# Why is "$A$ unless $B$" equivalent to $A \lor B$?

$A$ unless $B$ surely means, 'given that $B$ does not happen, $A$ will happen'.

So if $B$ happens, $A$ does not happen.

Yet I've read, by those officially accredited, that $A$ unless $B$ = $A$ or $B$. However, if $B$ happens, and $A$ still happens despite the fact, that this is apparently true.

Surely, the sheffer stroke is appropriate here? So that: $A$|$B$.

If $B$ occurs, $A$ does not, thus, $A$ unless $B$. If $B$ does not occur, $A$ does.

The meaning of "unless" in English is a bit ambiguous. "You won't pass the exam unless you study". Does that imply that if you study you are guaranteed to pass the exam? I would say no.

• Which is exactly why we use quantifiers and not English (or another human language) to write math. May 18, 2016 at 11:30

Your first line was correct. $A$ unless $B$ means 'given that $B$ does not happen, $A$ will happen. Thus, $A$ unless $B$ means $\neg B\implies A$. But, we can put this implication into its disjunctive to get $\neg(\neg B)\vee A$, which, in turn, is equivalent to $B\vee A$.

Your mistake comes in your second line when you claim that $\neg B\implies A$ is equivalent to $B\implies \neg A$. This is not true. The general contrapositive rule you might be confusing your assumption with is this: $P\implies Q\equiv \neg Q\implies \neg P$.

• But if B occurs, then A would not occur right? May 17, 2016 at 16:57
• No, that is not true. I have updated my answer.
– ervx
May 17, 2016 at 16:58
• But, doesn't 'A unless B mean': 'A (will happen) unless (it is stopped from happening by the occurrence of) B?'. If in the event that B occurs which impedes upon the occurrence of A, then surely, the mere fact that B occurs thus should mean that A does not. May 17, 2016 at 17:11
• Oh by the way, I know that A unless B does not incite B -> ¬A, I just don't see why A and B can occur simultaneously. May 17, 2016 at 17:13
• I will eat lunch unless I eat breakfast. This means that if I didn't eat breakfast I must eat lunch. However, if I did eat breakfast, I am still able to eat lunch without falsifying the statement. Read this to learn more if you wish. alphascore.com/posts/x-unless-y-formal-logic-for-lsat-prep
– ervx
May 17, 2016 at 19:34

"$A$ unless $B$" is read by logicians as "$A$ is guaranteed to be true, unless $B$ is true," which is not a claim that $A$ is false when $B$ is true ~ just that $B$ is true when $A$ is false.   In other words "If not $B$, then $A$", or "If not $A$, then $B$". $$\neg B\to A \\ B\leftarrow \neg A \\ A\vee B$$

You are reading it as an exclusive: "$A$ is true except in cases where $B$ is true." $$A\leftrightarrow\neg B \\ A\oplus B$$

Now many uses of "unless" in native English sentences do have the added context that $B$ being true will cause $A$ to be not true.   However, not all uses do so.

So, although, we might be able to infer more from the context of the statement, when told that "it will be $A$ unless it is $B$", we will always at least know that it means "if it is not $A$ then it is $B$".

And if told that "if it is not $A$ then it is $B$" then we can infer that "it will be $A$ unless it is $B$".

Thus we hold the statements to be equivalent.

"A unless B" is an inclusion/exclusion type statement. You start with all possible truths. Exclude those inconsistent with $A$. Then add those consistent with $B$.

Start, all possible truths: $$\{A \land B,~ A\land \lnot B,~ \lnot A \land B,~ \lnot A \land \lnot B\}$$

Truths inconsistent with $A$: $$\{\lnot A \land B,~ \lnot A \land \lnot B\}$$

Truths consistent with $B$: $$\{A \land B,~ \lnot A \land B\}$$

All possible truths excluding those inconsistent with $A$:

$$\{A \land B,~ A\land \lnot B,~ \lnot A \land B,~ \lnot A \land \lnot B\} - \{\lnot A \land B,~ \lnot A \land \lnot B\} = \{A \land B,~ A \land \lnot B\}$$

Add back all truths consistent with $B$:

$$\{A \land B,~ A \land \lnot B\} \cup \{A \land B,~ \lnot A \land B\} = \{A \land B,~ A\land \lnot B,~ \lnot A \land B\}$$

Which is exactly the truths consistent with $A \lor B$.

In casual English, when you say "X unless Y", it doesn't strictly speaking mean that X fails when Y holds. However, some people will infer that anyway because "why else would you say it that way". It is not correct, but it is not uncommon either, so use the word "unless" at your own risk.

It is like saying, "you can't have desert if you don't eat your vegetables". It was never actually said that you could have desert if you do eat your vegetables. But most people will infer that, and be very upset when you correctly inform them that you never promised them desert. Unless they study math.

• Even if they study math, they'll still be upset if you don't give them dessert. May 31, 2016 at 9:59