should do in ordinary English.
When is this claim true? Let's look at the form of it:
We don't know which of A, B, and C are true and which are false. We have to look
at all possibilities to decide when the compound claim is true. We can construct a
table:
In the table we first list all possible values for A, B, and C. Then we calculate
the value of A B. With the truth-value of A B we can use the truth-value of C
(to its left in the table) to calculate the truth-value of
We can see now that the original claim can be false only if both "Dick goes to
the movies" is true, and "Zoe visits her mother" is true, and "No one will walk Spot tonight" is false. For example, if Dick doesn't go to the movies (A is F) and Zoe
doesn't visit her mother (B is F), then the whole claim is true—the antecedent of
is false, so the claim is vacuously true.
Perhaps you could have figured out when this claim was true without using a
table. But it's equally routine to analyze a complex claim with the complicated form
Some compound claims are true for every way that their parts are true or false.
For example:
SECTION B The Truth-Value of a Compound Claim 363
Ralph is a dog or Ralph isn't a dog.
It doesn't matter whether A is true or false. Any claim with the form is true.
Tautology A compound claim is a tautology if it is true for every possible
assignment of truth-values to its parts.
The form is a tautology, which reflects that the order of the
parts of an "or" claim doesn't matter.
A claim is a tautology if in the table for its form
the last column of the table has only T.
364 APPENDIX: Truth-Tables
Exercises for Sections A and B
1. What are the four fundamental English words or phrases that we will analyze in studying
compound claims?
2. What is the first big assumption about claims we made when we decided to use the
symbols ?
3. What is a tautology?
4. What is the method for checking whether a claim is a tautology?
5. Explain the method for checking whether two forms of claims are equivalent.
Here's an example of a method Tom devised to check whether a claim is a tautology.
It's a little long-winded, but it made it clear to him.
Decide whether (A B) (A B) is a tautology.
Columns 1 and 2 are all the possible combinations of truth-values of the claims.
Columns 3 and 5 are just 1 and 2 repeated to see how to get column 4
(the table for A B ) .
Columns 7 and 9 are just 1 and 2 repeated so as to see how to get column 8
(the table for A B ).
Then column 6 is the table for i applied to column 8, which gives the table
for
Column 10 is just column 4 repeated. And column 12 is just column 6 again.
That lets us see how to get column 11 using the table for
Column 11 gives the truth-values for ( ) (A B). Since there's an F in
that column, this isn't the form of a tautology.
Use truth-tables to show that the following are tautologies:
6.
7.
8.
9.
Decide whether the following are tautologies using truth-tables. Then explain your answer
in your own words.
SECTION C Representing Claims 365
10.
11.
12.
13.
14.
15.
Using truth-tables, show that the following are equivalent.
16.
17.
18.
19.
C. Representing Claims
To use truth-tables we have to be able to represent ordinary claims and arguments.
E x a m p l e s Can the following be represented in a form that uses ?
Example 1 Spot is a dog or Puff is a cat and Zoe is not a student.
context, we have to guess. We analyze the argument on one reading, then on the
other, and see which is better. Our formal analyses help us see ambiguities.
Example 2 Puff is a cat or someone got swindled at the pet store.
Analysis This one's easy: Puff is a cat someone got swindled at the pet store.
Example 3 London is in England or Paris is in France.
Analysis We can represent this using exclusive "or":
(London is in England Paris is in France)
(London is in England A Paris is in France)
In Exercise 1 I ask you to show:
is true when exactly one of A is true or B is true.
Example 4 Harry is a football player if he plays any sport at all.
Analysis We're used to rewriting conditionals. This one is:
If Harry plays any sport at all, then Harry is a football player.
Harry plays any sport at all Harry is a football player
APPENDIX: Truth-Tables
Example 5 Zoe loves Dick although he's not a football player.
Analysis This is a compound claim, with parts "Zoe loves Dick" and "Dick is not a football player." But "although" isn't one of the words we're formalizing.
When is this compound claim true? If we stick to the classical abstraction, then
"although" doesn't do anything more than "and." It shows that the second part is
perhaps surprising, but that isn't what we're paying attention to. We can formalize
the claim as:
Zoe loves Dick Dick is not a football player
If all we're interested in is whether the argument in which this appears is valid, this
representation will do.
There are a lot of words or phrases that can sometimes be represented with :
and even if even though
but although despite that
Sometimes, though, these serve as indicator words, suggesting the roles of the claims
in the overall structure of the argument. "Even though" can indicate that the claim is
going to be used as part of a counterargument. We can represent these words or
phrases with , or we can just represent the parts of the sentence as separate claims.
