2.1ex I    Symbolizations in SL

Use the following symbolization key to symbolize 1 - 5 in the space provided.

A: Agnes will attend law school.
B: Bob will attend law school.
L: They will get a loan.

  1. Either Agnes or Bob will attend law school.
  2. If Agnes will attend law school, then so will Bob.
  3. It's not the case that they will get a loan.
  4. If it's not the case that they will get a loan, then Agnes will not attend law school.
  5. Agnes will attend law school if and only if Bob will attend.

2.1ex II        Basic SL

Multiple Choice: Click on the best answer and the page will jump forward to the next problem.

1. An atomic sentence of SL is a simplest sentence of our symbolic language.
a. True b. False

2. Any molecular sentence of SL is constructed by connecting two atomic sentences together.
a. True b. False

3. "Agnes and Bob will both attend law school" may be symbolized as
a. A&B b. AvB c. A>B d. A=B

4. "Either Agnes or Bob will attend law school" may be symbolized as
a. A&B b. AvB c. A>B d. A=B

5. "Agnes will attend law school just in case Bob will" may be symbolized as
a. A&B b. AvB c. A>B d. A=B

6. Which of the following is not a grammatically correct sentence of SL?
a. A&B b. AvB c. A~B d. A=B

7. A conjunction is made up of
a. A conjunct, a wedge, and a disjunct. b. Two conjuncts and an ampersand. c. Neither of the above.

8. A conditional is made up of
a. A conjunct, a wedge, and a disjunct. b. Two conjuncts and a horseshoe. c. An antecedent, a horseshoe, and a consequent. d. Neither of the above.

9. The components of a disjunction are called...
a. disjuncts. b. disjunctettes. c. conjuncts. d. None of the above.

10. "If Agnes won't attend law school, then neither will Bob" may be symbolized as
a. ~(A>B) b. ~A>B c. A>~B d. ~A>~B

 

2.2ex I    Drag to complete the following.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.2ex II           Semantics
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. A "metavariable" is
a. a variable ranging over all numbers. b. a variable ranging over all sentences of SL. c. a special kind of sentence of SL. d. none of the above.

2. Which of the following are metavariables:
a. #, $   b. P, Q   c. >, =   d. p, q

3. 'P&Q'
a. is a conjunction of SL. b. is nothing because of the quote marks. c. stands for any conjunction of SL.

4. Inclusive "or"
a. has nothing to do with wedge. b. means a disjunction assigned true when both its disjuncts are true. c. is really conjunction and should be symbolized with the '&'.

5. Exclusive "or"
a. may be symbolized with the wedge. b. is the English "or". c. is false when both disjuncts are true.

6. The wedge of SL is
a. inclusive "or". b. exclusive "or". c. neither of the above.

 

2.2ex III         More Semantics
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. A conditional statement P>Q is true if
a. P is true and Q is true. b. P is false and Q is true. c. P is false and Q is false. d. All of the above.

2. A conditional statement P>Q is false if
a. P is true. b. Q is false. c. P is true and Q is false. d. All of the above.

3. A biconditional P=Q is true if
a. P is true and Q is true. b. P is true and Q is false. c. Both of the above. d. None of the above.

4. A biconditional P=Q is logically equivalent to
a. P&Q b. P>Q c. (P>Q)&(Q>P) d. None of the above.

 

2.3ex I       Truth Functionality
Instructions: Just use your mouse pointer to click on what you think is the best answer. Then move on to the next exercise. After you've finished, click on the "get score" button to see how well you've done.

1. Which of the following is NOT a truth functional operator of SL?
&
~
None, each is truth functional. (All SL connectives are truth functional.)

2. Which of the following connectives of English is NOT a truth functional connective?
It's not the case that
It's possible that
None, each is truth functional. (All English connectives are truth functional!?)

3. Which of the following English phrases signals that a compound is not truth functional?
Both
If it were the case that
Although it's the case that

4. Which of the following English phrases signals that a compound is truth functional?
possibly
believes that
it's false that

 

2.3ex II        Truth Functional and non-Truth Functional Connectives
See if you can come up with some new examples from English. This isn't easy. To start this exercise, go back to the tutorial and think about the examples discussed there. That will help you come up with new ones.

1. Give an example of an English connective that is not truth functional. Then, briefly explain why the example is fails to be truth functional. (To explain, keep the definition of truth functionality firmly in mind.)

2. An example of a NON-truth functional connective and an explanation goes here:

3. Now, what might be just as hard, come up with a new example of an English connective which seems to you to be truth functional. Again explain why you think it is.

4. An example of a truth functional connective and an explanation goes here.

 

2.4ex I         Symbolizations in SL

Use the following interpretation and symbolize each of the following in the space provided:

A: Ames is a politician.
B: Bates is a politician.
C: Connors is a politician. D: Ames is disreputable.
E: Bates is emotional.
F: Connors is fastidious.


