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.

(a)“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.”

(b)“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.”

(c)“I am either clever or lucky.” “I am not lucky.” “If I am lucky, then I will win the lottery.”

(d)“Every computer science major has a personal computer.” “Ralph does not have a personal computer.” “Ann has a personal computer.”

(e)“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.”

(f)“All rodents gnaw their food.” “Mice are rodents.” “Rabbits do not gnaw their food.” “Bats are not rodents.”

Short answer

The conclusions can be drawn from given collection of premises. Also, different rules of inference are used to obtain each conclusion from the premises.

Step-by-step solution

Definition of Argument    

A rule of inference is a logical structure that consists of a function that accepts premises, interprets them, and provides a conclusion.

To find conclusion from the given premises using rules of inference (a)

(a) “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.”

Suppose,

\(p = \)“I take the day\(x\)off”,

\(q = \)“It rains on day\(x\)”,

\(r = \)“It snows on day\(x\).”

Prepare the table for premises and rules of inference.

	Steps	Reason
\(1\)	\(\forall x\left( {p \to \left( {q \vee r} \right)} \right)\)	Premise
\(2\)	\(p\left( {Tuesday} \right) \vee p\left( {Thursday} \right)\)	Premise
\(3\)	\(\neg \left( {q\left( {Tuesday} \right) \vee r\left( {Tuesday} \right)} \right)\)	Premise
\(4\)	\(\neg \left( {r\left( {Thursday} \right)} \right)\)	Premise
\(5\)	\(\left( {p\left( {Tuesday} \right)} \right) \to \left( {q\left( {Tuesday} \right)} \right) \vee \left( {r\left( {Tuesday} \right)} \right)\)	Universal instantiation from\(\left( 1 \right)\)
\(6\)	\(\left( {p\left( {Thursday} \right)} \right) \to \left( {q\left( {Thursday} \right)} \right) \vee \left( {r\left( {Thursday} \right)} \right)\)	Universal instantiation from\(\left( 1 \right)\)
\(7\)	\(\neg \left( {p\left( {Tuesday} \right)} \right)\)	Modus ponens from\(\left( 2 \right)\)and\(\left( 5 \right)\)
\(8\)	\(\left( {p\left( {Thursday} \right)} \right)\)	Disjunctive syllogism from\(\left( 2 \right)\)and\(\left( 7 \right)\)
\(9\)	\(\left( {q\left( {Thursday} \right)} \right) \vee \left( {r\left( {Thursday} \right)} \right)\)	Modus ponens from\(\left( 6 \right)\)and\(\left( 8 \right)\)
\(10\)	\(q\left( {Thursday} \right)\)	Disjunctive syllogism from \(\left( 4 \right)\)and \(\left( 9 \right)\)

Table 1

Step\(7\)indicates that “I did not take Tuesday off.”

Step\(8\)specifies that “I took Thursday off.”

Step \(10\) shows that “It rained on Thursday.”

To find conclusion from the given premises using rules of inference (b)

(b) “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.”

Suppose\(p = \)“I eat spicy food”,

\(q = \)“I have strange dreams”,

\(r = \)“There is thunder while I sleep.”

Prepare the table for premises and rules of inference.

	Step	Reason
\(1\)	\(p \to q\)	Premise
\(2\)	\(r \to q\)	Premise
\(3\)	\(\neg q\)	Premise
\(4\)	\(\neg p\)	Modus tollens from\(\left( 1 \right)\)and\(\left( 3 \right)\)
\(5\)	\(\neg r\)	Modus tollens from \(\left( 2 \right)\)and \(\left( 3 \right)\)

Table 2

Step\(4\)indicates that “I did not eat spicy food.”

Step \(5\) specifies that “There was no thunder while I slept.”

To find conclusion from the given premises using rules of inference (c)

(c) “I am either clever or lucky.” “I am not lucky.” “If I am lucky, then I will win the lottery.”

Suppose\(p = \)“I am clever”,

\(q = \)“I am lucky”,

\(r = \)“I win the lottery.”

