3.1ex I   One Row Truth Tables
Do one row truth tables for each of the following. Pick truth values at random for each of the components.

  1. A>~B
  2. C=B
  3. ~L&M
  4. (BvC)>D
  5. A>(B>C)
  6. ~~B
  7. ~(L&N)
  8. (G=~S)>(F&T)
  9. ~(~S&T)
  10. (F&G)=[S&(TvL)])

3.1ex II   One Row Truth Tables
Do one row truth tables for each of the following. Pick truth values at random for each of the components.

  1. ~(JvT)
  2. ~(JvT)>~J
  3. ~(L&M)v~K
  4. F&[(T>J)=~P]
  5. [A>(B>C)]vA
  6. ~(~BvL)~~BvL
  7. (~G>~S)&(F&T)
  8. F&(G>[L&(SvT)])
  9. (~F&G)=~[S&(TvG)

3.2ex I: Full Tables for Simple Sentences
Do full truth tables for each of the following sentences:

  1. A>~B
  2. ~A&~B
  3. C=(F&C)
  4. ~Cv(A>C)
  5. ~(LvM)

3.2 ex III: Full Tables
Do full truth tables for each of the following sentences:

  1. A&(BvC)
  2. A>~(BvC)
  3. F=(GvF)
  4. ~(A&C)v(L>S)
  5. ~(L&M)>~M

3.2ex IV Truth Tables
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. If you do a truth table for '~A>~B', then the last truth values will be filled in under the
a. first tilde b. the horseshoe c. the second tilde d. none of the above

2. When filling in a truth table for a sentence, the last column to be finished is always that under its main connective.
a. True b. False

3. In a truth table for '~A>~B', we usually have row 1 assigning both 'A' and 'B' true. This first row
a. indicates that truth comes before falsity. b. represents one possible assignment of truth values to the atomic components of '~A>~B'. c. is the only row we need to consider in order to determine the truth value of the whole sentence '~A>~B'.

4. If you do a truth table for '~A>~B', there will be
a. 2 rows. b. 4 rows. c. 8 rows. d. 16 rows.

5. If you do a truth table for '(~A>~B)=(A>B)', there will be
a. 2 rows. b. 4 rows. c. 8 rows. d. 16 rows.

6. If you do a truth table for '(~A>~B)=(CvD)', there will be
a. 2 rows. b. 4 rows. c. 8 rows. d. 16 rows.

7. 2 to the 4th power (24) =
a. 2 x 4 b. 2 x 2 x 2 x 2 c. 24

3.3ex I Logical Truth, Falsity, Indeterminacy
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. A truth table shows that a sentence is logically true if
a. that sentence is true on all rows of the table. b. that sentence is true on some but not all rows of the table. c. that sentence is true on no rows of the table

2. A truth table shows that a sentence is logically false if
a. that sentence is true on all rows of the table. b. that sentence is true on some but not all rows of the table. c. that sentence is true on no rows of the table.

3. A truth table shows that a sentence is logically indeterminate if
a. that sentence is true on all rows of the table. b. that sentence is true on some but not all rows of the table. c. that sentence is true on no rows of the table.

4. Truth tables show us that
a. a sentence can be both logically true and false by being true in one row and false in another. b. an argument is always logically true if it's valid. c. a sentence which is logically indeterminate can never be false. d. none of the above

3.3ex II Tables Judging Single Sentences: Logical Truth, Logical Falsity and Logical Indeterminacy in SL.
Do a full truth tables for each of the following sentences and use the table to show whether the sentence is logically true, false, or indeterminate.

  1. A=~B
  2. (A>A)vB
  3. (C>~A)&(A>C)
  4. (D&L)=~L
  5. (A&B)&~C

3.3ex III Tables Judging Single Sentences: Logical Truth, Logical Falsity and Logical Indeterminacy in SL.
Do a full truth tables for each of the following sentences and use the table to show whether the sentence is logically true, false, or indeterminate.

  1. ~(LvF)>(~L&~F)
  2. J
  3. ~Z&(BvO)
  4. (D&L)=[(L>F)&~F]
  5. (FvG)>(~G>F)

3.4ex I Logical Equivalence
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. If a truth table shows that a sentence is true on some rows but false on others, then
a. that sentence is not logically equivalent to any sentence. b. that sentence is logically equivalent to all sentences. c. that sentence is not logically equivalent to any logical truth. d. this tells us nothing about logical equivalence.

2. If the truth table for a pair of sentences shows they are logically equivalent, then
a. it shows that both sentences are logically true or logically false. b. it shows that both sentences are true on every row of the truth table. c. it shows that there is no row making both sentences false. d. none of the above

3. Suppose that only on the first row do the truth values of a pair of sentences differ. Then
a. those sentences cannot both be logically true. b. those sentences cannot both be logically false. c. those sentences cannot be logically equivalent. d. all of the above.

3.4ex II Tables Judging Pairs of Sentences: Equivalence.

Do a full truth tables for each of the following sentence pairs. For each pair, use the table to determine whether or not its members are logically equivalent.

  1. A>B, B>A
  2. (K=L)&M, (M&L)&K
  3. (C&D)vU, C&(DvU)
  4. S&(Bv~S), S&B
  5. A=G, (A>G)&(L>A)

3.4ex III Tables Judging Pairs of Sentences: Equivalence.

