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For a first-order signature with at least one constant symbol, its Herbrand universe is the set of all ground terms built from its constant and function symbols. If necessary add a fresh constant to make this set nonempty. A Herbrand structure has exactly this domain; each constant denotes itself and each function symbol forms the corresponding syntactic first-order term. Its remaining choices are the truth values of the ground predicate atoms. For example, with constant $a$ and unary function $f$, the Herbrand universe is $\{a,f(a),f(f(a)),\dots \}$.

One useful clausal form of Herbrand's theorem is the following. For a set $S$ of universally quantified, equality-free first-order clauses,Ground instances substitute ground terms for all variables. On the right, treat the ground predicate atoms as independent Boolean variables. Equivalently, $S$ is satisfiable if and only if it has a Herbrand model. A general first-order sentence can first be put into Skolem normal form and clausal form, with the new symbols included in the Herbrand universe.

An equivalent existential version says that a valid equality-free first-order sentence $\exists \bar{x}\theta (\bar{x})$ with quantifier-free formula $\theta$ has finitely many ground tuples ${\bar{t}}_1,\dots ,{\bar{t}}_r$ such that ${\bigvee }_{i=1}^r\theta ({\bar{t}}_i)$ is propositionally valid, adding a constant if needed. This explains why a finite ground resolution refutation can establish a consequence involving infinitely many possible ground terms.

The equality-free qualification matters. If first-order equality is present, one must handle its axioms and use a term model modulo a suitable congruence, or use the corresponding theorem with equality. Literal syntactic term interpretation alone cannot model a consistent requirement such as $a=b$ for distinct constant symbols.

This is a finite, discrete, partially observable, single-agent, sequential search problem with an unknown initial state. The cave is static while the agent deliberates. Given a complete world and an action, the described physical effects and ideal sensor readings are deterministic; ignorance of the world does not itself supply a probability distribution. The fixed monster and gold are environmental objects, not competing decision makers. A known action/sensor law with unknown object locations is different from having to learn an unknown search transition model.

Use partially observable search over belief states. A complete physical state can be writtenwhere $p$ is the agent’s room, $m$ the fixed monster room, $P$ the pit set, $g$ the fixed gold room, $\ell$ records whether the monster lives, $r$ whether the arrow remains, $k$ whether gold has been collected and $e$ whether the agent has exited or failed. The known room dimensions and exit are fixed data. Include death or entrapment as failure states. Movement, shooting toward an adjacent room, collection and departure have their physical meanings. In particular, a shot consumes the arrow, and kills the monster exactly when it targets the live monster’s room.

Let $T(s,a)$ be this physical search transition model. For a surviving agent, an ideal observation is the tuple of three truth valuesrecording sound, breeze and glitter. Here $\sim$ denotes geometric adjacency and $1$ an indicator function. Initially $B_0$ contains all worlds compatible with the known facts and any initial observation. After action $a$ and observation $o$, update the belief state byKnown success or failure observations are included as needed. No prior probabilities are required. A guaranteed-success goal test accepts $B$ only when every $s\in B$ has exited carrying gold. A policy must use the same action in all worlds still indistinguishable to the agent.

An AND-OR search has OR choices for the agent’s next action and AND branches for every possible resulting observation with nonempty $B_{a,o}^{\prime }$. Its output is a contingent plan, not merely a route. To demand safe success, reject any action whose possible outcomes include irreversible death or entrapment, and require a successful continuation on every observation branch. In an online execution the agent updates its belief state, follows the observation branch and can plan again. A strong plan need not exist: some admissible worlds have no safe route, and even solvable worlds can be observationally indistinguishable until a dangerous choice. Without an added prior and numerical rewards, expected-risk optimization is not determined by the description.

Two propositional formulas are logically equivalent if every Boolean valuation gives them the same truth value. A disjunctive normal form is a logical disjunction of logical conjunctions of literals, with a literal being a propositional variable or its negation.

Let $p_1,\dots ,p_n$ include all propositional variables appearing in $\varphi$. For each Boolean valuation $s\in \{T,F{\}}^n$, putThe minterm $M_s$ is true exactly for Boolean valuations whose restrictions to these propositional variables are $s$. ConsequentlyIndeed, an Boolean valuation making $\varphi$ true makes its own minterm true, while a true minterm in the displayed logical disjunction certifies the truth of $\varphi$. There are only finitely many Boolean valuations, so this is a finite disjunctive normal form. If there are no satisfying Boolean valuations, use $p\wedge \neg p$ for any propositional variable $p$; this is a disjunctive normal form for falsity without needing an empty logical disjunction convention.

