Translating Natural Language

∴Why This Matters

The hardest part of logic isn't the symbols—it's figuring out what someone actually means. Natural language is ambiguous, vague, and full of traps. Translation is where logic meets the real world.

This module teaches you to convert everyday English into precise logical notation—and to recognize when a statement can be interpreted multiple ways.

Basic Patterns

English	Logic	Notes
P and Q	P ∧ Q	Both required
P or Q	P ∨ Q	Inclusive or
not P	¬P	Negation
if P then Q	P → Q	Implication
P if and only if Q	P ↔ Q	Biconditional

Tricky Patterns

"Only if"

A only if B

Translates to A → B. "I'll go only if you come" means if I go, you must come. My going requires your coming.

!Common Mistake

Don't confuse "A only if B" with "A if B". They have opposite arrow directions!
• "A if B" = B → A (B is sufficient for A)
• "A only if B" = A → B (B is necessary for A)

"Unless"

A unless B

Translates to ¬B → A or equivalently ¬A → B. "Unless" means "if not".

"You can't enter unless you pay" = If you don't pay, you can't enter = ¬P → ¬E

"Neither...nor"

Neither A nor B

Translates to ¬A ∧ ¬B. Both are false. Equivalent to ¬(A ∨ B) by De Morgan's Law.

?Quick Check

Translate: 'You will pass only if you study and attend class.'

Ambiguity

Some sentences have multiple valid translations. Consider: "You can have cake or ice cream."

Inclusive: C ∨ I — at least one (maybe both)
Exclusive: (C ∨ I) ∧ ¬(C ∧ I) — exactly one

✦Tip

When in doubt, ask for clarification. In formal contexts, assume inclusive OR unless explicitly stated otherwise.