1B. Two ways that deductions can go wrong

1. What is Logic?



5



For any deduction, there are two ways that it could be weak:

1) One or more of the hypotheses might be false. A deduction gives you a reason to

believe its conclusion only if you believe its hypotheses.

2) The hypotheses might fail to entail the conclusion. Even if the hypotheses were true,

the form of the deduction might be weak.

The example we just considered is weak in both ways.

When a deduction is weak in the second way, there is something wrong with the logical

form of the deduction: hypotheses of the kind given do not necessarily lead to a conclusion of

the kind given. We will be interested primarily in the logical form of deductions.

Consider another example:

Hypotheses:

You are reading this book.

This is an undergraduate textbook.

Conclusion: You are an undergraduate student.

This is not a terrible deduction. Most people who read this book are undergraduate students.

Yet, it is possible for someone besides an undergraduate to read this book. If your mother

or father picked up the book and thumbed through it, they would not immediately become

an undergraduate. So the hypotheses of this deduction, even though they are true, do not

guarantee the truth of the conclusion. Its logical form is less than perfect.

A deduction that had no weakness of the second kind would have perfect logical form. If its

hypotheses were true, then its conclusion would necessarily be true. We call such a deduction

“deductively valid” or just “valid.”

Even though we might count the deduction above as a good deduction in some sense, it is

not valid; that is, it is “invalid.” The task of logic is to sort valid deductions from invalid ones.



1C. Deductive validity

A deduction is valid if and only if its conclusion is true whenever all of its hypotheses are

true. In other words, it is impossible for the hypotheses to be true at the same time that the

conclusion is false. Consider this example:

Hypotheses:

Oranges are either fruits or musical instruments.

Oranges are not fruits.

Conclusion: Oranges are musical instruments.

The conclusion of this deduction is ridiculous. Nevertheless, it follows validly from the

hypotheses. This is a valid deduction; that is, if both hypotheses were true, then the conclusion

would necessarily be true. For example, you might be able to imagine that, in some remote

river valley, there is a variety of orange that is not a fruit, because it is hollow inside, like a

gourd. Well, if the other hypothesis is also true in that valley, then the residents must use the

oranges to play music.

This shows that a deductively valid deduction does not need to have true hypotheses or

a true conclusion. Conversely, having true hypotheses and a true conclusion is not enough to

make a deduction valid. Consider this example:



6



1. What is Logic?

Hypotheses:

London is in England.

Beijing is in China.

Conclusion: Paris is in France.



The hypotheses and conclusion of this deduction are, as a matter of fact, all true. This is

a terrible deduction, however, because the hypotheses have nothing to do with the conclusion.

Imagine what would happen if Paris declared independence from the rest of France. Then the

conclusion would be false, even though the hypotheses would both still be true. Thus, it is

logically possible for the hypotheses of this deduction to be true and the conclusion false. The

deduction is invalid.

The important thing to remember is that validity is not about the actual truth or falsity

of the assertions in the deduction. Instead, it is about the form of the deduction: The truth of

the hypotheses is incompatible with the falsity of the conclusion.

EXERCISES 1.3. Which of the following is possible? If it is possible, give an example. If it is

not possible, explain why.

1) A valid deduction that has one false hypothesis and one true hypothesis.

2) A valid deduction that has a false conclusion.

3) A valid deduction that has at least one false hypothesis, and a true conclusion.

4) A valid deduction that has all true hypotheses, and a false conclusion.

5) An invalid deduction that has at least one false hypothesis, and a true conclusion.

1D. Other logical notions

In addition to deductive validity, we will be interested in some other logical concepts.

1D.1. Truth-values. True or false is said to be the truth-value of an assertion. We

defined assertions as sentences that are either true or false; we could have said instead that

assertions are sentences that have truth-values.

1D.2. Logical truth. In considering deductions formally, we care about what would be

true if the hypotheses were true. Generally, we are not concerned with the actual truth value of

any particular assertions—whether they are actually true or false. Yet there are some assertions

that must be true, just as a matter of logic.

Consider these assertions:

1. It is raining.

2. Either it is raining, or it is not.

3. It is both raining and not raining.

In order to know if Assertion 1 is true, you would need to look outside or check the weather

channel. Logically speaking, it might be either true or false. Assertions like this are called

contingent assertions.

Assertion 2 is different. You do not need to look outside to know that it is true. Regardless

of what the weather is like, it is either raining or not. This assertion is logically true; it is

true merely as a matter of logic, regardless of what the world is actually like. A logically true

assertion is called a tautology.

