For each of the following collections of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. (Hint: Convert sentences to propositions and use inference rules for finding possible conclusion(s)) “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.” “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.” “I am either clever or lucky.” “I am not lucky.” “If I am lucky, then I will win the lottery.” “Every computer science major has a personal computer.” “Ralph does not have a personal computer.” “Ann has a personal computer.” “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.” “All rodents gnaw their food.” “Mice are rodents.” “Rabbits do not gnaw their food.” “Bats are not rodents.”
For each of the following collections of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. (Hint: Convert sentences to propositions and use inference rules for finding possible conclusion(s))
“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.”
“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.”
“I am either clever or lucky.” “I am not lucky.” “If I am lucky, then I will win the lottery.”
“Every computer science major has a personal computer.” “Ralph does not have a personal computer.” “Ann has a personal computer.”
“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.”
“All rodents gnaw their food.” “Mice are rodents.” “Rabbits do not gnaw their food.” “Bats are not rodents.”
- "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."
The first collection of premises is an example of propositional logic. The first premise can be written as P → (Q ∨ R),
where P represents taking the day off, Q represents raining, and R represents snowing. The second premise can be
written as (P ∨ S), where S represents taking Thursday off. The third premise can be written as ¬Q, and the fourth
premise can be written as ¬R.
Using the premise P → (Q ∨ R), we can conclude that P must be true because ¬Q and ¬R.
With the premise (P ∨ S) and ¬Q, we can conclude that P must be true.
And with the premise P and ¬R, we can conclude that Q must be true.
- Conclusion: I took Thursday off.
- Inference Rule Used: Modus Ponens
- "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."
The second collection of premises is an example of propositional logic. The first premise can be written as P → Q, where
P represents eating spicy foods and Q represents having strange dreams. The second premise can be written as R → Q,
where R represents thunder while sleeping. The third premise can be written as ¬Q.
Using the premise P → Q and ¬Q, we can conclude that P must be false.
And with the premise R → Q and ¬Q, we can conclude that R must be false.
- Conclusion: I did not eat spicy foods and there was no thunder while I slept.
- Inference Rule Used: Negation of antecedent.
- "I am either clever or lucky." "I am not lucky." "If I am lucky, then I will win the lottery."
The third collection of premises is an example of propositional logic. The first premise can be written as (P ∨ Q), where P
represents being clever and Q represents being lucky. The second premise can be written as ¬Q. The third premise can
be written as Q → R, where R represents winning the lottery.
Using the premise (P ∨ Q) and ¬Q, we can conclude that P must be true.
And with the premise ¬Q and Q → R, we can conclude that R must be false.
- Conclusion: I am clever.
- Inference Rule Used: Modus Tollens
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