CHAPTER 7

BASIC LOGIC

7.1 SOME BASIC CONCEPTS

Many people associate logic with brain teasers and mathematical puzzles, which seem to have little relevance to real life. The truth is that logic is of great practical significance. If your friend is in New Zealand, you know she is not in Japan. This piece of everyday reasoning involves logic. The core of logic is about consistency and deduction, both of which are indispensable for everyday thinking, not to mention scientific research and legal reasoning. Logic also plays a special role in computer technology. Computers are good at processing information because their processors can perform a huge number of logical operations very quickly. Obviously, normal people are capable of logical reasoning to some extent, or else we would not be able to survive very long! But making the effort to study some logic can improve our understanding of what good reasoning is like so we can become even better. In this chapter we shall look at some basic concepts of logic.

7.1.1 Consistency

A set of statements is consistent when and only when it is logically possible for all of them to, be true in the same situation. Otherwise they are inconsistent. So for example, “Adrian is happy” and “Adrian is married” are consistent with each other since there is no reason why a married person cannot be happy. On the other hand, “Visanna is 30 years old” and “Visanna is 20 years old” are obviously inconsistent. Here are a few more points to remember about consistency:

• Inconsistent statements are also known as contraries.
• We can also speak of a single statement as consistent or inconsistent, depending on whether it is logically possible for it to be true. “There are round squares” is inconsistent and false. “Paris is in France” is consistent and true. “Nobody lives in Paris” is consistent but false.
• Whether a set of statements is consistent depends on whether it is logically possible for all of them to be true in the same situation. It is not necessary that they are actually true. “Paris is in Italy” and “Nobody lives in Paris” are consistent with each other, even though both are actually false.
• To show that a set of statements is consistent, we can either show that they are actually true or describe a logically possible situation in which they are all true. Consider the two previous statements about Paris. Imagine that Italy conquers France with chemical weapons and takes over Paris. But Paris became contaminated and everyone leaves. This imaginary situation is farfetched but coherent, and shows that the statements are consistent.
• Statements that are actually true are consistent with each other, but false statements might or might not be consistent with each other. The two previous statements about Paris are false but consistent. “Nobody lives in Paris” and “Only 10 people live in Paris” are false and inconsistent with each other.
• If a set of statements is inconsistent, the statements will entail a contradiction of the form:

P and it is not the case that P.

Take “Nobody lives in Paris” and “Only 10 people live in Paris.” The second statement entails “It is not the case that nobody lives in Paris,” and together with the first statement they entail the blatant contradiction:

Nobody lives in Paris and it is not the case that nobody lives in Paris.

Many inconsistencies are easy to detect, but not always. Suppose someone says we should be cautious in making general claims. This seems like good advice because sweeping generalizations like “Every Italian loves pizza” and “All Belgian chocolates are good” are bound to have exceptions. So we might be tempted to conclude that all general claims have exceptions. But the claim “All general claims have exceptions” is actually inconsistent. It is itself a general claim, and if it were true, it should also have an exception. But this implies that not all general claims have exceptions. In other words, the claim cannot possibly be true and is therefore inconsistent!

If we want to speak truly, we should avoid inconsistent statements. But sometimes ordinary speakers use sentences that seem to be inconsistent, such as, “I am happy and I am not happy.” Why do people say things that cannot be true? One answer is that these sentences have incomplete meaning. When we fully specify their meaning, they are no longer inconsistent. For example, perhaps the speaker is happy that she is getting married, but she is also not happy that her ex-boyfriend showed up at the wedding. She is happy about one thing and not happy about a different thing, so there is no real inconsistency.

7.1.2 Entailment

A set of statements P1 … Pn entails (or implies) a statement Q if and only if Q follows logically from P1 … Pn. In other words, if P1 … Pn are all true, then Q must also be true. For example, consider these statements:

P: A bomb exploded in London.

Q: Something exploded somewhere.

Here, P entails Q, but not the other way round. Just because there was an explosion does not mean that a bomb was involved. Perhaps it was it was an egg exploding in a microwave oven. When P entails Q, we say that Q is a logical consequence of P. In symbolic notation, it is P Q. Here are two important points about entailment:

• A set of true statements cannot have false consequences.
• A set of false statements can have true consequences.

