The Logic Problem, taken from WFF’N PROOF, The Game of Logic, has these two assumptions:
i. Logic is difficult or not many students like logic.
ii. If mathematics is easy, then logic is not difficult.
By translating these assumptions into statements involving propositional variables and logical connectives, determine whether CSCI 2610 Spring 2017 Cotterell, Peng, Subramani each of the following are valid conclusions of these assumptions and state the reason for your conclusion:
a) That mathematics is not easy, if many students like logic.
b) That not many students like logic, if mathematics is not easy.
c) That mathematics is not easy or logic is difficult.
d) That logic is not difficult or mathematics is not easy.
e) That if not many students like logic, then either mathematics is not easy or logic is not difficult.
The Correct Answer and Explanation is :
Let’s first translate the assumptions into logical propositions:
- Assumption i: “Logic is difficult or not many students like logic.”
This can be represented as:
[
L \vee \neg S
]
where:
- ( L ) means “Logic is difficult”
- ( S ) means “Many students like logic”
- Assumption ii: “If mathematics is easy, then logic is not difficult.”
This is a conditional statement and can be written as:
[
E \rightarrow \neg L
]
where:
- ( E ) means “Mathematics is easy”
Now, let’s evaluate each proposed conclusion:
a) “That mathematics is not easy, if many students like logic.”
This is a conditional statement. We need to express this as:
[
S \rightarrow \neg E
]
This would mean, “If many students like logic, then mathematics is not easy.”
To check if this is a valid conclusion, let’s look at the assumptions. Assumption ii (( E \rightarrow \neg L )) provides a connection between ( E ) (mathematics is easy) and ( L ) (logic is difficult). However, this doesn’t directly suggest that ( S ) (many students like logic) implies ( \neg E ) (mathematics is not easy). Therefore, this conclusion is not valid based on the given assumptions.
b) “That not many students like logic, if mathematics is not easy.”
This can be represented as:
[
\neg E \rightarrow \neg S
]
which means “If mathematics is not easy, then not many students like logic.”
We cannot directly infer this from the assumptions. While assumption i involves ( L ) or ( \neg S ), and assumption ii links ( E ) and ( L ), there is no direct implication that ( \neg E ) leads to ( \neg S ). Therefore, this conclusion is not valid.
c) “That mathematics is not easy or logic is difficult.”
This statement can be written as:
[
\neg E \vee L
]
This is logically equivalent to the contrapositive of assumption ii (( E \rightarrow \neg L )), which states that if mathematics is easy, logic is not difficult. The contrapositive of this is ( \neg E \vee L ), which directly matches this conclusion. Therefore, this conclusion is valid.
d) “That logic is not difficult or mathematics is not easy.”
This is written as:
[
\neg L \vee \neg E
]
This is the contrapositive of assumption ii (( E \rightarrow \neg L )), which is logically equivalent to ( \neg L \vee \neg E ). Therefore, this conclusion is valid.
e) “That if not many students like logic, then either mathematics is not easy or logic is not difficult.”
This can be written as:
[
\neg S \rightarrow (\neg E \vee \neg L)
]
There’s no direct evidence in the assumptions that ( \neg S ) implies ( \neg E \vee \neg L ). Assumption i relates ( L ) and ( \neg S ), but it doesn’t directly support this implication. Therefore, this conclusion is not valid.
Summary of Valid Conclusions:
- Valid: (c) “That mathematics is not easy or logic is difficult.”
- Valid: (d) “That logic is not difficult or mathematics is not easy.”
- Not Valid: (a), (b), (e).