## Propositional Logic and Symbolization

Read the introduction and section 3.1. The material reviews and elaborates upon procedures for translating ordinary statements into the language of symbolic logic, which the text calls propositional logic.

Complete the exercises to test your understanding.

# Propositional Logic

In this chapter, we study the second system of deduction, Propositional Logic. We are going to learn how to use the truth table to determine the validity of some deductive arguments, whose validity is much more difficult to determine using the system of Categorical Logic. For example, it would feel unnatural to paraphrase the following argument into a categorical syllogism and then decide its validity.

Either interest rates will be raised or inflation will get worse. Since inflation will get worse, interest rates will be raised.

So instead, we symbolize the argument in a different way to see its logical form. First of all, we use the capital letters “R” and “W” to stand for the sentences “Interest rates will be raised” and “Inflation will get worse.” We then use the symbol “∨” for the phrase “Either … or …” and symbolize the first sentence as “R ∨ W”. As a result, we get the following argument form:

 R ∨ W W R

____________________

R: Interest rates will be raised.

W: Inflation will get worse.

To decide whether the argument form is valid, we will learn a formal procedure called the truth table method. By using the method, we can find out that the form is invalid. The argument is therefore not a good argument.

## 3.1 Symbolization

In Propositional Logic, we use a capital letter to represent a simple sentence. Simple sentences are relatively short and do not contain any other sentence as a component. Here are three examples of simple sentences:

• Snow is white.
• The sky is blue.
• Nancy will go to the party.

We can use the letter “S” to symbolize “Snow is white” and “B” for “The sky is blue.”

In contrast with simple sentences, a sentence like

• John will not go to the party.

is a shortened form of the more wordy sentence

• It is not the case that John will go to the party.

which contains the simple sentence “John will go to the party” as a clause. Sentences that contain at least one simple sentence as a component are called compound sentences.

### 3.1.1 Five Types of Compound Sentences

To symbolize compound sentences we need to use symbols. There are five different types of compound sentences.

• #### Negations

The sentence

Inflation is not getting worse.

can be rewritten as

It is not the case that inflation is getting worse.

This makes it clear that it is a negation of the simple sentence “Inflation is getting worse.” Accordingly, we can symbolize it as ∼I. Here the symbol “∼” (tilde) stands for “It is not the case that …” and the letter “I” symbolizes the simple sentence “Inflation is getting worse.”

 Inflation is not getting worse. = ∼I

Notice that there is no space between “∼” and “I”. The equal sign “=” is used to indicate that the sentence and its symbolized expression “∼I” are logically equivalent.

• #### Conjunctions

Here is an example of conjunction and its symbolized expression:

 Globalization and free trade are good for business corporations. = G • F

It isier to see how the symbolization works by expanding the original sentence as

The letter “G” stands for “Globalization is good for business corporations,” and “F” represents “Free trade is good for business corporations.” The symbol “•” (dot) is used to symbolize the English connective “and”. Notice that there is a space between “G” and “•” and between “•” and “F”. Both G and F are called the conjuncts of the conjunction. The symbolization is fairly intuitive and there is a one-to-one correspondence to the English sentence.1

In the next conjunction,

 The cause of schizophrenia is not demonic possession, but defective genes. = The cause of schizophrenia is not demonic possession, but the cause of schizophrenia is defective genes. = ∼P • G

The letter “P” is used to stand for “The cause of schizophrenia is demonic possession,” and “G” for “The cause of schizophrenia is defective genes.” Notice that the first conjunct ∼P is a negation.

• #### Disjunctions

In the next example, two simple sentences are connected with the connective “or” to form a disjuction.

 The economy will slow down or inflation will get worse. = E ∨ I

The letter “E” is used to symbolize “The economy will slow down,” and “I” symbolizes “Inflation will get worse.” Both E and I are called the disjuncts of the disjunction. The symbol “∨” (wedge) is used to symbolize common English connectives such as “or” and “either … or …”.

The sentence

It won’t rain or snow.

is a negation of a disjunction. We can see this by rewriting it as

 It is not the case that it will rain or it will snow. = ∼(R ∨ S) R: It will rain. S: It will snow.