That's what we did in Chapter 6, and we can do that because the table for says that
the compound will be true exactly when both parts are true.
Example 6 Spot thinks that Dick is his master because Zoe doesn't take him
for walks.
Analysis Can we represent "because" using ? Consider the following
two claims:
Spot is a dog because Las Vegas is in the desert.
Spot is a dog because Las Vegas is not in the desert.
Both of these are false. Spot is a dog, and that's true whether Las Vegas is or is not
in the desert. The truth-value of "Las Vegas is in the desert" is irrelevant to the
truth-value of the whole compound. Yet all we've got to work with in representing
"because" are compounds that depend on whether the parts are true or false. We
can't represent this example as a compound claim.
Example 7 Zoe took off her clothes and went to bed.
Analysis We shouldn't represent this compound as:
Zoe took off her clothes Zoe went to bed
That has the same truth-value as:
Zoe went to bed A Zoe took off her clothes
EXERCISES for Section C 367
Example 7 is true most nights, but "Zoe went to bed and took off her clothes" is
false. In this example "and" has the meaning "and then next," so that when the claims become true is important. But if we use these symbols we can only consider
whether the claims are true, not when they become true. So we can't represent this
claim.
Example 8 (On the playground): Hit me and I'll hit you.
Analysis We don't represent this as: You hit me I hit you. The example is a
conditional, and we represent it as:
You hit me I hit you
We can't blindly represent every use of "and", "or", "not", and
" i f . . . then . . ." as . We have to ask what the words mean in
the way they're used. Does the use accord with the classical abstraction?
Exercises for Section C
1. Make up the table for and show that it is true when exactly one
of A, B is true.
For each of the following, either represent it using , or explain why it can't be
represented.
2. If critical thinking is hard, then mathematics is impossible.
3. If you don't apologize, I'll never talk to you again.
4. Dick prefers steak, while Zoe prefers spaghetti.
5. Dick was shaving while Zoe was preparing dinner.
6. Either Dick loves Zoe best, or he loves Spot best.
7. Even if you do whine all the time, I love you.
8. Spot is a good dog even though he scared the living bejabbers out of your cat.
9. Spot is a good dog because he scared the living bejabbers out of your cat.
10. We're going to go to the movies or go out for dinner tonight.
11. Since 2 + 2 is 4, and 4 times 2 is 8,1 should be ahead $8, not $7, in blackjack.
12. If Dick has a class and Zoe is working, there's no point in calling their home to ask
them over for dinner.
13. If it's really true that if Dick takes Spot for a walk he'll do the dishes, then Dick won't
take Spot for a walk.
14. If Dick goes to the basketball game, then he either got a free ticket or he borrowed
money from somebody.
368 APPENDIX: Truth-Tables
15. Either we'll go to the movies or visit your mom if I get home from work by 6.
16. Whenever Spot barks like that, there's a skunk or raccoon in the yard.
17. I'm not going to visit your mother and I'm not going to do the dishes, regardless
of whether you get mad at me or try to cajole me.
18. Every student in Dr. E's class is over 18 or is taking the course while in high school.
19. No matter whether the movie gets out early or late, we're going to go out for pizza.
20. Suggest ways to represent:
a. A only if B d. A if and only if B
b. A unless B e. B just in case A
c. When A, B f. Neither A nor B
D. Checking for Validity
An argument is valid if for every possible way the premises could be true, the
conclusion is true, too. So suppose we have an argument of the form:
For an argument of this form to be valid, it has to be impossible that A -> B and
lA—>B are both true, and B is false. We need to look at all ways that A—>B and
~iA—>B could be true:
We list all the values of A and B. Then we calculate the truth-values of A B
and . In the first row both of those are true, and so is the conclusion, B.
Ditto for the third row. In the second row is false, and we don't care about
that. In the last row is false, and we can ignore that. So whenever both
are true, so is B. Any argument of this form is valid.
Valid argument form An argument form is valid if every argument of
that form is valid.
We can show that an argument form is valid by making a table that
includes all the premises and the conclusion. If in every row in which all
the premises are true, the conclusion is true, too, then the form is valid.
SECTION D Checking for Validity 369
Let's look at the indirect way of reasoning with conditionals:
I've drawn a line to indicate the conclusion, rather than write "so" or "therefore."
Again, we have to look at every way the premises could be true.
