Chapter 5


1. From the following premises: 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: 5. From the following premises: (L>M)&(M>S) T&L derive: S 
5.1ex III 1. From the following premises:
2. From the following premises:
3. From the following premises: 4. From the following premises: ~(AvF)&~(G=U) ~(G=U)>~F derive: ~F
5. From the following premises: 
1. From the following premise: 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: 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: 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


1. From the following premise: 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: 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 
1. From the following premise: 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: 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

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. 
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, 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. 

1. From the following premises: 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: 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: 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:
2. From the following premise: 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, 4. Show the following set: { F=(A&~A), 5. Show the following set: { ~CvA,

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 ~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, 5. Show that ~(A>C) is logically equivalent to A&~C by providing two appropriate derivations. 
1. From the following premises: 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: 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: 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

Still More Derivations (using all rules) 1. From the set of premises: A>B ~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, 5. Show that ~(A>C) is logically equivalent to A&~C by providing two appropriate derivations. 
1. From the following premise: 2. From the following premises: Mv~S S M>~(W&T) derive: ~Wv~T
3. From no premise (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, (to show the sentence is a logical truth).

5.8ex II 1. From the following premises: 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 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 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), 4. Show the following set: { ~(A>~B)&T, 
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