• #### Conditionals

The most commonly seen conditional sentences are sentences combined together using the phrase “If …, then …”.

 If the worldwide demand for oil continues to grow, then gas prices will keep on rising. = D ⊃ G

The symbol “⊃” (horseshoe) is used to symbolize the phrase “If …, then …”. In doing so, we treat conditionals as material conditionals, even though it should be noted that there are differences between material conditionals and other types of conditionals such as indicative conditionals and counterfactual conditionals.

• #### Biconditionals

Sentences such as

 The defendant is guilty if and only if the jury reaches such a verdict. = D ≡ J

are biconditional sentences. The phrase “if and only if” is symbolized using “≡” (triple bar). We will study conditionals and biconditionals in more details later on in this section.

The following table lists the five connectives and the types of compound sentences they are used to form.

### 3.1.2 The Use of Parentheses

When we symbolize compound statements, it is important to use parentheses properly when they are required. Notice the difference between sentences (3.1a) and (3.1b).

 Mike will go to college or find a job, and get married. 3.1a = (C ∨ J) • M
 Mike will go to college, or find a job and get married. 3.1b = C ∨ (J • M)

Without the parentheses, the symbolized expression C ∨ J • M amounts to the sentence without the comma “,”.

Mike will go to college or find a job and get married.

The sentence is ambiguous because we would not be able to tell whether it means (3.1a) or (3.1b).

Early on, we saw the sentence

 It won’t rain or snow. = ∼(R ∨ S)

Since it is a negation of the disjunction R ∨ S, we need to put parentheses around the disjunction.

### 3.1.3 Conditional Sentences

Conditional sentences such as

• If it rains, then the game will be postponed.
• Interest rates will be raised if inflation gets worse.

are used frequently in Propositional Logic. They have the common form of

If p, then q.

or

q if p.

Here the italic letters “p” and “ q” are used as variables for sentences. Notice that “ q if p” is just another way of writing “If p, then q.”

• #### Symbolizing Conditionals

As stated above, we treat conditional sentences as material conditionals and symbolize them using the connective “⊃”.

 If p, then q. = q if p. = p ⊃ q If p, then q. = p only if q. = p ⊃ q

Using the above formula, we can see that the following three sentences are simply different ways of writing the same conditional.

 If Tim is a football player, then Tim is an athlete. = Tim is an athlete if Tim is a football player. = Tim is a football player only if Tim is an athlete. = F ⊃ A F: Tim is a football player. A: Tim is an athlete.

Pay special attention to the difference between the word “if” and the phrase “only if” when they are used in the middle of sentences.

• #### The Sufficient Condition and the Necessary Condition

In a conditional pq, p is called the antecedent, and q is the consequent. The antecedent is a sufficient condition for the consequent, and the consequent is a necessary condition for the antecedent.

To say that p is a sufficient condition for q means that p is all that is needed for q to be the case.

 p is a sufficient condition for q. = p is all that is needed for q to be the case.

For example, in the conditional

 If it rains, then the game will be postponed. = R ⊃ P R: It rains. P: The game will be postponed.

R is a sufficient condition for P. This means that raining is all that is needed for the game to be postponed. In other words, raining alone can result in the game being postponed. In the next example,

• If it is a dolphin, then it is a mammal.

being a dolphin is a sufficient condition for an animal being a mammal.

If p is a sufficient condition for q, then this also means that q is a necessary condition for p. So being a mammal is a necessary condition for being a dolphin. That is, without being a mammal, an animal cannot be a dolphin.

 q is a necessary condition for p. = Without q, it cannot be the case that p.

In the conditional

Alex can earn a high school diploma only if he passes the Exit Examination.

passing the exam is a necessary condition for earning the diploma, that is, without passing the exam, Alex would not be able to earn the high school diploma.

In 1.3, we learn about certain concepts such as validity and soundness used in argument evaluation. For a deductive argument to be sound, it needs to be valid and have true/acceptable premises. So if a deductive argument is not valid, it cannot be sound. This means that being a valid argument is a necessary condition for being a sound argument.

