Sorry, I didn't post my argument the last time. Write two arguments in English, one in the form of modus ponens and one in the form of modus tollens. Then, write the arguments in symbols using sentence letters and truth-functional connectives.

Question

Sorry, I didn't post my argument the last time. Write two arguments in English, one in the form of modus ponens and one in the form of modus tollens. Then, write the arguments in symbols using sentence letters and truth-functional connectives.

The following is one of the arguments and I believe it can be used for both modus ponens and modus tollens depending on how I word it but I don't know how to write it in symbols using sentence letters and truth-functional connectives.

"I have a sump pump in my basement. When it rains the sump pump cuts on and stops
my basement from flooding. It’s raining today but my basement won’t flood
because I have a sump pump."

Answer 1

Modus Ponens is Latin for "the mode that affirms".

P: If you have a new key to your home, then you can open the doors.
P: You have a new key to your home
Therefore:
C: You can open the doors.
P→Q
P ∴ Q

Modus Tollens is Latin for "the mode that denies".
P: You can't open the front door.
P: If you have a new key to your home, then you can open the doors.
Therefore:
C: You don't have a new key.
¬Q
P→Q
∴ ¬P

Answer 2

Modus ponens is a valid argument form that goes as follows:

1. If A, then B.
2. A.
3. Therefore, B.

So, in this case, we can represent the modus ponens argument as follows:

Let A represent "It is raining today."
Let B represent "My basement won't flood because I have a sump pump."

1. If A, then B. (If it is raining today, then my basement won't flood because I have a sump pump.)
2. A. (It is raining today.)
3. Therefore, B. (Therefore, my basement won't flood because I have a sump pump.)

Now, let's represent the modus tollens argument, which is another valid argument form structured as follows:

1. If A, then B.
2. Not B.
3. Therefore, Not A.

In symbols, we can represent the modus tollens argument as:

Let A represent "It is raining today."
Let B represent "My basement won't flood because I have a sump pump."

1. If A, then B. (If it is raining today, then my basement won't flood because I have a sump pump.)
2. Not B. (My basement will flood.)
3. Therefore, Not A. (Therefore, it is not raining today.)

Note that for modus tollens to be valid, both the conditional statement and the negation of the consequent must be true. So, if the basement does flood, it can be concluded that it is not raining today.