1. Ames is a politician but he's disreputable.
2. Either Ames is a politician or Bates is one.
3. Ames is a politician only if he's not disreputable.
4. Ames is a politician if he's not disreputable.
5. Ames is a politician if and only if Connors is a politician.

 

2.4ex II        Symbolizations
Symbolize each of the following. Once you have finished with any answer and moved on — it's best to TAB to move on — the program will check your work. As long as your answer is logically equivalent to what I deem to be a correct answer, your answer will be counted as correct too.

Hints: If you just can't get an answer, you can get help. Enter "?" instead of a symbolization, and the correct answer will pop up. Finally, you can enter text without the shift-key. The program will understand.

Use the following interpretation:

L: Lamb works for OU.
M: Moss works for OU.
N: Nute works for OU.
G: Lamb tends the OU golf course.
H: Moss manages the OU hotel.
A: Nute is an administrator.

  1. Both Lamb and Moss work for OU.
  2. Neither Lamb nor Moss work for OU.
  3. Either Lamb or Moss does not work for OU.
  4. Lamb works for OU unless Nute is an administrator.
  5. Moss and Nute don't both work for OU.
  6. If Lamb works for OU then neither Moss nor Nute does.
  7. Lamb works for OU only if he tends their golf course.
  8. Being an administrator is a necessary condition for Nute to work for OU.
  9. Moss works for OU if she manages the OU hotel.
  10. Nute works for OU however he is an administrator.
  11. Being an administrator is a sufficient condition for Nute to work for OU.
  12. Moss works for OU if and only if she manages the OU hotel.
  13. Moss manages the OU hotel just in case Nute is an administrator.
  14. Neither Moss nor Lamb works for OU only if Nute is an administrator.
  15. Lamb works for OU if he tends the OU golf course and, also, only if he tends that course.

 

2.4ex III          Symbolizations
Symbolize each of the following. Once you have finished with any answer and moved on, the program will check your work. As long as your answer is logically equivalent to what I deem to be a correct answer, your answer will be counted as correct too.

Move on...after you have filled in an answer, you need to tell the computer you're finished. To do so, press the tab key, or click outside your answers field, or press enter (on some systems).
You can enter text without the shift-key. The program will understand -- but it will show the intended answer only after you move on.

Use the following interpretation:

L: Lamb works for OU.
M: Moss works for OU.
N: Nute works for OU. G: Lamb tends the OU golf course.
H: Moss manages the OU hotel.
A: Nute is an administrator.

  1. Either Lamb or Moss works for OU.
  2. Lamb works for OU only if Moss does.
  3. Lamb works for OU unless Moss does.
  4. Lamb works for OU if Moss does.
  5. If Nute works for OU, then she's an administrator.
  6. If Lamb works for OU then both Moss and Nute do too.
  7. Lamb's working for OU implies that he tends their golf course.
  8. Being an administrator is a necessary condition for Nute to work for OU.
  9. Being an administrator is a sufficient condition for Nute to work for OU.
  10. Being an administrator is a necessary and sufficient condition for Nute to work for OU.
  11. Nute works for OU but she is an administrator.
  12. Moss works for OU but doesn't manages the OU hotel.
  13. Moss manages the OU hotel if Nute is an administrator.
  14. Neither Moss nor Lamb works for OU.
  15. Moss and Lamb don't both work for OU.

 

2.5ex I
Syntax

Which of the following are sentences of SL? For this exercise outside parentheses may be dropped and brackets may be used instead of parentheses. Select all correct answers.

  1. ~(AvB)
  2. ~AvB
  3. ~AvA~
  4. ~Av(A&)
  5. ~(~A&~B)>(B)
  6. ~(~A&~B)>B
  7. ~~(~B)=C
  8. ~F&(GvS)>~L
  9. ~F&[(GvS)>~L]
  10. [(Av~S)>D&T]=(S&T)
  11. [(Av~S)>(D&T)]=(S&T)]
  12. [(Av~S)>(D&T)]=(S&T)
  13. ~[[(Av~S)>(D&T)]=(S&T)]
  14. ~~~[[(Av~S)>(D&T)]=(S&T)]
  15. A>[~~J&U]v[T&S]
  16. A>([~~J&U]v[T&S])
  17. (L&S)>~[T&(U>J)]
  18. (L&S)~>~[T&(U>J)]
  19. S
  20. (S>&G)v[~J>(P&Q)]

Here are the ten sentences from above.
Now, click on all main connectives.

  1. ~(AvB)
  2. ~AvB
  3. ~(~A&~B)>B
  4. ~F&[(GvS)>~L]
  5. [(Av~S)>(D&T)]=(S&T)
  6. ~[[(Av~S)>(D&T)]=(S&T)]
  7. ~~~[[(Av~S)>(D&T)]=(S&T)]
  8. A>([~~J&U]v[T&S])
  9. (L&S)>~[T&(U>J)]
  10. S

Finally, here are some of the longer sentences from above. For each, determine all immediate components.

