Lesson 8.4 — NOT-all and NOT-some

The Move

What happens when you put NOT in front of an ALL or a SOME?

"Not all dogs bark."
"No dog is purple."

The first one is a NOT in front of an ALL. The second is a NOT in front of a SOME (notice: "no" is just a tighter way of saying "not some"). In both cases, the NOT moves through the quantifier — but it changes the quantifier as it goes.

This is the flip-and-swap rule:

• NOT all → some NOT. "Not all dogs bark" = "Some dog doesn't bark."
• NOT some → all NOT. "No dog is purple" = "All dogs are not purple."

The NOT slides past the quantifier, and the quantifier swaps to its opposite. ALL becomes SOME. SOME becomes ALL. The property inside gets negated.

Why It's Two Rules in One Trick

It's the same idea as Module 2's NOT-distributes (De Morgan's Law) — just for quantifiers instead of AND/OR. Remember:

• NOT (P AND Q) = NOT P OR NOT Q   (NOT distributes, AND flips to OR)
• NOT (P OR Q) = NOT P AND NOT Q   (NOT distributes, OR flips to AND)

From Lesson 8.2, ALL is just AND stretched across a group. From 8.3, SOME is OR stretched across a group. So when NOT moves through ALL or SOME, it does exactly what De Morgan said it would — distributes inside, and the quantifier flips.

See the Pairs

Click each kid to toggle whether they play soccer. Watch how the four claims relate.

The Common Misread

People constantly misread "not all" as "none." That's the biggest predicate-logic mistake in everyday English.

"Not all my friends came to the party."

This does NOT mean "none of them came." It means "at least one didn't come." Maybe nine out of ten came; maybe one out of ten came. Either way, "not all" is TRUE as soon as ONE is missing.

If you want to say nobody came, say "none of my friends came" — that's the NOT-some / no-one claim, which is way stronger.

How Logic Writes It

The flip-and-swap rule in symbols:

• ¬∀x P(x) ≡ ∃x ¬P(x)   "Not all x have property P" = "There exists x that doesn't have P."
• ¬∃x P(x) ≡ ∀x ¬P(x)   "No x has property P" = "Every x lacks P."

The ≡ symbol means "is equivalent to" — same truth value in every situation.

Real-World Use

This rule is how you ATTACK universal claims. To disprove "all swans are white," you don't have to inspect every swan in the world. You just have to find ONE non-white swan. That's the SOME-NOT form of NOT-ALL. Hunting counterexamples is the most efficient way to break a universal claim.

The same trick works in reverse: to disprove a "no" or "none" claim, you just need one example. "No dog is friendly" — disprove it with one friendly dog. That's the SOME form of NOT-NONE.

Tricky Bits

1. "All ... not" is dangerous in English. "All that glitters is not gold" is famously ambiguous — does it mean "no glittery thing is gold" (∀-NOT) or "not all glittery things are gold" (NOT-∀)? Shakespeare meant the second. English ALL-NOT can swing either way depending on emphasis.

2. "Any" can be confusing too. "Anyone can do it" sounds like a SOME claim (someone can) but actually means an ALL claim (everyone can). The word "any" is one of English's slipperier quantifiers.

3. Double negation cancels. "Not no dog barks" = "Some dog barks." Two NOTs cancel out, like minuses in algebra. If you ever find yourself nesting NOTs, flip and swap twice — you end up where you started.

Recap

• NOT ALL = SOME NOT. Not all dogs bark = some dog doesn't bark.
• NOT SOME = ALL NOT. No dog is purple = every dog is not purple.
• The flip-and-swap rule is De Morgan's Law for quantifiers.
• "Not all" does NOT mean "none." It means "at least one exception."
• To disprove an ALL claim, find ONE counterexample.