Sample Logic Problems (see end for Section 2.5 problems)

Sudoku

7 / 3 / 1
2 / 5 / 9
6 / 1 / 7 / 5
7 / 9 / 5 / 4
4 / 2 / 3 / 7 / 8
2 / 5 / 9 / 1 / 3 / 6
4 / 3 / 5
9
2

Prove that R6,C4 is a 4

4 is not repeated on R4Premise

If R4,C4 is a 4 it will be a repeatPremise

R4,C4 ~= 4Modus Tollens

If R4,C6 is a 4 it will be a repeatPremise

R4,C6 ~= 4Modus Tollens

One of the cells in center block has a 4Premise

Every cell but R6,C4 ~= 4Premise

R6,C4 = 4Disjunctive Syllogism

Prove that R6,C3 is an 8

Prove that R7,C2 is a 2

Logic Proofs

Translate the following into symbolic form and give the justification for each step in the reasoning process:

When Alexis attends math class, her sorority sisters Guppy and Desmorelda also attend. Since Desmorelda is in love with Luke, Luke's attendance at class is a sufficient condition for her to attend as well. On the other hand, for Desmorelda to attend class it is necessary that Alexis also be there (as she needs someone to talk to during the boring portions of the class). Therefore, Luke won't attend class unless Guppy also attends.

  1. A → (G  D)Premise
  2. L → DPremise
  3. D → APremise
  4. L → ATransitivity of b) and c)
  5. L → (G  D)Transitivity of a) and d)
  6. ~L  (G  D)Switcheroo on e)
  7. (~L  G)  (~L  D)Distribution on f)
  8. ~L  GSimplification on g)
  9. L → GSwitcheroo on h)

You cannot be both happy and rich. Therefore, you are either not happy, or not rich. Now you do appear to be happy. Therefore, you must not be rich.

H – You are happy

R – You are rich

  1. Premise
  2. DeMorgans on a)
  3. Premise
  4. Disjunctive Syllogism on b) and c)

If I were smart or good-looking, I would be happy and rich. But I am not rich. So it's true that either I'm not happy or I'm not rich. In other words, I am not both happy and rich. Therefore I am not smart or good-looking,. In other words I am not smart and neither am I good-looking. In particular, I am not smart.

S – I am smart

G – I am good looking

H – I am happy

R – I am rich

  1. Premise
  2. Premise
  3. Addition on b)
  4. DeMorgans on c)
  5. Modus Tollens on a) and d)
  6. DeMorgans on e)
  7. Simplification on f)

If interest rates fall, then the stock market will rise. If interest rates do not fall, then housing starts and consumer spending will fall. Now, consumer spending is not falling. So, it's true that housing starts are not falling or consumer spending is not falling; that is, it is false that housing starts and consumer spending are both falling. This means that interest rates are falling, so the stock market will rise.

R – Interest rates are falling

S – The stock market will rise

H – Housing starts are falling

C – Consumer spending is falling

  1. Premise
  2. Premise
  3. Premise
  4. Addition on c)
  5. DeMorgans on d)
  6. Modus Tollens on b) and d)
  7. Modus Ponens on a) and g)

I am wise if you are a fool; either you are a fool or the sun sets in the east; but the sun cannot both set and rise in the east; observe that the sun rises in the east. Therefore I am wise and you are a fool.

W – I am wise
F – You are a fool
S – Sun sets in the east
R – Sun rises in the east

  1. F WPremise
  2. F  SPremise
  3. ~(S  R)Premise
  4. R Premise
  5. ~S  ~RDeMorgans on c)
  6. ~S Disjunctive Syllogism on d) and e)
  7. F Disjunctive Syllogism on b) and f)
  8. W Modus Ponens on a) and g)
  9. F  WConjunction of g) and h)

You are chairing an important committee at the UN, and are faced with the following predicament. Upper Volta refuses to sign your new peace accord unless both Costa Rica and Bosnia sign as well. Since Bosnia has a lucrative trade agreement with Iraq, Iraq's signing the peace accord is a sufficient condition for Bosnia to sign the accord. On the other hand, Bosnia, fearful of Upper Volta's recent military buildup, refuses to sign the accord unless Upper Volta also signs. You conclude that Iraq won't sign unless Costa Rica also signs. (I am interpreting “not A unless B” as “if not B then not A” but it could cut out several steps to recognize that it means the same as “if A then B” – the contrapositive)

U – Upper Volta signs

C – Costa Rica signs

B – Bosnia signs

I – Iraq signs

  1. Premise
  2. Premise
  3. Premise
  4. Contrapositive of c)
  5. Transitivity of b) and d)
  6. Contrapositive of a)
  7. Transitivity of e) and f)
  8. Simplification of g)
  9. Contrapositive of h)

If no violets are red and some roses are blue,
Then nobody loves you; that is certainly true.
But Lucille does love you,
And Susan and Joy.
Some roses are blue
Or you're just a boy.
You assure me you're grown up
And that I believe;
(The brew in your cup
Is so strong, I perceive.)
Now this surely implies
Violets red as your eyes.

