Predicate Logic · step 2 of 13

Translating English with predicates

Predicate translation extends propositional translation with a domain, predicates like Student(x), and relations like Takes(x, y). The recurring difficulty is scope: "Everyone loves someone" and "Someone is loved by everyone" use the same three symbols in a different order and mean completely different things. Getting the quantifier order right is worth more marks than getting the predicates elegant.

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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.

Write the dictionaryFix the domain and list each predicate with its arity and reading: Takes(x, y) = "x takes course y".
Identify the outermost quantifierThe subject of the English sentence usually becomes the outermost quantifier. "Every student takes a course" starts ∀x.
Restrict with the right connectiveUniversal restrictions use →; existential ones use ∧. ∀x (Student(x) → ∃y (Course(y) ∧ Takes(x, y))).
Read your formula back into EnglishTranslate the symbols back literally and compare with the original sentence. Mismatched scope is obvious under this check.

Worked example

Translate "Every student has taken some course that no lecturer teaches", with domain: people and courses.

Dictionary: S(x) = x is a student, C(y) = y is a course, L(z) = z is a lecturer, T(x, y) = x has taken y, Teaches(z, y) = z teaches y.
Outer structure: for every student … so ∀x (S(x) → …).
Inner claim: there is a course y taken by x … ∃y (C(y) ∧ T(x, y) ∧ …).
Extra condition on y: no lecturer teaches it … ¬∃z (L(z) ∧ Teaches(z, y)).

Answer. ∀x (S(x) → ∃y (C(y) ∧ T(x, y) ∧ ¬∃z (L(z) ∧ Teaches(z, y)))).

Where marks get dropped

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

Swapping the quantifier order for "everyone … someone" sentences. ∀x ∃y and ∃y ∀x differ, and the second is almost always the stronger, wrong reading.
Restricting a universal with ∧. ∀x (S(x) ∧ P(x)) says everything in the domain is a student, which is not what "every student" means.
Reusing a bound variable name for two different roles, which makes the scopes overlap and changes the meaning.

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Translating English with predicates — frequently asked questions

Should I use a restricted domain or predicates?

Either is acceptable if declared. Setting the domain to "students" simplifies formulas but stops you talking about lecturers, so multi-sort sentences usually need predicates.

How do I translate "only"?

Reverse the implication. "Only students may enter" is ∀x (Enters(x) → Student(x)), matching the propositional rule that "only if" points away from the subject.

How is a translation marked?

Typically: correct dictionary, correct quantifier order, correct restricting connective, correct scope of negation. QED scores those separately, so a near-miss still earns partial credit.

The rest of Predicate Logic

Quantifiers, predicates, binding, and validity. Each subtopic below has its own method, worked example and mark-losing traps.

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