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Classical first-order logic has the quantifier and logical equality rules of natural deduction together with unrestricted double-negation elimination, or equivalently the law of excluded middle. It is interpreted in ordinary two-valued first-order structures.

Classical propositional logic interprets propositions as true or false and validates principles such as the law of excluded middle and double-negation elimination.

Intuitionistic first-order logic has the usual introduction and elimination rules for logical connectives, quantifiers and logical equality, but no unrestricted double-negation elimination or law of excluded middle. Its proofs retain the witness and logical disjunction information that can be lost in classical first-order logic.

Mathematical logic studies formal languages, formal proofs, model theory, computability theory, and the foundations of mathematics. It makes the rules of reasoning explicit enough that reasoning itself becomes an object of mathematics. Proof theory studies deductions; model theory studies interpretations in which statements hold.

Three questions organize the subject: is an expression a well-formed first-order formula, is it true in a specified first-order structure, and does it have a formal proof from specified axioms? These questions are distinct. For example, $\exists n(n+n=1)$ is a first-order sentence in an appropriate first-order language; it is false in the natural numbers but true in the rational numbers, with $n=1/2$.

The beauty of mathematical logic is that the same logical connectives and rules of universal quantification and existential quantification organize arguments across mathematics. Classical propositional logic uses two truth values; Intuitionistic propositional logic interprets logical implication through constructions on proofs, without unrestricted law of excluded middle. This proof-based framework is developed in Frank Pfenning, Natural Deduction, sections 1–3.

Work in ZF with the law of excluded middle, without the axiom of choice. The paper's terminology is an infinite Dedekind-finite set. The useful common principle is countable union of explicitly ordered finite lists without choice: given an actual sequence of finite lists, enumerate their entries by list number and position using a Cantor pairing function. If the union of entries is infinite, repeatedly take the entry with the least code not already selected. This produces an injection from $\mathbb{N}$ without choosing any enumerations of unordered sets.

Consequently, if a family of objects has an explicitly ordered finite list of labels for each object, and only finitely many objects can be made from any given finite pool of labels, then a countably infinite list of distinct objects forces a countably infinite subset of the label set. For a Dedekind-finite label set this is impossible. If the label set embeds into the object family as well, infinitude is preserved. ThusThe distinction between ordered lists supplied by the data and arbitrary finite subsets is essential: a blanket countable-union theorem for unordered finite sets would introduce a choice principle.

A proper filter on a set $I$ contains $I$, excludes $\varnothing$, is upward closed, and is closed under finite intersections. Order its proper extensions by inclusion. The union of a nonempty chain is again a proper filter: finitely many members lie together in one chain member. Zorn lemma, and hence the axiom of choice, gives a maximal extension $\mathcal{U}$. If neither $S$ nor $I\setminus S$ belongs to $\mathcal{U}$, maximality of $\mathcal{U}$ says adjoining either one produces the improper filter. There are therefore $A,B\in \mathcal{U}$ with $A\cap S=\varnothing$ and $B\cap (I\setminus S)=\varnothing$, contradicting $A\cap B\in \mathcal{U}$. Thus $\mathcal{U}$ is an ultrafilter.

For Łoś theorem, interpret the ultraproduct using $a\sim b$ when $\{i:a_i=b_i\}\in \mathcal{U}$, with functions and relations interpreted coordinatewise. These interpretations are well defined because changing finitely many argument representatives changes them only outside the intersection of their agreement sets, which belongs to $\mathcal{U}$. For any first-order formula and sequences of parameters,Prove this by structural induction. Atomic formulas follow from the interpretation of terms. Conjunction uses finite intersections; negation uses the ultrafilter dichotomy between a set and its complement. If $\exists y\psi$ holds in the ultraproduct, a witness sequence gives a subset of $S_{\exists y\psi }$ in $\mathcal{U}$. Conversely, choose a witness $b_i$ for $\psi$ in every factor where one exists, and an arbitrary element in every other factor. This is the second explicit use of the axiom of choice; all factors have nonempty carriers. The inductive hypothesis applied to $\psi (\mathbf{a},\mathbf{b})$ supplies the ultraproduct witness. The other connectives and universal quantification follow from these cases.

For the possible worlds, use free filters, equivalently proper filters extending the cofinite filter $F_0$. Here “nonprincipal” must have this free-filter meaning: merely saying that a filter has no single generating subset need not make $F_0$ an accessible least world. The designated world is $F_0$, and all worlds have carrier $D={\prod }_i{M}_i$. Atoms, including equality, are forced byEquality may identify distinct sequences at a world; it is a persistent congruence, and at an ultrafilter world its quotient is the ordinary ultraproduct.

The required equivalence with all ultraproducts uses dense-extension filter semantics. Its recursive clauses areEvery extension in these clauses is a free proper filter. The dense clause for disjunction is essential; for propositional formulas, local disjunction would instead give a Kripke model for intuitionistic propositional logic and would invalidate the claimed equivalence.

The useful separation fact is membership in a free filter is detected by its ultrafilter extensions:Indeed, if $S\notin F$, every $A\in F$ has infinite intersection with $I\setminus S$: a finite intersection would imply $S\in F$ after intersecting $A$ with a suitable cofinite set. Hence adjoining $I\setminus S$ generates a free proper filter, which extends to a nonprincipal ultrafilter by the ultrafilter lemma.

Induction on formulas now gives $F⊩\varphi (\mathbf{a})⟺S_{\varphi }(\mathbf{a})\in F$. Negation and implication follow by adjoining the positive set on which their classical truth conditions fail. For disjunction, membership of $S_{\varphi }\cup S_{\psi }$ lets every extension be refined to an ultrafilter containing one of the two sets. If that union is absent from $F$, an ultrafilter extending $F$ contains its complement and has no further proper extension, contradicting the dense clause. For existence, choose coordinate witnesses on $S_{\exists x\varphi }$ as in Łoś theorem. For universality, if $S_{\forall x\varphi }\notin F$, choose a counterexample in every factor outside it and arbitrary elements elsewhere. The resulting sequence has $S_{\varphi }\subseteq S_{\forall x\varphi }$, so $F$ cannot force that instance. The converse follows from upward closure and persistence.

At the designated world, therefore,This semantics validates classical first-order logic, including law of excluded middle. For example, every free filter has an ultrafilter refinement deciding an atomic formula, so its excluded middle is forced even when neither disjunct is forced at the original world. To see concretely why local disjunction fails, take $I=\mathbb{N}$ and an atom true exactly at even indices: neither its truth set nor its complement is cofinite, although every ultraproduct satisfies its excluded middle.

A tautology is a propositional formula true under every Boolean valuation in classical propositional logic, such as the law of excluded middle $p\vee \neg p$. A primitive proposition $p$ is not a tautology, even if it represents a true statement. See forall x: Calgary, section 12.1.