Use the laws of logic to prove the conclusion from the hypotheses. Give proposit

Question

Use the laws of logic to prove the conclusion from the hypotheses. Give propositions and predicate variable names in your proof. The hypotheses are:

If I drive on the freeway, I will see the fire.
I will either drive on the freeway or take surface streets.
I am not going to take surface streets.

Conclude that I will see the fire.

part 2

Which of the following arguments are valid? Explain your reasoning.

I have a student in my class who is getting an A. Therefore, John, a student in my class is getting an A.
Every girl scouts who sells at least 50 boxes of cookies will get a prize. Suzy, a girl scout, got a prize. Therefore Suzy sold 50 boxes of cookies.

Explanation / Answer

1. Let A: I will drvie on the freeway B: I will see the fire C: I will take surface streets

The given statements can be translated as:

If I drive on the freeway, I will see the fire : A -> B

I will either drive on the freeway or take surface streets : A v C

I am not going to take surface streets : ~C

The three statements can be combined as

(A -> B) ^ (A v C) ^ ~C

= (A -> B) ^ (~A -> C) ^ ~C (Law of material implication)

= (A -> B) ^ A (Modus tollens)

= B (Modus ponens)

Hence the conclusion is I will see the fire.

2. Let s: Student of my class and A(x): Getting an A. Let j: John

I have a student in my class who is getting an A : s, A(s)

John need not be the student i.e j s could be true

=> ~A(j) could be true and the given sentence is invalid.

Let G: girl scout. C: Selling 50 boxes of cookies. P: Getting a prize.

s: Suzy

Every girl scouts who sells at least 50 boxes of cookies will get a prize: x G, C(x) -> P(x)

Suzy, a girl scout, got a prize : s G, P(s)

Since P(s) is true, C(s) need not be true. Reason: false -> true is also true.

Therefore the statement is invalid.

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