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://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:
- if not B, then A : ¬B → A
- 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)
Best Answer
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 https://english.stackexchange.com/, and I'll say no more about it here.