6.1ex I Symbolizations

The key: j: John Lennon r: Ringo Starr b: Bob Dylan s: Sonny Jurgenson
Bx: x was a Beetle. Ax: x was an athlete.
Sxy: x was a better singer than y. Qxy: x was a better quarterback than y.

  1. John Lennon was a Beetle.
  2. Bob Dylan was not a Beetle.
  3. John and Ringo were Beetles.
  4. Sonny Jurgenson was not a Beetle but was an athlete.
  5. John Lennon was a better singer than Bob Dylan.
  6. Sonny wasn't a better singer than Ringo.
  7. If Sonny was a Beetle, then Bob was an athlete.
  8. Sonny was a better quarterback than any of the others.
  9. Bob Dylan was a better singer than Ringo Starr unless Sonny Jurgenson was an athlete.

6.1ex II Introduction to Symbolization
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. Which of the following expressions of English is a UNIVERSAL QUANTIFIER?
a. Agnes b. is working on her novel c. there is at least one d. everyone

2. Which of the following expressions of English is an EXISTENTIAL QUANTIFIER?
a. Agnes b. is working on her novel c. there is at least one d. everyone

3. Which of the following expressions of English is a PREDICATE?
a. Agnes b. is working on her novel c. there is at least one d. everyone

4. Which of the following expressions of English is a NAME?
a. Agnes b. is working on her novel c. there is at least one d. everyone

5. Which to the following expressions of English is a two-place predicate or relation?
a. Agnes and Bob b. is seated behind c. is green with envy d. is the sum of

6. Which to the following expressions of English is a three-place predicate or relation?
a. Agnes and Bob b. is seated behind c. is green with envy d. is the sum of

7. Which to the following expressions of English is a one-place predicate or relation?
a. Agnes and Bob b. is seated behind c. is green with envy d. is the sum of

8. The universal quantifier of our new language is...
a. ^ b. % c. \ d. None of the above

9. The existential quantifier of our new language is...
a. ^ b. % c. \ d. None of the above

6.1ex III Symbolizations

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

Symbolization Key:
UD = the set of students including exactly Aimee, Marty, Pam and Vesna;
a: Aimee, m: Marty, p: Pam, e: Vesna ('v' is not a PL name!);
Mx: x is male, Cx: x is a communications major, Txy: x is taller than y.

1. Which of the following is the best symbolizaiton for "Marty is male but Vesna is not".
a. Mav~Ve b. Mm>~Me c. Mm&~Vv d. Mm&~Me

2. Which of the following is the best symbolizaiton for "Both Amy and Pam are communications majors".
a. Cap b. Ca&p c. Ca&Cp d. &Ca&Cp

3. Which of the following is the best symbolizaiton for "Neither Pam nor Marty are communications majors".
a. ~(CpvCm) b. ~(Cp&Cm) c. Both are correct. d. Both are incorrect.

4. Which of the following is the best symbolizaiton for "Pam and Marty are not both communications majors".
a. ~Cpv~Cm b. ~(Cp&Cm) c. Both are correct. d. Both are incorrect.

5. Which of the following is the best symbolizaiton for "Aimee is taller than Marty unless Marty is male".
a. Tam>Ma b. Tam>~Ma c. ~Ma>Tam d. Ma>~Tam

6. Which of the following is the best symbolizaiton for "Marty is taller than Pam only if Pam is not taller than Marty".
a. Tmp>~Tmp b. Tmp>~Tpm c. Tpm>~Tpm d. None of the above.

6.1ex IV More PL Symbolizations

Consider the group of people we considered earlier, students in a logic class including — but not limited to — Agnes, Bob, and Carola. (We call this group our "universe of discourse" because these are the people under discussion.) Now, symbolize the sentences below using quantifiers (as well as truth functional connectives, '&', 'v', '>', '=', '~', and the names and predicates from the following key).

UD: Agnes, Bob, and Carola
a: Agnes, b: Bob, c: Carola
Wx: x will attend law school
Nx: x will need a loan
Sxy: x scored as well as y on the LSAT's.

  1. Someone (in the group) will need a loan.
  2. Someone will attend law school but not need a loan.
  3. Everyone will attend law school, but not everyone will need a loan.
  4. Someone needs a loan but will not attend law school.
  5. Someone scored as well on the LSAT's as Bob, but will need a loan.
  6. Everyone scored as well on the LSAT's as Agnes.
  7. Print then back to chapter 6 exercises...
6.2ex I

For this exercise, simply select the best symbolization. Assume the UD is the set of all students in MI and that Ann (a) and Bob (b) are two of these people. Hxy: x has higher grades than y; Sx: x is a law student.
1. Ann is a law student only if her grades are higher than Bob's.
a) Sa>Hba
b) Hab>Sa
c) Sa>Hab

2. If everyone is a law student, then both Bob and Ann are.
a) (^x)(Sx>(Sab))
b) (%y)Sx > Sa & Sb
c) (^x)Sx>(Sa&Lb)

3. Someone is a law student, but not Bob.
a) (%y)Ly & ~Lb
b) (^y)Ly v ~Lb
c) La & ~Lb

4. Ann has higher grades than Bob, still she's not a law student.
a) ~La > Hab
b) Hab > ~La
c) Hab & ~La

6.2ex II Matching

Drag symbolizations from the right to the correct location on the left.

