Question

Conditional proposition 1: If it is sunny, then I’ll go.

Conditional proposition 2: I will go unless it is not sunny.

Let’s decompose them as simple propositions.

A: It is sunny.

B: I will go.

Thus re-write the previous 2 conditional propositions:

1: If A, then B

2: B, unless not A

In my opinion, the truth table for each of them are:

1:

A--------B--------Proposition 1

T--------T-------------T

T--------F-------------F

F--------T-------------T

F--------F-------------T

2:

A--------B--------Proposition 2

T--------T-------------T

T--------F-------------F

F--------T-------------F   <---- here is the difference.

F--------F-------------T

So I think these 2 statements are not equivalent, but the famous Discrete Mathematics and its Applications by Kenneth H. Rosen indicates that they are equivalent.

Could someone shed some light on this?

Another post is made here:

https://math.stackexchange.com/questions/129691/are-these-two-statements-equivalent

Answer 1

I think the issue is the word “unless.” Unless is really describing when something is not true.

Conditional proposition 1: If it is sunny, then I’ll go.

Conditional proposition 2: I will go unless it is not sunny. I.E. If it is not sunny, I will not go.

1: If A, then B

2: If Not B, then Not A

A ⇒ B is the same as ¬B ⇒ ¬A. I can’t remember the exact name of the law, but it’s easy to derive. Use the implication law to convert it to ¬A ∨ B and B ∨ ¬A and the commutative law will change B ∨ ¬A to ¬A ∨ B