91影视

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) 鈥淚f I play hockey, then I am sore the next day.鈥� 鈥淚 use the whirlpool if I am sore.鈥� 鈥淚 did not use the whirlpool.鈥� b) 鈥淚f I work, it is either sunny or partly sunny.鈥� 鈥淚 worked last Monday or I worked last Friday.鈥� 鈥淚t was not sunny on Tuesday.鈥� 鈥淚t was not partly sunny on Friday.鈥� c) 鈥淎ll insects have six legs.鈥� 鈥淒ragonflies are insects.鈥� 鈥淪piders do not have six legs.鈥� 鈥淪piders eat dragon-flies.鈥� d) 鈥淓very student has an Internet account.鈥� 鈥淗omer does not have an Internet account.鈥� 鈥淢aggie has an Internet account.鈥� e) 鈥淎ll foods that are healthy to eat do not taste good.鈥� 鈥淭ofu is healthy to eat.鈥� 鈥淵ou only eat what tastes good.鈥� 鈥淵ou do not eat tofu.鈥� 鈥淐heeseburgers are not healthy to eat.鈥� f ) 鈥淚 am either dreaming or hallucinating.鈥� 鈥淚 am not dreaming.鈥� 鈥淚f I am hallucinating, I see elephants running down the road.鈥�

Step by step solution

01

Part a - Analyze Premises

Premises:1. If I play hockey, then I am sore the next day. (H 鈫� S)2. I use the whirlpool if I am sore. (S 鈫� W)3. I did not use the whirlpool. (卢W)

02

Part a - Derive Intermediate Conclusion

From (3) 卢W and (2) S 鈫� W, apply Modus Tollens to conclude 卢S.

03

Part a - Final Conclusion

From (1) H 鈫� S and 卢S, apply Modus Tollens to conclude 卢H.

04

Part b - Analyze Premises

Premises:1. If I work, it is either sunny or partly sunny. (W 鈫� (S 鈭� P))2. I worked last Monday or I worked last Friday. (M 鈭� F)3. It was not sunny on Tuesday. (卢S虏)4. It was not partly sunny on Friday. (卢P岫�)

05

Part b - Derive Intermediate Conclusion

Since (4) 卢P岫�, if F, 卢(S 鈭� P) at F. Hence, from (1), W on Friday would be false.

06

Part b - Final Conclusion

Since W on Friday is false, from (2), if F is false, then M is true. Hence, I worked last Monday.

07

Part c - Analyze Premises

Premises:1. All insects have six legs. (I 鈫� L)2. Dragonflies are insects. (D 鈫� I)3. Spiders do not have six legs. (S 鈫� 卢L)4. Spiders eat dragonflies. (S 鈫� E)

08

Part c - Derive Intermediate Conclusion

From (1) and (2), Dragonflies have six legs (D 鈫� L).

09

Part c - Final Conclusion

No further meaningful conclusion linking spiders and dragonflies directly.

10

Part d - Analyze Premises

Premises:1. Every student has an Internet account. (S 鈫� I)2. Homer does not have an Internet account. (卢I拾)3. Maggie has an Internet account. (I岬�)

11

Part d - Final Conclusion

From (1) and (2), Homer is not a student (卢S拾). Maggie is consistent with every student having internet as per premise (1).

12

Part e - Analyze Premises

Premises:1. All foods that are healthy do not taste good. (H 鈫� 卢T)2. Tofu is healthy. (T岬�)3. You only eat what tastes good. (E 鈫� T)4. You do not eat tofu. (卢E岬�)5. Cheeseburgers are not healthy. (卢H岫�)

13

Part e - Derive Intermediate Conclusion

From (2) and (1), Tofu does not taste good (T岬� 鈫� 卢T岬�). From (3) and (4), You do not eat Tofu (卢E岬�).

14

Part e - Final Conclusion

From combining (2), (3), and (4), the premises are consistent with each other.

15

Part f - Analyze Premises

Premises:1. I am either dreaming or hallucinating. (D 鈭� H)2. I am not dreaming. (卢D)3. If I am hallucinating, I see elephants running down the road. (H 鈫� E)

16

Part f - Final Conclusion

From (1) and (2), I am hallucinating (H). From (H) and (3), I see elephants running down the road (E).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Modus Tollens

Modus Tollens is a rule of inference in propositional logic. It helps us make logical deductions when we know that a certain conditional statement (if-then statement) is true and the conclusion of that statement is false.

The formal structure of Modus Tollens is:

• If P then Q (P 鈫� Q)
• Not Q (卢Q)
• Therefore, not P (卢P)

Let's look at an example from the exercise:

In part (a), the premises are:

• If I play hockey, then I am sore the next day. (H 鈫� S)
• I use the whirlpool if I am sore. (S 鈫� W)
• I did not use the whirlpool. (卢W)

From these premises, we can derive:

• Since 卢W and S 鈫� W, using Modus Tollens, we conclude 卢S.
• Then, using H 鈫� S and 卢S, using Modus Tollens again, we conclude 卢H.

Therefore, I did not play hockey. That's how Modus Tollens helps us draw conclusions by eliminating possibilities.

Logic

Logic is a systematic way of thinking that allows us to derive valid conclusions from premises. It helps us analyze and form coherent arguments.

In the context of this exercise, logic involves using rules of inference to derive conclusions. For example:

• In part (d), the premises state:
• Every student has an Internet account. (S 鈫� I)
• Homer does not have an Internet account. (卢I拾)
• Therefore, Homer is not a student (卢S拾).

This logical deduction is based on understanding how the premises relate and applying logical rules. By using logic, we form consistent conclusions from different sets of premises, ensuring that all statements are coherent with each other.

Premises and Conclusions

In any logical argument, we start with premises and arrive at conclusions through reasoning.

Premises are statements or facts that we accept as true, which serve as the foundation for the argument. Conclusions are the inferences we draw from those premises.

For instance, in part (e) of the exercise:

• Premises:
• All foods that are healthy to eat do not taste good. (H 鈫� 卢T)
• Tofu is healthy to eat (T岬�)
• You only eat what tastes good (E 鈫� T)
• Therefore, you do not eat tofu (卢E岬�).

Here, each premise leads logically to the next, and the conclusion follows from the premises. It's essential to ensure that the premises provided are accurate and logically lead to the conclusion. This careful analysis of premises and conclusions ensures our reasoning is sound and our arguments are persuasive.