Lesson 8.3 — Some

The Word

"Some" is the lazy cousin of "all." It says less, but it says enough:

"Some kids play piano."
"There's a quiet dog on this block."
"At least one of my socks has a hole."

These all mean the same thing: at least one. Some, there is, at least one, exists — different words, same idea. You don't have to say which one, or how many. Just that there's a match.

Where ALL was an AND stretched across a group, SOME is an OR stretched across a group. From Module 1: an OR is true when AT LEAST ONE half is true. SOME just scales that up.

The Truth Test

"Some kids play piano" is TRUE when at least one kid plays piano. It's FALSE only when no kid plays — when the property holds for nobody in the group.

Try it. Click each kid to flip whether they play piano. Watch the claim's truth value.

How Logic Writes It

The symbol for "some" or "there exists" is ∃ — a flipped capital E. It's called the existential quantifier.

"Some kids play piano" becomes:

∃x (Kid(x) AND PlaysPiano(x))

Read it as: "There exists some thing x such that x is a kid AND x plays piano." We use AND inside, not IF — we're saying both parts hold for the same x.

Compare with ALL from the last lesson, which used IF inside. This contrast — AND for SOME, IF for ALL — is one of the most-confused parts of formal predicate logic. We won't lean on it heavily, but it's the reason ALL and SOME behave so differently under negation (next lesson).

How Few Counts as "Some"?

One is enough. In logic, "some" doesn't tell you how many — just at least one. "Some kids play piano" is TRUE even if exactly one kid plays. It's also TRUE if half play, or all play.

That last bit catches people. If ALL kids play piano, then SOME kids play piano is ALSO true — because ALL includes SOME. The reverse isn't true: SOME doesn't imply ALL.

• ALL → SOME (if every kid plays, then at least one kid plays — automatically)
• SOME does NOT imply ALL (if one kid plays, the others might not)

A Real Example

"There's a quiet dog on this block."

This is just SOME in a different costume. The claim is TRUE if there's at least one quiet dog on the block. One is enough. You don't have to count them; you just have to find one.

This is how detective work, science, and lawyer-talk uses SOME all day long: you don't need to know everyone — you just need one example, one witness, one specimen.

Tricky Bits

1. English "some" sometimes implies "not all." "Some of my friends came" usually means "some but not all of them came." In strict logic, "some" includes the case where all of them came. The English implication is a politeness thing, not a logic thing.

2. SOME of nothing is FALSE. "Some unicorns can fly" is FALSE in the real world — because there are no unicorns at all, there can't be any flying ones. Compare with the previous lesson: ALL of nothing is TRUE; SOME of nothing is FALSE. They flip on the empty case.

3. "At least one" is the cleanest way to read SOME. If "some" feels fuzzy, mentally swap in "at least one." That's exactly what logic SOME means.

Recap

• SOME means AT LEAST ONE member of the group has the property.
• One match is enough to make a SOME claim TRUE.
• SOME is FALSE only when nobody has the property.
• Logic symbol: ∃ (the existential quantifier).
• SOME is like an OR stretched across a whole group.
• ALL → SOME automatically, but not the other way around.