Kenneth Rosen Edition 7 Exercise 1.6 Question 9 (Page No. 78)

Question

For each of these collections of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises.

1. “If I take the day off, it either rains or snows.” “I took Tuesday off or I took Thursday off.” “It was sunny on Tuesday.” “It did not snow on Thursday.”
2. “If I eat spicy foods, then I have strange dreams.” “I have strange dreams if there is thunder while I sleep.”“I did not have strange dreams.”
3. “I am either clever or lucky.” “I am not lucky.” “If I am lucky, then I will win the lottery.”
4. “Every computer science major has a personal computer.” “Ralph does not have a personal computer.”“Ann has a personal computer.”
5. “What is good for corporations is good for the United States.” “What is good for the United States is good for you.” “What is good for corporations is for you to buy lots of stuff.”
6. “All rodents gnaw their food.” “Mice are rodents.”“Rabbits do not gnaw their food.” “Bats are not ro-dents.”

Answer 1

a) 

Assume, 

O(d) : I take the day(d) off. 

R(d) : it rains on the day(d). 

S(d) : it snows on the day(d). 

s(d) : it was sunny on the day (d). 

Premises:

1.$\forall_{d}$(O(d)$\rightarrow$(R(d)$\vee$S(d))) 

2. O(Tuesday)$\vee$O(Thursday) 

3. s(Tuesday) $\equiv$ $\neg$(R(Tuesday)$\vee$S(Tuesday))

4. $\neg$S(Thursday) $\equiv$ R(Thursday) 

Using 1 by applying Universal Instantiation. If Tuesday is a particular element of the domain.we get, 

O(Tuesday)$\rightarrow$(R(Tuesday)$\vee$S(Tuesday)) --------(5) 

If Thursday is a particular element of the domain. we get,

O(Thursday)$\rightarrow$(R(Thursday)$\vee$S(Thursday)) --------(6) 

Using 5 & 3 by applying modus tollens, we get $\neg$O(Tuesday) -------(7) . 

Using 7 & 2 by applying Disjunctive syllogism, we get O(Thursday) ----------(8) 

Hence, from 4, 7 and 8 we can conclude the conclusion is R(Thursday) ^ $\neg$O(Tuesday) ^ O(Thursday). 

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b) 

Assume, 

E: I eat spicy foods. 

D: I have strange dreams. 

T: There is a thunder while I sleep. 

Premises:

1. E$\rightarrow$D

2. T$\rightarrow$D

3. $\neg$D

Using 1 and 3 by applying modus tollens, we get $\neg$E -------(4) 

Using 2 and 3 by applying modus tollens, we get $\neg$T --------(5) 

Hence, from 4 and 5 we can conclude conclusion is $\neg$E ^ $\neg$T.

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c)  

Assume, 

C: I am clever. 

L: I am lucky. 

W: I will win the lottery. 

Premises:

1. C$\vee$L

2. $\neg$L

3. L$\rightarrow$W $\equiv$ $\neg$L$\vee$W

Using 1 & 2 by applying disjunctive syllogism, we get C -------(4) 

Using 1 & 3 by applying Resolution rule, we get C$\vee$W -----(5) 

Hence, conclusion is C$\vee$W. 

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d) 

Assume, 

C(x): x is a computer science major. 

P(x): x has a personal computer. 

Premises:

1. $\forall_{x}$(C(x)$\rightarrow$P(x)) 

2. $\neg$P(Ralph) 

3. P(Ann) 

Using 1 by applying universal Instantiation, if Ralph is a particular element of the domain.we get C(Ralph)$\rightarrow$P(Ralph) --------(4) 

Using 4 and 2 by applying $\neg$C(Ralph) 

Hence, we can conclude the conclusion is $\neg$C(Ralph) . 

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e) 

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f)

Assume, 

R(x): x is a Rodent. 

G(x): x gnaw their food. 

Premises:

1. $\forall_{x}$(R(x)$\rightarrow$G(x)) 

2. R(Mice) 

3. $\neg$G(Rabbit) 

4. $\neg$R(Bats) 

Using 1 by applying Universal Instantiation. If Mice is a particular element of the domain, we get R(Mice)$\rightarrow$G(Mice) -----(5) 

If Rabbit is a particular element of the domain,we get R(Rabbit)$\rightarrow$G(Rabbit) -----(6) 

Using 2 & 5 by applying modus ponnes, we get G(Mice) -----(7) 

Using 3 & 6 by applying modus tollens, we get $\neg$R(Rabbit) -----(8) 

Hence, from 7 and 8 we can conclude the conclusion is G(Mice) ^ $\neg$R(Rabbit). 

Comments

Bhai yes it is true but $\forall x U(x)$ is not a premise here so how do you know that it is true ?

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As we know that  1st premise is true,by consider c is "for you to buy a lot of stuff" we apply Universal Instantiation on (1) and we get,

C(c)$\rightarrow$U(c) 

Now because of (3) premise is true LHS is true and if LHS is than RHS can't be false Hence, U(c) is true. 

I did a mistake in my answer, there is not $\forall_{x}$ in (4) and (5). 

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Now, the things which you have written is correct and also whatever I had solved using both Universal Generalization and Universal Instantiation is also correct.

So, both ways are correct.