1.1ex I
Getting Started with the Logic Cafe
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. This course will be concerned with
a. symbolic logic. b. travel to South America. c. Chris's grades.

2. Disjunctive syllogism is
a. a principle of logic involving different possibilities. b. a principle of logic relating only to higher mathematics. c. a principle of the law.

3. Did you notice that the "status bar" below is giving hints when you guide the mouse pointer over the choices?
a. Yes. b. No. c. I don't know.

4. What does "1.1ex I" mean? (It's at the top of this page.)
a. "1.1" means "Chapter One, section or tutorial one". "ex I" indicates the first group of exercises in 1.1. b. It doesn't mean anything. c. I don't know.

5. How will I keep track of which exercises, quizzes, and tutorials I've done and which ones I should be doing?
a. Go to the top-left of the Café and click on "resultsTrack" to see a list of finished problems. b. Go to the top-left of the Café and click on "notepad" to keep my own notes on my work and what I should do. c. Both of the above.


1.2ex I Which of the following passages are arguments? Put those which are in standard form.

1. Because Sam has attended each class up until today and since she is here today, it follows that she has perfect attendance.

2. Sports teams like the Atlanta Braves and the Washington Redskins should change their names. This change should be made because the use of native American names for a sports team is degrading.

3. I was born on a poor tobacco farm in Southern Virginia just after the turn of the century. Indoor plumbing didn't come to our community until I was in my teens. By then, we had more serious affairs of war to be worried about.

1.2ex II Which of the following passages are arguments? Put those which are in standard form.

1. Fire is oxidation, the combination of a substance with oxygen. So, when a substance is burned, one can expect that oxygen in the air will be decreased.

2. The computer is down again so you should reload Windows because it, Windows, is the most likely cause of your problems.

3. Over the last two days tornados have resulted in the deaths of 23 individuals. In addition, countless homes were damaged or destroyed. In Oklahoma City alone, damage is estimated to be in the hundreds of millions of dollars.

4. Attendance at law school is justified only for people with a real interest in the law. Hart, however, is primarily interested in a six figure income. Thus, law school is not right for Hart.

5. God, by definition, is a being greater than any other possible being. Since, such a great being must exist, it follows that God exists.

6. Law school is not right for Hart. Attendance at law school is justified only for people with a real interest in the law. Hart, however, is primarily interested in a six figure income though also has some special interests in the opposite sex.

1.2ex III Which of the following passages are arguments? Put those which are in standard form.

1. We are becoming a "Prozac Nation". We need to put a stop to this for a nation of drug dependent robots is unthinkable. Thank God a little depression is not yet seen as intolerable!

2. Look around you. Not only are our bodies well tuned for life, but so are all the rest of the organisms we find. Think about the eagle's eyesight or the rodent's teeth. But it is not just a question of evolution: the cosmos too are well tuned for life. Astronomers and physicists tell us that if the universe were only a little different in density or if the size or proportion of subatomic particles were even slightly altered, then life could not exist at all. It all goes to show that the universe must have a designer: God.

3. God does not exist for by definition God is an all powerful and supremely good being. But a supremely good being would not (if it was powerful enough to help it) allow a universe with so much suffering and evil.

4. Depleted uranium should be used in weaponry even though it's dangerous both to the defense workers who handle it and to those in the line of fire. There are two main reasons for this use: we need to get rid of depleted uranium and it's very dense and thus can penetrate armor.

5. It makes my blood boil! Abortion is very wrong. For one thing, killing a fetus is taking a human life. We are not fit to say that this life is any less worthy than our own -- might it not have just as much soul as we have? But taking a human life is murder!

6. Perhaps abortion should be seen as contrary to some of the world's major religions. But a legal ban on abortion is another matter and should not be countenanced in the United States. In the US, there is freedom of religion. Yet the main reason to think that abortion is wrong presupposes religious principles about the status of the fetus: has it a soul or not?

