above conditional “if it is raining then the ground is wet” when it is both raining and the ground is wet (i.e., the first line of the truth table below), then the conditional statement would be true in that scenario. However, if I assert it and it is raining but the ground isn’t wet (i.e., the second line of the truth table below), then my statement has been shown to be false. Why? Because I’m asserting that any time it is raining, the ground is wet. But if it is raining but the ground isn’t wet, then this scenario is a counterexample to my claim—it shows that my claim is false. Now consider the scenario in which it is not raining but the ground is wet. Would this scenario show that my conditional statement is false? No, it wouldn’t. The reason is that the conditional statement R ⊃ G is only asserting something about what is the case when it is raining. So this conditional statement isn’t asserting anything about those scenarios in which it isn’t raining. I’m only saying that when it is raining, the ground is wet. But that doesn’t mean that the ground couldn’t be wet for other reasons (e.g., a sprinkler watering the grass). So the meaning of the material conditional should count a statement true whenever its antecedent is false. Thus, in a scenario in which it is neither raining nor the ground is wet (i.e., the fourth line of the truth table), the conditional statement should still be true. Would the fact of a sunny day and dry ground show that the conditional R ⊃ G is false? Of course not! Thus, as we’ve seen, the material conditional is false only when the antecedent is true and the consequent is false.
|R||G||R ⊃ G|
It is sometimes helpful to think of the material conditional as a rule. For example, suppose that I tell my class:
If you pass all the exams, you will pass the course.
Let’s symbolize “you pass all the exams” as “E” and “you pass the course” as “C.” We would then symbolize the conditional as:
E ⊃ C
Under what conditions would my statement E ⊃ C be shown to be false? There are four possible scenarios:
|E||C||E ⊃ C|
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).
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)
2 Actually, there is one more truth functional connective that we will not be learning and that is what is called the “biconditional” or “material equivalence.” However, since the biconditional is equivalent to a conjunction of two different conditionals, we don’t actually need it. Although I will discuss material equivalence in section 2.9, we will not be regularly using it.