A Boolean valuation is an arbitrary function $v$ assigning one of the two truth values to each propositional variable. It extends uniquely to all propositional formulas by following the construction of the propositional formula:For logical implication, $\tilde{v}(A\to B)$ is false exactly when $\tilde{v}(A)$ is true and $\tilde{v}(B)$ is false; otherwise it is true. Here the operations on the right are Boolean operations on truth values. Structural recursion gives existence, and structural induction gives uniqueness: once the immediate subformulas have their values, the outer connective fixes the resulting value.

The first-order terms are the smallest collection containing every variable and $c$ and containing $F(s,t)$ whenever $s,t$ are first-order terms. The atomic formulas are exactly $s\approx t$ for first-order terms $s,t$. Logical formulas are then obtained from atomic formulas by the logical connectives and quantifiers: if $\theta ,\psi$ are logical formulas and $x$ a variable, then $\neg \theta$, $(\theta \wedge \psi )$, $(\theta \vee \psi )$, $(\theta \to \psi )$, $\forall x\theta$ and $\exists x\theta$ are logical formulas. These closure rules, with no additional strings admitted, define the syntax; a presentation using fewer primitive connectives treats the others as abbreviations.

Semantic entailment is the conditionThe same first-order variable assignment is used for all premises and the conclusion. This matters when free variables occur; for first-order sentences the truth value is independent of the first-order variable assignment.

A useful consequence of the formation rules is that logical formulas form a prefix-free set of strings. To see this, describe a deterministic parser that reads one logical formula from its starting position. On a logical variable it consumes that one symbol. On a negation symbol it recursively reads one following logical formula. On an opening parenthesis it recursively reads a left logical formula, requires a logical implication symbol, recursively reads a right logical formula, and requires the closing parenthesis. Induction on formation proves that, if a valid logical formula begins at the current position, this procedure consumes exactly that logical formula and stops at its end.

Suppose now that both $\alpha$ and $\varphi$ are logical formulas and $\varphi =\alpha ϵ$. The same parser, started at their common first symbol, must stop both at the end of $\alpha$ and at the end of $\varphi$. Its deterministic stopping position cannot differ, so $ϵ$ is empty. HenceThis argument applies to symbol strings; it does not use truth values or logical equivalence.

Fix any Boolean valuation $v$. Logical equivalence of $\alpha$ and $\beta$ means $v(\alpha )=v(\beta )$ for this Boolean valuation and every other one. Follow the unique path from the replaced node to the root of the formula syntax tree. At a negation node, equal input truth values have equal negations. At a logical implication node, the unaffected child has the same truth value before and after subformula replacement, while the affected child has the same truth value by induction; the resulting logical implications therefore agree as well.

Induction along that finite path yields $v(\varphi )=v({\varphi }^{\prime })$. Since $v$ was arbitrary,This is the replacement theorem for logical equivalence: the semantics of each logical connective depends only on the truth values of its immediate inputs.

A collection of logical connectives is functionally complete, or adequate, if every Boolean function of finitely many propositional variables is represented by a logical formula using only those logical connectives. More explicitly, for every $n\ge 1$ and every $F:\{0,1{\}}^n\to \{0,1\}$ there is a logical formula $\theta (p_1,\dots ,p_n)$ whose truth value under each Boolean valuation is $F(v(p_1),\dots ,v(p_n))$. Constant functions with Boolean values may use an otherwise irrelevant logical variable, for example $p\to p$ for truth and $\neg (p\to p)$ for falsity.

Translate logical formulas recursively, leaving propositional variables unchanged and translating negation by applying negation to the translation. For the other two logical connectives setThe first logical implication is false exactly when both original inputs are false; the second displayed logical formula is true exactly when both inputs are true. Hence each clause preserves truth values. Structural induction gives $v(T(\theta ))=v(\theta )$ for every Boolean valuation and every logical formula $\theta$ in the larger language.

By the assumed functional completeness of negation, logical conjunction and logical disjunction, any finite Boolean function has such a logical formula $\theta$. Its translation $T(\theta )$ uses only negation and logical implication and represents the same function. Thus $\{\neg ,\to \}$ is adequate.

