Lesson 9.1 — Must and Might

A Different Kind of Claim

Every claim we've worked with so far describes what IS:

"It is raining."
"Pedro is happy."
"All dogs bark."

These claims line up against the actual world. They're TRUE or FALSE based on what's really there.

But you've also used a whole other kind of claim your whole life — claims about what MUST be or what COULD be:

"I MUST eat my vegetables."
"It MIGHT rain tomorrow."
"She HAS TO be home by 9."
"We COULD go to the park."

These don't describe what is right now. They describe how the world is constrained — what's necessary, what's possible, what's allowed, what's required.

Words like must, might, could, has to, may, should, possibly, certainly are called modal words. They modify a claim with a strength of commitment.

Two Big Strengths

Modal words split into two main strengths:

• Necessity — must, has to, certainly, definitely, will surely. The claim is forced — no way around it.
• Possibility — might, could, may, possibly, perhaps. The claim is open — it could go either way.

"I MUST eat my vegetables" is a necessity claim — no choice. "It MIGHT rain" is a possibility claim — maybe it will, maybe it won't.

Click each modal word to see which strength it carries:

The Same Sentence at Three Strengths

Watch how the strength changes meaning. Same base claim, three modal frames:

• "It IS raining." — Plain claim. The world is one way: raining.
• "It MIGHT be raining." — Possibility. Could be raining, could not. We don't know.
• "It MUST be raining." — Necessity. Given what we know, it can't be anything else. The umbrella's wet, the puddles are forming — it has to be rain.

Three different commitments to the same claim. The plain "is" is what you SEE. "Might" is what's POSSIBLE. "Must" is what's FORCED.

How Logic Writes It

Modal logic uses two new symbols:

• □P — "necessarily P" or "must P." The box symbol means necessity.
• ◇P — "possibly P" or "might P." The diamond symbol means possibility.

So "It must be raining" becomes □Raining. "It might be raining" becomes ◇Raining.

The Quantifier Connection

If Module 8 felt useful, you've already got the framework for modal logic.

• Necessity (□) acts like ALL (∀) — but stretched across all possible situations instead of all members of a group. "Must P" = "in every possible situation, P holds."
• Possibility (◇) acts like SOME (∃) — but across possible situations. "Might P" = "in at least one possible situation, P holds."

Same machinery, different domain. Where Module 8 quantified over things (dogs, kids), Module 9 quantifies over ways things could be. The lessons that follow will lean on this parallel hard.

Real Uses

Detective talk: "The window is broken from the outside. The intruder MUST have come in this way." Necessity from evidence.

Weather: "It COULD storm later." Possibility based on what's in the forecast.

Rules: "You MUST wear a helmet." Necessity from a regulation.

Math: "If x is even, x MUST be divisible by 2." Necessity from a definition.

Modal claims show up everywhere there are rules, evidence, hypotheticals, or uncertainty.

Tricky Bits

1. "Have to" and "must" mean the same thing. English has many synonyms for necessity (must, has to, needs to, gotta, is required to) and possibility (might, could, may, maybe, perhaps). Logic just has two — □ and ◇ — and lumps the synonyms together.

2. "Should" is a softer necessity. "You should eat vegetables" is weaker than "you must eat vegetables." Logic doesn't usually distinguish degrees, so we'll treat both as necessity here — but in real life, "should" allows exceptions while "must" doesn't.

3. "May" is two-faced. "It may rain" is possibility (could happen). "You may leave" is permission (allowed). Same word, different meanings — context tells you which.

Recap

• Modal words don't describe what IS — they describe what MUST or COULD be.
• Two big categories: necessity (must, has to) and possibility (might, could).
• Logic symbols: □ for necessity, ◇ for possibility.
• Necessity is like ALL across possible situations; possibility is like SOME across possible situations.
• Modal claims unlock reasoning about evidence, rules, hypotheticals, and uncertainty.