You do not need to check the weather to know about Assertion 3, either. It must be false,

simply as a matter of logic. It might be raining here and not raining across town, it might be

raining now but stop raining even as you read this, but it is impossible for it to be both raining



1. What is Logic?



7



and not raining here at this moment. The third assertion is logically false; it is false regardless

of what the world is like. A logically false assertion is called a contradiction.

To be precise, we can define a contingent assertion as an assertion that is neither a tautology

nor a contradiction.

Remark 1.4. An assertion might always be true and still be contingent. For instance, if there

never were a time when the universe contained fewer than seven things, then the assertion “At

least seven things exist” would always be true. Yet the assertion is contingent; its truth is not

a matter of logic. There is no contradiction in considering a possible world in which there are

fewer than seven things. The important question is whether the assertion must be true, just

on account of logic.

EXERCISES 1.5. For each of the following: Is it a tautology, a contradiction, or a contingent

assertion?

1) Caesar crossed the Rubicon.

2) Someone once crossed the Rubicon.

3) No one has ever crossed the Rubicon.

4) If Caesar crossed the Rubicon, then someone has.

5) Even though Caesar crossed the Rubicon, no one has ever crossed the Rubicon.

6) If anyone has ever crossed the Rubicon, it was Caesar.

EXERCISES 1.6. Which of the following is possible? If it is possible, give an example. If it is

not possible, explain why.

1) A valid deduction, the conclusion of which is a contradiction.

2) A valid deduction, the conclusion of which is a tautology.

3) A valid deduction, the conclusion of which is contingent.

4) An invalid deduction, the conclusion of which is a contradiction.

5) An invalid deduction, the conclusion of which is a tautology.

6) An invalid deduction, the conclusion of which is a contingent.

7) A tautology that is contingent.

1D.3. Logical equivalence. We can also ask about the logical relations between two

assertions. For example:

John went to the store after he washed the dishes.

John washed the dishes before he went to the store.

These two assertions are both contingent, since John might not have gone to the store or washed

dishes at all. Yet they must have the same truth-value. If either of the assertions is true, then

they both are; if either of the assertions is false, then they both are. When two assertions

necessarily have the same truth value, we say that they are logically equivalent.

EXERCISES 1.7. Which of the following is possible? If it is possible, give an example. If it is

not possible, explain why.

1) Two logically equivalent assertions, both of which are tautologies.

2) Two logically equivalent assertions, one of which is a tautology and one of which is

contingent.

3) Two logically equivalent assertions, neither of which is a tautology.



8



1. What is Logic?

4) Two tautologies that are not logically equivalent.

5) Two contradictions that are not logically equivalent.

6) Two contingent sentences that are not logically equivalent.

1E. Logic puzzles



Clear thinking (or logic) is important not only in mathematics, but in everyday life, and can

also be fun; many logic puzzles (or games), such as Sudoku, can be found on the internet or in

bookstores. Here are just a few.

EXERCISE 1.8 (found online at http://philosophy.hku.hk/think/logic/puzzles.php). There was a

robbery in which a lot of goods were stolen. The robber(s) left in a truck. It is known that:

1) No one other than A, B and C was involved in the robbery.

2) C never commits a crime without inviting A to be his accomplice.

3) B does not know how to drive.

So, can you tell whether A is innocent?

EXERCISES 1.9. On the island of Knights and Knaves∗, every resident is either a Knight or a

Knave (and they all know the status of everyone else). It’s important to know that:

• Knights always tell the truth.

• Knaves always lie.

You will meet some residents of the island, and your job is to figure out whether each of them

is a Knight or a Knave.

1) You meet Alice and Bob on the island. Alice says “Bob and I are Knights.” Bob says,

“That’s a lie — she’s a Knave!” What are they?

2) You meet Charlie, Diane, and Ed on the island. Charlie says, “Be careful, not all three

of us are Knights.” Diane says, “But not all of us are Knaves, either.” Ed says, “Don’t

listen to them, I’m the only Knight.” What are they?

3) You meet Frances and George on the island. Frances mumbles something, but you

can’t understand it. George says, “She said she’s a Knave. And she sure is — don’t

trust her!” What are they?

Here is a version of a famous difficult problem that is said to have been made up by Albert

Einstein when he was a boy, but, according to Wikipedia, there is no evidence for this.

EXERCISE 1.10 (“Zebra Puzzle” or “Einstein’s Riddle”). There are 5 houses, all in a row, and

each of a different colour. One person lives in each house, and each person has a different

nationality, a different type of pet, a different model of car, and a different drink than the

others. Also:

• The Englishman lives in the red house.

• The Spaniard owns the dog.

