# Need guidance solving this particular propositional logic question

I've been trying to get my head around this logic question posed on an exam I took some weeks ago, but wasn't able to figure it out by myself, so I've come here for some help.

The question reads as follows:

Below you'll find two distinct arguments and their conclusions have been omitted:

Arg #1:

If the car is not blue or the ball is green, then the building is not high.

The building is high, and the water is not cold

Conclusion: XXXX

Arg #2:

If the ball is not green or the water is cold, then the car is neither blue nor green.

The car is green, and the ball is not red.

Conclusion: XXXX

Which of the following conclusions would turn both arguments simultaneously valid?

a) The car is blue, and the water is not cold

b) The car is green, and the water is not cold

c) The ball is red, or the water is not cold

d) The building is high, the car is green, and the ball is red

e) The building is high, the car is not blue, and the ball is not red

So, apparently, the correct answer is C, but as much as I try to I just can not find my way around to get there. I really need some explanation for this.

Any help would be much appreciated.

## 2 Answers

From the first argument, we may infer that

• the building is high (2nd line)

• hence, by contradiction, the car is blue and the ball is not green (1st line and 2nd line combined)
• the water is not cold

from the second argument, we conclude that

• the car is green (1st line)
• hence, by contradiction, the ball must be green and the water not cold (2nd line)
• the ball is not red

Now note that the two arguments are in fact contradictory (e.g. the colour of the car is different), so the task is to find conclusions that follow from both arguments. This is the case for c), because it suffices (by definition of the logical "or") that one of the two statements is valid, and the water is never cold.

HINT

We can show that (c) is correct by proving:

• If the premises of Arg #1 are true, then (c) is true
• If the premises of Arg #2 are true, then (c) is true

We can show that (a) is incorrect by proving:

• If the premises of Arg #2 are true then (a) is false.

Assume that if a car is green then it is not blue.

Similarly, you can show that the other answers are incorrect.