Do a full truth tables for each of the following sentence pairs. For each pair, use the table to determine whether or not its members are logically equivalent.

  1. ~A, ~~~A
  2. S>T, ~SvT
  3. (C&D)vU, U&(C&D)
  4. ~M&~(N>O), ~[M&(N>O)]
  5. ~(S=T)&T, (S=T)>(TvS)

3.5ex I   Entailment and Validity

Do a full truth tables for each of the following arguments (the conclusion comes after the slash). Is the argument valid?

  1. AvS, ~S / A
  2. L=S, S / ~L
  3. A>B, B>C / A>C
  4. AvB, ~Bv~A / A&~B
  5. A&B, B=C / A&C

3.5ex II   Testing for Validity with Truth Tables
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. A truth table for an argument includes
a. truth value assignments for all atomic sentences b. truth value assignments for premises only c. truth value assignments for the conclusion but not the premises. d. truth value assignments for premises and conclusion.

2. A truth table for an argument shows that argument valid if
a. it shows the premises and conclusion true on some row. b. it includes no row with premises true and conclusion false. c. every row makes the premises true. d. it includes no row with conclusion true and premises false.

3. A truth table for an argument shows that that argument is valid IF it also shows the argument's conclusion is logically true.
a. true b. false

4. A truth table for an argument shows that argument invalid if
a. it has at least one row making premises true and conclusion false. b. it has at least one row making premises false and conclusion false. c. it has at least one row making premises false and conclusion true.

3.5ex III   Validity

Do a full truth tables for each of the following arguments. Is the argument valid? (The premises are written before the slash, the conclusion after.)

  1. ~L&~B / ~(LvB)
  2. ~Lv~B / ~(LvB)
  3. T=(Sv~U), T&S / U
  4. ~(M>P), ~(M&P) / X
  5. (U&W)>(YvZ), ~Y>W / ~U
  6. F>~G, G / ~F
  7. F>G, ~G / ~FvX
  8. Y=(X&T), ~X / ~T

3.5ex V: Consistency

Check for consistency by doing a full truth table for each of the sets of sentences.

  1. {AvS, ~S>A, ~A}
  2. {U&~W, ~(UvW)}
  3. {A>B, B>C, A>C}
  4. {(GvI)>(S&I), ~~(IvG)}
  5. {[~Cv(A&B), B=~C, A&C}

3.5ex VI   Consistent and Inconsistent Sets of SL Sentences
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. If each member of a set of sentences is false on the first row of its table, then
a. that set is consistent. b. that set is inconsistent. c. that set might or might not be consistent.

2. One can tell that a set of sentences is consistent if
a. on some row each member of the set is true. b. some member of the set is true on all rows. c. some member of the set is true on some row.

3. One can tell that a set is inconsistent if
a. all members of that set are false on some row. b. on some row all members of the set are true. c. on no row are all members true.

4. A set that has one member which is logically false must be inconsistent.
a. true b. false

5. A set that has one member which is logically true must be consistent.
a. true b. false

3.5ex VII: Consistency

Do a full truth tables for each of the sets of sentences for consistency

  1. {A=B, C=A}
  2. {W>~G, W>G}
  3. {A, B, C}
  4. {~(T>S), ~(JvS)}
  5. {Cv(~D&J), D, D>~C}
P and Q are logically true sentences of SL.
by definition of logically true in SL, both P and Q are true on every truth value assignment.
there can be no truth value assignment making one of P and Q true and the other false. (None makes either false.)
by definition of equivalence, P and Q are logically equivalent in SL.
Suppose             __________
Then,          __________
So,          __________
So, finally,        __________
Q.E.D.

Chapter 3
Review Exercises II
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 two sentences are logically true in SL, then they are logically equivalent in SL.

 

 

 

 

 

Proof by example:

       __________

'AvB' and 'BvA' are logically indeterminate but not logically equivalent.
'A' and 'B' are logically indeterminate but not logically equivalent.
'A>A' and 'A&~A' are logically indeterminate but not logically equivalent.
Q.E.D.

Now, show that

2. Two sentences may be logically indeterminate in SL yet fail to be logically equivalent in SL.

 

 

 

 

Suppose         __________

A is any argument with an inconsistent set of premises.
A is invalid in SL.
by the definition of invalidity in SL, there must be a truth value assignment on which A has true premises and false conclusion.
Q.E.D.
then A's premises are true on this truth value assignment, so A's premises are not inconsistent in SL.
because this last contradicts our original supposition about A, the assumption in red must be false and A must be valid.

Assume (for contradiction) that

        ____________   

So,        ________
But,        ________
So,        ________

Now, show that

3. If the set of premises of an SL argument is inconsistent in SL, then the argument is valid in SL.

 

 

 

 

 

Chapter 3   Review Exercises III
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. (Make sure you print or otherwise save your work; their is no resultsTracking for this exercise.)

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. If P=Q is logically true in SL (for any SL sentences P and Q), then P and Q are logically equivalent in SL.

2. If P>Q is logically true in SL, then the argument with single premise P and conclusion Q is valid in SL.

3. If P is logically false in SL, then P&Q is also logically false in SL.

4. Two SL sentences P and Q may both be logically indeterminate in SL yet their disjunction PvQ may fail to be logically indeterminate in SL.