Kenneth Rosen Edition 7 Exercise 1.6 Question 10 (Page No. 79)

Question

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

a) “If I play hockey, then I am sore the next day.” “I use the whirlpool if I am sore.” “I did not use the whirlpool.”

b) “If I work, it is either sunny or partly sunny.” “I worked last Monday or I worked last Friday.” “It was not sunny on Tuesday.” “It was not partly sunny on Friday.”

c) “All insects have six legs.” “Dragonflies are insects.” “Spiders do not have six legs.” “Spiders eat dragonflies.”

d) “Every student has an Internet account.” “Homer does not have an Internet account.” “Maggie has an Internet account.”

e) “All foods that are healthy to eat do not taste good.” “Tofu is healthy to eat.” “You only eat what tastes good.” “You do not eat tofu.” “Cheeseburgers are not healthy to eat.”

f) “I am either dreaming or hallucinating.” “I am not dreaming.” “If I am hallucinating, I see elephants running down the road.”

Answer 1

a) 

Assume, 

H: I play hockey. 

S: I am sore. 

W: I use the whirlpool. 

Premises:

1) H$\rightarrow$S

2) S$\rightarrow$W

3) $\neg$W 

Using 1 & 2 by applying Hypothetical Syllogism. we get, 

H$\rightarrow$W  --------(4) 

Using 3 & 4 by applying modes tollens. we get, 

$\neg$H

Hence, we can conclude the conclusion is I didn't play hockey. 

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b)  

Assume, 

W(x) : I work on  x day. 

S(x) : It's sunny on x day. 

P(x) : It's partly sunny on x day. 

Premises:

1. $\forall_{x}$W(x)$\rightarrow$(S(x)$\vee$P(x))

2. W(M)$\vee$W(F) 

3. $\neg$S(T) 

4. $\neg$P(F) 

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

W(M)$\rightarrow$(S(M)$\vee$P(M)) ------(5) 

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

W(F)$\rightarrow$(S(F)$\vee$P(F)) -------(6) 

Using 5, 6 & 2 by applying Constructive Dilemma. we get, 

(S(M)$\vee$P(M))$\vee$(S(F)$\vee$P(F)) ----(7) 

Using 3 & 4 by applying Conjunction Rule. we get, 

$\neg$S(T)$\wedge\neg$P(F) ------(8) 

Hence, from 7 & 8 we can conclude that the conclusion is $\neg$S(T)$\wedge\neg$P(F) ^ (S(M)$\vee$P(M))$\vee$(S(F)$\vee$P(F)). 

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c)

Assume, 

I(x) : x is a insect. 

L(x) : x has six legs. 

D(x) : x is a dragonfly. 

S(x) : x is a Spider. 

E(x, y) : x eats y. 

Premises:

1. $\forall_{x}$(I(x)$\rightarrow$L(x)) 

2. $\forall_{x}$(D(x)$\rightarrow$I(x))

3. $\forall_{x}$(S(x)$\rightarrow\neg$L(x))

4. $\forall_{x}\forall_{y}$(S(x)$\wedge$D(y)$\rightarrow$E(x,y))

Applying Universal Instantiation on 1,2 & 3. we get, 

I(c)$\rightarrow$L(c)  ------------(5) 

D(c)$\rightarrow$I(c)  ------------(6) 

S(c)$\rightarrow\neg$L(c))  ------(7) 

Using 6 & 5 by applying hypothetical syllogism. we get, 

D(c)$\rightarrow$L(c)  -----------(8) 

we can rewrite 7 as, 

L(c)$\rightarrow\neg$S(c) 

Using 5 & 7 by applying hypothetical syllogism. we get, 

I(c)$\rightarrow\neg$S(c) ----------(9) 

Hence, from 8 & 9 we can conclude that the conclusion is "if c is a dragonfly then it has six legs" and "if c is a insect then it is not a spider". 

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d)

Assume, 

S(x) : x is a student. 

I(x) : x has an Internet account. 

Premises:

1. $\forall_{x}$(S(x)$\rightarrow$I(x)) 

2. $\neg$I(H) 

3. I(M) 

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

S(H)$\rightarrow$I(H) ------(4) 

Using 4 & 2 by applying modus tollens. we get, 

$\neg$S(H)  ----------(5) 

Hence, from 5 we can conclude that the conclusion is $\neg$S(H) . 

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e)

Assume, 

H(x) : x is healthy to eat. 

T(x) : x has taste good. 

E(x) : You eat x. 

Premises:

1. $\forall_{x}$(H(x)$\rightarrow\neg$T(x)) 

2. H(T) 

3. $\forall_{x}$(T(x)$\rightarrow$E(x)) 

4.$\neg$E(T) 

5. $\neg$H(C) 

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

H(Tofu)$\rightarrow\neg$T(Tofu) -----(6) 

Using 6 & 2 by applying modus ponnes. we get, 

$\neg$T(Tofu)   -------(7) 

Hence, from 7 we can conclude that the conclusion is "Tofu hasn't taste good".

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f)

Assume, 

D : I am dreaming. 

H : I am hallucinating. 

E : I see elephants running down the road. 

Premises:

1. D$\vee$H

2. $\neg$D

3. H$\rightarrow$E

Using 1 & 2 by applying Disjunctive syllogism. we get, 

H  --------(4) 

Using 4 & 3 by applying modus ponnes. we get, 

E  ------(5) 

Hence, from 5 we can conclude that the conclusion is "I see elephants running down the road".

Comments

So, which rule of inference will applicable on above premise and how? & what will be the conclusion for this?

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It is not necessary that you have to use all the premises and similarly you can make multiple conclusions based on the given facts.

So, here, $\forall x (D(x) \rightarrow I(x))$ and  $\forall x (I(x) \rightarrow L(x))$ gives $\forall x (D(x) \rightarrow L(x))$

It means all dragonflies have 6 legs.

Now, from (1), you can write $I(c) \rightarrow L(c)$ and from (3), you can write $L(c) \rightarrow \neg S(c)$ and from these two, you can write $I (c) \rightarrow \neg S(c)$ and so you can say, If c is an insect then it is not a spider.

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@ankitgupta.1729 i have edited my answer. Thanks for your effort.