Chapter 5
Derivations

Multiple Choice problems are further down the page.

Derivation Exercises: The following are statements of the derivation problems in chapter 5 of the Logic Café.

5.1ex I

1. From the following premises:
      (L&N)>P
      L
      N

 derive: P

 

2. From the following premises:
      A&B
      A>C
      C>D

 derive: D&B

 

3. From the following premise:
      (A&B)&(B>C)

 derive: C

4. From the following premises:
      (J&H)>(L&S)
      H&J

 derive: S&J

 

5. From the following premises:
      (L>M)&(M>S)
      T&L

 derive: S

5.1ex III

1. From the following premises:
      (A>B)&(B>C)
      A&L

 derive: C

 

2. From the following premises:
      (L&M)>S
      M&L

 derive: S

 

3. From the following premises:
      T&P
      R&S

 derive: S&T

 

4. From the following premises:
      ~(AvF)&~(G=U)
      ~(G=U)>~F

 derive: ~F

 

5. From the following premises:
      ~F&~G
      ~G>L

 derive: L&~F

5.2ex I

1. From the following premise:
       L&S

derive: (L&S)v(F>T)

 

2. From the following premises:
      (FvG)=(H>~I)
      (~I>H)&F

derive: ~I=H

 

3. From the following premises:
      (A&B)&C
      B>D

derive: D

 

4. From the following premises:
      (AvB)>L
      A=F
      C&F

derive: L

 

5. From the following premises:
      AvB
      A>(L>S)
      B>(L>S)
      S>L

derive: L=S

5.2ex III

1. From the following premises:
      A=(I&W)
      A

derive: W

 

2. From the following premises:
      A>(S&T)
      L&[(S&T)>A]

derive: A=(S&T)

 

3. From the following premises:
      MvN
      (M>S)&(N>S)

derive: S

 

4. From the following premises:
      (AvS)=(D&U)
      S

derive: U

 

5. From the following premises:
      T>U
      U>T
      (U=T)>(LvS)
      (LvS)=X

derive: XvT

 

6. From the following premises:
      ~MvO
      ~M>(ZvY)
      O>(ZvY)

derive: (ZvY)vX

5.2ex IV

1. From the following premises:
      ~Dv~L
      (~D>F)&G
      ~L>F

derive: F&G

 

2. From the following premises:
      (A>B)>(B>A)
      A>B

derive: (A=B)&(B=A)

 

3. From the following premise:
      A&(B&C)

derive: (AvB)vC

 

4. From the following premises:
      (A=B)=(B=C)
      B>C
      C>B

derive: A=B

 

5. From the following premises:
      (~A>B)>(LvS)
      (A>B)&(B>A)
      (LvS)=(B=A)

derive: LvS

 

6. From the following premises:
      (~A>B)>(LvS)
      (~A>B)&(B>A)
      (LvS)=(B=A)

derive: LvS

 

5.3ex I

1. From the following premise:
      A>(B&C)

 derive: A>(BvD)

 

2. From the following premises:
      B>C
      A>B

 derive: A>C

 

3. From the following premise:
      A=C

 derive: C>A

 

4. From the following premises:
      A=C
      L

 derive: C>(A&L)

 

5. From the following premises:
      (AvL)>F
      B>L

 derive: B>F

5.3ex II

1. From the following premises:
      A=C
      A>L

 derive: C>L

 

2. From the following premises:
      (A&J)>(D&S)
      A

 derive: J>D

 

3. From the following premises:
      M>L
      L>(M&J)

 derive: L=M

 

4. From the following premise:
      M=L

 derive: L=M

 

5. From the following premises:
      AvB
      A=C
      B>C

 derive: C

5.4ex I

1. From the following premise:
      ~(AvB)

 derive: ~A

 

2. From the following premises:
      B=A
      ~B

 derive: ~A

 

3. From the following premises:
      A
      ~A

 derive: B

 

4. From the following premise:
      ~~A

 derive: A

 

5. From the following premises:
      (A&B)>(L&S)
      ~L

 derive: ~(A&B)

 

5.4ex II

1. From the following premises:
      (AvL)>(T&U)
      ~T

 derive: ~(AvL)

 

2. From the following premise:
      ~(~Lv~M)

 derive: M

 

3. From the following premises:
      ~(A&B)
      A

 derive: ~B

 

4. From the following premises:
      ~(A&~B)
      A

 derive: B

 

5. From the following premises:
      J
      ~J

 derive: X

 

5.5ex I

1. Show that the set {~A>B,~A&~B} is inconsistent by giving a derivation.

2. Show that '(A&~D)>(AvX)' is logically true by giving a derivation.

3. Show that '(A>~A)&A' is logically false by giving a derivation.

5.5ex II
More Derivation Tests

1. Show that

      L>(MvL)

   is logically true by giving a derivation.