1.2ex IV
Validity and Soundness
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. If an argument is valid, then it MUST
a. have true premises. b. have a true conclusion. c. be sound. d. none of the above.

2. If the premises of an argument are true and its conclusion is also true, then
a. the argument must be valid. b. the argument must be sound. c. the argument must be valid and sound. d. none of the above

3. The argument "All and only gophers dig holes in the ground. The animals in my yard dig holes in the ground. So, the animals in my yard are gophers" is
a. valid. b. sound. c. valid and sound. d. none of the above.

4. The argument "Every one of Tell's books has been a best seller. So, his next book is likely to be a best seller." is
a. valid. b. sound. c. valid and sound. d. none of the above.

5. An invalid argument MAY have
a. false premises and false conclusion. b. false premises and true conclusion. c. true premises and true conclusion. d. all of the above.

6. A valid argument MAY have
a. false premises and false conclusion. b. false premises and true conclusion. c. true premises and true conclusion. d. all of the above.

7. A valid argument MAY NOT
a. be sound. b. have true premises and a false conclusion. c. neither of the above d. both of the above.

 

1.2ex V                   Valid Arguments

All ravens are birds.
No birds are mammals.
_____________
No ravens are mammals.
Some monkeys are human.
All humans have tails.
_____________
Some monkeys have tails.
All whales are fish.
All fish fly.
_______
All whales fly.
Valid, True Premises,
False Conclusion
Valid, True Premises,
True Conclusion
Valid, False Premises,
False Conclusion
Valid, False Premises,
True Conclusion
Invalid, True Premises,
False Conclusion

Drag the description (in red) to the appropriate argument below.

 

 

 

 

 

 

 

 

 

1.3ex I

Drag the description (in red) to the appropriate argument below. (Place them in the white space above the argument.)

All students of logic use
the Logic Cafe.
Al Gore is a logic student.
Al Gore uses the Logic Cafe.
All students I've talked to said
they liked using the Logic Cafe.
_________________
So, probably, students all like
using the Logic Cafe.
No whales are fish.
No fish are whales.
Sara and Beth are
identical twins.
Sara is very tall.
Beth too is tall.
Deductive, Sound, not Valid
Deductive, Sound, Valid
Inductive, Sound, Valid
Inductive, not Strong
Deductive, Valid, not Sound
Inductive, Strong
Inductive, not Strong, Cogent

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.3ex II
Validity, Soundness, Inductive, Deductive
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. If an argument is valid then
a. it must be sound. b. it cannot be sound. c. it is sound only if all its premises are true. d. it is sound but has false premises.

2. If an argument is valid then
a. it is impossible for its premises to be true when the conclusion is false. b. it cannot be sound. c. it is possible for its premises to be true when its conclusion is false. d. it is possible for its premises to be true when its conclusion is false but only if it is not sound.

3. If a statement is logically impossible, then
a. that statement is true. b. that statement is known to be false. c. it could not have turned out that the statement is true.

4. If an argument is inductive and inductively strong, then
a. that argument has true premises but a false conclusion because it is not deductive. b. that argument is valid but not sound. c. that argument has a conclusion which is probably true if the premises are true. d. that argument must be cogent.

5. If an argument is sound, then
a. it may have a false premise. b. it may fail to be valid. c. it may have a false conclusion. d. none of the above.

6. If an argument is deductive, then
a. it is intended that it be valid. b. it is probably valid. c. it is sound if it is valid. d. it is valid.

by the definition of sound, A is both (1) valid and (2) has only true premises.
because A has only true premises, A has no false premises.
an argument A has false premises.
by the definition of "valid", A cannot have true premises and a false conclusion.
Suppose       __________
Then,      __________
So, finally,      __________
Q.E.D.

1.3ex III          Informal Proofs

Matching. Drag sentences from the right to the correct location in the proof box. Don't print until you've come to the final page of this exercise.

First show that...