Under any Boolean valuation there are two possibilities. If $\neg (\psi \to \chi )$ is false, the outer logical implication is true by vacuous truth. If it is true, then $\psi \to \chi$ is false, which forces $\psi$ true and $\chi$ false. In that case $\neg \chi$ is true, so the outer logical implication is again true. These possibilities exhaust the truth values of its antecedent; therefore

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.

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.

Neither constants alone nor functions alone gives the same uniform adequacy on all nonempty carriers. A one-element set has exactly one interpretation of each function symbol of any finite arity and of every constant symbol. In contrast it has two interpretations of a unary predicate: empty or full. Thus a fixed purely functional or constant-only language has only one structure on that singleton, and cannot encode both truth values of $\exists xP(x)$ by a same-carrier truth-preserving translation with reconstruction. This is the singleton truth-preservation obstruction to expressive completeness, even when logical equality is available.

Constants alone also cannot represent general variable-dependent relations on larger infinite carriers: a fixed formula mentions only finitely many named elements, and its remaining equality tests do not encode an arbitrary binary relation. Functions offer more flexibility. On carriers with at least two elements, predicates can be encoded by characteristic functions taking two distinguished values, and constants can be represented by constant-valued unary functions if explicit constant symbols are forbidden. Such a construction can work after adding axioms for distinct truth markers, but those axioms exclude the singleton case. Enlarging the carrier or adding a Boolean sort is another possible encoding; it changes the same-carrier sense established in the preceding part and must not be silently identified with it.

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.

Logical equivalence means that the two propositional formulas receive the same truth value under every Boolean valuation of the propositional variables. Equivalently, $\alpha \leftrightarrow \beta$ is a tautology:

Semantic consequence under an assignment means that for every first-order structure $\mathcal{A}$ with the relevant first-order variable assignment $v$ into its domain,More explicitly, in a fixed structure this is the satisfaction implication at $v$; logical consequence requires that implication in all structures and assignments. An assignment takes values in a particular domain, so a single set map is not silently shared between unrelated domains.

By induction on first-order formulas, their truth values depend only on the values of their free variables. A first-order sentence has none. If all formulas here are sentences, every displayed satisfaction statement, and therefore the consequence condition, is independent of $v$.

A prime model of a complete theory $T$ is a first-order model admitting an elementary embedding of first-order structures into every first-order model of $T$. Let $\mathcal{A}\models T$ be canonical and $\mathcal{B}\models T$. For each closed term $t$, defineEvery element of $A$ has a name, so this defines the desired map on all of $A$. If two first-order terms agree in $\mathcal{A}$, their logical equality first-order sentence holds in one first-order model of the complete theory and therefore in every first-order model of it; they agree in $\mathcal{B}$. This proves well-definedness. If they are distinct in $\mathcal{A}$, their inequality holds throughout the complete theory as well, proving injectivity.

For every logical formula $\varphi (\vec{x})$ and named tuple ${\vec{t}}^{\mathcal{A}}$, the substituted first-order sentence $\varphi (\vec{t})$ has the same truth value in the two first-order models by completeness of $T$. ConsequentlyThus $e$ is elementary. Since $\mathcal{B}$ was arbitrary, every such canonical first-order structure is prime. This formal proof also handles closed-term generation when function symbols are present; under the constant-name convention one simply uses constants as the first-order terms.

If $\mathcal{A}\subseteq \mathcal{B}$, interpret each new constant $c_a$ in $\mathcal{B}$ as the same element $a$. Atomic logical formulas and their negations then have exactly their truth values in ${\mathcal{A}}_A$, so the expansion satisfies the diagram of a structure. If the inclusion is elementary, every logical formula with those named parameters has the same truth value, so the expansion satisfies the elementary diagram.

Conversely, from any expansion ${\mathcal{B}}^{+}$ satisfying the diagram of a structure defineThe diagram of a structure contains $c_a\ne c_{a^{\prime }}$ whenever $a\ne a^{\prime }$, so $e$ is injective. It contains every true function-value equation, so $e$ commutes with all operations and constants. Positive and negative relation instances ensure both preservation and reflection of the relations. Thus $e$ is an embedding of first-order structures. If the expansion satisfies the elementary diagram, for every logical formula $\varphi$ and tuple $\vec{a}$ it also givesso $e$ is an elementary embedding of first-order structures.