• Coffee is drunk in the green house.

• The Ukrainian drinks tea.

• The green house is immediately to the right of the ivory house.

• The Oldsmobile driver owns snails.

• A Cadillac is driven by the owner of the yellow house.

∗



http://en.wikipedia.org/wiki/Knights and knaves



1. What is Logic?



9



• Milk is drunk in the middle house.

• The Norwegian lives in the first house.

• The person who drives a Honda lives in a house next to the person with the fox.

• The person who drives a Cadillac lives next-door to the house where the horse is kept.

• The person with a Ford drinks orange juice.

• The Japanese drives a Toyota.

• The Norwegian lives next to the blue house.

Who owns the zebra? (Assume that one of the people does have a zebra!)



SUMMARY:

• Important definitions:

◦ assertion

◦ deduction

◦ valid, invalid

◦ tautology, contradiction

◦ contingent assertion

◦ logical equivalence



Chapter 2



Propositional Logic

You can get assent to almost any proposition so long as you are not going to

do anything about it.

Nathaniel Hawthorne (1804–1864), American author



This chapter introduces a logical language called Propositional Logic. It provides a convenient

way to describe the logical relationship between two (or more) assertions.

2A. Using letters to symbolize assertions

In Propositional Logic, capital letters are used to represent assertions. Considered only as a

symbol of Propositional Logic, the letter A could mean any assertion. So, when translating

from English into Propositional Logic, it is important to provide a symbolization key that

specifies what assertion is represented by each letter.

For example, consider this deduction:

Hypotheses:

There is an apple on the desk.

If there is an apple on the desk, then Jenny made it to class.

Conclusion: Jenny made it to class.

This is obviously a valid deduction in English. In symbolizing it, we want to preserve the

structure of the deduction that makes it valid. What happens if we replace each assertion with

a letter? Our symbolization key would look like this:

A: There is an apple on the desk.

B: If there is an apple on the desk, then Jenny made it to class.

C: Jenny made it to class.

We would then symbolize the deduction in this way:

Hypotheses:

A

B

Conclusion: C

There is no necessary connection between some assertion A, which could be any assertion,

and some other assertions B and C, which could be any assertions. The structure of the

deduction has been completely lost in this translation.

The important thing about the deduction is that the second hypothesis is not merely any

assertion, logically divorced from the other assertions in the deduction. The second hypothesis

11



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2. Propositional Logic



contains the first hypothesis and the conclusion as parts. Our symbolization key for the deduction only needs to include meanings for A and C, and we can build the second hypothesis from

those pieces. So we symbolize the deduction this way:

Hypotheses:

A

If A, then C.

Conclusion: C

This preserves the structure of the deduction that makes it valid, but it still makes use of

the English expression “If. . . then. . ..” Although we ultimately want to replace all of the English

expressions with mathematical notation, this is a good start.

The assertions that are symbolized with a single letter are called atomic assertions, because

they are the basic building blocks out of which more complex assertions are built. Whatever

logical structure an assertion might have is lost when it is translated as an atomic assertion.

From the point of view of Propositional Logic, the assertion is just a letter. It can be used to

build more complex assertions, but it cannot be taken apart.

There are only twenty-six letters in the English alphabet, but there is no logical limit to

the number of atomic assertions. We can use the same English letter to symbolize different

atomic assertions by adding a subscript (that is, a small number written after the letter). For

example, we could have a symbolization key that looks like this:

A1 : The apple is under the armoire.

A2 : Deductions always contain atomic assertions.

A3 : Adam Ant is taking an airplane from Anchorage to Albany.

.

.

:

.

A294 : Alliteration angers all astronauts.

Keep in mind that A1 ,A2 ,A3 ,. . . are all considered to be different letters—when there are

subscripts in the symbolization key, it is important to keep track of them.

2B. Connectives

Logical connectives are used to build complex assertions from atomic components. There are

five logical connectives in Propositional Logic. This table summarizes them, and they are

explained below.

symbol nickname

what it means

¬

not

“It is not the case that

”

&

and

“Both

and

”

∨

or

“Either

or

”

⇒

implies

“If

then

”

⇔

iff

“

if and only if

”

As we learn to write proofs, it will be important to be able to produce a deduction in

Propositional Logic from a sequence of assertions in English. It will also be important to be

able to retrieve the English meaning from a sequence of assertions in Propositional Logic, given

a symbolization key. The table above should prove useful in both of these tasks.

NOTATION 2.1. The symbol “.˙. ” means “therefore,” and we sometimes use

A1 , A2 , . . . , An , .˙. B

as an abbreviation for the deduction