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I am particularly confused at the the difference(s) when translating conditional statements into english.

For instance the statements below from a question I found online.

I will go to the movie if I complete my assignment.
I will go to the movie only if I complete my assignment.
I will not go to the movie if I do not complete my assignment.

I understand the differences between the first two in simple english, but how can I express this in propositional logic using the conditional (or bi-conditional) arrows?

djkasn
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  • You can use $\implies$ in one of them and use $\impliedby$ in the other if you want to keep the order of the conditions the same. In the third, consider also using $\neg$'s. – JMoravitz Oct 02 '20 at 16:12
  • What does your attempt look like, and which part are you unsure about? – Natalie Clarius Oct 02 '20 at 16:18

1 Answers1

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Define M = I go to the movie A = I complete my assignment

I will go to the movie if I complete my assignment.

We can write this sentence as "if I complete my assignment, i will go to the movie" This can be translated using an implication. A → M This statement indicates that every time condition M is true, A will always be true.

I will go to the movie only if I complete my assignment.

Logically, the meaning of "only if" is different than "if". This is important to learn when making translations. In this statement "only if" indicates that the if I went to the movie, I must have completed my assignment. If the assignment is not complete then it is impossible that i go to the movie. This means the following sentence is true "If I go to the movie, I completed my assignment." We can translate this as M → A

This might seem confusing at first but it's just a very technical way of understanding english. If you ever see "A only if B" or "Only if B, A", you can translate it as A → B.

Just to further understand this concept I'll give you more depth to the explanation. Let's say Bob wants to rob a bank and get away with it. Bob knows there is a security camera system, and being seen on the camera would ensure he cannot get away with it. He needs to disable the cameras if he wants to successfully get away with the crime. In this case I could say "Bob can successfully rob the bank only if he disables the security cameras" If bob disables the cameras, he may or may not succeed in robbing the bank. But if Bob does succeed in robbing the bank, we know he must have disabled the security camera. This can be translated to

(successfully robbing bank) → (disabling security camera)

NOT (disabling security camera) → (successfully robbing bank)

I will not go to the movie if I do not complete my assignment.

Rewrite this as "If I do not complete my assignment, I will not go to the movie."

-A -> -M

(this is logically equivalent to A → B, you will learn that later)

nokia2008
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