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Propositional Logic · step 6 of 13

Translating English into propositional logic

Translation questions test whether you can map ordinary English onto the connectives without being misled by word order. The hard cases are a small, fixed list: "only if" is the arrow pointing the other way from "if", "unless" behaves like "or", "necessary" and "sufficient" swap the antecedent and consequent, and "neither … nor" is a negated disjunction. Learn these six patterns and translation questions become mechanical.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

  1. Define the atoms explicitlyWrite a dictionary: let p = "it rains", q = "the match is cancelled". Every exam scheme awards marks for this line alone.
  2. Find the main connectiveIdentify the sentence’s top-level structure before worrying about the parts — the comma and the word order usually reveal it.
  3. Apply the standard patterns"A if B" is B → A. "A only if B" is A → B. "A unless B" is ¬B → A, equivalently A ∨ B. "B is necessary for A" is A → B. "B is sufficient for A" is B → A.
  4. Sanity-check with a false casePick the situation where the English is clearly false and confirm your formula is false there too.

Worked example

Translate: "The alarm sounds only if the door is open, and it does not sound unless the power is on."

  1. Let a = "the alarm sounds", d = "the door is open", w = "the power is on".
  2. "a only if d" gives a → d.
  3. "not a unless w" means: if the power is not on, the alarm does not sound, i.e. ¬w → ¬a, equivalently a → w.
  4. Join the two clauses with ∧.

Answer. (a → d) ∧ (a → w), which can also be written a → (d ∧ w).

Where marks get dropped

These are the specific errors that cost credit on translating english into propositional logic questions — QED's rubric penalises each of them separately.

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Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

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Translating English into propositional logic — frequently asked questions

Is "unless" the same as "or"?

In classical logic, yes: "A unless B" is standardly rendered ¬B → A, which is equivalent to A ∨ B. Some textbooks treat it as a biconditional, so follow the convention your course states.

How do I handle "necessary and sufficient"?

That is the biconditional. "B is necessary and sufficient for A" translates to A ↔ B, combining A → B and B → A.

Does the choice of letters matter?

Not mathematically, but the dictionary must be stated and used consistently. Markers deduct for a formula whose atoms are never defined.

The rest of Propositional Logic

Connectives, truth tables, equivalences and normal forms. Each subtopic below has its own method, worked example and mark-losing traps.

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