# 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)

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$B$ 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. – Matthew 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 $A$".

Under this interpretation, if one were being mathematically pedantic, one would have to interpret the statement, "I will go unless it is not sunny" as, "I will go. Unless it is not sunny -- in which case, who can say?"

I actually think this is valid in a wider nonmathematical context, too, but that's a question for http://english.stackexchange.com/, and I'll say no more about it here.

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This is a case where the translation from logic to standard English is flawed. Special notations exist to avoid this kind of confusion. Travel Agent: I will charge you the full price, unless you cancel at least a week in advance. You (later): I'm not able to go. Give me my refund. Travel Agent: What refund? My statement only specified what would happen if you did not cancel a week in advance. You: What?! TA: You should have interpreted what I said as "I will charge you full price. Unless you cancel a week in advance -- in which case, who can say?" You: Go to hell. – Hank Apr 9 '12 at 17:06
Honestly, the discrepancy between logical operators and English conjunctions is as old as "Your money or your life!" – Rahul Apr 9 '12 at 17:26
"Unless it is not sunny -- in which case, who can say?" , it is to say "in case it is not sunny, no one can tell whether I will go or not." But I believe I feel the implication that "in case it is not sunny, I will definitely NOT go." If I will go is my money, it is not sunny will be my life. – smwikipedia Apr 11 '12 at 6:03
@smwikipedia: I've added the words "in a mathematical context" to my answer. Does that help? The correspondence between mathematical terms (including words like or, if, unless) and their usual English meaning is not exact. – Rahul Apr 11 '12 at 6:20
@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