In the sentence

 Tim will go camping only if it does not rain or snow. = C ⊃ ∼(R ∨ S)

the negation ∼(R ∨ S) is the consequent. This means that a necessary condition for Tim to go camping is that it does not rain or snow. So if it rains or snows, Tim will not go camping.

• #### Biconditionals

The following sentence is a biconditional sentence and is symbolized with the connective “≡” (triple bar).

 Maria will receive a pay raise if and only if she gets the TV commercial account. 3.1c = P ≡ A P: Maria will receive a pay raise. A: Maria gets the TV commercial account.

A biconditional is really a conjunction of two conditionals. The connective “≡” typically stands for the English phrase “ … if and only if …”. The formula

p if and only if q

is an abbreviation for

 (p if q) and (p only if q) = (q ⊃ p) and (p ⊃ q)

Fully symbolized, it becomes

(qp) • (pq)

Now in the first conditional qp, q is the antecedent. This means that q is a sufficient condition for p. In the second condition pq, q is the consequent, and hence is a necessary condition for p. The two conditionals together says that q is both a sufficient condition and a necessary condition for p.

Moreover, since

(qp) and ( pq)

is logically equivalent to

(pq) and (qp)

this means that p is also both a sufficient condition and a necessary condition for q.

In the example we just saw

 Maria will receive a pay raise if and only if she gets the TV commercial account. 3.1c = P ≡ A P: Maria will receive a pay raise. A: Maria gets the TV commercial account.

Maria’s getting the TV commercial account is both a sufficient condition and a necessary condition for her receiving a pay raise. This means that getting the account is the only thing she needs to do to receive a pay raise. Moreover, without getting the account, she won’t receive a pay raise. That is, getting the account is also the only way she can receive a pay raise:

 getting the account → receiving a pay raise not getting the account → not receiving a pay raise

Compare (3.1c) with the next example:

 Maria will receive a pay raise if she gets the TV commercial account. 3.1d = A ⊃ P P: Maria will receive a pay raise. A: Maria gets the TV commercial account.

The conditional says that Maria’s getting the TV commercial account is a sufficient condition for her receiving a pay raise. This means that all she needs to do to receive a pay raise is to get the account. Notice the conditional does not say that getting the account is a necessary condition for the pay raise. So it leaves it open whether Maria will receive a pay raise if she does not get the account. This means that it is possible for Maria to receive a pay raise even though she fails to land the account. One possible scenario is that Maria does not get the account, but manages to sell the commercial to another company and as a result receives a pay raise.

 getting the account → receiving a pay raise not getting the account → may or may not receive a pay raise

The important difference between (3.1c) and (3.1d) is that (3.1c) specifically says that getting the account is required (that is, a necessary condition) for Maria to receive a pay raise whereas (3.1d) does not say that it is required. This makes (3.1d) less demanding than (3.1c). Now compare the above two examples with another one:

 Maria will receive a pay raise only if she gets the TV commercial account. 3.1e = P ⊃ A P: Maria will receive a pay raise. A: Maria gets the TV commercial account.

The sentence (3.1e) says that Maria’s getting the TV commercial account is a necessary condition for her receiving a pay raise. This means that without getting the account, she won’t receive a pay raise. But (3.1e) does not specify that getting the account is a sufficient condition for receiving a pay raise. So Maria may or may not receive a pay raise even if she does get the account.

 getting the account → may or may not receive a pay raise not getting the account → not receiving a pay raise

It is important to notice the differences among the three examples here. The specification of condition(s) makes (3.1d) the best deal among the three for Maria, and (3.1e) the least favorable.

### 3.1.4 Exclusive vs. Nonexclusive Disjunctions

The connectives “or” and “either … or” are used in two distinct ways in daily discourses. When a host asks you “Coffee or Tea?”, it is implicitly implied that you should choose coffee or tea, but not both. The connective “or” is used in the exclusive sense to mean “one or the other, but not both.” Afterwards, when the host asks you again “Cream or sugar?”, you can respond by saying “Both, please.” Now the connective is used in the nonexclusive sense of “one or the other, or both.”

