Question

“If you can press a button to get $1M and a random person dies somewhere in the world would you press the button?”

A = press button

B = get $1M

C = random person dies

Here is what I think it should be:

If A, then B AND c

According to the original statement is it:

(If A, then B) AND C 

or

If A, then (B AND C)

Answer 1

You’ve correctly identified the three propositional variables:

• P₁(x): “x presses a button.”
• P₂(x): “x receives one million dollars.”
• P₃(x): “x causes the death of a random person.”

You want to express the sentence Q: “if someone presses the button, then they receive a million dollars and a person dies.” At first glance, it seems like P₁(x) ⇒ P₂(x) ∧ P₃(x) correctly expresses this. How can we be sure? Let’s draw a truth table:

 P1   P2   P3   P2 ^ P3   P1 --> P2 ^ P3
---- ---- ---- --------- ----------------
 T    T    T       T            T
 T    T    F       F            F
 T    F    T       F            F
 T    F    F       F            F
 F    T    T       T            T
 F    T    F       F            T
 F    F    T       F            T
 F    F    F       F            T

Notice that “you receive a million dollars and cause a death” is true only when both of the constituent parts are true. This makes sense; if both parts don’t come true, the whole is not also true.

Notice also the truth values for the entire statement Q: it’s false whenever the second part is false and the first part is true. This makes sense: if you press the button but either (1) the million dollars doesn’t appear or (2) nobody dies, the prediction implied by Q is not true. So our assertion is correct.