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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. a) 鈥淚f I take the day off, it either rains or snows.鈥� 鈥淚 took Tuesday off or I took Thursday off.鈥� 鈥淚t was sunny on Tuesday.鈥� 鈥淚t did not snow on Thursday.鈥� b) 鈥淚f I eat spicy foods, then I have strange dreams.鈥� 鈥淚 have strange dreams if there is thunder while I sleep.鈥� 鈥淚 did not have strange dreams.鈥� c) 鈥淚 am either clever or lucky.鈥� 鈥淚 am not lucky.鈥� 鈥淚f I am lucky, then I will win the lottery.鈥� d) 鈥淓very computer science major has a personal computer.鈥� 鈥淩alph does not have a personal computer.鈥� 鈥淎nn has a personal computer.鈥� e) 鈥淲hat is good for corporations is good for the United States.鈥� 鈥淲hat is good for the United States is good for you.鈥� 鈥淲hat is good for corporations is for you to buy lots of stuff.鈥� f ) 鈥淎ll rodents gnaw their food.鈥� 鈥淢ice are rodents.鈥� 鈥淩abbits do not gnaw their food.鈥� 鈥淏ats are not ro- dents.鈥�

Step by step solution

01

Title - Premises for Part A

Identify the premises: 1. If I take the day off, it either rains or snows.2. I took Tuesday off or I took Thursday off.3. It was sunny on Tuesday.4. It did not snow on Thursday.

02

Title - Applying rules of inference to Part A

Using the premises: If I took Tuesday off and it was sunny, then it did not rain or snow on Tuesday. Therefore, I must have taken Thursday off and since it did not snow on Thursday, it must have rained.

03

Title - Premises for Part B

Identify the premises: 1. If I eat spicy foods, then I have strange dreams.2. I have strange dreams if there is thunder while I sleep.3. I did not have strange dreams.

04

Title - Applying rules of inference to Part B

Using Modus Tollens on the premises: If no strange dreams occurred, then no spicy foods were eaten, and there was no thunder while I slept.

05

Title - Premises for Part C

Identify the premises: 1. I am either clever or lucky.2. I am not lucky.3. If I am lucky, then I will win the lottery.

06

Title - Applying rules of inference to Part C

By Disjunctive Syllogism: Since I am not lucky, I must be clever.

07

Title - Premises for Part D

Identify the premises: 1. Every computer science major has a personal computer.2. Ralph does not have a personal computer.3. Ann has a personal computer.

08

Title - Applying rules of inference to Part D

Using Universal Instantiation and Modus Tollens: Ralph is not a computer science major, while Ann may or may not be because she has a personal computer.

09

Title - Premises for Part E

Identify the premises: 1. What is good for corporations is good for the United States.2. What is good for the United States is good for you.3. What is good for corporations is for you to buy lots of stuff.

10

Title - Applying rules of inference to Part E

Using Hypothetical Syllogism: Since buying lots of stuff is good for corporations, it is good for the United States and ultimately good for you.

11

Title - Premises for Part F

Identify the premises: 1. All rodents gnaw their food.2. Mice are rodents.3. Rabbits do not gnaw their food.4. Bats are not rodents.

12

Title - Applying rules of inference to Part F

Using Universal Instantiation and Modus Tollens: Mice gnaw their food, rabbits are not rodents, and the premise about bats is irrelevant.

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Key Concepts

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

Logical Reasoning

Logical reasoning is the process we use to make decisions and reach conclusions based on the available information. It helps us move from premises (the starting points or assumptions) to a conclusion.
When solving logical problems, we often apply specific rules of inference that provide a structured approach to derive conclusions.
Understanding these rules can help you solve complex problems by breaking them down into simpler, more manageable parts.

Premises and Conclusions

Premises are the statements or facts that we start with in logical reasoning. They are the foundation upon which we base our arguments. Conclusions are the statements that we derive from the premises using logical rules.
To properly apply logical rules, it's crucial to clearly identify all the premises first.
For example, in problem (a), we have premises such as 'It was sunny on Tuesday'. Starting from these premises, we use logical steps to reach conclusions like 'It must have rained on Thursday.' Identifying premises and conclusions helps in structuring your logical arguments effectively.

Modus Tollens

Modus Tollens is a rule of inference that allows us to derive a conclusion by negating the consequent and the antecedent. Put simply, it follows this pattern:

• Premise 1: If P, then Q.
• Premise 2: Not Q.
• Conclusion: Therefore, not P.

In problem (b), 'If I eat spicy foods, then I have strange dreams'. From Premise 2 ('I did not have strange dreams'), we conclude using Modus Tollens that 'I did not eat spicy foods'.

Disjunctive Syllogism

Disjunctive Syllogism helps us infer conclusions from statements that involve 'or' conditions. It takes the form:

• Premise 1: P or Q.
• Premise 2: Not P.
• Conclusion: Therefore, Q.

In problem (c), we use premises like 'I am either clever or lucky' and 'I am not lucky'. By Disjunctive Syllogism, we conclude, 'I must be clever.'

Universal Instantiation

Universal Instantiation allows us to move from general statements to specific instances. It works like this: If a statement is true for all things in a particular set, then it is also true for any specific thing in that set.
For instance, in problem (d), we have a general statement: 'Every computer science major has a personal computer.' From this, if Ralph does not have a personal computer, we can conclude that Ralph is not a computer science major.
Likewise, because mice are classified as rodents and all rodents gnaw their food, we can conclude that mice gnaw their food as highlighted in problem (f).

Hypothetical Syllogism

Hypothetical Syllogism involves chaining together conditional statements to reach a conclusion. It follows this pattern:

• Premise 1: If P, then Q.
• Premise 2: If Q, then R.
• Conclusion: Therefore, if P, then R.

In problem (e), you see conditional statements like 'What is good for corporations is good for the United States', and 'What is good for the United States is good for you'. Using Hypothetical Syllogism, we conclude that 'What is good for corporations is good for you.'