How can middle school students intuit 'if not" = "except if'?

We use the conjunction unless to mean ‘except if’.

Unless means 'except if'

"Unless" cannot be used to replace "because ... not". It is logically equivalent to "if ... not".

They're pretty much the same thing. Unless is more common and combines condition with exception (except + if)

Dan Christensen proved "A is true except if B is true" ≡ A ⊻ B (Exclusive OR).. But rule out propositional logic proofs and truth tables, too complicated for middle school!

How do I teach middle school students that "if not" = "except if"? What's the simplest, most intuitive explanation?

†Hurley, Patrick. A Concise Introduction to Logic, 14 edn, 2023, p 221.

• If your students cannot handle truth tables, you might instead introduce them to conditional (IF-THEN-ELSE) statements in a simple programming language, e.g. Python or Ruby. Commented Aug 7 at 16:01

I would start out with a simple concrete example that clearly demonstrates the intuition:

"If you do not show up to class, you will be marked absent"

"Unless you show up to class, you will be marked absent."

If you want to use the specific phrase "except if" instead of the word "unless," then I would still start out with the example above. Once everyone accepts that the two sentences above mean the same thing, you can present the following:

"You will be marked absent if you do not show up to class"

"You will be marked absent unless you show up to class"

And finally, the following:

"You will be marked absent if you do not show up to class"

"You will be marked absent except if you show up to class"

• With the usual interpretation of "if", all but the last of these example sentences amount to the inclusive disjunction of "you will be marked absent" and "you show up for class. They allow the possibility that both are true (you show up for class, sleep through the whole period, and are marked absent). But the question, citing an answer by @Dan Christenden, says exclusive disjunction. I think "except" admits both interpretations; the English language strikes again. Commented Aug 7 at 18:41

Example: You will race unless ("except if") it snows tomorrow.

Results:

• If it doesn't snow tomorrow, you will race; (if not = it happens)

• If it does snow tomorrow, you might race and you might not. (If = ?)

If no snow, to the race you go.

But if it snows, who knows?

[I'm not allowed to comment on other answers, but it leaves out the weird possibility: even if you show up to class, it's possible to be marked absent based on the rules offered. That's why I prefer showing something that includes this other possibility.]

• If I was told, "you will race unless it snows tomorrow", then I would NOT believe that "if it does snow tomorrow, you might race and you might not".
– None
Commented Aug 7 at 12:54
• Then it's not "except if." It's "If and only if"/IFF (or the negation). Which is more specific and different than "except if." But you're right, this isn't the most intuitive example then. But the other happens to be logically inequivalent to "except if" or "unless". Commented Aug 7 at 23:09
• @None I agree with you. this answer baffles me. Commented Aug 16 at 14:56
• "I'm not allowed to comment on other answers," why not? you have enough reputation now. Commented Aug 16 at 14:56
• @user16249: They didn't when they wrote that.
– None
Commented Aug 17 at 16:09