In first-order logic, which of the following is a term but NOT an atomic formula?
AP(x), where P is a unary predicate and x is a variable
Bf(a, b), where f is a binary function symbol and a, b are constants
Cx = y, where x and y are variables
D∀x P(x), where P is a unary predicate
f(a, b) is a complex term: a function symbol applied to two term arguments. It refers to an object in the domain — it denotes something but makes no claim that could be true or false. P(x) is an atomic formula (predicate applied to a term). x = y is also an atomic formula (the built-in equality predicate applied to two terms). ∀x P(x) is a quantified formula. The key distinction is that terms denote objects while formulas make claims.
Question 2 Multiple Choice
Consider g(f(x), c), where g is a binary function symbol, f is a unary function symbol, x is a variable, and c is a constant. What is this expression?
AAn atomic formula, because it contains predicate-like symbols applied to terms
BA complex term, because it is built from function symbols applied to other terms
CA quantified formula, because it contains a variable
DAn atomic formula when g is interpreted as a predicate in a specific structure
g(f(x), c) is a complex term built inductively: x is a variable (term), c is a constant (term), f(x) is a complex term, and g(f(x), c) is a complex term (binary function applied to two terms). No predicate symbol appears here. Function symbols build terms (noun phrases referring to objects); predicate symbols build atomic formulas (sentences making claims). The presence of a variable does not make something a quantified formula — quantification requires explicit ∀ or ∃.
Question 3 True / False
An atomic formula is the simplest kind of expression in first-order logic that can be evaluated as true or false.
TTrue
FFalse
Answer: True
Terms (variables, constants, complex terms) refer to objects but cannot be evaluated as true or false — they denote, they do not claim. Atomic formulas are the base case of the formula definition: P(t₁,...,tₙ) applies a predicate to terms, making the simplest possible claim. All compound formulas (conjunctions, negations, quantified statements) are built from atomic formulas by applying connectives and quantifiers. Atomic formulas are where truth values first enter.
Question 4 True / False
A variable in first-order logic is a type of atomic formula.
TTrue
FFalse
Answer: False
A variable is a term, not a formula. Terms and formulas are distinct syntactic categories. Terms refer to objects (noun phrases); formulas make claims that can be true or false (sentences). A variable like x denotes whatever object is assigned to it in an interpretation — it does not assert anything and cannot be evaluated as true or false. Confusing terms with formulas is a fundamental syntax error in first-order logic.
Question 5 Short Answer
What is the difference between a term and an atomic formula, and why does this distinction matter for interpreting first-order logic?
Think about your answer, then reveal below.
Model answer: A term is a syntactic expression that denotes an object: a variable, constant, or function symbol applied to terms. An atomic formula is a predicate symbol applied to a sequence of terms, making the simplest possible claim that can be true or false. Terms answer 'which object?' while atomic formulas answer 'what is true of that object?' The distinction matters because truth values only apply to formulas — you cannot negate or quantify a term, only a formula.
This separation between the naming layer (terms) and the claiming layer (formulas) is fundamental to how first-order logic is interpreted in structures. When evaluating a formula in a model, terms are mapped to domain elements, then the predicate is checked on those elements. Understanding that f(x) is a term (object reference) while P(f(x)) is a formula (claim) clarifies what can be negated, quantified over, or combined with logical connectives — all operations that apply to formulas, not terms.