Translation Examples

Mathematical Logic · Axiom Academy

Step-by-step English-to-logic translations in predicate logic Excellent work! You've completed all translation examples. Here's what we learned: Negation patterns: "No one" translates to negating an existential quantifier: ¬∃x. Remember this is logically equivalent to ∀x¬. Handling "else": When a statement says "everyone else," you must add a condition (x ≠ y) to exclude the subject from the quantified variable. "Only" statements: The word "only" reverses the logical direction. "Only A are B" means "If B then A" (∀x: B(x) → A(x)), not "If A then B." Nested quantifiers: Pay careful attention to the order and scope of quantifiers. ∃x∀y is very different from ∀y∃x. Domain restrictions: When translating statements about specific types of objects (like prime numbers), use conjunctions in the antecedent of conditionals. These translation patterns appear frequently in mathematical logic. Practice identifying keywords like "no one," "everyone else," and "only" to build fluency in converting English to formal logic!

This is the written version of the interactive lesson above. See the full Mathematical Logic course.