2. Show that

      A=A

   is logically true by giving a derivation.

3. Show that

      F=~F

   is logically false by giving a derivation.

4. Show the following set:

     { J=K,
      J&~K }

   is inconsistent by giving a derivation.

5. Show that

      A

   is logically equivalent to

      A&A

   by providing two appropriate derivations.

5.5ex III

1. Show that '(A&B)>(~J>B)' is a logical truth by giving a derivation.

2. Show that '(A&B)&~(C>A)' is logically false by giving a derivation.

3. Show that the set { J&(~F>T) , ~(FvT) } is inconsistent by doing a derivation.

4. Show that 'A&B' and 'B&A' are logically equivalent by giving a derivation.

5. Show that '(A=B)>(B=A)' is logically true by giving a derivation.

 

5.6ex I

1. From the following premises:
      L=(B&C)
      B

 derive: C>L

 

2. From the following premises:
      A=(Lv~S)
      L&(~A=T)

 derive: ~T

 

3. From the following premises:
      SvT
      S>(W&Z)
      T>Z

 derive: ZvL

 

4. From the following premises:
      [J>(Y&L)]&K
      K>~(Y&L)

 derive: ~J

 

5. From the following premises:
      ~X>(Y&T)
      ~(LvX)

 derive: Y=T

 

5.6ex II

1. From the following premise:
      ~(O&J)

 derive: O>~J

 

2. From the following premises:
      (L&M)>~(JvK)
      J&L

 derive: ~M

 

3. From the following premises:
      AvB
      ~B

 derive: A

 

4. From the following premise:
      B>~A

 derive: A>~B

 

5. From the following premise:
      Z

 derive: ~M>[(W&~K)>Z]

 

5.6ex III

1. From the following premises:
      (GvD)>B
      A=~L
      L
      B>A

 derive: ~B&~G

 

2. From the following premise:
      (AvB)>C

 derive: A>~~C

 

3. From the no premise, derive:   (A&B)>(C>B)
This is to show the sentence is a logical truth.

 

4. From the following premise:
      ~(A>B)

 derive: ~B

 

5. From the following premise:
      ~(~A&~B)

 derive: AvB

5.6ex IV

1. From the following premises:
      A=B
      B>C

 derive: A>(CvD)

 

2. From the following premise:
      A=B
      B=C

 derive: A>C

 

3. From the following premise:
      Av(B&C)
      A>L
      B>L

 derive: L

 

 

 

5.6ex VII
Logical Truth

1. Show that

      L>(J>L)

   is logically true by giving a derivation.

2. Show that

      A=(A&A)

   is logically true by giving a derivation.

3. Show that

      (L&S)>(L=S)

   is logically true by giving a derivation.

4. Show that

      E>~~E

   is logically true by giving a derivation.

5. Show that

      Rv~R

   is logically true by giving a derivation.

5.6ex VIII
Logical Falsehood and Inconsistency

1. Show that

      ~(Jv~T)&(~T&L)

   is logically false by giving a derivation.

2. Show that

      J&~~~J

   is logically false by giving a derivation.

3. Show the following set:

      { A>B,
      (~C>~B)&(~C>~D),
      A&~C
           }
   is inconsistent by giving a derivation.

4. Show the following set:

      { F=(A&~A),
      ~F>F
           }
   is inconsistent by giving a derivation.

5. Show the following set:

      { ~CvA,
      C&~A
           }
   is inconsistent by giving a derivation.

5.6ex IX
Logical Equivalence

1. Show that

      A

   is logically equivalent to

      A&(B>B)

   by providing two appropriate derivations.

2. Show that

      L>W

   is logically equivalent to

      ~W>~L

   by providing two appropriate derivations.

3. Show that

      J

   is logically equivalent to

      ~~J

   by providing two appropriate derivations.

4. Show that

      ~(A>A)

   is logically equivalent to

      A&~A

   by providing two appropriate derivations.

