Lesson 2.4 — Translating Ambiguous English

English is sloppy. Logic is precise. The translation skill is everything.

The Setup

English is sloppy. The same sentence can mean different things depending on context, tone, who's saying it, and what came before. Logic doesn't allow that sloppiness — every sentence must mean exactly one thing.

This lesson is about spotting the sloppiness and pinning down the precise version. Four common ambiguity patterns, then a drill.

1. The Inclusive vs. Exclusive OR

"You can have ice cream or cake."

Does the speaker mean:

• Inclusive OR: at least one — maybe both?
• Exclusive OR: exactly one — not both?

You have to ask, or guess from context. In a fancy restaurant menu, probably exclusive. From a generous grandparent, probably inclusive.

In logic, OR defaults to inclusive. If we mean exclusive, we say so explicitly — usually by spelling it out as (X OR Y) AND NOT (X AND Y).

2. The Hidden NOT (De Morgan in disguise)

"I don't like vanilla or chocolate."

Two readings:

• (NOT vanilla) OR (NOT chocolate) — at least one of them I don't like.
• NOT (vanilla OR chocolate) — neither of them I like.

Most English speakers mean the second: "I dislike both." Notice the De Morgan flip from Lesson 2.2 — NOT (vanilla OR chocolate) equals (NOT vanilla) AND (NOT chocolate). The first reading and the second reading aren't even logically equivalent; they describe different states of the world.

So when someone says "I don't like X or Y," they almost always mean: NOT X AND NOT Y. The everyday-English "or" is doing AND's work after a NOT moves in front of it.

3. The IF That's Really an IFF

"You can stay up late if you finish your homework."

Two readings:

• One-way IF: finishing homework guarantees staying up late. Other paths to staying up late might also exist.
• Two-way IFF: staying up late happens only if homework is finished. No homework = no late night.

Most parents mean the second. Logic's "IF" is one-way, but everyday "if" usually carries the silent "only" with it. The IF chapter (Lesson 1.4) covered the IFF form in detail; the point here is just that you need to listen for it.

4. The Cause-vs-Truth-Relationship Slip

"If it's raining, the street is wet."

Two readings:

• Truth-relationship: when raining is TRUE, wet is also TRUE. (Logic's IF.)
• Causal claim: raining causes the street to be wet.

Logic only cares about the truth-relationship. Real life cares about causes. They line up most of the time, but not always — sprinklers also make streets wet, so the "if raining, then wet" relationship holds via correlation even when the rain isn't the cause.

When you translate to logic, drop the cause-talk. Logic doesn't have "because." It only has "in every world where the first part is true, the second part must also be true."

Translation Drill

Pick the best logical translation for each English sentence. Click any option to see why.

Tricky Bits

1. "Either / or" doesn't always mean exclusive. Context decides. "Either we leave now or we miss the train" is probably XOR (you can't both leave and miss). "Either way is fine" is inclusive. Listen carefully.

2. Triple-negative English is a trap.

"It's not the case that I don't dislike that."

That's three NOTs. Pair them off: NOT (NOT (NOT dislike)) = NOT dislike. The speaker dislikes it. English makes you work for this; logic does the cancellation cleanly.

3. "Unless" is "IF NOT."

"You can't have dessert unless you eat your veggies."

= "If NOT veggies, then NOT dessert" = (NOT veggies) → (NOT dessert). Equivalent to "veggies IFF dessert" if the speaker also means the reverse (which they probably do). Logic loves "unless" because it's just a sneaky IF NOT.

Practice

Translate each English sentence into precise logic. Watch for the four patterns.

1. "I'll have soup or salad — pick one, not both."
2. "You can't go out unless your room is clean."
3. "I don't drink coffee or tea." (Means: neither.)
4. "If you study, you'll pass." (Most teachers mean: only if you study, you'll pass.)
5. "It's raining, so the streets are wet."

Answers:

1. (soup OR salad) AND NOT (soup AND salad) — exclusive OR.
2. (NOT clean_room) → (NOT go_out), or equivalently go_out → clean_room. Most parents also mean the reverse, so go_out IFF clean_room.
3. (NOT coffee) AND (NOT tea) — De Morgan.
4. study IFF pass — two-way IF, where the teacher means studying is required.
5. The "so" is causal English, but the logical content is just raining → wet. Logic doesn't say why; it only says when one is true, the other follows.

Recap

• English is ambiguous; logic is precise. The translation skill is what makes logic useful in the real world.
• Watch for inclusive vs. exclusive OR — logic defaults to inclusive.
• Watch for the hidden NOT — "I don't like X or Y" usually means neither (De Morgan).
• Watch for IF that's really IFF — most everyday "if" carries a silent "only if."
• Drop the cause-talk when translating. Logic doesn't have "because"; it only has truth-relationships.
• "Unless" = "IF NOT."