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An assignment in a first-order structure chooses a domain element for each variable. It determines the value of every first-order term and the truth of every first-order formula, once the interpretations of symbols are fixed. A first-order sentence has no free variables, so its truth is independent of the assignment in a first-order structure. Quantification changes the value assigned to the bound variable while retaining the other values.

A logical variable is a symbol whose value is supplied by the semantics of a logical language. A propositional variable receives a truth value under a Boolean valuation. An individual variable of first-order logic receives an element of a structure under an assignment. These are different kinds of variable and should not be identified.

An $L$-structure consists of a nonempty domain $A$, an element $c^{\mathcal{A}}\in A$ for each constant symbol, a function $f^{\mathcal{A}}:A^n\to A$ for each $n$-ary function symbol, and a relation $R^{\mathcal{A}}\subseteq A^n$ for each $n$-ary predicate symbol. When equality is part of the logic it is interpreted as actual identity on $A$.

An assignment in a first-order structure is a function $s$ from logical variables to $A$. It induces a term valuation of first-order terms recursively:For an atomic formula, substitute these first-order term values into the interpreted relation, or compare them for equality. Extend satisfaction for a set structure through the truth tables of the logical connectives, and through the quantifier clausesThus the first-order structure and assignment in a first-order structure together determine truth values for all logical formulas. A first-order sentence has no free variables, so its truth does not depend on the assignment in a first-order structure.

The assertion that $\mathcal{A}$ is a first-order model of $\Gamma$ means that every first-order sentence belonging to $\Gamma$ is true in the first-order structure:Equivalently each first-order sentence is true under every assignment in a first-order structure, since first-order sentences have no free variables. The first-order sentences must all hold in the same interpretation of the language; one cannot select a different first-order structure for each member of $\Gamma$.

A first-order structure consists of a nonempty domain $A$, an interpretation $c^A\in A$ of each constant symbol, a total function $f^A:A^n\to A$ for each $n$-ary function symbol, and a relation $R^A\subseteq A^n$ for each $n$-ary predicate symbol. Logical equality is interpreted as actual identity.

An assignment in a first-order structure is a map $s$ from the variables into $A$. It extends recursively to terms: $x^A[s]=s(x)$, $c^A[s]=c^A$, and $f(t_1,\dots ,t_n{)}^A[s]=f^A(t_1^A[s],\dots ,t_n^A[s])$. Atomic truth is determined by membership of the resulting tuple in $R^A$, or equality of the two term values. Negation reverses the truth value, and an implication is false exactly when its antecedent is true and its consequent false.

For a universal quantifier,These clauses define a valuation on every formula by structural recursion. Only free-variable values affect a formula's truth; consequently a sentence has the same truth value under every assignment. This is a semantic assignment of domain elements, not a programming assignment operation.

The first-order terms are the least collection containing every variable and constant and closed under formation of $f(t_1,\dots ,t_k)$ whenever $f$ has arity $k$ and the $t_j$ are terms. A closed term has no variables.

The first-order formulas are the least collection containing the atomic formulas $t_1\dot{=}{t}_2$ and $P(t_1,\dots ,t_k)$ of the appropriate arities, and closed under negation, logical implication, and universal quantification: from formulas $\varphi ,\psi$ one may form $\neg \varphi$, $(\varphi \to \psi )$, and $\forall x_i\varphi$, with the chosen fully parenthesized grammar. Only finite expressions obtained by these rules are formulas. No additional strings belong to either collection.

Terms denote elements under an assignment in a first-order structure; formulas receive truth values in a first-order structure. The two syntactic categories have different formation rules even when they use some of the same punctuation.

No. The full language can express $\neg (x_0\dot{=}{x}_0)$, which is false in the singleton structure from part (i). Every formula of the reduced language is true in that structure under every assignment in a first-order structure. Therefore no formula of the reduced language can be logically equivalent to this negated equality.

This is a singleton truth-preservation obstruction to expressive completeness. Adding both quantifiers to logical implication does not repair the failure: on this particular nonempty singleton model, quantification cannot create a false formula from true atomic formulas.

A set of sentences is satisfiable if some nonempty first-order structure in its language makes every sentence in the set true. Because the formulas are sentences, their truth is independent of the assignment in a first-order structure.

This is a semantic existence condition. Its equivalence with syntactic consistency is the content of the classical Gödel's completeness theorem, together with soundness.

A set of first-order sentences is satisfiable when it has a nonempty structure in which all its sentences are true. Truth of a sentence is independent of assignment in a first-order structure. The soundness theorem for first-order logic gives consistency from satisfiability, and Gödel's completeness theorem gives the converse for the standard classical deductive system.

A first-order structure and an assignment determine the value of every term recursively: a variable has its assigned value, a constant has its interpreted value, and a compound function term is evaluated by applying the interpreted function to the recursively evaluated arguments. This operation supplies the input values for the satisfaction clauses of atomic formulas.