Symbolization Key:
UD = the set of students including Aimee, Marty, Pam and Vesna;
a: Aimee, m: Marty, p: Pam, e: Vesna ('v' is not a PL name!);
Mx: x is male, Cx: x is a communications major, Txy: x is taller than y.

(^x)Mx
~(^w)Mw
(%x)Cx>(^x)Mx
(%x)Txm
(%y)(My&Cy)
(%z)Tmz
(^x)Tmx
~(%y)My
(^x)Txm
 
Everyone is male.
Not everyone is male.
If someone is a communications major then everyone is male.
Someone is taller than Marty.
Someone is male and a communications major.
Marty is taller than someone.
Marty is taller than everyone.
No one is male.

 

6.2ex III Symbolizations

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

Symbolization Key:
UD = the set of students including exactly Aimee, Marty, Pam and Vesna;
a: Aimee, m: Marty, p: Pam, e: Vesna ('v' is not a PL name!);
Mx: x is male, Cx: x is a communications major, Txy: x is taller than y.

1. Which of the following best symbolizes "No one is a communications major"?
a. ~(%y)Cy b. (^z)~Cz c. Both of the above. d. None of the above.

2. Which of the following best symbolizes "Everyone is both male and a communications major"?
a. (^x)(Mx&Cx) b. (%y)~(My&Cy) c. Both of the above. d. None of the above.

3. Which of the following best symbolizes "Pam is taller than someone"?
a. (%x)Tpx b. (%x)Txp c. (%x)Txx d. None of the above.

4. Which of the following best symbolizes "Pam is taller than no one"?
a. ~(%x)Tpx b. ~(%x)Txp c. ~(%x)Txx d. None of the above.

5. Which of the following best symbolizes "Pam is taller than everyone"?
a. (^x)Tpx b. ~(%x)Txp c. ~(^x)~Tpx d. None of the above.

6. Which of the following best symbolizes "Pam is taller than anyone in the group"?
a. (^x)Tpx b. (^x)Txp c. (%x)Tpx d. None of the above.

7. Which of the following best symbolizes "Anyone is taller than Pam"?
a. (^x)Tpx b. (^x)Txp c. (%x)Tpx d. None of the above.

8. Which of the following best symbolizes "a communications major is male"?
a. (^x)(Cx&Mx) b. (%x)(Cx&Mx) c. Both of the above. d. None of the above.

6.2ex IV Symbolizations

UD: people in a symbolic logic class
j: John (a student in the UD)
Lx: x is late for class
Kxy: x knows y.

  1. Someone is late.
  2. Everyone is late.
  3. John knows everyone in class.
  4. Everyone knows John.
  5. John knows someone in class.
  6. Everyone is late but not everyone knows John.
  7. No one is late.
  8. No one knows John.
  9. There is a tardy student who knows John.
  10. Each student is late.
  11. John knows a student.
  12. Not everyone knows John.

6.3ex I Drag

Drag four of the following parts...

 

 

 

...to make formulas of the following:

 

 

 

 

 

 

 

 

 

 

 

 

 

6.3ex II   PL Syntax: Formulas

Correct the following. Make changes to each of the following form entries so that the end result is both a formula of PL and is at least as long as the original entry.

 

6.3ex III    PL Syntax: Sentences

Correct the following. Make changes to each of the following form entries so that the end result is both a sentence of PL and is at least as long as the original entry.

 

6.3ex IV    Substitution Instances

Rewrite each of the following sentences. Take the quantified sentence and replace it with a substitution instance.

6.3ex V
Below is a standard multiple choice quiz on the syntax (i.e., grammar) of PL.

1. Which of the following is a formula of PL?
a) (%y)[Bxy&(Lxy=(^y)&Ty)]
b) (^x)[Pxy>(%z)(Txz&Uy)]
c) (%y)[Pxy(^y)Rxy]

2. Which of the following is a sentence of PL? (I.e., which s a fromula of PL with no free variables.)
a) (^x)(%y)Lxy>Lyx
b) (^x)(%y)Lxy>(%y)(^x)Lyx
c) (^x)(%y)Lxy>Lyx&(^z)Dz

3. Which of the following is a sentence of PL with the universal quantifier, the upside-down A, as main logical operator.
a) (^x)(%y)(Lxy>Lyx)
b) (^x)(%y)Lxy>Lyx
c) (^x)(%y)Lxy>(%y)(^x)Lyx

4. Which of the following is a substitution instance of (%x)(Bx&Lxb)
a) (%a)(Ba&Lab)
b) (Ba&Lbb)
c) (Ba&Lab)

5. Which of the following is a substitution instance of (^y)(%z)(Tyz&(By>Ly))
a) (Tab&(Ba>La))
b) (^y)(Tyr&(By>Ly))
c) (%z)(Taz&(Ba>La))