If we look at the example carefully, we can see that if P entails Q, and Q turns out to be false, then we should conclude that P must also be false. This point is worth remembering because we often decide that a hypothesis or a theory is false because it entails something false. However, if P entails Q, and P is false, it does not follow that Q is also false. A false theory can have true consequences, perhaps as a lucky accident. Suppose someone believes that the Earth is shaped like a banana. This false belief entails that the Earth is not like a pyramid, which is true. This example tells us we should avoid arguments of the following kind:

Your theory entails Q.

Your theory is wrong.

Therefore, Q must be wrong.

Entailment is related to the logical strength of statements. If a statement P entails another statement Q but not the other way round, then P is stronger than Q, or equivalently, Q is weaker than P. Thus “That is a Boeing 747 airplane” is stronger than “That is an airplane.” As you can see, a stronger statement provides more information, but at the same time it runs a higher risk of being false. Here are some typical ways to qualify a statement to make it weaker:

Although qualifying a statement might make it more plausible, do not overdo it. “The government should increase taxes now” makes a definite and substantive claim. “The government should increase taxes when it is appropriate to do so” is so weak as to say nothing. Highly qualified writing can appear to be boring and wishy-washy. A well-argued but interesting claim is much preferable.

We should also be careful of the opposite situation in which people fail to qualify their claims for various reasons. For example, some people believe shark cartilage can cure cancer, and there is even a book called Sharks Don’t Get Cancer. But it turns out that sharks do get cancer. The book does acknowledge that, but it says that a qualified claim such as “almost no sharks get cancer” would not make a good book title. Guess what, sharks can even get cancers in their cartilage!

7.1.3 Logical Equivalence

If P entails Q and Q entails P, then P and Q are logically equivalent—for example, “Superman is more powerful than Batman” is logically equivalent to “Batman is less powerful than Superman.” When two statements are logically equivalent, they necessarily have the same truth value—it is not possible for one of them to be true and the other one to be false.

• In formal logic, P Q means that P and Q are logically equivalent.
• If P Q, then Q P. Every statement is logically equivalent to itself.

7.2 LOGICAL CONNECTIVES

A logical connective is a logical term that can be attached to statements to form more complex statements.

7.2.1 Conjunction

Given two statements P and Q, their conjunction is the complex statement “P and Q”. P is the left conjunct, Q the right conjunct. Examples:

• Jack died, and Jill went to a party.
• Protons are positively charged, and electrons are negatively charged.

The logical behavior of a conjunction is quite simple. A conjunction “P and Q” is true when both conjuncts P, Q are true. Otherwise the conjunction is false. But be careful of possible ambiguity when and is used to join phrases:

• Ravel studied the philosophy of music and literature. (Literature and philosophy of music, or philosophy of music together with philosophy of literature?)
• We should hire more temporary and part-time drivers. (Temporary drivers and part-time drivers, or part-time drivers who work on a temporary basis?)
• You must use screws, nuts, and bolts of stainless steel. (Are the screws and nuts also made of stainless steel?)

7.2.2 Disjunction

Disjunction is expressed by the word or in English, but it is useful to bear in mind two types of disjunction. When “P or Q” is used in the exclusive sense, this is equivalent to “either P or Q, but not both.” An example might be when a girl issues an ultimatum to her two-timing boyfriend: “Either you stay with me, or you go out with her.” Presumably she is not saying that her boyfriend can do both!

On the other hand, under the inclusive reading, “P or Q” is consistent with the possibility where both P and Q obtains. Suppose your computer is not working, and your friend says, “The hard drive is broken or the motherboard is not working.” We might not want to say that your friend is wrong if it turns out that both components are not working.

The two possible interpretations presents a potential problem in drafting and interpreting legal documents.1 To avoid disputes and unintended consequences, it might be a good idea to be more explicit when disjunction is used, by adding “or both”, or “but not both.” Also, like and, the use of or can lead to syntactic ambiguity:

• You should use white glue or tape. (Does the tape have to be white?)
• No hunting of turtles, fish, or birds on the endangered list. (All turtles and fish, or just those on the list?)