Here is an example of “either … or …” used in the nonexclusive sense:

 Either fire or smoke can damage the paintings. 3.1f = F ∨ S F: Fire can damage the paintings. S: Smoke can damage the paintings.

If either fire or smoke alone can damage the paintings, then the two together can damage the paintings. In Propositional Logic, the wedge “∨” is used to symbolize nonexclusive disjunctions. So the sentence (3.1f) is symbolized as F ∨ S. By contrast, in the next sentence “either … or …” is used in the exclusive sense.

 The Federal Reserve will either raise interest rates or leave them intact. 3.1g = (R ∨ L) • ∼(R • L) R: The Federal Reserve will raise interest rates. L: The Federal Reserve will leave interest rates intact.

Obviously the Federal Reserve cannot both raise interest rates and keep them intact. Consequently, we have to symbolize (3.1g) as (R ∨ L) • ∼(R • L). The first conjunct R ∨ L means that the Federal Reserve will do one or the other, or both. But the second conjunct ∼(R • L) says that the Federal Reserve will not do both. So together, they capture the meaning of “one or the other, but not both.”

• #### How to Symbolize “unless”

A compound sentence formed with the connective “unless” can be symbolized as a conditional or a biconditional, depending on the meaning of the sentence. It can also be symbolized as a disjunction. But in doing so, we need to pay attention to whether it is the exclusive or the nonexclusive disjunction. The sentence

 Jeff cannot graduate unless he completes all the GE requirements. 3.1h

states clearly that fulfilling the GE requirements is a necessary condition for Jeff to graduate. That is, without fulfilling the GE requirement, Jeff cannot graduate. So we can rewrite (3.1h) as

 Jeff can graduate only if he completes all the GE requirements. 3.1i = G ⊃ C G: Jeff can graduate. C: Jeff completes all the GE requirements.

The sentence (3.1i) is also logically equivalent to

 If Jeff does not complete all the GE requirements, then he cannot graduate. 3.1j = ∼C ⊃ ∼G G: Jeff can graduate. C: Jeff completes all the GE requirements.

We can also rewrite (3.1h) as

 Either Jeff completes all the GE requirements or he cannot graduate. 3.1k = C ∨ ∼G

Notice (3.1k) is symbolized as a nonexclusive disjunction C ∨ ∼G, because it is possible that Jeff completes all the GE requirements but still cannot graduate due to, say, having not yet met the total unit requirement.

Since p ∨ q is logically equivalent to q ∨ p, now if we let p = C and q = ∼G, then we can see clearly that C ∨ ∼G is logically equivalent to ∼G ∨ C. As a result, we can symbolize (3.1h) as ∼G ∨ C.

 Jeff cannot graduate unless he completes all the GE requirements. 3.1h = ∼G ∨ C

This shows us that we can symbolize “unless” in the nonexclusive sense with the symbol “∨”. Now compare (3.1h) with the next sentence:

 Mike will remain single unless he marries Katie. 3.1l = S ≡ ∼K S: Mike will remain single. K: Mike marries Katie.

(3.1l) means that if Mike does not marry Katie, then he will remain single, and if he marries Katie, he would not remain single. (The second conditional is normally left unstated because there is no need to say the obvious.) The two conditionals together say that marrying Katie is both a sufficient condition and a necessary condition for Mike’s not staying single. So we can rewrite it as

 Mike will remain single if and only if he does not marry Katie. 3.1m = S ≡ ∼K S: Mike will remain single. K: Mike marries Katie.

The example shows that we can symbolize “unless” in the exclusive sense with the triple bar “≡” and the tilde “∼”.

### 3.1.5 “Not … both …” and “Both … not …”

It is important not to conflate “Not … both …” and “Both … not …”. Compare these two sentences:

 Not both Monet and Chopin are painters. 3.1n = It is not the case that Monet is a painter and Chopin is a painter. = ∼(M • C) M: Monet is a painter. C: Chopin is a painter.
 Both Dvořák and Schubert are not painters. 3.1o = Dvořák is not a painter and Schubert is not a painter. = ∼D • ∼S D: Dvořák is a painter. S: Schubert is a painter.