The precise conclusion for an arbitrary expansion is an embedding of first-order structures, rather than a literal inclusion of the original carriers. The source's inclusion notation is correct when one uses the customary identification of $\mathcal{A}$ with its embedded copy, or requires the expansion to interpret $c_a$ as the actual element $a$ of a carrier already containing $A$. With that naming convention, $e$ is the inclusion and the requested two equivalences follow exactly.

Without the convention the literal wording fails, even in the first-order language of logical equality: singleton first-order structures with carriers $\{0\}$ and $\{1\}$ have a diagram-satisfying expansion by sending the first first-order structure's named point to $1$, but the first carrier is not a subset of the second. The embedding of first-order structures formulation proved above states the invariant result and makes the intended inclusion formulation precise.

The cardinality of a vector space is central: here “infinite” must continue to refer to the underlying set. Over a countably infinite field, every nonzero finite-dimensional space is already infinite. Thus the model class now consists of all nonzero $K$-vector spaces, not only infinite-dimensional ones.

Since independence of a pair is not definable over an infinite field, for (i), independence of a single vector is still $x\ne 0$, but independence of tuples of length at least two is not definable by one formula in this one-sorted language. Quantifier elimination follows from the linear-equation argument in the preceding part for all these infinite domains. Any candidate two-variable formula therefore involves only finitely many linear equations. Take an independent pair, and compare it with $(x,{\lambda }_{a}x)$, where $x\ne 0$ and $a\in K$ avoids the finitely many ratios making one of those nontrivial equations vanish. Both pairs satisfy the same listed atomic formulas, contradicting a proposed definition of independence. Longer tuple independence would define pair independence by existentially completing a pair inside an infinite-dimensional model, giving the same obstruction.

For (ii), the vector-space axiom schema together with $\exists x(x\ne 0)$ axiomatizes the infinite underlying spaces. For (iii), there is again no universal axiomatization, because the zero subspace is a substructure.

For finite axiomatizability with infinitely many scalar symbols, in (iv) there is no finite axiomatization in the full language: any finite axiom set mentions only finitely many scalar symbols. In a nonzero vector space alter an unused nonzero ${\lambda }_a$ to send zero to a nonzero vector, retaining all mentioned interpretations. The finite sentences keep their truth values, but a vector-space axiom is now false. Relative to the already imposed vector-space axioms, the extra condition is finitely axiomatized by the single nonzero-vector sentence. These absolute and relative answers must be distinguished.

For (v), the one-variable algebra still has two atoms and four Elements. In two variables the isolated atoms are $(0,0)$ and the countably many rank-one directions, represented by $x\ne 0,y={\lambda }_{a}x$ and by $x=0,y\ne 0$. There is also a nonisolated complete type of an independent pair. A formula using finitely many linear equations either picks finitely many isolated atoms, or contains that independent type and all but finitely many isolated atoms. Therefore $F_2/E_2(T)$ is the finite-cofinite Boolean algebra on a countably infinite set of atoms, with the independent type the extra nonisolated point in its Stone space. In particular this algebra is countably infinite, rather than the full power set of its atoms.

For (vi), dimension can be $1,2,3,\dots$ or ${\aleph }_0$ while the domain is countable, producing pairwise nonisomorphic countable models. Thus $T$ is not countably categorical. For an uncountable $\kappa$, a vector space over this countable field has cardinality $\kappa$ exactly when its dimension is $\kappa$, since in general a nonzero space has size $\max (|K|,\dim V)$ when that maximum is infinite. Basis bijections then give categoricity in every uncountable cardinal. Quantifier elimination also shows that these different positive dimensions are elementarily equivalent; independence of an arbitrarily long tuple is not being silently treated as definable over this infinite field.

Use structural induction on the propositional formula. The induction claim is that its truth value under the all-one Boolean valuation is one. A variable has this truth value by definition. For the induction step, suppose the immediate subformulas $\alpha ,\beta$ each have truth value one. ThenThese are all permitted construction cases. Thus every such propositional formula evaluates to one. Equivalently, the three logical connectives define one-preserving Boolean functions, and composition preserves this property. The variable-only base grammar matters: admitting a constant false formula would invalidate the claim.