1. What are the immediate components for '~F&[(GvS)>~L]'?

2. What are the immediate components for '~[[(Av~S)>(D&T)]=(S&T)]'? 3. What are the immediate components for 'A>([~~J&U]v[T&S]) '?

 

2.5ex II        Logical Form

In this exercise, you are to pick out those sentences from a list which have a given logical form. For instance, on this first page, you are asked to pick out sentences with the form ~P&Q. All this form means is that the main connective of the given sentence is '&' and the first conjunct of this sentence is a negation. (Thus '~(AvS)&M' counts but '~Av(S&M)' does not. The latter has 'v' as its main connective.)

Which of the following sentences have the form '~P&Q'?

  1. ~A&B
  2. ~A&~B
  3. ~[(L&S)&G]
  4. ~(L&S)&G
  5. ~(L&S)&(GvS)
  6. ~~(L&S)&(GvS)
  7. (~A&S)&M
  8. ~A&(S&M)

Which of the following sentences are of the form 'P&(QvR)'?

  1. (L&S)v(QvR)
  2. (L&S)&(QvR)
  3. (L&S)&~(QvR)
  4. (L&S)&[~(QvR)v(M>N)]

Chapter 2 Review Exercises ex I

Instructions: Just use your mouse pointer to click on what you think is the best answer. Then move on to the next exercise. After you've finished, click on the "get score" button to see how well you've done.
1. Let M stand for "Halpin is male" (true) and "F" stand for "Hogs fly" (false). Then which of the following is true?
M&F
~MvF
F>M
F=M

2. Which of the following has main connective the horseshoe, >?
(A>B)vC
~(A>B)>(D=C)
~(D>C)&(L>~C)
T


3. Which of the following connectives of English should NOT be symbolized with the horseshoe, >?
only if
then
implies
moreover

Chapter 2 Review Exercises ex II

Instructions: Just use your mouse pointer to click on what you think is the best answer. Then move on to the next exercise. After you've finished, click on the "get score" button to see how well you've done.
1. Which of the following best symbolizes "Bob jogs regularly only if Carol does"?
B>C
C>B
B=C
B<C


2. Which of the following best symbolizes "Bob jogs regularly if Carol does"?
B>C
C>B
B=C
B<C


3. Which of the following best symbolizes "If Carol jogs regularly, then Bob does unless Albert doesn't"?
(C>B)v~A
C>(B>~A)
(C>B)>~A
C>(Bv~A)

Chapter 2 Review Exercises ex III

Some of these are a bit harder and require some original thought. See what you think...

1. Which of the following pairs of sentences is a logically equivalent pair?
AvB , A&B
~(AvB) , ~A&~B
~(A&B) , ~~A&~~B
~(AvB) , ~Av~B

2. Which of the following is NOT true?
'John' has four letters.
'John' is the name of John.
'John' is equal to John.
'John' is equal to 'John'.


3. Which of the following is NOT a grammatically correct sentence of SL? (You may drop outside parentheses or use brackets instead of parentheses.) Note: We use ">" for our horseshoe and "=" in place of the triple-bar -- these may fail to display properly on some computers.
(Av(B>C))
~~Av(B>C)
(AvB>C)
(A>C)&(A>C)


4. Which of the following is NOT of the form ~P>~Q
~[(AvB)>~(CvD)]
~(AvB)>~(Cvd)
~(AvB)>~~(Cvd)
~~(AvB)>~(Cvd)


5. Which of the following is a good symbolization of "Neither the French nor the Germans wins a match". (Symbolizing in the obvious way.)
F>~G
~F&G
~(FvG)
~(F&G)


6. Which of the following is a good symbolization of "If one of the teams (form France, Germany and Denmark) wins, the other two lose."?
G>~(FvD)
[Gv(FvD)]>[~G&(~F&~D)]
(G>~(FvD)&[(F>~(DvG)&(D>~(FvG)]
[(Gv(FvD))>(~G&(~F&~D))]&[G>~(FvD)]


7. Which of the following is a good symbolization of "At most one of the teams wins"?
G>~(FvD)
(Gv(FvD))>(~G&(~F&~D))
(G>~(FvD)&[(F>~(DvG)&(D>~(FvG)]
[(Gv(FvD))>(~G&(~F&~D))]&[G>~(FvD)]


8. What thought do you need to add to make the English statement of question 7 say "Exactly one of the teams win"? That is, which of the following do you need to add to "At most one wins" to get the idea that exactly one wins?
"the Germans didn't win"
"at least one team wins"
"no team loses"
"at most two teams win"


9. There is a nice simple way to express "At most two teams win". How would you symbolize that?
~F&(G&D)
~F&~(G&D)
(~F&~G)&~D
(~Fv~G)v~D


10. If "A" is true and "B" is false, which of the following is false?
A&~B
(~~Av~B)
(~Av~~B)
~A=B