The sentence (3.1n) denies that both Monet and Chopin are painters. That is, it says that at least one of them is not a painter. It can be rephrased as

 Either Monet or Chopin is not a painter. 3.1p = Either Monet is not a painter or Chopin is not a painter. = ∼M ∨ ∼C

It is probably easier to see why (3.1p) is symbolized as ∼M ∨ ∼C if we expand it fully as

Either Monet is not a painter or Chopin is not a painter.

Since (3.1n) and (3.1p) are logically equivalent, we have ∼(M • C) = ∼M ∨ ∼C.

By contrast, the sentence (3.1o) is a conjunction.

 Both Dvořák and Schubert are not painters. 3.1o = Dvořák is not a painter and Schubert is not a painter. = ∼D • ∼S D: Dvořák is a painter. S: Schubert is a painter.

Moreover, (3.1o) is logically equivalent to both sentences below:

 Not either Dvořák or Schubert is a painter. 3.1q = It is not the case that either Dvořák is a painter or Schubert is a painter. = ∼(D ∨ S)
 Neither Dvořák nor Schubert is a painter. 3.1r = It is not the case that either Dvořák is a painter or Schubert is a painter. = ∼(D ∨ S)

Both (3.1q) and (3.1r) deny that at least one of them is a painter. So ∼D • ∼S is logically equivalent to ∼(D ∨ S).

The following formulas sum up the differences between “Not … both …” and “Both … not …” and show how to symbolize them:

 Not both p and q = ∼(p • q) = Either ∼p or ∼q = ∼p ∨ ∼ q
 Both ∼p and ∼q = ∼p • ∼q = Not either p or q = ∼(p ∨ q) = Neither p nor q = ∼(p ∨ q)

As a matter of fact, these two formulas are called De Morgan’s laws in logic.

1. In Propositional Logic, a capital letter without quotation marks represents a simple statement. When we want to talk about the letter itself, we put quotation marks around the letter. Following this convention, we would say that “B” is the second letter in the symbolized expression “A • B”. By contrast, we would say that B is the second conjunct in the conjunction A • B. This is important because it would be a mistake to say that “B” is the second conjunct in the conjunction A • B.

Following the same convention, we say “T” and “F” are two capital letters, but T (true) and F (false) are two possible truth values of a statement in Propositional Logic.

##### Exercise #1

Use the concepts of the sufficient condition and the necessary condition to decide whether the inferences are correct.

1. Suppose Nick’s mother tells Nick that he can play video games only after he finishes his homework. This means that Nick is entitled to play video games after he finishes his homework.
2. Ryan needs to pass the exit examination to graduate from high school. If we find out that Ryan has passed the test, then we know that he can graduate.
3. Ashley will buy a new car if she is promoted to the manager position. After learning that she gets the promotion we can infer that she will buy a new car.
4. The Miller family will buy a house unless their offer is rejected. The house owner accepts their offer. So they are going to buy the house.
5. A reasonable person cannot be closed-minded. From the fact that a person is open-minded it follows that she or he is reasonable.
6. Lowering interest rates can boost the stock market. Upon learning that the Federal Reserve just lowered the interest rates, we can be confident that the stock market will go up.
7. We learn from Alzheimer’s disease that memory requires a healthy and functional brain. This means that without a brain, we would not have memory.
8. A college degree has increasingly become a required qualification for a good-pay job. So by making college education affordable and accessible we can achieve economic equality.
9. Many moral philosophers maintain that we are morally responsible for our actions because we have free will. So if our free will is curtailed under certain circumstances, then we would be less morally responsible in such situations.
10. Without stem cell researches, it would be much harder and take longer time to find cures for many diseases. So if we want good progress in finding cures, we need to permit and fund stem cell researches.
11. Reducing the number of migrant workers would lead to labor shortage or higher labor cost, which would in turn cause higher consumer price index and weaken the economy. So to keep the economy strong, we need to find a way to accommodate migrant workers.
12. There is no religious freedom if we have a state religion. This is why the separation of church and state is essential for freedom of religion.

Source: Wu Wei-Ming, http://www.butte.edu/resources/interim/wmwu//iLogic/3.1/iLogic_3.html