Lesson 8.2 — All

The Word

You use it constantly:

"All dogs bark."
"All my friends came to the party."
"Everyone in this class did the homework."

"All" and "every" and "everyone" make a universal claim. They reach across a whole group and say: every single one has the property.

From Module 1 we know an AND sentence is true only when BOTH parts are true. ALL is like an AND stretched out across a whole group — every member has to check out, not just two.

The Truth Test

"All dogs bark" is TRUE when every dog in the group barks. It's FALSE the moment one dog doesn't bark. Even one exception breaks an ALL claim.

Try it. Click each dog to flip whether it barks. Watch the claim's truth value.

How Logic Writes It

The symbol for "all" or "for every" is ∀ — a flipped capital A. It's called the universal quantifier.

"All dogs bark" becomes:

∀x (Dog(x) → Barks(x))

Read it as: "For every thing x, IF x is a dog, THEN x barks." The x is a placeholder — it stands for "any object you pick." The IF restricts the claim to dogs only.

For clarity, we'll usually just say "for all dogs, Barks(dog)" — but underneath, the formal version is the IF-form above.

Why One Exception Kills It

From Module 1, we know AND is picky — one false half ruins the whole sentence. ALL is the same idea on a bigger scale. Imagine spelling it out for our four-dog group:

Barks(Spot) AND Barks(Rex) AND Barks(Buddy) AND Barks(Lucky)

That's an AND of four claims. Module 1 told us: ONE false claim makes the whole AND false. So ONE quiet dog makes "All dogs bark" false.

ALL is just AND across a whole group, however big.

A Real Example

"All my friends came to the party."

For this to be TRUE, every friend came. If you have ten friends and nine came but Lisa didn't — the claim is FALSE. Lisa is the counterexample. One missing person is enough to break the universal.

This matters for promises and rules. "All students must wear a badge" is broken the moment ONE student walks in without a badge. The rule doesn't say "most students" — it says ALL.

Tricky Bits

1. "All" in English sometimes means "most." "All my friends love that movie" might really mean "most of them." Logic ALL is strict: every single one, no exceptions. Be careful with how loose the everyday "all" is.

2. ALL of nothing is TRUE (the weird case). "All unicorns can fly" is TRUE in the real world — because there are no unicorns to check, there are no counterexamples. This is called vacuous truth. It feels odd but matches the rule: the claim is only FALSE if you find a non-flying unicorn, which you can't, because there aren't any.

3. ALL is about the WHOLE group, not just the ones you see. "All swans are white" was thought true for centuries in Europe — until explorers found black swans in Australia. The claim was always FALSE; people just hadn't seen the counterexample yet.

Recap

• ALL means EVERY member of the group has the property.
• One counterexample makes an ALL claim FALSE.
• Logic symbol: ∀ (the universal quantifier).
• ALL is like an AND stretched across a whole group.
• If the group is empty, ALL is technically TRUE (vacuous).