Natural Language

As an exercise in working with Propositional Logic, let's look at the encoding of various English sentences as formal sentences in Propositional Logic. As we shall see, the structure of English sentences, along with various key words, such as if and no, determine how such sentences should be translated.

The following examples concern three properties of people, and we assign a different proposition constant to each of these properties. We use the constant c to mean that a person is cool. We use the constant f to mean that a person is funny. And we use the constant p to mean that a person is popular.

As our first example, consider the English sentence If a person is cool or funny, then he is popular. Translating this sentence into the language of Propositional Logic is straightforward. The use of the words if and then suggests an implication. The condition (cool or funny) is clearly a disjunction, and the conclusion (popular) is just a simple fact. Using the vocabulary from the last paragraph, this leads to the Propositional Logic sentence shown below.

Next, we have the sentence A person is popular only if he is either cool or funny. This is similar to the previous sentence, but the presence of the phrase only if suggests that the conditionality goes the other way. It is equivalent to the sentence If a person is popular, then he is either cool or funny. And this sentence can be translated directly into Propositional Logic as shown below.

A person is popular if and only if he is either cool or funny. The use of the phrase if and only if suggests a biconditional, as in the translation shown below. Note that this is the equivalent to the conjunction of the two implications shown above. The biconditional captures this conjunction in a more compact form.

Finally, we have a negative sentence. There is no one who is both cool and funny. The word no here suggests a negation. To make it easier to translate into Propositional Logic, we can first rephrase this as It is not the case that there is a person who is both cool and funny. This leads directly to the following encoding.

Note that, just because we can translate sentences into the language of Propositional Logic does not mean that they are true. The good news is that we can use our evaluation procedure to determine which sentences are true and which are false.

Suppose we were to imagine a person who is cool and funny and popular, i.e. the proposition constants c and f and p are all true. Which of our sentences are true and which are false?

Using the evaluation procedure described earlier, we can see that, for this person, the first sentence is true.

The second sentence is also true.

Since the third sentence is really just the conjunction of the first two sentences, it is also true, which we can confirm directly as shown below.

Unfortunately, the fourth sentence is not true, since the person in this case is both cool and funny.

In this particular case, three of the sentences are true, while one is false. The upshot of this is that there is no such person (assuming that the theory expressed in our sentences is correct). The good news is that there are cases where all four sentences are true, e.g. a person who is cool and popular but not funny or the case of a person who is funny and popular but not cool. Question to consider: What about a person is neither cool nor funny nor popular? Is this possible according to our theory? Which of the sentences would be true and which would be false?