Chapter B.2, Problem 14E

Solution

(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 $p\to q$ and $q\to r$ are true, then $p\to r$ is true.

According to Modus Ponens, if $p\to q$ and $p$ are true, then $q$ is true.

According to Modus Tollens, if $p\to q$ is true and if $q$ is false, then $p$ is false.

Calculation:

Let $p$ denote “being a woman” and $q$ denote “being mortal”.

It is given that “All women are mortal”.

Then, $p\to q$ is true.

It is also given that “Hypatia was a woman”.

Then, for Hypatia, $p$ is true.

Now, for Hypatia, $p\to q$ and $p$ are both true.

Then, according to Modus Ponens, it follows that $q$ is true for Hypatia.

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 $p\to q$ and $q\to r$ are true, then $p\to r$ is true.

According to Modus Ponens, if $p\to q$ and $p$ are true, then $q$ is true.

According to Modus Tollens, if $p\to q$ is true and if $q$ is false, then $p$ is false.

Calculation:

Let $p$ denote “being a square”, $q$ denote “being a quadrilateral” and $r$ denote “being a polygon”.

It is given that “All squares are quadrilaterals”.

Then, $p\to q$ is true.

It is also given that “All quadrilaterals are polygons”.

Then, $q\to r$ is true.

Now, $p\to q$ and $q\to r$ are both true.

Then, according to Chain rule, it follows that $p\to r$ is true.

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 $p\to q$ and $q\to r$ are true, then $p\to r$ is true.

According to Modus Ponens, if $p\to q$ and $p$ are true, then $q$ is true.

According to Modus Tollens, if $p\to q$ is true and if $q$ is false, then $p$ is false.

Calculation:

Let $T$ denote the set of all teachers, $I$ denote the set of all intelligent people and $R$ denote the set of all rich people.

It is given that:

All teachers are intelligent.

This implies that $T\subset I$ .

It is also given that:

Some teachers are rich.

This implies that $T\cap R\ne \varnothing$ such that $T\not\subset R$ .

Now, $T\subset I$ and $T\cap R\ne \varnothing$ together implies that $I\cap R\ne \varnothing$ .

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 $p\to q$ and $q\to r$ are true, then $p\to r$ is true.

According to Modus Ponens, if $p\to q$ and $p$ are true, then $q$ is true.

According to Modus Tollens, if $p\to q$ is true and if $q$ is false, then $p$ is false.

Calculation:

Let $p$ denote “being a freshman” and $q$ denote “taking mathematics”.

It is given that “If a student is a freshman, then she takes mathematics”.

Then, $p\to q$ is true.

It is also given that “Jane is a sophomore”.

Then, for Jane, $p$ is false.

Now, for Jane $p\to q$ is true and $p$ is false.

However, it does not logically follow that $q$ is false for Jane.

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.