Questions: Predicates and Relations in First-Order Logic

Question 1

A student claims: 'The formula Tall(Alice) is true because Tall means tall and Alice is clearly a tall person.' What is wrong with this reasoning?

• A. Nothing — predicates in FOL are defined by their English-language meanings
• B. It is correct for unary predicates but does not generalize to binary predicates
• C. The truth of Tall(Alice) depends on the interpretation — without a specified domain and assignment of Tall to an extension, the formula has no truth value at all
• D. The student should have written Tall(x) with a variable, not a constant term

Answer: (C)

This is the core insight of model-theoretic semantics: predicate symbols have no inherent meaning. The symbol 'Tall' is just syntax until an interpretation assigns it a set of domain objects (its extension). In an interpretation where Alice refers to a 5-foot person and Tall picks out people over 6 feet, Tall(Alice) is false. Truth is relative to an interpretation, not to the name of the symbol.

Question 2

Which of the following best captures what the 'extension' of a binary predicate like GreaterThan(x, y) is?

• A. A function that takes two number names as input and returns a number
• B. The set of ordered pairs (a, b) from the domain for which GreaterThan(a, b) holds true
• C. The meaning of the predicate symbol, fixed independently of any interpretation
• D. The result of applying the predicate to specific terms, which yields a truth value

Answer: (B)

An n-ary predicate's extension is the set of n-tuples from the domain satisfying the predicate. For a binary predicate, the extension is a set of ordered pairs. This connects directly to your prerequisite on binary relations: a relation is a set of ordered pairs, and a binary predicate in FOL denotes exactly such a relation under a given interpretation. Option C is wrong: the extension is given by the interpretation, not by the symbol itself.

Question 3

In FOL, the predicate symbol Red has a fixed meaning: it picks out the set of most red things in the domain.

• T. True
• F. False

Answer: False

Predicate symbols in FOL are uninterpreted — they are just names. The symbol Red gains meaning only when an interpretation assigns it an extension: a specific set of domain objects. Under one interpretation, Red might pick out fire engines; under another, it might pick out nothing at all. This is the fundamental distinction between the formal language (syntax) and its interpretations (semantics).

Question 4

A unary predicate in FOL can be understood as picking out a subset of the domain — the set of all objects for which the predicate holds true under a given interpretation.

• T. True
• F. False

Answer: True

A unary predicate of arity 1 is semantically equivalent to a characteristic function: it classifies each domain object as either in or out of the set it defines. The extension of a unary predicate is exactly this subset. This view generalizes: an n-ary predicate's extension is a set of n-tuples (a relation). The subset/relation view makes the connection between FOL predicates and set-theoretic relations precise.

Question 5

Why does first-order logic require a separate 'interpretation' for formulas to have truth values, and what does an interpretation specify?

Model answer: FOL is a formal language: its symbols — predicate names, function symbols, constants — carry no inherent meaning. An interpretation gives them meaning by specifying: (1) a non-empty domain of objects, (2) an assignment of each constant symbol to a specific domain object, (3) an assignment of each predicate symbol to a relation (set of tuples) on the domain, and (4) an assignment of each function symbol to a function on the domain. Without these assignments, formulas are just syntax — neither true nor false. With them, every closed formula has a definite truth value.

This interpretation-dependence is the core insight of model-theoretic semantics: truth is not a property of formulas alone but of formulas relative to models. The same formula can be true in one model and false in another. This framework allows logicians to study validity (true in all interpretations), satisfiability (true in some), and consequence (true in all interpretations satisfying the premises) with mathematical precision.