Lesson 9.3 — Possible

The Word

"Could" and "might" are the modal words for keeping options open:

"It MIGHT rain later."
"She COULD be at the library."
"There MIGHT be one cookie left in the jar."

A possibility claim says: there's at least one way this is true. The claim isn't committed — it doesn't say "yes, it's true" or "no, it's false." It just says: this is on the menu.

The Truth Test

Imagine all the ways things could be — every possible situation. A possibility claim is TRUE when the claim holds in at least one of those situations. It's FALSE only when no situation works.

Click each situation to flip whether it has rain. Watch how "It could be raining" tracks.

The Quantifier Parallel

This should also feel familiar. From Lesson 8.3:

• SOME needs the property to hold in at least one member of a group.
• POSSIBLE (◇) needs the property to hold in at least one possible situation.

Same OR-across-a-domain logic. One match is enough. SOME ranges over things; POSSIBLE ranges over situations.

How Few Counts as "Possible"?

One situation is enough. If you can think of even ONE way the claim could be true, the possibility holds. You don't need to know which way is the real one.

And — just like SOME implies nothing about ALL — possibility implies nothing about necessity. "It could rain" doesn't mean it will. The claim is open, not committed.

• NECESSARY → POSSIBLE (if it MUST be raining in every situation, then it CERTAINLY could be raining in at least one)
• POSSIBLE does NOT imply NECESSARY (just because it could rain doesn't mean it has to)

Real Uses

Hypotheses. "She might be at the library, or at her friend's house, or stuck in traffic." Each "might" is a possibility — at least one situation where she's there. We're listing live options.

Forecasts. "It could storm later." There are situations where it does — even if there are also situations where it doesn't.

Permission and ability. "You can sit anywhere" means it's possible for you to sit anywhere — every option is on the menu. Possibility shows up in legal and social rules constantly.

Disproving necessity. If someone claims "the killer MUST have come through the window," you challenge it by showing the killer COULD have come through the back door. One alternative possibility kills the necessity claim.

How Logic Writes It

The diamond: ◇. Read it as "possibly" or "might" or "could."

◇Raining — it could be raining; in at least one possible situation, it's raining.

◇(door = locked) — the door might be locked; among the situations we're considering, at least one has it locked.

Tricky Bits

1. "Could" doesn't tell you the odds. "It could rain" is TRUE whether the chance is 1% or 99%. Possibility is binary: yes-it's-on-the-menu or no-it's-not. For finer-grained guessing, you need probability — a different tool.

2. Possibility is FALSE only when no situation works. "Unicorns might exist" is TRUE if you grant any possible situation with unicorns (even a fictional one). FALSE if you restrict to situations where physics forbids them. The truth of possibility claims depends on what counts as a "relevant" situation.

3. "Can" mixes possibility and ability. "I can swim" is ability (a skill). "It can rain in July" is possibility (sometimes does). English uses "can" for both; logic separates them.

Recap

• POSSIBLE (◇) means TRUE IN AT LEAST ONE POSSIBLE SITUATION.
• One example-situation makes a possibility claim TRUE.
• Logic symbol: ◇.
• Possibility is like SOME, but over situations instead of group members.
• Necessary → Possible. Possible does NOT imply Necessary.