1. A sound argument has no false premises.

 

 

 

 

 
"sound", for A to be sound it would need to be (1) valid and (2) have only true premises.
because A does not meet the first requirement for soundness (because A is not valid), it follows that A is not sound.
A is any valid argument.
"sound", for A to be sound it would need to be (1) inductive and (2) have a true conclusion.
Suppose     __________
By the definition of          __________
So,     __________
Q.E.D.
A is any invalid argument.

 

2. An invalid argument is not sound.

Proof:

 

 

 

 

 

 

A is any sound argument.
by the definition of sound, A both (1) is valid and (2) has only true premises.
by (1), A is valid which means that it's impossible for it to have only true premises and a false conclusion.
by (2), all A's premises are true.
putting the last two statements together, we see that it's impossible for A to have a false conclusion. Thus, A's conclusion is true.
Suppose         ___________
Then,         ___________
So,         ___________
But,         ___________
So,         ___________
Q.E.D.

This one is a little more involved; it makes you work with two definitions.

3. If an argument is sound, then it has a true conclusion.

Proof:

 

 

 

 

 

 

 

Either China or Costa Rica is a country in Europe.
China is not a country in Europe.
Costa Rica is a country in Europe.
China is a country in Europe.
(premise 2 here)
______________
(conclusion here)

Here's a different sort of proof. We show that a type of argument is possible by displaying an example of one. So, just drag sentences into place below so that it's obvious that:

4. It is possible for a valid argument to have a false conclusion.

Proof by example:

 

If a country is in Europe, then it's in the Northern Hemisphere.
Costa Rica is a country in Europe.
 
If a country is not in Europe, then it's not in the Northern Hemisphere.
______________________
If a country is not in the Northern Hemisphere, then it's not in Europe.
    (the premise goes here)

Last one. Notice that 5 might seem like the older kind of proof — about an arbitrary subject — is in order. But because of the word "may" it is not!

Again, drag to give an example.

5. A sound argument may have a single premise.

Proof by example:

 

1.4ex I

Drag the description (in red) to the appropriate sentence, pair, or set below.

 
 
 
 
 
All logic students are married.
No logic students are married.
All logic students are unmarried.
All logic students are students.
Some logic students are
not logic students.
All logic students are married.
All math students are unmarried.
{All logic students are male, All math students are female.}
{All logic students are male.,
Some non-males are logic
students.}
  Consistent:
Inconsistent:
Logically equivalent:
Not logically equivalent:
Logically true:
 
Logically false:
Logically indeterminate:
Valid:
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.4ex II
Further Concepts
Multiple Choice: Click on the best answer and the page will jump forward to the next problem.

1. Which of the following is a logically equivalent pair?
a. Neither Bob nor Carol is now a student. Either Bob is not now a student or Carol is not.
b. Neither Bob nor Carol is now a student. Both Bob and Carol are not now students.
c. Neither Bob nor Carol is now a student. If Bob is not a student, then neither is Carol.

2. Which of the following is NOT logically equivalent to "No monkeys are gorillas"?
a. All monkeys are non-gorillas.
b. No gorillas are monkeys.
c. All gorillas are non-monkeys.
d. none of the above

3. If one premise of an argument is logically equivalent to that argument's conclusion, then that argument
a. must be valid.
b. must be sound.
c. must be both valid and sound.
d. none of the above

4. The sentence "All physicians trained in cardiology are physicians" is
a. logically true.
b. logically false.
c. logically indeterminate.

5. The sentence "All physicians are trained in cardiology " is
a. logically true.
b. logically false.
c. logically indeterminate.

6. The sentence "Not all physicians are physicians" is
a. logically true.
b. logically false.
c. logically indeterminate.

7. A logically true sentence may
a. be true.
b. be false.
c. all of the above.
d. none of the above.

8. A logically indeterminate sentence may
a. be true.
b. be false.
c. all of the above.
d. none of the above.

9. The members of a pair of logically true sentences are
a. logically equivalent.
b. both true.
c. all of the above.
d. none of the above.