Only in the last row are both premises true. There we find that A is
true, too. So every argument of this form is valid.
The third row of this table also shows that, in contrast, denying the antecedent
is invalid:
Both are true, but is false. It is possible to have the premises true
and the conclusion false.
Reasoning in a chain provides a more complicated example:
We have the table:
I've circled the rows in which both premises are true. In each of them the conclusion
is also true. So every argument of this form is valid.
370 APPENDIX: Truth-Tables
This last table also shows that the following form isn't valid:
The third row from the bottom has both A B and A C true, with B C false.
So far this has been just a game, playing with symbols. It's only when we can
apply these tables to real arguments that we're doing critical thinking. Consider:
If Tom knows some logic, Tom is either very bright or he studies hard.
Tom is bright. Tom studies hard. So Tom knows some logic.
First we represent these claims. Only the first is a compound claim:
Tom knows some logic (Tom is very bright Tom studies hard)
So this argument has the form:
I've circled a row in which all of A (B C), B, and C are true, yet the conclusion
A is false. So the argument isn't valid.
That alone does not make it a bad argument. We still have to see if it could be
strong. But this argument isn't even strong: Though Tom is very bright and studies
hard, and the first premise is true too, it's not at all implausible that Tom could have
been majoring in art history and knows no logic at all.
You might not have needed a table to figure out this last one. But you will for
some of the exercises. Have fun.
Exercises for Section D
1. What does it mean to say an argument form is valid?
2. If an argument has a form that is not valid, is it necessarily a bad argument?
EXERCISES for Section D 371
Use truth-tables to decide whether the following argument forms are valid.
3.
4.
5.
6.
7.
8.
9.
10.
Represent the arguments in the following exercises and decide whether they are valid.
Use truth-tables or not as you wish.
11. If Spot is a cat, then Spot meows. Spot is not a cat. So Spot doesn't meow.
12. Either the moon is made of green cheese or 2 + 2 = 4. But the moon is not made of
green cheese. So 2 + 2 = 4.
13. Either the moon is made of green cheese or 2 + 2 = 5. But the moon is not made of
green cheese. So 2 + 2 = 5.
14. The students are happy if and only if no test is given. If the students are happy, the
professor feels good. But if the professor feels good, he won't feel like lecturing, and
if he doesn't feel like lecturing, he'll give a test. So the students aren't happy.
15. If Dick and Zoe visit his family at Christmas, then they will fly. If Dick and Zoe visit
Zoe's mother at Christmas, then they will fly. But Dick and Zoe have to visit his family
or her mother. So Dick and Zoe will travel by plane.
16. Tom is not from New York or Virginia. But Tom is from the East Coast. If Tom is
from Syracuse, he is from New York or Virginia. So Tom is not from Syracuse.
17. The government is going to spend less on health and welfare. If the government is going
to spend
less on health and welfare, then either the government is going to cut the
Medicare budget or the government is going to slash spending on housing. If the
372 APPENDIX: Truth-Tables
government is going to cut the Medicare budget, the elderly will protest. If the
government is going to slash spending on housing, then advocates of the poor will
protest. So the elderly will protest or advocates of the poor will protest.
Summary By concentrating on just whether claims are true and the structure of
arguments that involve compound claims, we can devise a method for checking the
validity of arguments. We introduced symbols for the words "and", "or", "not", and "if. . . then . . ." and made precise their meaning through truth-tables. We
learned how to use the symbols and tables in representing claims. Then we saw how
to use truth-tables to check whether the structure of an argument relative to the
compound claims in it is enough to guarantee that the argument is valid.
Key Words classical abstraction conjunction
truth-table negation
disjunction
conditional
tautology
valid argument form
Further Study For a fuller study of the formal logic of reasoning with compound
claims, see my Propositional Logics, also published by Wadsworth.
Aristotelian Logic
A. The Tradition 373
B. Categorical Claims 374
• Exercises for Section B 375
C. Contradictories, Contraries, and Subcontraries 378
• Exercises for Section C 380
D. Syllogisms 381
• Exercises for Section D 383
A. The Tradition
Aristotle, over 2,300 years ago in his Prior Analytics, focused his study on
arguments built from claims of the forms:
All S are P. No S is (are) P.
Some S is (are) P. Some S is (are) not P.
The following argument, for example, uses only claims of these forms:
No police officers are thieves.
Some thieves are sent to prison.
So no police officers are sent to prison.
Aristotle developed a method for determining whether such an argument is valid by
inspection of its form. From then until the early 1900s his work was the basis for