Prepare the table for premises and rules of inference.

	Step	Reason
\(1\)	\(p \vee q\)	Premise
\(2\)	\(\neg q\)	Premise
\(3\)	\(q \to r\)	Premise
\(4\)	\(p\)	Disjunctive syllogism from \(\left( 1 \right)\)and \(\left( 2 \right)\)

Table 3

Step \(4\) indicates that “I am clever.”

To find conclusion from the given premises using rules of inference (d)

(d) “Every computer science major has a personal computer.” “Ralph does not have a personal computer.” “Ann has a personal computer.”

Suppose\(p = \)“\(x\)is a computer science major”,

\(q = \)“\(x\)has a personal computer”,

Prepare the table for premises and rules of inference

	Step	Reason
\(1\)	\(\forall x\left( {p \to q} \right)\)	Premise
\(2\)	\(\neg q\left( {Ralph} \right)\)	Premise
\(3\)	\(q\left( {Ann} \right)\)	Premise
\(4\)	\(p\left( {Ralph} \right) \to q\left( {Ralph} \right)\)	Universal instantiation from \(\left( 1 \right)\)
\(5\)	\(\neg p\left( {Ralph} \right)\)	Modus tollens from \(\left( 2 \right)\)and \(\left( 4 \right)\)

Table 4

Step \(5\) specifies that “Ralph is not a computer science major.”

To find conclusion from the given premises using rules of inference (e)

(e) “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.”

Suppose\(p = \)“Good for corporations”,

\(q = \)“Good for united states”,

\(r = \)“Good for you”,

\(s = \)“You buy lots of stuff”,

Prepare the table for premises and rules of inference.

	Step	Reason
\(1\)	\(p \to q\)	Premise
\(2\)	\(q \to r\)	Premise
\(3\)	\(s \to p\)	Premise
\(4\)	\(s \to q\)	Hypothetical Syllogism from\(\left( 1 \right)\)and\(\left( 3 \right)\)
\(5\)	\(s \to r\)	Hypothetical Syllogism from\(\left( 2 \right)\)and\(\left( 4 \right)\)
6	\(p \to r\)	Hypothetical Syllogism from \(\left( 1 \right)\)and \(\left( 2 \right)\)

Table 5

Step\(4\)specifies that “You buying lots of stuff is good for the United States.”

Step\(5\)specifies that “You buying lots of stuff is good for the you.”

Step \(6\) specifies that “What is good for corporations is good for you.”

To find conclusion from the given premises using rules of inference (f)

(f) “All rodents gnaw their food.” “Mice are rodents.” “Rabbits do not gnaw their food.” “Bats are not rodents.”

Suppose\(p = \)“\(x\)are rodents”,

\(q = \)“\(x\)gnaws their food.”

Prepare the table for premises and rules of inference.

	Step	Reason
\(1\)	\(\forall x\left( {p \to q} \right)\)	Premise
\(2\)	\(p\left( {Mice} \right)\)	Premise
\(3\)	\(\neg q\left( {Rabbit} \right)\)	Premise
\(4\)	\(\neg p\left( {Bats} \right)\)	Premise
\(5\)	\(p\left( {Mice} \right) \to q\left( {Mice} \right)\)	Universal instantiation from\(\left( 1 \right)\)
\(6\)	\(p\left( {Rabbit} \right) \to q\left( {Rabbit} \right)\)	Universal instantiation from\(\left( 1 \right)\)
\(7\)	\(q\left( {Mice} \right)\)	Modus ponens from\(\left( 2 \right)\)and\(\left( 5 \right)\)
\(8\)	\(\neg p\left( {Rabbit} \right)\)	Modus tollens from \(\left( 3 \right)\)and \(\left( 6 \right)\)

Table 6

Step\(7\)specifies that “Mice gnaw their food.”

Step\(8\)specifies that “Rabbits are not rodents.”

Therefore, the conclusions can be drawn from given collection of premises. Also, different rules of inference are used to obtain each conclusion from the premises.