10. If a set of sentences is inconsistent, then
a. at least one member is logically false.
b. it's not possible for all members of the set to be true.
c. it's not possible for all members of the set to be false.
d. none of the above.

 

\ is a set of sentences including the logically false sentence P.
(for contradiction) that \ is consistent.
by the definition of consistency, it is possible that all members of \ be true together.
P, a member of \, is possibly true. But that means that P is not logically false. However, we are supposing from the beginning that it is logically false!
our assumption (in red) leads to a contradiction. That assumption is wrong, so \ is inconsistent after all.
Suppose that      __________
Assume       __________
Then,      __________
So, it follows that      __________
Thus       __________

1.4ex III         More Informal Proofs

Matching. Drag sentences from the right to the correct location in the proof box. Don't print until you've come to the final page of this exercise.

First show that...

1. If some member of a set of sentences is logically false, then that set is inconsistent.

Prove this by our indirect means of "reductio ad absurdum". (That is, you will show an assumption wrong by showing it leads to contradiction.)

Q.E.D.

 

argument A has a logically true conclusion P.
A is invalid.
to the definition of "invalid", it's possible for all of A's premises to be true while P is false.
(we supposed at the outset) that P is logically true, it can't be false! So, our assumption (in red) has led to a contradiction and must be wrong. Instead A is valid.
Suppose that       ___________
Assume for contradiction
that      ___________
According      ___________
But      ___________
Q.E.D.

2. If the conclusion of an argument is logically true, then that argument is valid.

Proof:

 

 

 

 

 

 

P and Q are both logically false sentences.
P and Q are not logically equivalent.
by the definition of logical equivalence, it is possible for one of P and Q to be true while the other is false.
this "it's possible for one to be true" contradicts are beginning supposition that both are logically false. Hence, the assumption (in red) must be wrong and so P and Q are logically equivalent.
Suppose that       ___________
Assume for contradiction
that       ___________
Then       ___________
But       ___________
Q.E.D.

3. If a pair of sentences are both logically false, then they are logically equivalent.

Proof:

 

 

 

 

 

 

 

 

1.4ex IV         Proofs by Example

Drag groups of sentences from the right and use them as examples to prove the statements on the left. (Place them inside the empty, white boxes.)
Two false sentences can be logically equivalent.
Proof by example:
It's false to claim that two logically indeterminate sentences must be logically equivalent.
Proof by example:

A valid argument may have logically indeterminate premises and conclusion.
Proof by example:

Not all valid arguments have logically indeterminate conclusions.
Proof by example:

Lieberman or McCane won the 2000 presidential election.

Either McCaine or Lieberman won the presidential election of 2000.

Lieberman or McCane will win the 2004 presidential election.

Nader or Gore will win the 2004 presidential election.

Neither Nader nor Buchanan will win the 2004 presidential election.
Alan Keyes will also not win this election.
__________________
Neither Nader nor Keyes will win.

Either Gore or Bush will win the 2004 presidential election.
__________________
Gore will win or he won't

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.4ex V
Yet More Informal Proofs

Write out proofs for each of the following propositions. This work is yours and in your own words. So, the computer can give only limited help.

Try to follow the methods developed in earlier problems. You may want to begin with a supposition about an arbitrary member of a class and prove that a claim holds in general. In other cases, the proposition to prove only requires an example. How do you tell the difference? (a) The general case will ask you to prove a proposition about all or any member of a class. Begin with a supposition about an arbitrary member, and show it must satisfy the claim. (b) The cases for which an example is sufficient will ask you to prove a proposition that some sort of thing is possible. Construct an example to show it is possible!

1. All pairs of logically true sentences are logically equivalent.
This requires a proof by example a general proof.

 

 

2. No invalid arguments are sound.
This requires a proof by example a general proof.

 

3. Not all unsound arguments are invalid.
This requires a proof by example a general proof.

4. If two sentences logically entail each other, then they are logically equivalent.
This requires a proof by example a general proof.