# Are these two statements equivalent?

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?

http://stackoverflow.com/questions/10075846/are-these-2-statments-equivalent

## Update

(Below is my latest thought on the shuttle to my company this morning.)

As a normal human being, we come to the following 2 conclustions without doubt.

"A unless B" implies that:

1. if not B, then A : ¬B → A
2. if B then not A : B → ¬A

Though these 2 implications are acceptable to a human, they are not consistent with each other as logic is concerned. Because they are logic inverse of each other. And logoic inverse leads to different truth table.

Though we cannot tolerate ambiguity in math/logic, we shouldn't live with only one of the the 2 implications. Because either of them cannot hold the complete meaning of the original statement without the other.

I think we should translate the A unless B into:

(¬B → A)^(B → ¬A)

that is:

A ↔ ¬B (A is equivalent with ¬B)

-
$B$ unless not $A$ only indicates that $B$ will always be true when $A$ is. If $A$ is false, all you know is that it is possible for $B$ to be also false, not that it definitely has to be. So you should still have a T in the third row of your second truth table. –  Matt Pressland Apr 9 '12 at 16:20
I have a feeling that there should be some "order" here. Not sure yet, I will take a look at cr.yp.to/2005-261/bender1/Lo.pdf –  smwikipedia Apr 10 '12 at 5:18
There's several pages about this construction and translations in general here: math.stackexchange.com/questions/121388/… –  Rachel Apr 10 '12 at 5:48
Many thanks Rachel. I will read that in detail. –  smwikipedia Apr 10 '12 at 9:33
I think the correct conclusion is Never use "unless...not" in a mathematical context. –  JeffE Apr 11 '12 at 7:57

You're interpreting "$B$ unless $A$" as "$B$ if and only if not $A$", whereas in a mathematical context it usually means "$B$ if not $A$". A mathematical statement of this form in a book on discrete mathematics certainly means "$B$ if not $A$".
@smwikipedia: We can argue all day about which meaning is more intuitive or more acceptable, but I don't see what good will come out of that. Let's just say it is better to be have a way to make a weaker statement ("$B$ if not $A$") and optionally strengthen it afterwards, than to only have a way to make the stronger statement ("$B$ if and only if not $A$") and be unable to express the weaker one. –  Rahul Apr 11 '12 at 8:13