 

5.6ex X
Harder Problems

1. From the set of premises:

      A>B
      ~B

   derive the conclusion:

      ~A

2. Show that

      ~(LvS)&(~L>S)

   is logically false by giving a derivation.

3. Show that

      (A>C)=(~AvC)

   is logically true by giving a derivation.

4. Show the following set:

      { A=B,
      ~(A>B)
           }
   is inconsistent by giving a derivation.

5. Show that

      ~(A>C)

   is logically equivalent to

      A&~C

   by providing two appropriate derivations.

5.7ex I

1. From the following premises:
      A>B
      B>C
      (A>C)>(LvS)
      ~L

 derive: S

 

2. From the following premise:
      (AvS)&~A

 derive: S

 

3. From the following premises:
      (L&J)vT
      ~T
      L>F
      

 derive: F

 

4. From the following premises:
      (T>U)vF
      (F>G)&~G
      U>~L

 derive: T>~L

 

5. From the following premises:
      L>(T=Y)
      (T=Y)>~M
      ~~M

 derive: ~L&~~M

 

5.7ex III

1. From the following premise:
      ~A&(B&C)

 derive: (B&~A)&C

 

2. From the following premise:
      Lv~O

 derive: O>L

 

3. From the following premise:
      ~(~Av~B)

 derive: A&B

 

4. From the following premises:
      ~Av~B
      (A&B)v(~LvS)

 derive: L>S

 

5. From the following premise:
      ~(L>T)

 derive: L&~T

 

5.7ex IV

1. From the following premises:
      ~S>~T
      ~(T>S)vL

 derive: L

 

2. From the following premise:
      A>(A>E)

 derive: A>E

 

3. From the following premises:
      ~(P=Q)
      ~(P&Q)>(~R&~S)

 derive: ~(RvS)

 

4. From the following premise:
      (~S&G)v(~S&K)

 derive: (KvG)&~S

 

5. From the following premises:
      L&(S>T)
      ~(L&T)

 derive: L&~S

 

5.7ex V
Still More Derivations (using all rules)

1. From the set of premises:

      A>B
      ~B

   derive the conclusion:

      ~A

2. Show that

      ~(LvS)&(~L>S)

   is logically false by giving a derivation.

3. Show that

      (A>C)=(~AvC)

   is logically true by giving a derivation.

4. Show the following set:

      { A=B,
      ~(A>B)
           }
   is inconsistent by giving a derivation.

5. Show that

      ~(A>C)

   is logically equivalent to

      A&~C

   by providing two appropriate derivations.

5.8ex I

1. From the following premise:
      ~(A>B)

 derive: ~B

 

2. From the following premises:
      Mv~S
      S
      M>~(W&T)

 derive: ~Wv~T

 

3. From no premise
  derive: (Bv~A)>(A>B)

(to show the sentence is a logical truth).

 

4. From the following premises:
      A&(B&C)
      (A&B)>(LvS)
      ~(SvM)

 derive: L

 

5. From no premises,
 derive: Rv~R

(to show the sentence is a logical truth).

 

5.8ex II

1. From the following premises:
      (L&A)>~J
      (A>~J)>S

 derive: L>S

 

2. From the following premise:
      (A>(B&B))&(A>C)

 derive: A>(B&C)

 

3. From the following premise:
      ~L>(J=~K)

 derive: (~L&F)>(~J=K)

 

4. Show that the set including the following two sentences:
      [(~L>K)v(L>K)]>A
      ~(Av(~J>T))

 is inconsistent by giving a derivation.

 

5. From the following premises:
      L>T
      (L&T)>U
      K>(LvU)

 derive: ~KvU

 

5.8ex III
Logical Truth

Show that each of the following are logical truths by giving a derivation in SD+

1. (~A>G)=(~G>A)

2. (J=~K)>(~K=J)

3. Rv~R

4. ~~~~~(R&~R)

5. [A>(B>L)]v[(A&B)&~L]

 

5.8ex IV
Logical Falsehood and Inconsistency

1. Show that

      [(T>L)&(~T>L)]&~L

   is logically false by giving a derivation.

2. Show that

      [A>(C&D)]&~(A>C)

   is logically false by giving a derivation.

3. Show the following set:

      { A>(G>L),
      ~(B>L)vK,
      ~(~K>~A)
           }
   is inconsistent by giving a derivation.

4. Show the following set:

      { ~(A>~B)&T,
      (A=~B)v(S&~T)
           }
   is inconsistent by giving a derivation.

