Exercises Negotiation Strategies

1. Exercises Negotiation Strategies (Logic) [8 points]
[0] Exercise:
Poof or disproof the following statements by using truth tables:
[a] ¬ (A ˄ B) = ¬ A v ¬ B
[b] ¬ (A v B) = ¬ (¬ A ˄ ¬ B)
[c ] A XOR B = (A  B)  (A B)
[d] A ↔ B = (A → B)  ¬ (B → A)
[1] Exercise: [10 points]
Which of the following are statements (propositions)?
Statement Yes or No?
1. All swans are white.
2. Look in thy glass and tell whose face thou viewest.
3. 1,000,000,000 is the largest number.
4. There is no largest number.
5. There may or may not be a largest number.
6. This definitely not a statement.
7. The speaker is lying.
8. Stop at the red light!
9. Good Morning!
10. I wish you a merry Christmas!
[2] Exercise: [5 points]
Let be
p: “Our mayor is honest.”
q: “Our mayor is a good speaker.”
r = “Our mayor is a patriot.”
Express each of the following statements in logical form:
[1] Although our mayor is not honest, he is a good speaker.
[2] Our mayor is honest or he is a good speaker.
[3] Our mayor is an honest patriot who speaks well.
[4] It may or may not be the case that our mayor is honest.
[5] Our major is neither a good speaker nor a patriot.
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[3] Exercise: [5 points]
Let
p: “Anton is a good teacher.”
q: “Berta is a good teacher.”
r: “Anton’s students hate logic.”
s: “Berta’s students hate logic.”
Write the following formulas as English sentences.
[1] p ˄ (¬r)
[2] p v (r ˄ (¬q))
[3] q v (¬q)
[4] r ˄ (¬r)
[5] ¬ (q v s)
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[4] Exercise [6 points]
Assume that it is true that “Anton sings well,” it is false that “Berta writes well,” and it is true that “Carl is good at math.” Determine the truth of each of the

following statements.
[1] Anton sings well and Berta writes well.
[2] Anton sings poorly and Berta writes well.
[3] Either Anton sings well and Berta writes poorly, or Carl is good at math.
[4] Anton signs well or Berta writes well, or Carl is good at math.
[5] Anton sings well, and either Berta writes well or Carl is good at math.
[6] Neither Carl is good in maths nor does Anton sing well.
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[5] Exercise: [4 points]
Apply the stated logical equivalence to each of the following statements:
[1] p ˄ (¬p); the Commutative law; Contradiction
[2] ¬p v (¬q); De Morgan’s Law
[3] p v ¬ (p ˄ q); De Morgan’s Law; Contradiction
[4] p v (¬p ˄ q); the Distributive Law; Tautology
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[4] [6] Exercise: [5 points]
Determine the truth values of following statements:
[1] If “1 = 0”, then “1 = 1”.
[2] If “1 = 1”, then “1 = 0”.
[3] A sufficient condition for “1 = 2” is “1 = 3”.
[4] “1 = 2“ is a necessary condition for “1 to be unequal to 2”.
[5] “0 = 0” XOR “2=2”.
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[5] [7] Exercise (Symbolic reasoning) [5 points]
Determine a smaller set of proposition by application of natural deduction.
[1] {q →p, ¬ p} ⊢ ?
[2] {(¬p∨q) → ¬ (q∧r), ¬(p∧ ¬ q) } ⊢ ?
[3] {(p∧q) → r, p, q} ⊢ ?
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[8] Exercise [10 points]
Convert each of the following statements into a symbolic proof, and supply the justifications for each step.
[1a] For me to carry my umbrella it is necessary that it rain. When it rains I always wear my hat. Today I did not wear my hat. Therefore, it must not be raining, and

so I am not carrying my umbrella.
[1b] For me to take my umbrella it is sufficient that it rain. For me to wear my hat it is necessary that it rain. I am wearing my hat today. Therefore, it must be

raining, and so I must have taken my umbrella.
[2] You cannot be both happy and rich. Now be happy. Therefore, you must not be rich.
[3a] 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.
[3b] If interest rates go down, inflation will rise. If inflation rises, so will the bond market. Therefore, if interest rates go down, the bond market will rise.
[4] Carl is chairing an important committee at the UN, and is 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. Carl concludes that Iraq

won’t sign unless Costa Rica also signs. Is he right?
[5] A Shnirch is a Shnarch or a Shnurch. If a Shnerch is not a Shnirch then 1=0. Either is a Shnerch a Shnirch, or a Shnarch is a Shnirch. Is a Shnirch a Shnurch?
[9] Exercise [22 points]
Formalize
[1] All cows eat grass.
[2] Some cows eat grass.
[3] Some cows are not birds and some are.
[4] Although some city drivers are insane, Dorothy is a very sane city driver.
[5] If one or more lives are lost, then all lives are lost.
[6] Some numbers are larger than two; others are not.
[7] There is only one Elvis.
[8] There exists a smart student.
[9] Every student loves some student.
[10] Every student loves some other student.
[11] There is no direct connection from Kleve to Frankfurt.
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[10] Deformalize [8 points]
[1] ∀x*Rx→Sx+ R = “is a raindrop,” S = “makes a splash”
[2] ∃x[Dx∧Wx] D = “is a dog,” W = “whimpers.”
[3] ∀x*Dx→¬Wx+ D = “is a dog,” W = “whimpers.”
[4] ∃x,y[Dx∧Wx] ∧ [Dy ∧ ¬Wy] D = “is a dog,” W = “whimpers.”
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[11] Which statements are valid? [4 points]
[1] ∀ x ∃ y Loves(x, y) ⇔ ∃ x ∀ y Loves(x, y)
[2] ∀ x Loves(x, Snoopy) ⇔ ¬ ∃ x ¬Loves(x, Snoopy)
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[12] Brains in a vat: [8 points]
Hypothesis: Are humans not real but brains in a vat?
v(x) := x is a brain in a vat
h(x) := x is human
t(x,y) := The thought of x refers to a real externality y, not to a simulation of y. (*)
c(x) := x has a causal relation with the world
(*) Explanation: If x is a human the thought to a tree refers to a real tree, the thought to a vat refers to a real vat. If x is a brain in a vat the thought to a tree

refers to a simulation of a real tree, the thought to a vat refers to a simulation of real vat. Hilary Putnams1 argument against the brain in the vat hypothesis

formalized:
[1] t(x,y) → c(x)
[2] ¬c(x) → ¬t(x,y);
[3] v(x) → ¬c(x)
[4] v(x) → ¬t(x,y)
[5] v(x) → ¬t(x,v(x))
[6] h(x) → t(x,y)
[7] t(x,v(x)) → ¬v(x)
[8] h(x) → ¬v(x)
Translate this argument to English statements.
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