(a.)
The validity of the given argument.
It has been determined that the given argument is valid.
Given:
All women are mortal.
Hypatia was a woman.
Therefore, Hypatia was mortal.
Concept used:
According to Chain rule, if
According to Modus Ponens, if
According to Modus Tollens, if
Calculation:
Let
It is given that “All women are mortal”.
Then,
It is also given that “Hypatia was a woman”.
Then, for Hypatia,
Now, for Hypatia,
Then, according to Modus Ponens, it follows that
Thus, it logically follows that “Hypatia was mortal”.
Hence, the given argument is valid.
Conclusion:
It has been determined that the given argument is valid.
(b.)
The validity of the given argument.
It has been determined that the given argument is valid.
Given:
All squares are quadrilaterals.
All quadrilaterals are polygons.
Therefore, all squares are polygons.
Concept used:
According to Chain rule, if
According to Modus Ponens, if
According to Modus Tollens, if
Calculation:
Let
It is given that “All squares are quadrilaterals”.
Then,
It is also given that “All quadrilaterals are polygons”.
Then,
Now,
Then, according to Chain rule, it follows that
Thus, it logically follows that “all squares are polygons”.
Hence, the given argument is valid.
Conclusion:
It has been determined that the given argument is valid.
(c.)
The validity of the given argument.
It has been determined that the given argument is valid.
Given:
All teachers are intelligent.
Some teachers are rich.
Therefore, some intelligent people are rich.
Concept used:
According to Chain rule, if
According to Modus Ponens, if
According to Modus Tollens, if
Calculation:
Let
It is given that:
All teachers are intelligent.
This implies that
It is also given that:
Some teachers are rich.
This implies that
Now,
Thus, it logically follows that:
Some intelligent people are rich.
Hence, the given argument is valid.
Conclusion:
It has been determined that the given argument is valid.
(d.)
The validity of the given argument.
It has been determined that the given argument is not valid.
Given:
If a student is a freshman, then she takes mathematics.
Jane is a sophomore.
Therefore, Jane does not take mathematics.
Concept used:
According to Chain rule, if
According to Modus Ponens, if
According to Modus Tollens, if
Calculation:
Let
It is given that “If a student is a freshman, then she takes mathematics”.
Then,
It is also given that “Jane is a sophomore”.
Then, for Jane,
Now, for Jane
However, it does not logically follow that
Thus, it does not logically follow that “Jane does not take mathematics”.
Hence, the given argument is not valid.
Conclusion:
It has been determined that the given argument is not valid.
Chapter B.2 Solutions
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