5.8ex V
Logical Equivalence

1. Show that

      ~GvL

   is logically equivalent to

      ~(G&~L)

   by providing two appropriate derivations.

2. Show that

      (~A>A)>B

   is logically equivalent to

      ~B>~A

   by providing two appropriate derivations.

3. Show that

      ~(A>(B>~C))

   is logically equivalent to

      (A&B)&C

   by providing two appropriate derivations.

4. Show that

      ~A&(BvC)

   is logically equivalent to

      (B>A)>(~A&C)

   by providing two appropriate derivations.

 

5.1ex II    Derivation Basics
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. When we use the rule "&E", we cite how many line numbers in the justifictaion?
a. 1 b. 2 c. 3

2. When we use the rule "&I", we cite how many line numbers in the justifictaion?
a. 1 b. 2 c. 3

3. When we use the rule ">E", we cite how many line numbers in the justifictaion?
a. 1 b. 2 c. 3

4. In order to break "A&B" down into its parts, a derivation could use
a. &I b. &E c. >E

5. In order to derive "(A>B)&(DvL)" from two premises, "A>B" and "DvL", one can always use which rule?
a. &I b. &E c. >E

6. Which of the following rules is not defined in tutorial 5.1?
a. &I b. >I c. >E d. &E

7. If one's premises on lines 1 and 2 are 'A>B' and 'A&N', then on line 3, which rule could not be used.
a. &I b. &E c. >E

5.2ex II     Derivations
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. Suppose a derivation's premises are 'A>B', 'B>A', and 'AvB'. Which of the following rules could be applied on line 4?
a. >E b. vE c. =I d. None of the above
2. Suppose a derivation's premises are 'A>C', 'B>C', and 'AvB'. Which of the following rules could be applied on line 4?
a. >E b. vE c. =I d. None of the above

3. Suppose a derivation's premises are 'A>C', 'C>B', and 'AvB'. Which of the following rules could be applied on line 4?
a. >E b. vE c. =I d. None of the above

4. The rule of &E requires an input line with main connective ampersand: a conjunction. What about vI? vI requires an input which
a. may be any sentence. b. may be any disjunction. c. may be any conjuction. d. may be any conditional.

5. When one justifies a line using vE, one needs to cite
a. 1 line number. b. 2 line numbers. c. 3 line numbers.

6. When one justifies a line using vI, one needs to cite
a. 1 line number. b. 2 line numbers. c. 3 line numbers.

7. When one justifies a line using =E, one needs to cite
a. 1 line number. b. 2 line numbers. c. 3 line numbers.

8. When one justifies a line using =I, one needs to cite
a. 1 line number. b. 2 line numbers. c. 3 line numbers.

5.3ex III
Derivations and Subderivations
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. If your goal is to derive 'J', then (because it has no main connective) all you can do to begin the derivation is assume 'J'.
a. True b. False

2. Once a subderivation is terminated, each line is inacessible. (That is to say, each line is off limits and cannot be cited.)
a. True b. False

3. If your goal sentence is of the form 'P>Q', then
a. You will always use >I to derive it. b. You will often use >I to derive it. c. You will always use >E to derive it. d. You will often use >E to derive it.

4. If your goal sentence is of the form 'P=Q', then you may need to derive this by using =I in the end. But in order to do this one should first...
a. try to prepare for =I by first deriving two conditionals. This may require two applications of >I. b. try to prepare for =I by first using ~I to prove ~(P=Q). c. try to prepare for =I by assuming P=Q.

5.7ex II
The first rules of replacement:
DN, AS, CM, DM, IM
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. A rule of replacement is different from a rule of inference because...
a. It may be used to make an inference in two directions, thus the double arrow in the statement of the rule. b. It may be used to replace a component of a sentence. c. All of the above. d. None of the above.

2. One may use DN to derive 'AvB' from
a. ~~(AvB) b. ~(AvB) c. ~(Av~B) d. ~Av~B

3. One may use AS to derive 'A&(BvC)' from
a. (A&B)vC b. (BvC)&A c. All of the above. d. None of the above.

4. One may use CM to derive 'A&(BvC)' from
a. (A&B)vC b. (BvC)&A c. All of the above. d. None of the above.

5. One may use DM to derive ~(Av~B) from...
a. ~Av~B b. ~AvB c. ~A&~B d. ~A&~~B

6. One may use IM to derive A>B from ...
a. B>A b. ~A>~B c. ~AvB d. ~BvA