7.2.3 Negation

The negation of a statement P is any statement whose truth-value is the opposite of P. Given any statement in English, we can form its negation by appending the expression “it is not the case that.” So the negation of “it is raining” is “it is not the case that it is raining,” or, in other words, “it is not raining.” Here are some facts about negation:

• A statement and its negation are always inconsistent with each other.
• A statement and its negation form a pair of exhaustive and exclusive alternatives, e.g. Santa Claus exists; Santa Claus does not exist. They cannot both be true and they cannot both be false.2
• Negation involving modal verbs in English can be tricky. “You must leave” and “you must not leave” are inconsistent. But they are not exhaustive alternatives because it is also possible that there is nothing you must do. Perhaps it is up to you whether you stay or leave. The negation of “you must leave” is “it is not the case that you must leave,” not “you must not leave.” However, the negation of “you may leave” is “you may not leave”!
• In formal logic, the negation of P can be symbolized as P, - P, or not - P.

7.2.4 The conditional

A conditional statement (or aconditional) is any statement of the form “If P then Q“—for example, “If you are a member, then you can get a discount.” Conditionals are of special importance because they can be used to formulate rules and general laws:

• Computer programs contain lots of rules about what to do in some given situation. A rule for removing spam messages might be: “If an email contains the words viagra and sex, put it in the trash folder.”
• Many universal scientific laws are conditionals in disguise. “All electrons have negative charge” is equivalent to “For any object x, if x is an electron, then x has negative charge.”
• A lot of legal rules are conditionals describing the legal consequences of specific situations—for example, if you are in a moving vehicle equipped with seat-belts, then you are required to wear one.

Given a conditional “if P then Q“, P is the antecedent of the conditional, and Q the consequent. To accept a conditional is to accept a certain logical or evidential connection between P and Q. But you don’t have to accept that P and Q are both true. For example, you might agree with this statement:

If the sun explodes tomorrow, then we shall all die a sudden death.

But you can consistently agree that the statement is true, even if you do not believe that the sun will explode tomorrow, and you also do not believe that we shall all die suddenly. Here are some additional points about the conditional:

• These claims are correct:

  – When P is true but Q is false, “If P then Q” is false. “If you drink coffee you won’t be able to sleep” is false when you still manage to sleep after drinking coffee.

  – “If P then Q” is logically equivalent to “If not-Q, then not-P“.

  – P. If P then Q. Q.

  – Not-Q. If P then Q. not-P.

• But please note that the two claims below are false:

  – Not-P. If P then Q. not-Q.

  – Q. If P then Q. P.

• The converse of “If P then Q” is “If Q then P“. (In other words, the antecedent and the consequent are swapped.) Normally, a conditional does not entail its converse.3

7.2.5 The biconditional

A biconditional is any statement of the form “P if and only if Q.” This is logically equivalent to:

If P, then Q, and if Q, then P.

In other words, a biconditional is a conjunction of a conditional and its converse. Here are some equivalent formulations:

• P iff Q.
• P when and only when Q.
• P Q (in formal logic)

Here is a more technical point that you might skip if you want: P Q is not the same as P Q. If P Q is true, then P Q does follow. But the converse is not correct: P Q does not imply P Q. For example, it might be true that in a particular course, passing the exam involves getting at least 50 marks. In this situation, a teacher would be speaking the truth when she says, “You pass the exam if and only if you get at least 50 marks.” However, “you pass the exam” is not logically equivalent to “you get at least 50 marks.” It just so happens that in this particular situation, one sentence is true if and only if the other one is. But this is not logically necessary, since a different pass mark is possible, and so equivalence fails.

EXERCISES

1 Some linguists and philosophers argue that or has only the inclusive meaning. Any impression otherwise is due to conversational implicatures or other pragmatic effects. Whatever the case may be, there is no harm in making things clearer when we want to avoid misunderstanding.

2 Recall Section 4.6 on exclusive and exhaustive possibilities.

3 Unless for conditionals such as “If P then P”!