R – Some violets are red

B – Some roses are blue

L – Someone loves you

G – You are grown up

  1. Premise
  2. Premise
  3. Premise
  4. Premise
  5. Disjunctive Syllogism on c) and d)
  6. Modus Tollens on a) and b)
  7. DeMorgans on f)
  8. Disjunctive Syllogism on e) and g)

If you glue, then Wu glues
And Golly glues too.
If Golly glues, Molly glues
And Solly, you too!
But Holly, not Solly glues
With green gooey glue.
Thus Dolly or Holly glues
But not you nor Wu!

Interpret this last statement as “not you and not Wu”

Y – You glue

W – Wu glues

G – Golly glues

M – Molly glues

S – Solly glues

H – Holly glues

D – Dolly glues

  1. Premise
  2. Premise
  3. Premise
  4. Premise
  5. Addition on c)
  6. Addition on d)
  7. DeMorgans on f)
  8. Modus Tollens on b) and g)
  9. Addition on h)
  10. DeMorgans on i)
  11. Modus Tollens on a) and j)

Note that we CAN conclude that you don’t glue. However, having Wu glue is perfectly consistent with all the premises logically but violates the conclusion, which is that Wu does not glue. Therefore the conclusion can NOT be inferred from the premises.

Using a broader approach, similar to truth tables, to solve probems from Section 2.5

  1. Broken window, who did it? A says ~C, B. B says ~B, C. C says ~C, A. If one statement is all true, one is all false and one is half true, which one is guilty? Assume each one is guilty and identify the correct scenario (keep the statements separate).

A saysB saysC says

A guilty:T, FT, FT, T

B guilty:T, TF, FT, F

C guilty:F, FT, TF, F

  1. Marble players, what order and age and how many marbles? Start by ruling out things that are explicitly forbidden. This actually gives us the order of play.

OrderAgeMarbles

G1 2 3 4, 3 10 17 18, 9 12 12 15

H123 4, 3 10 17 18, 9 12 12 15

I1234, 3 10 17 18, 9 12 12 15

J12 3 4, 310 17 18, 9 12 12 15

Gary and Jack either both have 12 or Jack has 9 and Gary 15.

If they both have 12 then Iggy has 15, in which case Iggy can’t be the youngest or even 10. Since we already ruled out 17, then Iggy would be the oldest. But since Jack is older than the one with 15, that is impossible.

The other alternative is Jack has 9 and Gary 15. This means that Gary is not 3 and we already ruled out 10. Since Jack is older than Gary, Jack must be 18 and Gary 17. Since Harry has 12 marbles he is not the youngest and must be 10. This leaves Iggy as the 3 year old, also with 12 marbles. No contradictions result.

  1. Another Liar Problem. Who committed the murder? One way to do this is to progressively assume each one did it and count up the true statements (each should have three). The difficulty is that some are statements supporting or contradicting each other. It may be easier to see what you need to assume to get three statements out of everyone.

O saysC saysS saysM saysK says

O:1+A1+~D+E+F2+D1+F+~G+~A2+~G+~E

C:1+B+A~D+E(+F)2+D+B1+~G+~A(+F)2+~G+~E

S:1+B+A1+~D+E+F2+D+B1+F+~G+~A2+~G+~E

M:1+B+A1+~D+E+F2+D+B~G+~A2+~G+~E

K:2+B+A1+~D+E+F2+D+B1+F+~A~E

Basically, there is no way to get 3 truths out of Otto if he is guilty. Either the Kid did it or Mickey is lying about Otto(A) and Slim is not (B). The Kid could not have done it because he would have too many lies if he did. If Mickey is indeed lying about Otto he can’t also be lying about not being guilty, leaving Curly or Slim as the guilty party. We are lead to a dilemma here. Either we have to assume that Curly and Mickey were both in Detroitthat night (F) and Curly still managed to pull off the murder or we get too many truths out of Curly. It is necessary for Curly NOT to own a revolver (D) and the Kid NOT to have been in Pontiac (G) but lying about knowing not knowing Curly (E).

  1. Another Liar Problem. Here we just want to see who needs to pull a fruit out of their bag to determine which bag is which. Since each one is lying we list the possible true options.

A says:2 peachesPossible optionsA has: 2 plums,1 each

B says:2 plumsB has:2 peachs,1 each

C says:1 eachC has:2 plums,2 peaches

If A pulls out a plum we won’t know for sure what we have

If B pulls out a peach we won’t know for sure what we have

If C pulls out a plum we know she has 2 plums, which means A has 1 each and B has 2 peaches

If C pulls out a peach we know she has 2 peaches, which means B has 1 each and A has 2 plums