10.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 a 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.
a) From the premises, the relevant conclusion that can be drawn is: "I did not play hockey."
The rules of inference used to obtain this conclusion are:
1. Modus ponens: If we have a conditional statement "If p, then q" and we also have p, then we can assert that q is true. In this case, the premises "If I play hockey, then I am sore the next day" and "I played hockey" imply that "I am sore the next day" is true.
2. Modus tollens: If we have a conditional statement "If p, then q" and we also have "not q", then we can conclude "not p". In this case, the premises "I use the whirlpool if I am sore" and "I did not use the whirlpool" imply that "I am not sore" is true, which can be restated as "I did not play hockey."
b) From the premises, the relevant conclusion that can be drawn is: "I worked last Monday."
The rules of inference used to obtain this conclusion are:
1. Disjunctive syllogism: If we have a statement "p or q" and we also have "not q", then we can infer "p". In this case, the premises "I worked last Monday or I worked last Friday" and "It was not partly sunny on Friday" imply that "I worked last Monday" is true.
c) From the premises, the relevant conclusion that can be drawn is: "Spiders are not insects."
The rules of inference used to obtain this conclusion are:
1. Modus tollens: If we have a conditional statement "If p, then q" and we also have "not q", then we can conclude "not p". In this case, the premises "All insects have six legs" and "Spiders do not have six legs" imply that "Spiders are not insects" is true.
d) From the premises, no specific relevant conclusion can be drawn, as the second premise contradicts the first premise.
e) From the premises, the relevant conclusion that can be drawn is: "You do not eat what tastes good."
The rules of inference used to obtain this conclusion are:
1. Modus tollens: If we have a conditional statement "If p, then q" and we also have "not q", then we can conclude "not p". In this case, the premises "All foods that are healthy to eat do not taste good" and "You only eat what tastes good" imply that "You do not eat tofu" is true.
f) From the premises, the relevant conclusion that can be drawn is: "I am hallucinating."
The rules of inference used to obtain this conclusion are:
1. Disjunctive syllogism: If we have a statement "p or q" and we also have "not p", then we can infer "q". In this case, the premises "I am either dreaming or hallucinating" and "I am not dreaming" imply that "I am hallucinating" is true.
a) Premises:
1) If I play hockey, then I am sore the next day.
2) I use the whirlpool if I am sore.
3) I did not use the whirlpool.
Conclusion: It can be concluded that I did not play hockey.
Explanation: To obtain this conclusion, we can follow the rule of inference called Modus Tollens. Modus Tollens states that if we have a conditional statement "if P, then Q" and we have the negation of Q (not Q), we can infer the negation of P (not P). In this case, the premises include a conditional statement "if I play hockey, then I am sore the next day" (P → Q). We also have the negation of Q (not Q), which is "I did not use the whirlpool." Applying Modus Tollens, we can conclude that I did not play hockey (not P).
b) Premises:
1) If I work, it is either sunny or partly sunny.
2) I worked last Monday or I worked last Friday.
3) It was not sunny on Tuesday.
4) It was not partly sunny on Friday.
Conclusion: It can be concluded that I did not work last Monday.
Explanation: To obtain this conclusion, we can use the rule of inference called Disjunctive Syllogism. Disjunctive Syllogism states that if we have a disjunction "P or Q" and we know that Q is false (not Q), we can infer that P must be true (P). In this case, the premises include a disjunction "I worked last Monday or I worked last Friday" (P or Q). We also have the negations of Q (not Q), which are "It was not sunny on Tuesday" and "It was not partly sunny on Friday." Applying Disjunctive Syllogism, we can conclude that I did not work last Monday (P).
c) Premises:
1) All insects have six legs.
2) Dragonflies are insects.
3) Spiders do not have six legs.
4) Spiders eat dragonflies.
Conclusion: It can be concluded that spiders are not insects.
Explanation: To obtain this conclusion, we can use the rule of inference called Modus Tollens again. The first two premises state that all insects have six legs (P → Q) and dragonflies are insects (P). The third premise states that spiders do not have six legs (not Q). Applying Modus Tollens, we can infer that spiders are not insects (not P).
d) Premises:
1) Every student has an Internet account.
2) Homer does not have an Internet account.
3) Maggie has an Internet account.
Conclusion: It can be concluded that Homer is not a student.
Explanation: This conclusion can be obtained by the rule of inference called Modus Tollens. The first premise states that every student has an Internet account (P → Q). The second premise states that Homer does not have an Internet account (not Q). Applying Modus Tollens, we can infer that Homer is not a student (not P).
e) Premises:
1) All foods that are healthy to eat do not taste good.
2) Tofu is healthy to eat.
3) You only eat what tastes good.
4) You do not eat tofu.
5) Cheeseburgers are not healthy to eat.
Conclusion: It can be concluded that cheeseburgers taste good.
Explanation: The conclusion can be obtained by applying the rule of inference called Disjunctive Syllogism. Premise 1 states that all foods that are healthy to eat do not taste good (P → not Q). Premise 2 states that tofu is healthy to eat (P). Premise 3 states that you only eat what tastes good (Q). Premise 4 states that you do not eat tofu (not P). Premise 5 states that cheeseburgers are not healthy to eat (not P). Applying Disjunctive Syllogism, we can infer that cheeseburgers taste good (Q).
f) Premises:
1) I am either dreaming or hallucinating.
2) I am not dreaming.
3) If I am hallucinating, I see elephants running down the road.
Conclusion: It can be concluded that I am not hallucinating elephants running down the road.
Explanation: This conclusion can be obtained through the process of Elimination. The first premise states that I am either dreaming or hallucinating (P or Q). The second premise states that I am not dreaming (not P). The third premise states that if I am hallucinating, I see elephants running down the road (Q → R). By eliminating the possibility of dreaming (not P), we can infer that I am not hallucinating elephants running down the road (not R).
10.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 a 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.
a. You will continue to be sore.
b. All can be true.
c. All can be true.
d. Either one of the first two sentences is false, or Homer is not a student.
e. All can be true.
f. All can be true.